Data Types as Lattices

DATA TYPES AS LATTICES by Dana Scott Oxford University Computing Laboratory Programming Research Group \ 1~ j l I 1 ~ " "~._- - 30338&8&40 ..""-_.. __ ._._­­_. Technical Monograph PRO­5, September 1976 Oxford University Computing Laboratory Programming Research Group 45 Bllnbury Road. Oxford OX2 6PE Reprinted from the SIAM Journal on Computing. Volume 5, 1976, pages 522­587. Copyright 1976 by Society for Industrial and Applied Mathematics. Philadelphia. All rights reserved. r SIAM 1 COMPU1. Vol .\ Nfl l, Seplclllt>er 1\,176 0,-1", ..,." , f..,. "'.... r("o·.I.. . '11/ C...., \ DATA TYPES AS LATTICES To rhe Memor} of Chflstopher Strache", 19 I0­ J975 , .' " Jll,r'I.'.I"'l"J : ,,': U~.l J (j' (~ . /I '- ,:,.j I .lliU DANA SCOTT! Abstrad. The meaning of many k.inds of expressions In programming languages can be tak.en as elements of cerra in spaces of ··partial" objects. In this report these spaces are modeled in one universal domain Pw. the set of all subsets of the integers. This domain renders the connection of Ihis semantic theory with the ordinary theory of number theoretic (especially general recursive) funClIons clear and straightforward. Key worm. programming language semantics, lattice, continuous lattice, algebraic Jatlice, computabilily, retract. combinatory logk. lambda calculus, recursIon theorem, enume'rarion degrecs, continuous function, fixed­point theorem Introduction. Investigations begun in 1969 with Christopher Strachey led to the idea that the denotations of many kinds of expressions in programming languages could be taken as elements of certain kinds of spaces of "parrial" objects, As these spaces could be treated as function spaces, their structure at first seemed excessively complicated­­even impossible. But then the author discovered that there were many mOre spaces than we had first imagined­­even wanted. They could be presented as lattices (or as some prefer, semilattices). and the main technique was to employ topological ideas, in particular the notion of a continuous function. This approach and its applications have been presented in a number of publications, but that part of the foundation concerned with computability (in the sense of recursion theory) was never before adequately exposed. The purpose of the present paper is to provide such a foundation and to simplify the whole presentation by a de­emphasis of abstract idea•. An Appendix and tbe references provide a partial guide to the literature and an indication of connections with other work, The main innovation in this report is to model everything within one "universai" domaii, Pw ~ {xix ~ wi. the domain of all subsets of the .el w of nonnegative integers. The advantages are many: by the most elementary considerations Pw is recognized to be a lattice and a topological space. [n fact, Pw is a continuous lattice, even an algebraic lattice, but in the beginning we do not even need to define such an "advanced" concept; we can save these ideas for an analysis of what has been done here tn a more direct way. Next by taking the set of integers as the basis of the construction, the connection with the ordinary theory of number­theoretic (especially, general recursive) functions can be made clear and straightforward. The model Pw can be intuItively viewed as the domain of multIple-valued integers; what is new in the presentation is that functions are not only multiplevalued but also "multiple­argumented", This remark is given a precise sense in § 2 below, but the up.hot of the idea is that multiple­valued integers are regarded as "Receivt:d by thc editor~ May 23,1975, and in revised form March 17. 1916. of Malht:malll:.;' Depanmt:nt. Oxford Unl\'cr"ity. Oxford, Ellgland OX:: oPE 522 C'i, _"J-c"Jry nATA TYPES ,\S ',A'ITICLS 521 objects in themsdves­possibly inhmte­and as something more than Just the collection o[ single integers contained in Ihem. This combination o[ the finite wilh the mfinite into a single domain, together with the idea that a continuous function can be reduced to its graph (in the end, a set of integers). makes it possible to view an x E Pw at one time as a value, at another as an argument, then as ;m integer, then as a [unction, and still later as a [unctional (or combinator). The "paradox" o[ sd[­application (as in ~»xi is solved by allowing the same x to be used in two differeht ways. This is done in ordinary recursion theory via Gode! numbers (as in (eHe». but the advantage o[ the present theory is that not only is the function concept the extensional one, but it includes arbiuary continuous functions and not just the computable ones. Section 1 introduces the elementary ideas on the topology o[ Pw and the continuous [unctions including the fixed­point theorem. Section 2 has to do with computability and definability. The language LAMBDA IS introduced as an extension of the pure A­calculus by four arithmetical combinators; in fact, it is indicated in § 3 how the whole system could be based 00 Ohe combinator. What is shown is that computability in Pw according to the natural defioition (which assumes that we already know what a recursively enumerable set of integers is) ~i equivalent to LAMBDA­definability. The main tool is. not surprisingly, the first recursion theorem [ormulated with the aid of the so­called paradoxical comblhalOr Y. The plan is hardly original, but the point is to work out what it all means in the model. Along the way we have to show how to give every A­term a denotation in Pw; the resulting principles o[ A ­calculus that are thereby verified are summarized in Table I. O[ these the first threc, (n), ((3), and (~), arc indeed valid in the model; however, rule (11), which is a stronger version of extensionality, fails in the Pw model. This should not be regarded as ;] disadvantage since the import of (11) is to suppose every object is a function. A quick construction of these special models i!' indicated at the end o[ § 5, Since Pw is partially ordered by c::, there are also laws involving this relation. Law (';*) is an improvement of (E,; while IlL) is a form of mt)fiotnnicity for application. T >\BU Some l(1w~ (If A cafC/IIII.\ 52.+ DANA SCOTr Section 3 has to do with enumeration and degrees. Godel numbers for LAMBDA are defined in a very easy way which takes advantage of the notation of combinators. This leads to the second recursion theorem, and results on incompleteness and undecidability follow along standard lines. Relative recursiveness is also very easy to define in the system, and we make the tie-in with enumeration degrees which correspond to finitely generated combinatory subalgebras of Pw. Finally a theorem of Myhill and Shepherdson is interpreted as a most satisfactory completeness property for definability in the system. Sections 4 and 5 show how a calculus of retracts leads to quite simple definitions of a host of useful domains (as lattices). Section 6 investigates the classification of other subsets (nonlattices) of Pw; while § 7 contrasts partial (multiple-valued) functions with total functions, and interprets various theories of functionality. Connections with category theory are mentioned. What is demonstrated in this work is how the language LAMBDA, together with its interpretation in Pw, is an extremely convenient vehicle for definitions of computable functions on complex structures (all taken as subdomains of Pw). ft is a "high-level" programming language for recursion theory. It is applied combinatory logic, which in usefulness goes far beyond anything envisioned in the standard literature. What has been shown is how many interesting predicates can be expressed as equations between continuous functions. What is needed next is a development of the proof theory of the system along the lines of the work of Milner, which incorporates the author's extension of McCarthy's rule of recursion induction to this high-level language. Then we will have a flexible and practical "mathematical" theory of computation. 1. Continuous lunctions. The domain Pw of all subsets of the set w of nonnegative integers is a complete lattice under the partial ordering ~ of set inclusion, as is well known. We use the usual symbols U, n, U, nfor the finite and infinite lattice operations of union and intersection. Pw is of course also a Boolean algebra; and for complements we write -x ={nlnEx} where it is understood that such variables as i, j, k, I. m, n range over integers in w, while u, L', W, X, y, z range over subsets of w. The domain Pw can also be made into a topological space-in many ways. A common method is to match each x£; w with the corresponding characteristic function in {D, 1 and to take the induced product topology. In this way Pw is a totally disconnected compact Hausdorff space homeomorphic to the Cantor "middle-third" set. This is not the topology we want; it is a positive-and-negative topology which makes the function -x continuous. We want a weaker topology: the topology of positive "information", which has the advantage that al1 continuou, fundion; possess fixed points. (The equation x = --x is impossible.) The topology that we do want is exactly that appropriate to con;idering Pw to be a continuous lattice. But all this terminology of abstract mathematics IS quite un ec s~ary, since the required definitions can he given in very elementary terms. t, DATA TYPES AS LA1TICES {e"jn 525 To make the topology "visible ", we introduce a standard enumeration E w} of all finite subsets of w. Specifically we set provided that k o < k l <. , . < k",_1 and n = I."m 2". Thus n is the code number for e", and the elements of e" are the exponents in the binary expansion of the integer n. This is a one-to-one enumeration of finite SUbsets, where k E ell always implies k < n, the function max (e,,) is (primitive) recursive in n, and the relations kEen, en ~ e,m en = em U fie are all (primitive) recursive in k, m, n. Topologically speaking the finite sets e" are dense in the space Pw, for each x E Pw is the "limit" of its finite subsets in the sense that To make this precise we need a rigorous definition of open subset of Pw. DEFINITION. A basis for the neighborhoods of Pw consists of those sets of the form: for a given ell' An arbitr£lry open subser is then a union of basic neighhorhoods. It is easy to prove that an open subset U ~ Pw is just a set of "finite character"; that is, a set such that for all x E Pw we have x E U if and only if some finite subset of x also belongs to U. An alternate approach would define directly what we mean by a continuous function using the idea that such functions must preserve limits. DEFINITION. A functionf: Pw....:,. Pw is continuous iff for all x E Pw we have: Again it is an e£lsy exercise to prove that a function is continuous in the sense of this definition iff it is continuous in the usual topological sense (namely: inverse images of open sets are open). For giving proofs it is even more convenient to have the usual £-D formulation of continuity. THEOREM II (The characterization theorem), A function f: Pw ~ Pw is continuous iff for all x E Pw and all Em we have: em )x(f~ iff 3e" ~ x.e m ~f(e"). Note that open sets and continuous functions have a monolonicity property: whenever x 0;; y and x E U, then y E U; and whenever x ~ y, then f(x) ~ f(y)· This gives a precise expression to the "positive" character of our topology. However, note too that openness and continuity mean rather more than just monotonicity. In particular. a continuous function is completely determined by the pairs of integer... such that m E f(e n). as can be seen from the definition. (Hence, there are only a continuum number of continuous functions. but more than a continuum number of monotonic functions.) This brings us to the definition 01' the function. graph of a contilOu~ 526 DANA SCOTT To formulate the definition, we introduce a standard enumeration (n, m) of all pairs of integers. Specifically we set In, m) =~(n +m)(n ,. m + 1) +m. This is the enumeration along the "little diagonals" going from left to right, and it produces the ordering: (0,0), (1, 0), (0, 1), (2, 0), 0, 1), (0, 2), (3, 0), (2,1), ' , .. Note that n ~(n. m) and m ~(n, m) with equality possible only in the cases of (0,0) and (1,0). This is a one-to-one enumeration, and the inverse functions are (primitive) recursive-but we do not require at the present any notation for them. DEFI"tTloN. The graph of a continuous function f: Pw .... Pw is defined by the equation: graph(f)={(n, m)lmEfle,,)}: while the function determined by any set II ,; w is defined by the equation: fun(u)(x) = {mI3e" ,; x.(n, m) E u}. THEORE" 1.2 (The graph theorem). Every continuous function is uniquely determined by its graph in the sense that: (i) lun (graph(f)) = f. Conversely, every set of integers determines a conrinuous function and we have: Iii) u,; graph(lun(u», where equality hoLds just in case u satisfies: (iii) whenever(k. m)E uande" :::;e n , chen (n, m)E u. Besides functions of one variable we need to consider also functions of several variables. The qfficial definition for one variable given above can be extended simply by saying !(x, y, ... ) is continuous iff it is continuous in each of x, y, .... Those familiar with the product topology can prove that for our special positive topology on Pw this is equivalent to being continuous on the produd space (continuous in the variables jointly). Those interested only in elementary proofs can calculate nut directly from the definition Iwith the aid of 1.1) that continuity behaves under combinations by substitution [as 10: f(g(x, y), h(y, x, y»)] TIIEOREM 1.3 (The substitution theorem)'. Continuous functions of several variables on Pw are closed under substitution. The other general fact about continuous functions that we shall use constantly concerns fixed points. w'hose existence can be proved using a well-known method. TIIEOKFM 1A (The fixed-point theorem), Every crmtinllOus fftflcf;OI/ f; Pw""" Pw has a Least fixed point given hy the formuLa: fixil) where ,~_) = UU"(0)jn E w}, is the empiY set and!,' is the n-fold cO/llpwilflOn oj f wah itself. 527 DATA TYPES AS LATllCES Actually fix is a functional with continuity properties of its own. We shall not give the required definitions here because they can be more easily derived from tbc construction of tbe model given in the next section. For those familiar with the abstract theory of topological spaces we give in conclusion two general facts about continuous functions with values in Pw which indicate the scope and generality of our method THEOREM 1.5 (The extension theorem). Let X and Y be arbitrary topological spaces where X s; Yas a subspace. Then every continuous function f: X -. Pw can be extended to a continuous function f: Y -> Pw defined by the equation: fry) = U{(i{f(x)!x E X n v}IY E V}, where y E Y, and V ranges over lhe open sl,bsets of Y. THEOREM 1.6 (The embedding theorem). Every To-space X with a countable basis [V" In E w} for its topology can be embedded in Pw by the continuous function E : X -> Pw defined by the equation: Technically the To-hypothesis is what is needed to show that, is one-la-one. The upshot of these two theorems is that in looking for (reasonable) topological structures we can confine attention to the subspaces of Pw and to continuous functions defined on all of Pw. Thus the emphasis on a single space is justified structurally. What we shall see in the remainder of this work is that fhe use of a single space is also justified practically because the required subspaces and functions can be defined in very simple ways by a natural method of equations. In order to make the plan of the work clearer, the proofs of the theorems have been placed in an Appendix .....·hen they are more than simple exercises. 2. Computability and delinability. The purpose of the first section was to introduce in a simple-minded way the basic notions about the topology of Pw and its continuous functions. In this section we wish to present the details of a powerful language for defining particular functions-especially computable functionsand initiate the study of the use of these functions. This study is then extended in different ways in the following sections. Before looking at the language, a short discussion of the "meaning" of the elements x E Pw will be helpful from the point of view of motivation. Now in itself x E Pw is a set, but this does not reveal its meaning. Actually x has no "fixed" meaning, because it can be used in strikingly different ways; we look for meaning here solely in terms of use, Nevertheless it is possible to give some coherent guidelines. In the first place it is convenient to let the singleton subsets {n} E Pw stand for the corresponding integers. In fact, we shall enforce by convention the equation n = in} as a wayof simplifying notation. In this way, w s; Pw as a subset. (Note that our convention conflicts with such set-theoretical equations as 5 = {D, 1,2,3, 4}. What we have done is to abandon the usual laci~eroht-s conventions in favor of a slight redefinition of set of inregers which produces a more helpful convention for prescnt purposes.) So, if we choose, a singleton "means" a single integer. Tht: next 52X DANA scon question is what a "large" set x E Pw could mean. Here is an answer: if we write: x = {a, 5, 17} = aU 5 U 17, we are thinking of x as a multiple integer. This is especially useful in the case of multiple-valued function where we can write: [(a)=aU5U 17. Then" mE [(a)" can be interpreted as "m is one value of [at a." Now a E Pw, too, and so it is a multiple integer also. This brings us to an important point. A multiple integer is (often) more than just the (random) collection of its elements. From the definition of continuity, mE [(a) is equivalent to mE [(en) with e" sa. We may not be able to reduce this to mE [({n}) with n E a without additional assumptions on f. Indeed we shall take advantage of the feature of continuous functions whereby the elements of an argument a can join in cooperation in determining [(a). Needless to say, continuity implies that the cooperation cannot extend beyond fillite configurations, and so we can say that a is the union (or limit) of its finite subsets. However, finitary cooperation will be found to be quite a powerful notion. Where does this interpretation leave the empty set 0? When we write "[(a) = 0" we can read this as "[ has no value at a", or "[ is undefined at a". In this case [(a) exists (as a set), but it is not "defined" as an integer. Single- (or singleton) valued functions are "well-defined", but multiple-valued functions are rather "over-defined". How does this interpretation fit in with mono tonicity? In case as band mE[(a), then we must have mE[(b). We can read "asb" as"b isan improvement of a" is beuer-defined than a". The point of monotonicity is that the better we define an argument, the better we define a value. 'Better" does not imply "well" (that is, singleton-valuedness), and overdefinedness may well creep in. This is not the fault of the function; it is our fault for not choosing a different function. As a suhspace (I'.C;:: Pw is discrete. This implies that arbitrary functions -I' ware continuous. Note that p: w ~ Pw as well, because w £; Pw. By Theorem 1.5 we can ""tend p continuously to p: Pw -> Pw. The formula given produce,", this function: p :w (2.1) p(x) ~ { n{p(n)ln p(n) E w w} if x ~ 0; if x = nEw; otherwise. This is a rather abrupt extension of p (the maximal extension); a more gradual, continuous extension (the minimal extension) is determined by this equation: (2.2) p(x) = U{p(n)ln EX}. The same formulae work for all multiple-valued functions p: w -> Pw. Functions like [= p are exactly characterized as being those continuous functions [: Pw-> Pw which in addition are distributive in the sense of these equations: [(xUy)=[(x)U[(y) and [(0)~ . 529 DATA TYPES AS LATTICES The sets 0 and w play special roles. When we consider them as elemel1ts of Pw we shall employ the notation: 1=0 and T=w. The element 1 is the most "undefined" integer, and T is the most "overdefined". All others are in between. One last general point on meaning: suppose x E Pw and k E x. Then k = (11, m) for suitable (uniquely determined) integers 11 and m. That is to say, every element of x can be regarded as an ordered pair; thus, x can be used as a reLation. Such an operation as (2.3) x; y ~ {(n, 1)13m.(n, mJ E x, (m, I) Ey} is then a continuous function of two variables that treats both x and y as relations. On the other hand we could define a quite different continuous function such as (2.4) x + Y = In + min E x, mEl') which treats both x and y arithmeticaLLy. The only reason we shall probably never write (x + y); x again is that the values of this perfectly well-defined continuous function are, for the most part, quite uninteresting. There is. however, no theoretical reason why we cannot use the same set with several different "meaning~" in the same formula. Of Course if we do so. it is to he expected that we will show the point of doing this in the special case. We turn now to the definition of the general language for defining all such functions. The syntax and semantics of the language LAMBDA are set out in Table 2. The syntax is indicated on the left, and the meanings of the combinations are shown on the right as subsets of Pw. This is the basic language and could have been given (less understandably) in terms of combinators (see Theorem 2.4). It is. however. a very primitive language, and we shall require many definitions before we can see why such functions as in (2.3) and (2.4) are themselves definable TABLE 2 The language LAMBDA O~{O) I x + I ={n+ l!nEx} x-l={ldn+1EX} z:Jx.y=:nE.l!OEz}U{mcyj3k.k+lf,.::) u(x) '" Imf3e" ';: X.(II, Ax.• - l(11. ~,m!l m} f II} ;[(',,/ x_I_I ~ __J_~ I I 530 DANA scan The definition has been left somewhat informal in hopes that it will be more understandable. In the above, T is any term of the language. LAMBDA is type-free and allows any combination to be made by substitution into the given functions. There is one primitive constant (0); therc are twO unary functions (x + I, x -1); there is one binary function (u(x) and one ternary function (z =>x, y); finally there is one variable binding operator (Ax.T). The first three equations have obvious sense. In the fourth, z => x, y is McCarthy's conditional expression (a test for zero). Next u(x) defines application (u is treated as a graph and x as a set), and Ax.T is functional abstraction (compare the definition of fun). In defining Ax.T, we use T[e,!x] as a shorthand for evaluating the term T when the variable x is given the value en_ Note that the functions are all multiple-valued. Thus we have such a result as: (2.5) (6UlO)+1=7UII. The partial character of subtraction has expression as: 0-1=.L. (2.6) We shall see how to define + and - in LAMBDA later. The conditional could also have been defined by cases: (2.7) z=>x, y=!; xUy We say that a LAMBDA-term T ifz=1-; if z =0; if 0" z r' 1-; if OEl r' O. defines a function of iTS free variables (at least). Other results depend on this fundamental proposition: THEOREM 2.1 (The continuity theorem). All LAMBDA definable functions are continuous. Once that is proved. we can tl<e Theorem 1.2 to establish: THEOREM 2.2 (The conversion theorem). The three basic principles (aj, ({3), (,.;) of A-conversion are ail valid in the model. By "model" here we of course understand the interpretation of the language where the semantics gives terms denotations in Pw according to the stated definition. Through this interpretation, more properly speaking, Pw becomes a model for the axioms (a), ({3), (";). Two well-known results of the calculus of A-conversion allow the reduction of functions of several variables to those of one, and the reduction of all the primitives to combinators (constants)-all this with the aid of the binary operation of application. THEOREM 2.3 (The reduction theorem). Any continuous function of k- variables can be writren as where u is a suitably chosen element of Pw. THHlRFM::,4 (The comhinator Iheorem). The LAMBDA-definable func- DATA TYPES AS LATrICES 531 tions can be generated (from variables) by iterated applicacion wlch the aid of these six constants: 0=0 SDc=Ax.x+ 1 pred = Ax.x - 1 cond = AxAyAz.z :0 x, y K = AxAy.x S = AUAvAx.u(x)(v(x)) But the result that makes all this model building and combinatory logic mathematically interesting concerns the so-called paradoxicaL comhinator defined by the equation: (2.8) Y = Au.( Ax. u(x (x )))(Ax. u(x (x»). THEOREM 2.5 (The first recursion theorem). If u is the graph of a continuous function f, then Y(u) = fix(f), the least fixed point of f. There are two points to note here: the fixed point is LAMBDA-definahle if f is; and Y defines a continuous operator. The word "recursion" is attached to the It would theorem because fixed points are employed to solve recursion equation~. not be correct to call the fixed-point theorem (Theorem 1.4) the recursion theorem since it only shows that fixed points exist and not how they are definabLe in a language. The second recursion theorem (in Kleene's terminology) IS related, but it involves Godel numbers as introduced in § 3. From this point on we sc:e no need to distinguish continuous functions from elements of Pw; a continuous function will be identified with irs graph. Note that u is a graph iff u = Ax.u(x), which is equivalent to Theorem 1.2 (Iii). For thiS reason (functions are graphs) we propose the name Graph Model for this model of the A-calculus. (There is more to LAMBDA than just A, howeveL) The identification of functions with graphs entails that the function space of all continuous functions from Pw into Pw is to be identified (one-to-one) with the subspace FUN = {ulu = Ax.u(x));; Pw. The identification is topological in that the subspace topology agree with the product topology on the function space. This is the topology of pointwise convergence and is closely connected with the lattice structure on lhe function space which is also defined pointwise (that is, argumentwisel. In the notation of A-ab~tracti()n we can express this as the extension of the axiom of extensionality in Table I of the Introduction. The laws in Tahle I are nol the only ones called (~*) valid in the model, however. We may also note such argumentwi:-.c distribution 532 DANA SCOTT laws as: (fU g)(x) = fIx) U g(x); (2.9) (Ax. T) U (Ax.lT) (2.10) = Ax.( T U IT) ; (fn g){x) = f(x)n g(x); (2.11 ) (2.12) (Ax. T) n (Ax.lT) = Ax.( Tn IT). In the above f and g must be graphs. It is also true that if :Ji ~ Pw, then (2.13) UUlf E :Ji}(x) = UU(x)!f E n but the same does not hold for (l. We state now a sequence of minor results which show why some simple functions and constants are LAMBDA-definable. (2.14) ~ (Ax.x(x))(Ax.x(x)); xU y = (Az.O) => x, y; (2.15) (Hint: 0, 1 E AZ.O) T = Y{Ax.OU (x + 1)); (2.16) (2.17) = x n y = Y{AfAxAy.x => (y => 0, f(x - l){y -1) + l)(x)(y). ,)~ The elements ~ and T are graphs, by the way, and we can characterize them as the only fixed points of the combinator K: a = Ax.a (2.18) Next we use the notation (xo, defined by: Xl>" iff " X"_I) a or ~ T. for the function fJ where p: w -> Pw is if i < n; P(i)~{: ifi?-;n. () = ~ (2.19) (2.20) (XI = Az.z => x, 1- (x, y) = Az.z => x, (z - I => y, (221) (222) a = 1- (x", XI, •.. , x") = Az.z => x", )~ (XI> .•• , x"){z - 1). Obviously we should formalize the subscript notation so that u. = u(x); then we find: (2.23) (xo,Xj,"',Xrt 1);= { X, if i < 11; .1 I 'f' I 2': n. This gives us the method of LAMBDA-defining finite sequenc", (in a quite natural way), and the next step is to consider infinite sequences. But these are just p where p: w ~ Pw is arbitrary. What we need then is a condition the functio~ expressible in the language equivalent to saying tl ~ for some (J. This is the same as r u = AxU{u,li E xl, 533 DATA TYPES AS LATTICES but the U- and set-notation is not part of LAMBDA. We arc forced into a recursive definition: (2.24) $ = Y(AsAuAz.z ::::> uo, s(At.u,+ ,)(z -I). This equation generalizes (2.22) and we have: (2.25) Thus the combinator $ "revalues" an element as a distributive function. This suggests introducing the A-notation for such functions by the equation: (2.26) An E W.T =$(Az. T[zl n]). With all these conventions LAMBDA-notation becomes very much like ordinary mathematicaJ notation without too much strain. Suppose that [ is any continuous function and a E Pw. We can define p: w ~ Pw in the ordinary way by primitive recursion where: p(O) = pIn a: + I) = [(n)(p(n». The question is: can we give a LAMBDA-definition for constants, say)? The answer is clear, for we can prove: (2.27) p (in terms of [ and a as p = Y(AuAn E W.1l ::::>a, fIn -1)(u(Il-I»). This already shows that a large part of the definitions of recursion theory can be given in this special language. Of course, simultaneous (primitive) recursions C(in be transcribed into LAMBDA with the aid of the ordered tuples of (2.22), (2.23) above. But we can go further and connect with partial (and general) recursive functions. We state first a definition. DEFINtTION. A continuous function [ of k-variablcs is computable iff the relationship m E[(e")(e,,,) ... (e". ,) is recursively enumerable (Le.) in the integer variables m, no, nl,· .. ,nk .. l. If q is a partial recursive function in the usual sense, then we can regard it as a mapping q: W -> W U 1-L}, where q(ll) = ~ means that q is undefined at Il. Saying that q is partial recursive is just to say that mE q(n) is Le. as a relation in nand m. It is easy to see that this is in turn equivalent to the recursive enumerability of the relationship mE q(e,,); and so our definition is formally a generalization of the usual one. But it is also intuitively reasonable. To "compute" y ~ [(x), in the one-variable case, we proceed by enumeration. First we begin the enumeration of all finite subsets en ~ x. For each of these f starts up an enumemtion of the set [(e,,); so we sit back and observe which mE [(en) by enumeration. The totality of all such m for all ell :; x forms in the end the set y. THEolu',M 2.6 (The definability theorem). For a k-ary colllilluous [ullclion [, Ihe [ollowillg are equivalelll: (i) [is computable; (ii) AxoAx, ... Ax,. ,.[(xo)(x,)· .. (x. ,) as a sel is r.e.; (iii) AxoAx,' .. Ax•. ,.[(xo)(x, I ... (x.,) is LAMBDA-defillable. 534 DANA SCOTT As a hint for the proof we may note that the method of (2.27) shows that all primitive recursive functions p have the corresponding p LAMBDA-definable. Next we remark that a nonempty r.e. set is the range of a primitive recursive function; but the range of p is p(T). which is clearly LAMBDA-definable. That any LAMBDA-definable set (graph) is r.e. is obvious from the definition of the language itself. More details are given in the Appendix. We may draw some interesting conclusions from the definability theorem. In the first place, we see that the countable collection RE c!; Pw of r.e. sets is closed under application and LAMBDA-definability. Indeed it forms a model for the A-calculus (axioms (a), «(3), (e) at least) and it also contains the arithmetical combinators. (Clearly there will be many intermediate submodels.) In the second place, we can see now how very easy it is to interpret A-calculus in ordinary arithmetical recursion theory by means of quite elementary operations on r.e. sets. Thus the equivalence of A-definability with partial recursiveness seems not to be all that good a piece of evidence for Church's Thesis. In his 1936 paper (a footnote on p. 346) Church says about A-definability: The fact. however, that two such widely different and (in the opinion of the author) equally natural definitions or effective calculability turn out to be equivalent adds lo (he strength of the reasons adduced below for believing thai they constitute as general a characterization of this notion a!> is consiqent with the usual intuitive understanding of it. The point never struck the present author as an especially telling one, and the reduction of A-calculus to r.e. theory shows that the divergence between the theories is not at all wide. Of course it is a pleasant surprise to see how many complicated things can be defined in pure A-calculus (without arithmetical combinators), but this fact cuts the wrong way as evidence for the thesis (we want stronger theories, not weaker ones). Post systems (or even first-order theories) are much better to mention in this connection, since they are obviously more inclusive in giving enumerations than Turing machines or Herbrand-Godel recursion equations. But the equivalence proofs are all so easy! What one would like to see is a "natural" definition where the equivalence with Le. is not just a mechanical exerClse involving a few tricks of coding. In the course of the development in this section we have stated many equations which are not found in Table 1, and which involve new combinators. In conclusion we would like to mention an equation about Y which holds in the model, which can be stated in pure A-calculus, and which cannot be proved by ordinary reduction (though we shall not try to jUSll(y this last statement here). In order to shorten the calculatior!!., we note from definition (2.8) that Y{u) = Y(Ay.u{y»); so by Theorem 2.5 this also equals u(Y(u». (2.28) Call the left side YIA!Ax.g(x)(j(x)) r and the right f". Now f" = Ax.g(x)(Y{g(x») r = Ax. Y(g(x)) r ~Ax.g(x){f" x) , r thus c= f", because is a least fixed point. On the other hand ~ Ax.g(x)(j'(xl). so j'(x) = g(x)( j'(x)). Thus f"{x) c= j'(x), because f"{x) is a least fixed point. As this holds for all x, we Sec that f" c= and so they are equal. There must be many other such equations. r; DATA TYPES AS LAlTlCES 5.15 3. Enumeration and degrees. A great advantage of the combinators from the formal point of view is that (bound) variables are eliminated in favor of "algebraic" combinations. The disadvantage is that the algebra is not all that pretty, as the combinations tend to get rather long and general laws are rather few. Nevertheless as a technical device it is mildly remarkable that we can have a notation for all Le. sets requiring so few primitives. In the model defined here the reduction to one combinator rests on a lemma about conditionals: cond(x)(y)(cond(x)(y)) = y. (3.1) Recall that cond (or ::» is a test for zero, so that: cond(x)(y)(O) = x. (3.2) This suggests that we lump all combinators of Theorem 2.4 into this one: (33) G = cond«suc, pred, cond, K, S))(0). We can then readily prove: THEOREM 3.1 (The generator theorem). All LAMBDA-definable elements can be obtained from G by iterated application. A distributive function f is said to be lotal iff f(n) E w for all nEw. As they come from obvious primitive recursive functions, we do not stop to write out LAMBDA-definitions of these three total functions: apply = An (3.4) E w.Am E w.(n, m) (35) op«n, m» = n (3.6) arg«n, m» = m. +1 The point of these auxiliary combinators con!~ers Our Godel numbering of the Le. sets. The number 0 will correspond to the generator G: while (n, m)+ j will correspond to the application of the nth set to the mth. This is formalized in the combinator val which is defined as the least fixed point of the equation: (3.7) val = Ak E w.k::> G, val(op(k -l»(val(arg(k -1))). This function accomplishes the enumeration as follows: THEOREM 3.2 (The enumeration theorem). The combinator val enumerafes the LAMBDA-definable elements in that RE = {val(nJln E w}. Further: (i) val(O) = G, (ii) val(apply(n )(m») = val(n )(val( m)). As a principal application of the Enumeration Theorem we may mention the following: suppose u is given as LAMBDA-definable. We look at its definition and rewrite it in terms of combinators-eventually in terms of G alone. Then using 0 and apply we write down the name of an integer corresponding to the combination-say, n. By Theorem .1.2 we see that we have effectively found from 536 DANA SCOTT the definition an integer such that val n = u. This remark can be strengthened by some numerology. (3.8) apply(O)(O) = 1 and val(l) = 0; (3.9) apply(O)(l) = 3 and val(3) = (sue, ... ); (3.lOl apply(3)( I) = 12 and val(12) = sue. Thus, define as the least fixed point: (3.11) num = An E w.n::> I, apply(l2)(num(n -I)), and derive the equation for all nEw: val(num(n)) = n. (3.12) We note that num is a primitive recursive (total) function. The combinator num allows us now to effectively find a LAMBDA-definition, corresponding to a given LAMBDA-definable element u, of an element v such that uniformly in the integer variable n we have val(v(n») = u(n). Further, v is a primitive recursive (total) function. This is the technique involved in the proof of K1eene's wellknown result: THEOREM 3.3 (The second recursion theorem). Take a LAMBDA-definable element t1 such that: (i) val(v(n» = Am E w. val(n)(apply(m)(num(m))), and then define a combinator by: (ii) ree = An E w.apply(v(n»(num(v(n»). Then we have a primitive recursive function with this fixed-point property: (iii) val(ree(n) = val(n)(ree(n». Note that if u is LAMBDA-definable, then we find first an n such that val(n) ~ u. Next we calculate k =ree(n). This effectively gives us an integer such that val(k) = u(k). Gode! numbers represent expressions (combinations in G), and val maps the numbers to the values denoted by the expressions in the model. The k just found thus represents an expression whose value is defined in terms of its own Godel number. In recursion theory there are many applications of this result. Another familiar argument shows: THEOREM 3.4 (The incompleteness theorem). The set of integers n such that val(n) ~ 1- is not r.e.; hence, there can be no effectively given formal system for enumerating all true equations between LAMBDA-terms, (A critic may sense here an application of Church's thesis in stating the meta theoretic consequence of the nonresult.) A few details of the proof can be given to see how the notation works. First let v be a (total) primitive recursive function such that: val(v(n») ~ n n val(n). and note that: n val(n)~1- iff nEval(n). DATA TYPES AS I.ATTICES 5.17 Call the set in question in Theorem.1.4 the sel b. If it were Le., then so would be: {n E wlv(n) €b} = (An E w.vln) n b =0 n, n)(T). That would mean having an integer k such that: val(k) = {n E w[v(n) E b}. But then: kEval(k) iff V(k)Eb iff val(v(k») = liff ki'val(k), which gives us a contradiction. This is the usual diagonal argument. The relationship val(n) = val(m) means that the expressions with Godel numbers 11 and III have the same value in the model. (This is not only not Le. but is a complete n~-predicate.) A total mapping can be regarded as a syntactical transformation on expressions defined via Godel numbers. Such a mapping p is called extensional if it has the property: val(p(n))=val(p(m)) whenever val(n)= val(m). The Myhill-Shepherdson theorem shows that extensional, syntactical mappings really depend on the values of the expressions. Precisely we have: THEOREM 3.5 (The completeness theorem for definability). If a (total) extensional mapping p is ,elban~ed-ADBMAL then there is a LAMBDAde/irwble II meh 'hat val(p(n») ~ q (vall 11)) for aLI nEW. Of course q is uniquely determined (because the values of q are given at least on the finite sets). Thus any attempt to define something new by means of some strange mapping on GodeJ numbers is bound to fail as long as it ~.i effective and extensional. The main part of the argument is concentrated on showing these mappings to bc continuous; that is why q exists. The preceding results indicate that the expected results on Le. sets are forthcoming in a smooth and unified manner in this setting. Some knowledge of Le. theory was presupposed. but analysis shows that the knowledge required is slight. The notion of primitive recursive functions should certainly be well understood together with standard examples. Partial functions need not be introduced separately since they are naturally incorporated into LAMBDA (the theory of multiple-valued functions). As a working definition of r.e. one can take either ··empty or the range of a primitive recursive function" or, more uniformly, "a set of the form {mI3n.m + 1 ~ plll)j where p is primitive recursive". A few obvious closure properties of Le. sets should then be proved, and then an adequate foundation for the discussion of LAMBDA will have been provided. The point of introducing LAMBDA is that further closure properties are more eas.ily expressed in a theory where equations can be variously interpreted a!\ involving numbers, functionals, etc., without becoming too heavily involved in intricate G6del numbering and encodings. Another useful feature of the present theory concerns the ease with which we can introduce relative recursiveness. 538 DANA scon As we have seen, {val(nJln E w} is an enumeration of all Le. sets. Suppose we add a new set a as a new constant. What are the sets enumerable in a" Answer: {val(n)(a)ln E w}, since in combinatory logic a parametet can always be factored out as an extra argument. Another way to put the point is this: for b to be enumeration reducible to a it is necessary and sufficient that b = u(a) where u E RE. This is word for word the definition given by Rogers (1967, pp. 146-147). What we have done is to put the theory of enumeration operators (FriedbergRogers and Myhill-Shepherdson) into a general setting in which the language LAMBDA not only provides definitions but also the basis of a calculus for demonstrating properties of the operarors defined. The algebraic style of this language throws a little light on the notion of enumeration degree. In the first place we can identify the degree of an element with the set of all objects reducible to it (rather than just those equivalent to it) and write Deg(a) = {u(aJlu ERE}. The set-theoretical inclusion is then the same as the partial ordering of degrees. What kind of a partially ordered set do we have? THEOREM 3.6 (The subalgebra theorem). The enumeration degrees are exacrly the finitely generated combinatory subalgebras of Pw. By "subalgebras" here we of COurse mean subsets containing G and closed under application (hence, they contain all of RE, the least subalgebra). Part of the assertion is that every finitely generated subalgebra has a single generator (under application). This fact is an easy extension of Theorem 3.1. Not very much seems to be known about enumeration degrees. Joins can obviously be formed using the pairing function (x. y) on sets. Each degree is a countable set; hence, it is trivial to obtain the existence of a sequence of degrees Whose infinite join is not a degree (not finitely generated). The intersection of subalgebras is a subalgebra-but it may not be a degree even starting with degrees. There are no minimal degrees above RE. but there are minimal pairs of degrees. Also for a given degree there are only countably many degrees minimal over it: but the question of whether the partial ordering of enumeration degrees is dense seems still to be open. Theorem 3.6 shows that the semilattice of enumeration degrees is naturally extendable to a complete lattice (the lattice of all subalgebras of Pw), but whether there is anything interesting to say about this complete lattice from the point of view of structure is not at all c1eaL Rogers has shown (1967, pp. 151-153) that Turing degrees can be defined in terms of enumeration degrees by restricting to special elements. In our style of notation we would define the space: TOT ~ {ulu = $(u) and 'tin E w.u(n lEW}, the space of all graphs of total functions. Then the system {Deg(u)lu ETOT} is isomorphic to the system of Turing degrees. Now there are many other interesting subsets of Pw. Whether the degree structurc of these various subsets is worth investigation is a question whose answer awaits some new ideas. Among the subsets of Pw with natural mathematical structure, we of course have FUN, which is a semigroup under o ~ AuAvAx.u(v(x)). It is. however, a rather DATA TYPES AS LAITICES 539 complicated semigroup. We introduce for its study three new combinators: (3.13) R = Ax.(O, x); (3.14) L (3.15) u= = Ax.X,(X2); Ax.xu:O (l, u, x,), u(x, lex,). THEOREM 3.7 (The semigroup theorem). The countable semigroup REnFUN of computable enumeration operators is finitely generated by R, Land G. The proof rests on the verification of two equations which pcrmit an application of Theorem 3.1: u (3.16) Lo 0.17) UoVoR=u(v). 0 R = Ax.u(x) Certainly the word problem for RE nFUN is unsolvable, indeed, not even recursively enumerable. Can the semigroup be generated by two generators by the way? 4. Retracts and dais types. Data can be structured in many ways: ordered tuples, lists, arrays, trees, streams, and even operahons and functions. The last point becomes clear if one thinks of parameters. We would normally hardly consider the pairing function AxAy.(x, y) as being in itself a piece of data. But if we treat the first variable as a parameter, then it can be specialized to a fixed value, say the element a, producing the function Ay.(a, y). This function is mare likely to be the output of some process and in itself can be considered as a datum. It is rather like one whDle row of a matrix. If we were to regard a two-argument function f as being a matrix, then its ath row would be exactly AY.f(a)ty). If s were a selection function, then, for example, Ay.f(s(y»(y) would represent the selection of one element out of each column of the matrix. This selection could be taken as a specialization of parameters in the operator AuAvAy.u(v(y »)(y). We have not been very definite here about the exact nature of the fixed a, f, or s, or the range of thc variable y or the range of values of the function f. The point is only to recall a few elements of structure and to suggest an abstract view of data going beyond the usual iterated arrays and trees. What then is a dara rypeo Answer: a type of data. That is to say, a collection of data that have been grouped together for reasons of similarity of structure or perhaps mere convenience. Thus the collection may very well be a mixed bag, but mare often than not canons of taste or demands of simplicity dictate an adherence to regularity. The grouping may be formed to eliminate irrelevant objects and focus the attention in other ways. It is frequently a matter of good organization that aids the understanding of complex definitions. In programming languages, one of the major reasons for making an explicit declaration of a data type (that is, the restriction of certain variables to certain "modes") is that the computed objects of that type can enjoy a special represemation in the machine that allows the manipulation of thesc objects via the chosen representation to be reasonably efficient. This is a very critical matter for good language design and good compiler writing. In this report, however, we cannot discuss the problems of representation, 540 DANA SCOTT important as they may be. Our objective here is conceptual organization, and we wish to show how such ideas, in the language for computable functions used here, can find the proper expression. Which are the data types that can be defined in LAMBDA') No final answer can be given since the number is infinite and inexhaustible. From one point of view, however, there is only one: Pw itself. It is the universal type and all other lypes are subtypes of it; so Pw plays a primary role in this exposition. But in a way it is too big, or at least too complex, since each of its elements can be used in so many different ways. When we specify a subtype the intention is to restrict attention to a special use. But even the various subtypes overlap, and so the same elements still get different uses. Style in writing definitions will usually make the differentiation clear though. The main innovation to be described in this section is the use of LAMBDA expressions to define types as well as elements. Certain expressions define retracts (or better: retraction mappings), and it is the ranges (or as we shall see: sels of finite poinls) of such retracts that form the groupings into types. Thus LAMBDA provides a calculus of type definitions including recursive type definitions. Examples will be explained both here and in the following sections. Note that types as retracts turn out to be types as lattices, that is, types of partial and many-valued objects. The problem of cutting these lattice types down to the perfect or complete objects is discu.\sed in § 6. Another view of types and functionality of mappings is presented in § 7. The notion of a retract comes from (analytic) topology, but it seems almost an accident that the idea can be applied in the present context. The word is employed not because there is some deep tie-up with topology but because it is short and rather descriptive. Three easy examples will motivate the general plan: (4.1 ) fun = AuAx.u(x); (4.2) pair = Au.(uo, u ,); (4.3) bool = Au.u 5 0, l. Here 5 is ,he doubly strict conditional defined by (404) z 5 x, Y = z => (z => x, T), (z => T, y), which has the property that if z is both zero and positive, then it takes the value ~ instead of the value x U y. DEFlNlTlON. An element a E Pw is called a retract iff it satisfies the equation a = a 0 a. Of course the o-notation is used for functional composition in the standard way: (4.5) u' v = Ax.u(v(x)). And it is quite simple to prove that each of the three combinators in (4.1 \-(4.3) is a retract according to the definition. But what is the point? Consider fun. No matter what u E Pw we take, fun\u) is (the graph of) a function. And if u already is (the graph of) a function, then u =fun(uj. That is to say, the range of fun is the same as the set of fixed points of fun is the same as the set of all (graphs of) functions. Any mapping a whose range and fixed-point set 541 DATA TYPES AS LATTICJ::S coincide satisfies a = a a, and conversely. A retract IS a mappmg which "retracts" the whole space onto its range and which is the identity mapping on its range. That is the import of the equation a = a a. Strictly speaking, the range is the retract and the mapping is the retraction, but for us the mapping is more important. (Note, however, that distinct retracts can have the same range.) We let the mapping stand in for the range. Thus the combinator fun represents in itself the concept of a function (continuous function on Pw into Pw). Similarlypair,represents the idea of a pair and bool the idea of being a boolean value as an element of {1-, 0, I, Tj, since we must think in the multiple-valued mode. What is curious (and, as we shall see, useful) is that all these retracts which are defining subspaces are at the same time elements of Pw, DEFINITION, If a is a retract, we write u: a for u = a(u) and Au: a.T for Au.T[a(u)/u). Since retracts are sets in Pw, we cannot use the ordinary membership symbol to signify that u belongs to the range of a; so we write u : a. The other notation with the A-operator restricts a function to the range of a. For f to be so restricted simply means f ~ f a. For the range of f to be contained in that of the retract a means f = a f. These algebraic equations will be found to be quite handy. We are going to have a calculus of retracts and mappings between them involving many operators on retracb yet to be discovered. Before we turn to this calculus, we recall the well-known connection between lattices and fixeLi points. THEOREM 4.1 (The lattice theorem). The fixed points of any continuous function form a complete lal/ice (under <;;); while those of a retract form a continuous lattice. We note further that by the embedding theorem (Theorem 1.6), it follows that any separable (by which we mean countably-based) continuous lattice i" a retract of Pw; hence, our universal space is indeed rich in retracts. A very odd point is that a = a a is a fixed-point equation itself (Au.u u is obviously continuous). Thus the retraction mappings form a complete lattice. Is this a continuous lattice') (Ershov has proved it is not; see the Appendix for a sketch.) A related question is solved positively in the next section. Actually the ordering of retracts under £ doe:;; not seem to be all that interesting; a more algebraic ordering is given by: DEFINITION. For retracts a and b we write a ():; b for a = a b = boa. The idea here should be clear: a <k b means that a is a retract of b, It is easy to prove the: THEOREM 4.2 (The partial ordering theorem). The retracts are partially ordered by <k. There do not seem to be any lattice properties of <k of a general nature. Note, however, that if retracts commute, a b = boa, then a b is the greatest lower bound under <k of a and b. Also if we have a sequence where both a" ;~ a"+) anLi a" <;; U,,+ I for all nEw, then U{a" In E w j is the upper bound for the a" under "", as can easily be argued from the definition by continuity of o. Certainly there is no "least" retract under <k. One has 1- = 1- a (recall: 1- = Ax.1-), but not a 1- = 1-. This last equation means more simply that U (1-) ~ 1-; that is, a is strict. For retracts strictnt:ss is thus equivalent to -.L ~( a, so we can ~ay that there is a least strict retract. The combinator I = Au.lt clearly represents the 0 0 0 0 0 0 0 0 0 0 0 542 DANA SCOTI largest retract (the whole space), and it is strict also. In a certain sense strictness can be assumed without loss of generality. For if a is not strict, let b = Ax.{nja(x) '" aCl)}. This function takes values in {1-, T} and is continuous because {x E Pwla(x) '" a(L)} is open. Next define: a* = Au.a(u) n b(u). This is a strict retract whose range is homeomorphic (and lattice isomorphic) to that of a. Note, however, that the mapping from a to a * is not continuous (or even monotonic). To have a more uniform notation {or retracts we shall often write nil. id, and seq for the combinators 1-, I, $. Two further retracts of interest are (4.6) open = Au.{mI3e" <;:: em.n (4.7) E u}; int = Au.u 5 0, int(u - i) 5 u, u. The range of open is lattice isomorphic to the lattice of open subsets of Pw; definition (4.6) is not a LAMBDA-definition of the retract, but such can be given. In (4.7) we intend int to be the least fixed point of the equation. By induction on the least element of u (if any) one proves that: r1- int(u) = if u =1-; if u E w; lU otherwise. T This retract wipes out the distinctions between multiple values, moving all above the singletons up to T; its range thus has a very simple structure. The retract int clearly generalizes boo!. The range of fun is homeomorphic to the space of all continuous functions from Pw into Pw; the range of pair. to the space of all ordered pairs; the range of seq, to the space of all infinite sequences. A combination like Au.int () seq(u) is a retract whose range is homeomorphic to the space of infinite sequences of elements from the range of int We now wish to introduce some operators that provide systematic ways of forming new combinations of retracts. There are three principal ones: (4.8) a (4.9) a ® b = Au.(a(u,,), b(u,)); (4.10) a EB b ~ o-lo b = Au.b 0 u 0 a; Au.uo 5(0. a(lI,)), (1. b(u, ». These equation, clearly generalize (4.1)-(4.3). Before we explain our operators, note these three equations which hold for arbitrary a, b, a b ' E Pw: I, (411) (a o->b)o (a' 0-> b')=(a'o a) o->(h (4.12) (a ®b) 0 (a'® b') (4.13) (a EB b) 0 (a' ~ EB b') = 0 b'): (a 0 a')® (b (u 0 a') (f; ( bob'). 0 b'); The reversal of nrder (a' 0 a) on the right-hand side of 14. I I) 'hould he remarked. DATA TYPES AS LAITICES 543 These equations will be used not only for properties of types (ranges of retracts) but also for the mappings between the types. THEOREM 4.3 (The function space theorem). Suppose a, b, a', b', care retracts. Then we have: (i) a >-> b is a retract, and it is strict if b is; (ii) v: a >-> b iff u = Ax : a. u (x) and II x : a. u(x) : b ; (iii) if a e>;; a' and be>;; b', then a >-> b e>;; a' >-> b'; (iv) iff:a >->bandj':a'>->b', thenf>->j':(b >-> a') >->(a >->b'); (v) iff: a >-> b andj': b >-> c, thenj' 0 f: a >-> c. Parts (i), (iii), (iv), and (v) can be proved using (4.8) and (4.11) in an algebraic (formal) fashion. It is (ii) that tells us what it all means: the range of a >-> b consists exactly of those functions which are restricted to (the range of) a and which have values in b. So we can read u : a 0-+ b in the normal w'ay; u is a (continuous) mapping from a into b. In technical jargon, we can say that the (strict) retracts and continuous functions form a category. In fact, it is equivalent to the category of separable continuous lattices and continuous maps. In this context, (iv) shows that >-> operates not only on spaces (retracts) but also on maps: it is a functor contravariant in the first argument and covariant in the second. Further categori- cal properties will emerge. THEOREM 4.4 (The product theorem). Suppose a, b, a', b' are retracts. Then we have: (i) a ® b is a retract, and it is strict if a and bare; (ii) u:a®biffu=(uo,uJ)anduo:aanduJ:b; (iii) if a e>;; a' and b e>;; b', then a ® b e>;; a' ® b'; (iv) iff: a >-> band f: a' >-> b', thenf®j': a ® a' >-> b ® b'. Again the operator proves to be a functor, but what is stated in Theorem 4.4 is not quite enough for the standard identification of ® as the categorical product. For this we need some additional combinators: 1st = Au.uo; (4.14) ~dos (4.15) Au.uJ: diag = Au.(u, u). (4.16) Then we have these properties: (4.17) (4.18) (4.19) (4.20) (4.21) 1st (a ® b):(a ® b) >-> a; 0 sod 0 (a ® b): (a ® b) >-> b; diagoa:a.;,-..,a@a; 1st 0 (f® f') = foist; sod o (f® f) = f o sod. Here a and b arc retracts and f and j' are functions. Now suppose a, b, and care retracts and f: c- a and g: c .... b. Let h=(f®g)odiagoc. 544 DANA SCOTT We can readily prove that: h:c>-->(a0b), and 1st h = f and 0 snd h = g. 0 Furthermore, h is the unique such function. It is this uniqueness and existence property of functions into a 0 b that identifies the construct as a product. There are important connections between >--> and 0. ·To state these we require some additional combinators: (4.22) eva/ = Au.uo(u,); curry = AuAxAy.u«x, yl). (4.23) If a, b, and c are retracts, the mapping properties are: (4.24) (4.25) evaJ o «b>-->c)0b):«b""'cl0b)>-->c curry ° «a 0 b) >--> c): «a 0 b)"'" cl >--> (a >--> (b >--> c». Suppose next that f: (a 0 b) >--> c and g: a >--> (b >--> c). We find that eval (curry(f) 0 b) = f 0 and curry(eva/ 0 (g 0 b» = g. This shows that our category of retracts is a Carlesian closed calegory, which means roughly that product spaces and function spaces within the category interact harmoniously. THEOREM 4.5 (The sum theorem). Suppose a, b, a', b' are relracls. Then we have (i) a Eb> b is a retracI, and il is always slricl; Oi) u: a Eb> b iff u = 1- or u = Tor u = (0, u,) and u,: a or u=(1, u,landu,:b; (iii) if a Of; a' and b Of; b', Ihen a Eb> b Of; a' Eb> b'; iivl iff: a >--> band f: a' >--> b', thenftB f: aEb> a' >--> b Eb>b'. There are several combinators associated with EB: 14.26) inlell = Ax.(O, xl; (4.27) inright = Ax.(1, xl; (4.28) Budell = Au.uo '3 "I, 1-; 14.29) nutrighf =Au.Uo '31-, u,; 14.30) 14.31 ) which = Au. 110 ; nul = Au.u,. DATA TYPl::S AS LATTICES 545 (The last two arc the samc as 1st and snd. but they will be used differently.) Wc find: (4.321 (a (Bbl o inlelt 0 a: a (4.33) la (Bb) 0 inright o b: b (4.34) a 0 outlelt 0 (a(Bb): (a(Bb) ~ a; (4.35) b 0 outright 0 (a(Bb) :(a(Bb) ~ b; (4.311) which 0 (a(Bb): (a (Bb) ~ bool: (4.37) a 0 out 0 (a(Ba):(a(Ba) ~ a; ~ ~ (a(Bbj; (a(Bb); where a and b are retracts. Most of these facts as they stand arc trivial until one sets down the relations between all these maps; but there arc too many to put them down here. Note, however, if a, b, and c are retracts and f: a ~ c and g: b ~ c, then if we let we havc: h:(a Bb)~c, and h 0 inlelt = f and h 0 inright = g. BUI, though h exists, it is nol unique. So a (Bb is not the categorical sum (coproduct). The author does not know a neat categorical characterization of this uperator. There would be no difficulty in extending ® and EB to more factor~ by expanding the range of indices from n, I to n, I, .... n -1. The explicit formulae need not be given; but if we write au (2) al ® ... (2) an _ J, we intend this expanded meaning rather than the iterated binary product. To understand sums and other fact~ about retracls, consider the least fixed point of this equation: (4.3HJ tree ~ nil(B(tree <8 tree). To be certain that tree is a retract, we need a general theorem: THEOREM 4.6 (The limit theorem). Suppose F is a conlinuous func/lOtI that maps retracls 10 relracts atld leI c ~ YIP). Thetl c is also a relract. If ItI additiotl F maps strict retracts to strict retracts and is monotone in the sense that a 0:; b implies F(a) ex F(b) for all (slricl) relrac/s a atld b. Ihetllhe ratlge of c is homeomorphic 10 Ihe itlverse limil of the ranges of Ihe slricl relracls F~I1-J for tI E w. This can be applied lD the case of (4.38) where F ~ AZ.nil(B(z @ z). Thus we can analyze tree as an inverse limit. This approach has the great advantage over the earlier method of the author where limit> were required in showing that tree exists. Here we use Y to give existence at once, and then apply Theorems 4.3-4.5 to figure out the nature of the retract. In Theorem 4.6. the fact that c is a retract can be reasoned as follows: 1- is a 546 DA!'\A SCOTT retract. Thus each F"(~) is a retract. We compute' eo e ~ u{r(~)il1" ~ )~(r{u ~ 1li)~ r{u w} U{F"(~)il1 E 0 0 FOI~)in E E w} ~ w} (Note: same w} 11.) C. In case F IS monotone and preserves strictness, then we can argue that each oc c. The retracts FO(~) are the proJeetioMs of e onto the terms of the limit. Of course rll) oc )~(mF if 11;'" m. The u: e can he put into a one-to-one correspondence (homeomorphism, lattice isomorphism) with the infinite sequences (1'0, v,,' .. ,V o , ' • -), where 1'0: )~("F and v" ~ ')I+"V()~ r Indeed ~ lu()~ "F and u ~ U{v"jl1 E w}. This is exactly the inverse limit construction. Retreating from generalities back to the example of tree. we can grant that it exists and is provably a retract. Two things in its range are ~ and T by Theorem 4.5(ii), but they are not so interesting. Now ~: nil, so by Theorem 4.5(ii) we have (0) ~ (0, ~): tree. Let uS think of this as the atom. What else can we have'.' If x, y : tree, then (x, y): tree 0 tree and so (1, (x, y): tree. Thus (the range of) tree contains an atom and is closed under a binary operation . .Note that the atomic and nonatomic trees are distinguished by which and that suitable constructor and destructor functions are definable on tree. But the space also comains infinite trees since we can solve for the least fixed point of: FO(~) "0 t~ (I, «0), t» and t: tree. (Why?) And there are many other examples of infinite elements in tree. A point to stress in this conqruction is that tree being LA,r-vtBDA-definable is computable, and thf're are many computable functions definable on or to (rhe range of) tree. All the "strucrura!" functions, for example, are computahle. These are functions which in other languages would be called isatom or construct or node, and they are all easily LAMBDA-definable. Just as with 'E. 0, ~ , they are !lot explicit in the notarion, but they 3J': jennabk ncvcrthtless. In the case o~ node, we could use finite sequences of Boolean values to pick out or name nodes. Thus solve for name = nil@bool c.S name, and then give a recursive definition of: node: name ,,....,. l tree ,..>---) tree J. Any' combination of retract preserving functors can be used in this game. For example: 14.39) lamb ~ inlt.±J(lambr, lamb). This loob innocent. but the range of lamb would give a quite different and not unattractive model for the A-calculus Iplus arithmetic). What wc do to investigate this model is to modify LAMBDA slightly by ~nicalper the ternary conditional =~ x, y by a quarternary onc w ~ x, y, z; otherwi"e the syntax of thl' language remains the same. The semantics, however, is (-l little more comr1ex. Let us use T, (T, p, e as syntactical variables for expre:ssiuns in the modified Jangu'lge. The ~emantic., is provided ."») a function y{ that maps the expressions of the language to their values in Ithe range ,]f) lamb. To be completely rigorous \I.e 547 DATA nPES AS LATfiCES abo have to confront the question of free and bound variables. For simplicity let us indcx the variables of the language by integers, and let us take the variables to be Vo, l' I, V:" •..• V rl , • • • • \-Ve cannot simply evaluate out an expression T to its value :if/TI until we know the values of thc free variables in T. The values of these variables will be given by an "environment" r which can be construed as a sequence of values in lamb. We can restrict these environments to the retract: env = At.lamb 0 seq( r). (4.40) When t: env, then 10 : lamb is the value that the environment gives to the variable Vn . We also need to employ a transformation on environments as follows: (4.41) l[xlnJ= Am E w.eq(n)(m) ::>x, 1m . Here eq is the primitive recursive function that is 0, if n, m are equal, and is I, otherwise, for n, mEw. The eflect of I[xl n] is to replace the nth term of the sequence I by the value of x, otherwise to leave the rest of the sequence unchanged.' To correspond with our use of very simple variables we have selected a simple notion of environment: in the semantics of more general languages it is customary to regard an environment as a function from the set of variables into the domain of denotahle values. The correct way to evaluate a term T given an environment t is to find Je[ TI( I). We use the brackets [and I here simply as an aid to the eye in keeping the syntactical part separated from the rest. The environment erHers as a functionargument in the usual way; thus we shall have: (4.42) 7f[ TI: env ~ 14431 X[v.,n(r) = I" lamb. X[OI(1) ~ inlelt(O) ,i'1'[ T+ Iii t) = which(7f[ TI( r)) 5 inlelt(out(.i'1'[ d( t)) + I), X[ T- Iii I) = which(7f[ Tnu)) 5 inlelt(outi}('1Ir1(I)) - I), ~ ~ X[O::> T, cr, pl(1) = lamb(which(X[Ollitil::> (out(7f[ 01(1)) ::> X[TI( I), ,i'1'[ o-I(r)), X[p Ii I)) Je[ T(fr ,nu) = which(X[ TI(r)) 5.L, out(:it[TI(1) )(<K[crl(/)) :it[Avo . Till) = inright(Ax : lamb.X[TII(t[xl n]). A good question is: why does :Ie ?t~ixe The answer is: hecause of the fixed-point theorem. If we rewrite the semantic equations 7f[TI(r) = ( ... ) in (43) by the equation X[ TI = AI: env( ... ), then :if is seen to be a function from expressions to vaJues in lamb. As the range of lamb is contained in Pw, we can say more broadly that }{E PwEJ:P, where Exp is the syntactical set of expressions and the exponential notation designates the set of all functions from Exp into Pw. This function set i~ a complete lattice because Pw is. Therefore if we read (4.43) as a definition by cases on Exp, then we can find J{as a suitable fixed point in the complete lattice Pw bP . Inde~ it is the fixed point of a continuous operator. 54H DANA SCOTI Actually we can regard Exp as being a sub.1ft of Pw to avoid dragging in other lattices. What we need is another recursi·ve dennition of a data type: exp (4.44) ~ iol(±)oil(±)exp(±)expc±J iexp@exp @ exp@ exp) (±) iexp @ exp I(±) (iol@ exp) • :"otc that there arc as many,ummands in (4.441 as there are clauses in (4.43). We can think of exp as giving the "abstract" syntax of the language. We use the integers to index the variables and the nil element to stand for the individual constant. Read (..1.,44) as saying that every expression is either a variable or a constant or the successor of an expression or the predecessor of an expression or the conditional formed from a tuple of expressions or the abstraction formed from a pair of a variable and an expression. We do not need in (4.44J to introduce special "symbols" for the successor, application, etc., because the separation by cases given by the (±) operation is sufficient to makc the distinctions. (That i, why the syntax is "abstract",) The point is that for recursive definitions it does not matter how we make the distinctions as long as they can be made. From thi' ne" point of view, we could rewrite (4.43) so as to show: (4.45) :f{: exp ~ (en,· ~ lamb), which is clearly more satisfactory-especially a, it is now clear that :f{is complltahie. And this is a method that can be generalizcd to many other languages. The method also shows why it is useful to allow function spaces as particular data types. Another example of this method can be illustrated, if the reader will recall the Godel numbering of § 3. It will be seen that there are similarities with the tree construction: instead of 0 and apply(nl(m), tree uses (0) and (I, (x, Note, however, that Godel numbers are finite whik tree has infinite objects. But the infinite ohjects are always limits of finite objects, so there are connections. l\\e discuss this again in § 6.) In particular. recursive definitions on Godel numbers. like that of val, have analogues on tree. Here is the companion of (3.7). y». (.l ..1hl vaal- Ax. treewhichlx' :;> G, vao'iisl(oull x l))(vaal(snd(oul(x))) I. We have va. I : tre·~ id, where of course 14.46) is taken as denning vaal a' the least fixed point. This is an example of a computable function betv;:,:en effectlvel) given retracts The LAMBDA-dennable elements of Pw are the computable elements in the range of vaal. Vv'e have discussed the category of retracb and continuou"J maps, but if they" are all LAMBDA-dennable, then they fall within the countable model RE. Thus there is another category of effectively given retracts and effectively gl\'ell cnntinuous maps. (Example~: tree, id, vaal. and all those retracts and map" generated by 81, @, and 0-,.) ThIs category \)cems to deserve the status of a generalized recursion theory: though this i", not ru :-.ay that as yet very /7lUch IS known annut it- In fact, the proper formulation may require an enriched categ.ory rather than a restricted one, Thus instead of confining attention to the computable retracts and computable maps, it might be beller to use the full category with all maps and to 'ingle out the computable ones lalso maybe the nnite onesJ by special predicates. In ellect we have avoided any methodolngical decisions by wt)rking in DATA TYPFS AS LATI ICES 54~ the unive"al space Pw and by defilllllg a notion when required-if po"ihle with the aid of LAMBDA. This makes it possihle to give all the necessary definitions and to prove the theorems without at first having to worry ahout axiomatic problems. 5. Closure operations and algebraic lattices. Given any family of (finitary) operations on a set (say, w) there is a closure operation defined on the subsets of that set obtained by forming the least subset including the given elements and closed under Ihe operations. Examples are very familiar from algebra: the subgroup generated by a set of elements. the subspace spanned by a set of vectors, the convex hull of a set of geometric points. We simplify matters here by restricting attention to closures operating on sels in Pw, but the idea is quite general. The main point about these "algebraic" closure operations-as distinguished from topological closure operations-is that they are cOlltinuous. Thus, in the case of subgroups, if an element belongs to the subgroup generated by some clements, then it also belongs to the subgroup generated hy finitely many of them. In the context of Pw we can state the characteristic condition very simply. DEFINITION. An element Q E Pw is ealled a closure operation iff it satisfies: [ ~ Q = Q 0 Q. We see by definition that a closure operation IS not only continuous, but it is also a retract. This is reasonable since the closure of the closure of a subset must be equal to the closure. To say of a function that J <; a. means that x <; a(x) for all x E Pw. In other words, every set is contained in its closure. (Note that closures are opposite 10 the "projections", those retracts where a <; J.) Among examples of closure operations we find J and T; tbe first has the most closed sets (fixed points). Ihe second has the least. (Note that T = w always is a fixed point of a closure operation; T = Ax.T is thus the most trivial closure operation.) The examples iun, open, int of § 4 are all closure operations (cf. (.J.1), (4.6), 14.7)). We remarked that iun is a retract, but the reader should prove in addition: u (51) <; Ax.u(x). for all u E Pw (d. Theorcm 1.2). We note that thl' fact can be rewritten in the language of retracts as: (5.2) the significance of which will emerge after we develop a bit of the theory of closure operations. Unfortunately the natural definition of the retract bool does not yield a closure operation. In this section we adopt this modification: (5.3) boool = Au.u 2 0, T + 1. The closed sets of 00001 are 1, 0, T+ 1, and T. Note that with any closure operation a, the function value aCt) is the least closed set (fixed point of a I including as a subset the given set x. Thus given any family '(j <; Pw of "closed" sets which is closed under the intersection of sllhfamilies, if we define (5.41 550 DANA SCOTT then this will be a closure operation provided it is continuous. This r.emark makes it easy to check that certain functions are closure operations if we can spot easily the family 'e of fixed points. Alas, the "natural" definition of ordered pairs (ef. (2.21)) leads to projections rather than closures. Here we must choose another: (5.5) [x. y] = {2nln with these inverse functions: E x}U (2m + 11m E y}. (5.6) [u]0={nI2n (5.7) [u]]={mI2m+IEu}. E u}, We shall find that the main advantage of these equations lies in the obvious equation: (5.8) which is not true for the other pairing functions. Of course we have: (5.9) [[x, Y]]o = x, (5.10) (lx, y]]1 = y. We shall not extend the idea of these new functions to triples and sequenees, though it is clear what to do. Abstraetly, an algebraic lattice is a complete lattice ;n which the isolated points are dense. An isolated (sometimes called: compact) point in a lattice is one that is not the limit (sup or l.u.b.) of any directed family of its proper subelements. This definition works in continuous lattices, but more generally it is better to say that if the isolated point is contained in a sup, then it is also contained in a finite subsup (a sup of a finite selection of elements out of the gi",en sup). In the case of the lattice of subgroups of a group. the isolated ones are the finitely generated subgroups. The isolated points of Pw are the finite sets eo. To say that isolated points are dense means that every element in the lattice is the sup of the isolated points it contains. The sequel to Theorem 4.1 for closure operations relates them to algebraic lattices. THEOREM 5.1 IThe algebraic lattice theorem). The fixed points of anv closure operation form an algebraic laltice. The proof is very easy if one notes that the isolated points of {xix = a(x)}, where a is a closure operation, are exactly the images ale") of the fimte sets in Pw. What makes Theorem 5.1 more interesting is the converse. THEOREM 5.2 (The representation theorem for algebraic lattices). Every algebraic lattice with a countable number of isolated points is isomorphic to the range of some closure operation. By Theorem 1.6 we know that the algebraic lattice is a retract, but a more dlfeet argument makes the closure property clear. Thus. let D be the algebraic lattice with {d"in E w} as the set of all isolated points with the indieated enumeration. We shall use the square notation with symbols J;: and U for the lallice ordering and sup. The desired closure operation is defined by: a(x) = {mldm J;: U{d"!n EX}}. :;:;1 DATA TYPES AS LATTICES It is an easy exercise to show that from the definition of "isolated' It follows that a is continuous; and from density, it follows that D is in a one-to-one order preserving correspondence with the fixed points of a. In the last section we introduced an algebra of retracts, much of which carries over to closure operations given the proper definitions. Without any change we can use Theorem 4.3 on function spaces, provided we check that the required retracts are closures. THEOREM 5.3 (The function space theorem for algebraic lattices). Suppose that a and b are closure operations; then so is a 0-> b. The proof comcs down to showing that: u(x) c; b(u(a(x»), (5.11 ) whenever a and b are closure operations. But this is easy by mono tonicity. Note that (5.1) is needed. For those interested in topology, one can give a construction of the isolated points of the function space which is much more direct than just taking the functions b 0 e" 0 a, which on the face of it do not tell us too much. But we shall not need this explicit construction here. The reason for changing the pairing functions is to be able to form products and sums of closure operations. In the case of products, the analoguc of 09 is straightforward: (5.12) a ~b = Au.[a([u]o), b([u],)]; while for sums using a' = Ax.O U a(x -I) + 1 and similarly for b' we write: (5) 3) a ~ b = Au.([uo]o => 0,0) U ([U], => 1, I) 5 [a'([u ]0), 1-]. k b'l[ u] I Ij. We can then establish WIth the aid of (5.8)-(5.10): THEOREM 5,4 (The product and sum theorem for algebraic lattices). Suppose that a and b are closure operations; then so are a [><: b and a (±: b. Analogues of the results in Theorems 4,4 and 4.5 carryover. Following the discussion in § 4, we can also show that the closure operations form a Cartesian closed category, which in some ways is better than the category of all retracts. What makes it better is the existence of a "universe". Every continuous operation generates a closure operation by just closing up the sets under the continuous function (as a set operation). We can institutionalize this thought by means of this definition: 15.14) v = AaAx. Y(Ay.x U aly». Clearly V is LAMBDA-definable, continuous, etc. A more understandable characterization would define V(a)(x) by this equation: (5.151 V(aj(xj = nlylx c; y and aly) c; y). These two definitions are easily seen to be equivalent. What i~ unexpected is the discovery (due in a different form to Peter Hancock and Pcr Martin-LiiO that V itself is a closure operation. THEOREM 5.S (The universe theorem for algebraic lattices). The/unction V is a closure operation WId its fixed points comprisl! the set 0/ all clo:-.ure operation.\. DANA seOIi Thus to say a IS a closure operation, write a : V. To have a mapping on closure operations, write f: Vo->V, Remark that 5,5 allows us to write V: V, It all secms rather circular. but it is Cj ui te consisten t. The category of separa te algebraic lattices "contains itself"-if we are careful to work through retracts of Pw, The proof of Theorem 5,5 requires a few steps, We note first that for all x, a E Pw: (5,16) x£; V(a}lx), Let v = V( a )(x), This is the least y with x U a (y) £; y, What is the least z with y U a(z) £; z" The answer is of course y, which proves: (517) Via )lV(a)(x)) = Via )(x), Thus Via) is always a closure operation, If a is already a closure operation, thcn clearly V(aJ(x) = a(x), Therefore we have shown: a= (5,181 V(a) jff a is a closure operation. But then by (5,16) and (5,17) we have by (5,181: (5,19) V(a) = V(Vla I) From (5.16) by monotonicity we see: (5201 a(xl£;a(Vla}lx))£; Vla)(xl. Hence by (5.1) we can derive: (5.21) a £; AX.a(x) £; Ax. VI a)(x) = V( ai. From (5.191 and (5.21) it follows that V itsclf is a closure operation. The operation V form~ the least closure operation contaimng a given element. and it shows that the lattice of closure operations i~ not only a retract of Pw but also an algebraic lattice. Since we can now use V a~ a retract, the earlier results hecome formulas: (5.22) (Aa : V.Ab : V.a[><]b): Vo->(Vo->V): (5.23) (Aa : V.Ab : V.al±Jb): Vo->(Vo->V): we can also state such functorial properties as: (5.24) (Aa : V.Ab : V.ao->b): Vo->(Vo->V). Using this style of notation we have: THEOREM 5.6 (The limit theorem for algebraic lattices). (,If: Vo->V.YU)): (Vo->V)o->V. In words: if f is a mapping on closure operations, then its least fixed point is also a closure operation. The proof of course holds with any retract in place of V. but we are more interested in applications to V. For example, note that V( 1.) = I. Now let f = Aa : V.a 0-> a. The least fixed point of this fis the limit of the sequence: 1.. I, 10->1, (10->1)0->(10->11, (10->110->(10->1))0->((10->110->(10->1))," 55, DATA TYI>L:S AS 1.ATIICFS and we ,ee that all these retract' are strict. This means Y(j) is nontrivial 111 that It has at least two fixed points (viz., 1- and T). But d = YI II must be the lea,t closure operation satisfying (5.25) and we have thus proved that there are "o"trivial algebraic lattices isomorphic to their own function spaces. This construction (which rests on ha,dly more thi:tn 15.2), since we could take d = YiAa.1 U (a 0-; a))) is much quicker than the inverse limit construction originally found by the author to give A -calculus models satisfymg I ~ t. There are many other fixed points of 15.25) besides this least closure operation, but their connection with inverse limits is not fully investigated. We note in conclusion that most constructions by fixed points give algebraic lattices (like lamb in 9 4). and so we could just as well do them in V if we remember to use [2J and GJ The one-point space is T ("0' nil), and so the connection with inverse limits via Theorem 4.6 is not as clear when nonstrict functions are u~ed. For m£'lny ,~e opru this may not make any difference. 6. Subsels and Ibeir classification. Retract' produce vcry special subsets of Pw: a retract al\Nay~ has a nonempty range which forms a lattice under c;. For example, the range of inl is 11-, T} U w. We often wish to eliminate .c and T: and with £'I retr£'lC! like tree the situation is more complex. since com hi nations likt.' (I, (( I, (1.. (0»)), T) might require elimination. In these two cases the method Is simple. Consider these two functions: mid = Ax :inl. xS 0, (6.1 ) ° 16.21 perf·- Au : tree.which(u) 5 0, :>lperflfslloullu))))(perflsndloullu)111 where j, is a special combinator: :> = AxAdx => I y => 0, TI, T) U (y 16.3) We flnd that :>: (0 =- h: inCjmid(x 1-= ° :> 11. T 1- .L .L T 0 1- (J T T T T T The queslioll l~: what suh~ct i~ {u: treejperf( u.! pert i... dcflnt:'d recur')ivcly. Vv'c can ~LC perfI ,I ~ L perM!. (,. whell t, \,. tree. The aLolnic [rCL'I~ I~.\'LTY Ix => 0, TI. TI O} In the C3"e of (rec.... nok first thi ... behavior o( ------r- NilW co perffTI \'))1 c ~ 1. -=-= OJ? that perfl(O)1 ~ O. :>Iperll., )I(perlfyll trce. a"'ldc ff(l11l Tor !-. IS either ah)l1lic or cl pall' Ilt tree .... "pcrkct" (that i.... perfl(O))-OJ.;\ Jifllll' lrL'L' \\lllch dnl·"Il11t 554 DANA SCOTI' contain ~ O[ T is perfect-as ~'e can see inductively using the table above for ~. An infinite tree is never perfect: either some hraneh ends in T and perf maps it tD T, or T is never reached and perf maps it to L Thus the subset in question is Ihen seen to be the set of finite trees generated from the alom by pairing. This is clearly a desirable subset, and it is sorted out by a function with a simple recursive definition. The general question is: what subsets can be characterized by equations" The answer can be given by reference 10 the topology of Pw. DEFINITIOI". Let Cl:\ be the class of open subsets of Pw, and (\: be the class of closed subsets. Further let 'B be the class of all (finite) Boolean combinations of open sets. We recall from § 1 that V E elj just in case for all x E Pw, we have x E V if and only if some finite subset of x is in U The elass of open sets contains 0 and Pw and is closed under finite intersection and arbitrary union; in fact. it can be generated by these two closure conditions from subsets of the special form {x E Pwln E x) for the various nEw. An open set is always monotonic (whenever x E U and x <;; y, then y E V), so that every nonempty V E tij has T E U Another characterization of openness can be gjven by continuous functions. Suppose V E Cl:\. Define I: Pw -> {~, T} so that V={xl/(x) = T}; then I is continuous. Conversely, if such an I is continuous, then V is open. But if we do not assume the range of I is included in {l, T}, this is not true. For the case of general functions we know that I is continuous if and only if {xl/(x) E V} is open for all open V Tbis defines continuity in terms of openness, but we can turn it the other way around: THF.OREM 6.1 (The (~ theorem). The open subs:ets of Pw are exactly the St'ls of the form: (xl/(x) ;20}, where f: Pw ---) Pw is continuous. We could have written 0 E I{x) or the equation I(x) n IJ ~ 0 instead of l(x);2 O. Note tbat in case I: Pw -> {~, T}, then I(X);2 0 IS equivalent to I(x I ~ T. Also any otber integer could have been lIsed in place of O. We can say that {xiO EX} is the rypical open set, and that every other open set can be obtained as an inverse image of the typical set by a continuou"i function. We shall extend Ihis pattern to other classes, especially looking for equations. In the case of openness an inequality could also be used, giving as the typical set {xix '" .}~ BlIl since closed sets are just the complements of open sets, this remark gives us: TflEOR I'M 6.2 (The if theorem). nte closed subsets 01 Pw are exactly rhe sers 01 rhe lorm: {xl/lx)=H where f: Pw -J> Pw is continuous. Aside from ~xi = ~}, we eould have used Ixlx C ll} as the typical closed set where a E Pw is any element whatsoever aside from T. Thi"i T has. hy the way. a DATA TYPES AS LATfICES Thus IT} is a countable intersection of open sets, otherwise called a l~o-set. There are of course many other GJ,-sets, but {T} is the typical one: TlifORfM 6.3 (The 6l, theorem). The countable intersections of open suhsets of Pw are exactly the sets of the form: {xlf(x) = T}, where f: Pw -+ Pw is continuous. It may not be obvious that every (IJ.-set has this form. Certainly, as we have remarked, every CIl-set has this form. Thus if W is a OJ,. we have: and further, Un = {xIMx) = T), where the frl are suitably chosen continuous functions. Define the function g hy the equation: g(x)={(n, m)!mEfn(x)). Clearly g is continuous. and we have: W={xlg(x)=T}, as desired. We tet ii' (lj denote the class of all sets of the form C u. where CEil and U \0 (lj, Similarly fur iI (II,. Now {xix <; O} is closed and {xix ;2 O} is open. Thus {OJ E iI (I ,~C This set is typical. THEOREM 6.4 (The iI (Il theorem). The sets that are intersections of closed sets with open sets are exactly the sets of the form: n n n n {xlfix) = or, where f: Pw -+ Pw is continuous. Again it may not be obvious that every;\- n()i sd has this form. Vv'c can write: C= {xlflx) ~ ~}, and u={xlglx)=O}. where lE /\' and U ;e:_ }~( and the continuous hlx) = {2n + and h~ and g arc :-.uitably chosen_ DdlnL' lin E flxl} U {2nln \0 gi.r II, that It i" (,:ontinuou..,. We hllloC: rcmar~ dC"in:d. It f I" ca:-.y to "'Cl' that Id t ;\" n(~ if (' j<; linite, but ;11 gennal {a}, /\. (1 ('I)/i. In DANA SCOTT case a is infinite but not equal to T (say, a = (Il III > 0) = T + 1 ). then fa} is typical in its class, THEORHI 6.5 (The II' (II, theorem). The sets thai are illterseniolll of closed n sels wilh countable inlersections of open secs are exaclly Ihe sels of che jorm: {x!f(x)=a), where f: Pw -> Pw is conlinuous and a is a fixed infinice seC flOC equal 10 T. n n Note that (II' (IJ), is the same class as t5 (IJ" so we see by Theorem 6,4 that a good choice of a is An E w.O. There is no single subset ot Pw typical for ~\, which can be viewed as the finite unionI of sets from the class lI·n ill. THEOREM 6.6 (The Q:1 theorem) The sels thaC are Boolean combinaCions of open seCs are exactly the seCs of the form: (xlf(x) En where f: Pw -4- Pw is continuous and /.;' is a finite set of.finite elements of Pw. To see that every ~ set has this form, suppose that \/= II' nill. W"U W1U .. ·U w" where each W, E wheref,: Pw {lo 0, T} is continuous. Then define: -> I, We can write: g(xl=(21+jljEjilx)n(0, l}. i<lll. and note that g i~ continuous. Let t: = {y ~ {m:m < 2n)13i < n.li E y. 1; + I t. y}: we have: V={x!glxIE f'} as desired. THEOREM fl.? (The '..l\'i theorem). The sets that are countable intersections of Boolean combi,ratiom of open sets are exactly the sets of the form: (xIIIXI = g(xl), r where g; Pw ~ Pw are continuous. charucteriz<-ltion theorems, Thl:' is clearly the most interesting of thc~ : because equations like f(x) = g( x) turn up all the till1l' anu the collection is a very rich totJlitv of suhsets of Pw. It includes all the retract'l, ... inee the\' arc of the form {xix '-=- (dx)}. And mUl'h more. That every ~uch ~ct ill Thcnrem ().? is "\~ follows from lhese logical Ir_ansformations: {x': I (x) "- g I r \) ~ 1.r IV' Ii ,,= IJJ[ n .,:: ~ 1~r II--(l I {x I" " (i x ) .(,-'J. n F gl.\" JJI rr <I. " " gil II U { r," " II r I. "t gil III 557 DATA TYPES AS LATTICES On the other hand, we can see that (,f U(~)u is exactly lB" Because, in VIew o{ Theorem 6,6. ~'" the class of counlable unions of lB-sets, is exactly (I\' (11)" The remark we want to make then {allows by taking complements. Now leIS be an arbilrary lB.-set. We ean write: n r 5= n ({xlf,,(x)=OjU{xlgJrl=O)J, ,,=(j where f., g" : Pw -> {1-. O. T} are continuous. Now let u, v be continuous {unctions which on {1-, 0, T} realize these two tables: u 1- a T v 1- a T 1- 1- a 1- a a 0 T. 0 0 0 0' 0 0' 0' 0' 0' T where 0' = au 1. This x, y E {1-, 0, T): IS a T 0 0 0' 0' an exerCise in many-valued logic, and we find {or = v(x)(y) iff x = 0 or y ~ Thus define continuous (unctions f' and g' such f' = AxAn E w.u(f"lx»)(g,,(x». g' = AxAn E w.v(f,,(x»)(g,,(x), u(x)(y) O. and we find: 5 ~ (xlJ'(x) = g'(x)} as desired. This is as far as we can go with equations. More complicated sets can be or analytic sets can be fmr in thl.; defined using quantifiers, for example the (arm: '1; (x!3yI(x)(y) = g(x)(y)}, and their complements, the Il: sets, in the form (xj\ty 3z.h(x)(y)(z) = O}, with continuous f. g, h. For the three classes we then have as "typical" sets those shown in Table 3. It should be remarked that IB, contains all the closed sets in the Can lor space topology on Pw (that is, the topology obtained when it is regarded as the infinite product of discrete two-point spaces). Therefore the sets {or the two topologies on Pw are the same. Hence, since we know for Cantor space that ~ = L: nn; is the class of Borel sels, we can conclude that the two topologies on Pw have the same Borel sets. (That is. in both cases is the Boolean a-algebra gcnerated (rom the open scts.) L: u: 55H DANA s('un" TABLE J Classes and typical selS ----------Typical ~( {xIOE ""r'iru III IT! (~, n~; ~JO 'l..\,o; r: II: ~te; ,I {Of IT+t} {uluo"'=' utI (uI3y.uo(Y) ~ u,(yl} lulVy 3z.u(y)lzl ~ OJ Returning now to the example involving trees mentioned at the beginning of this section, we see that the set of perfect (finite) trees can be written in the form: {xix ~ tree(x), penix) ~ OJ ~ {xl(x, penix) = (tree(x), 0»); thus it is a !B,-ser. (Note that!B, are ohviously closed under finite intersection by the ordered pair method just illustrated; thal they are closed under finite union is a little messier to make explicit, but the essential idea is containeu in the proof of Theorem 6.7.1 As another example, we might wish to allow infinite trees hut not the strange tree T. Consider the following function; 16.4) top = All ; tree.whichl u) c;~, top(fst(out( I< I)) U toplsnd(out( u))). We can show that top: Pw -> ,~{ TJ. For a tree u the equation top( u) = ~ means that it does flol contain T, or as we might say: it is topless. The topless trees form a closed subset of the subspace of trees. (An interesting retract is the function AI<.tree( u) U top(") whose range consists exactly of the 'topless trees plus one exceptional tree T.) Such a closed suhset of (the range of) a retract is a kind of semi/alliee. (We shall not introduce a precise definition 'here.) Every directed subset has a limit (le"t upper bound) and every pair with an upper bound has a least upper hound. But generally least upper bounds do not have to exist within the semi lattice. The type of domains that interest us heeome continuous lattices with the addition of a rep element T large;- than all the other el ment~. The elimination of T i~ done with a function like top of our example. This is convincing evidence to the author that an independent theory of semilattices is quite unnecessary: they can all be derived from lattices. The prohlem is simply to define the top-culling operation, then restriction to the "topless" clements is indic()ted In this way all the eOrIstructions are kept within hy an equatiorI Ilike topl 1<) = ~). the control of a smooth-running theory ha!'cd on LAMBDA. Thi~ point seems to be important if one wants to keep track of which functions are computahle. An aspect ()f the problem of clas~if cation treated in ~iht -;ection which ha~ not heen given c!()se enough attention i~ the explicitly constructive way of veritying thl' clo~ure properties of the c1a~ cs. Consider the das~ \. !.\'i' for examplL'. Let B he the typical set as ~hown in Tahle 3. Then whatever r'~ Pw we choose, the ~d I'i/lx) ~ B} DATA TYPES AS LATTICES 559 is a llJ,-set and every such set has this form. Thus the {s index the elements of the class. Suppose f, g E Pw. What we should look for are two LAMBDA-definable combinators such that union(f)( g) and interi f)(g) give the functions that index the union and intersection of the sets determined by f and g. That is, we want: {xlunionif)(g)(x) E B} = {xlfix) E B} U {xlg(x) E B}. II should be possible to extract the precise definition from the outline of the proofs given above, but in general this matter needs more investigation. There may very well be certain classes where such operations are not constructive, even though the classes are simply defined. 7. Total functions and functionality. There is an inevitable conflict between the concepts of total and partial functions: we desire the former, but it is the latter we usually get. Total functions are better because they are "well-defined" at all their arguments, but the rub is that there is no general way of deciding when a definition is going to be well-defined in all its uses. In analysis we have singularities, and in recursion theory we have endless, nonfinishingcomputations. In the present theory we have in effect evaded the question in two ways. First we have embraced the partial function as the nOrm. But secondly, and possibly confusingly, the multiple-valued functions are normal, total functions from Pw into Pw. The point, of course, is that we are making a modelof the partial functions in terms of ordinary mathematical functions. But note that the success of the model lies in not using arbitrary functions: it is only the continuous functions that correspond to the kind of partial functions we wanted to study. It would be a mistake to think of the variables in A-calculus as ranging over arbitrary functions-and this mistake was made by both Church and Curry. The fixed-point operator Y shows that we must restrict attention to functions which do have fixed points. It is certainly the case that Pwis not the only model for the A-calculus, but it is a very satisfactory model and is rich enough to illustrate what can and what cannot be done with partial functions. Whatever the pleasures of partial [unctions (and the multiple-valued ones, too), the desire for total functions remains. Take the integers. We are more interested in w than w U{.l, T}. Since the multiple values.l and T are but two in number, it is easy to avoid them. The problem becomes tiresome in considering functions, however. The lattice represented by the retract int ~ int is much too large, in that there are as many nOn total functions in this domain as totai ones. The aim of the present section is to introduce an interpretation of a theory of functionality in the model Pw that provides a convenient way of restricting attention to the functions (or other objects) that are total in the desired sense. The theory of functionality is rather like proposals of Curry, but not quite the same for importlint reasons as we will see. In the theory of retracts of §§ 5 and 6, the plan of "restricting attention" was thl: very simple one of restricting to a Jubset. It was made notationally simple as the subsets in question could be parameterized by continuous functions. The retraction mappings stand in for their ranges. Even better, certain continuous functions act on these retractions as space-forming functors (such as EB and ?--:I'), which gives greater notational simplicity because one language is able to ~erv for several ,60 DANA scurr tasks. When we pass to the theory of total functions, this same kind of simplicity is no longer possible owing to an increase in quantifier complexity in the necessary deti'nitions. (This remark is made definite below.) Another point where there is some loss of simplicity concerns the representation of entities in Pw: subsets will no longer be enough, sincc we will need quotients of subsets. This is not a very startling point. Many constructions are affected in a natural way via equivalence classes. An equivalence relation makes you blind to certain distinctions. It may be easier also to remain a bit blind than to search for the most beautiful representative of an equivalence class: there may be nothing to choose between several candidates, and it can cost too much effort to attempt a choice. Thus our first agreement is that for many purposes a kind of object can be taken as a set of equivalence classes for an equivalence relation on a subset of Pw. Because Pw is closed under the pairing function AxAy.(x, y), we shalt construe relations on subsets of Pw as subsets of Pw all of whose elements are ordered pairs. That is, a relation A satisfies this inclusion: A,; {(x, y)lx, y E Pw). 17. I) DEFINITION. A (restricted) equivalence ,elation on Pw is a symmetric and transitive relation on Pw. Such relations are restricted because they are only reflexive on their domains-which are the same as their ranges-and these are the subsets with which the relations are concerned. We shall write x A y for (x, y) E A and x : A for . x A x. What we assume about these relations is the following: xA Y (7.2) x Ay (7.31 In case and implies yAz yA x. imply x A z. a is a retract, we introduce an equivalence relation to correspond: E a ~ {(x, x)lx: a). (7.4) This is the identity relation restricted to the range of a. Such relalions (for obvious reasons) and many others satisfy an additional intersection property: (7.5 ) xAy and xAz imply xA (ynz). We shall not generally assume 0.5) in this short discussion, but it is often convenient. Each equivalence relation represents a space: the space of all its equivalence classes. Such spaces form a category more extensive than the category of retracts studied above. The familiar functors can be extended to this larger category by these definitions: 17.hl 17.7) 17.X) A ~ B = {(Ax. ul x), Ax. vlx»1 ulx)Bvly) whenever x A y}, -A x B = {((x, x'), (y, y'»lx A y and x' B y'j, A +B ~ {((J, x), (0, y)!x A y}U {« 1, .'). (I, Y'lilx' B vl DATA TYPES AS LATTICES 561 THEOREM 7.1 (The closure theorem). If A and B are restricted equivalence relations, then so are A -> B, A x B and A + B. We find: (i) f:A->B iff f=Ax.fix) and whenever xAy, then f(x) Bfiy), in particular: (ii) iff: A -> B and x : A, then f(x) : B; furthermore, (iii) u: A x B iff u = ("'" u\) and Uo: A and u\ : B; (iv) u: A +B iff either u = (0, u\) and u,:A or u = (I, u\) and u\: B. It follows easily from 7.1 that the restricted equivalence relations form a Cartesian closed category which-in distinction to the category of retracts-has disjoint sums (or coprod"cts as they are usually called in category theory). This result is probably a special case of a more general theorem. The point is that Pw itself is a space in a Cartcsian closed category (that of continuous lattices and continuous maps) and it contains as subs paces the Boolean space and especially its own function space and Cartesian square. In this circumstance any such rich space must be such that its restricted equivalence relations again form a good category. Our construction is not strictly categorical in nature, as we have used the elements of Pw and have relied on being able to form arbitrary subsets (arbitrary relations). But a more abstract formulation must be possible, The connection with the category of retracts is indicated in the next theorem. THEOREM 7.2 (The isomorphism theorem). If a and b are retracts, we have the following isomorphisms and identities relating the spaces: (i) Ea,=(x,y)la(x)=a(y)}; (ii) E a_ b '= E a -> E,; (iii) E a0 , = E a x E,; (iv) Ea®b = E a + E b U {(L .1), (T, T)}. Part (iv) is not categorical in nature as it stands, but (ii) amd (iii) indicate that E is a functor from the category of retracts into the category of equivalence relations that shows that the former is a full sub-Cartesian-c1osed category of the latter. We cannot pursue the categorical questions here, but note that there are many subcategories that might be of interest; for example, the equivalence relations with the intersection property are closed under ->, x, and +. Returning to the question of total functions we introduce this notation: (7.9) N = {{n, n)ln E w}. This is the type of the integers without and 1. and T, i.e., the total integers. We note that: (7.10) Thus N is a (7.11) ~,-set. N = {ul u = (uo, uo), u" = inl(uo), mid(uo) = OJ. What is N N? We see: f: N-> N if! -> f:lunandf(nlE w whenever nEw. This N -> N is indeed the type of all total functions from w into w. It can be shown that N -> N is also a QI,-set: good. But what is (N -> N) -> N? This is no longcr a lB,-set, the best we can say is By Theorem 7.1 it corresponds to the type of all 11;. 562 DANA SCOTT (extensional) continuous total functions from N -> N into N. (The condition on A and B on rhe right side of (7.6) makes the concept of function embodied in A -> B extn~ioal, since the functions are meant to preserve the equi\'alence relations.) A more precise discussion identifies N -> N as a topological space, usually called the Baire space. If we introduce the finite discrete spaces by: (7.12) Nk = {tn, n)ln < k}, then N -> N, can also be identified with a topological space, usually called rhe CantDr space. In this identification we find at the next type, say either (N -> N) -> N or (N -> N,) -> N" that elements correspond to the usual notion of continuous function defined in topological terms. However, these higher type spaces are not at all conveniently taken as topological spaces. Certain of them can be identified as limit spaces according to the work of Hyland, and for these ->, x, and + have the natural interpretation. We cannot enter into these details here, but we can remark that the higher type spaces become ever more complicated. Thus ((N -> N) -> N)-> N is a n;-sct and each -> will add another quantifier to the cjefinition. This is reasonable, because to say that a function is total is to say that all its values arc well-behaved. But if its domam is a complex space, this statement of totality is even more complex. Despite this complexity, however, it is possible to sort out what kind of mapping properties many funclions have. We shall mention a few of the combinators. THEOREM 7.3 (The functionality theorem). The combinators J, K, and S enjoy the following functionality properties which hold for all equivalence relations A, B, C: (i) J: A -> A; (ii) K:A->(B->A); (iii) S: (A -> (B -> C» -> ((A -> B) -> (A -> C»). Furthermore, these combinators are wliquely determined by: these properties. Let us check that 5 satisfies (iii). Suppose that: f(A -> (B -> C»)f· We must show that: S(n((A -> B)-> (A -> C»S(f'). To this end suppose that: g(A -> B)g'. S([)(g) (A -> C) S([')(g'). We must show that: To this end suppose that: xA x'. We must show that: S([I(K )(x )CS([')(g')(x'). DATA TYPES AS LATTICES 563 Now by definition of the combinator S we have: Si!J(g)(x) = f(x)(g(x)), S(f)(g')(x') = ((x')ig'(x')). By assumptions on g, g' and on x, x', we know: g(x) B g'ix'l. By assumptions on f, f' and on x, x', we know: f(x)(B ... O{ix'). The desired conclusion now follows when we note such combinations as S(!J and S(f)(g) are indeed functions. (We are using Theorem 7.1(i) several times in this case.) In the ease of the converse. let us suppose by way of example that k E Pw is such that k:A ... (B ... A) holds for all equivalence relations A and B. By specializing to, say, the identity relation we see that whatever a E Pw we take, both k and k(a) are functions. To establish that k = K we need to show that the equation: k(a)(b) =a holds for all a, bE Pw. This is easy to prove, for we have only to set: A = {(a, and a)} B={(b, b)}, and the equation follows at once. Not all proofs are quite so easy, however. In the case of lhe combinator S it is not strictly true to say that Theorem 7.3(iii) determines it outright. The exact formulation s(f) ~ s(AxAy.f(x)(y)) and s(!J(g) IS this: if,. E Pw is such that: = s(f)(Ax.g(x)) for all f, & , Pw; and if s: (A ... (8 ... O)"'(IA ... B) ... (A ... ell for all A, B, C, then s ~ S. In other words, we need to know that s converts its first two arguments into functions with the right number of places before we can say that its explicit functionality identities as being the combinator S. In Hindley, Lercher and Seldin (1972) they show that the functionality property: (7.13) AfAg·fo g: (S ... 0'" IIA ... BI'" (A ... Cl). follows from Theorem 7. Uit) and Theorem 7.311i) and (iii) in view of the identlty: (7.14 ) (see Appendix AI. AfAg.f 0 g ~ S(K(S))(KI DANA SCOTT A more Interesting result concerns the Iteralor.\ JetlncJ as lollowOj: (7.15) Zo= Ajh.x, (7.16) Z"+I = AjAxIIZ"lfJ(xl). In other words, Z"If)(x) ~ Fix), These natural comblnators can be typed very easily, but Gordon Plotkin has sbown that tbe obvious ,typing actually characterize~ them. THEOREM ~,7 (Tbe iterator theorem), The combinators Z" enjoy the following functionality propert, which holds for all equivalence relations A: Z" : (A (i) -> A) -> (A -> A I. Further, If any eletrlenl IE Pw satisfies (i) for all A, then it must be one of the iterators, provided that l(f) = z(AxI(,)) holds jor all f E Pw. That eacb of the Z" satisfies Theorem 7.4li) is obvious. Suppose z were another such clement. Then clearly: I Suppose f anJ x are tlxeJ for = AfAr,zlflix) the moment. Let: A ={(f"(x), f"(x»ln where we call ~upose -> w}, in £tdditiol1 that: f Then f: A E ~ Ax.f(x), A is clear, and so zl f): A -> A also, But x: A, therefore z( [)(x) = Fix), I'm some n ~ w, because zr/)(x): A. Tbe trouble with this easy part of the <irglllllcllt j, th<it the integer n depends on I and x. ,",,'hat WI; must :-,how ~i independent of f and x, then z ~ Z" will follow. Plotkin's method for this case is to introduce functions: 17,1 jJ if, ~ Ax,{ij, k + I)I( j, ~()me independent successor k) ex). Note that: i7,I~) m((., O)=((/,m) J, I (T J ~ if/o-I'; .f ., '1""/. It then follows that: i7 19) Having these it.lentitlcs, we return 10 the argument. From what we saw before, given j e. w, there is an n/ such that: Take any two J, j' '= w. We also kno\\' there i" an fl'~ thai it i:-. OJ whl'rc: 5fiS Dr\1,\ TYPES AS LA rnCES in vie"! of (7. J <)). But since (f, IT, Il j ) £ (T) U fJi" and U (/, ni') c:: (j, (T/ r; u) U ur, we have: II) U (}', Ill. It follows that and so they are all equal. This determines the fixed nEw we want. Suppose that both! and x are finite sets in Pw. Choose f> max (!U x). Let A this rime bL: the least equivaicilce relation such that: rlx) Ar(x)Ulf, m) holds for JII III E w. We then check that: Ax.[(xi(A ~ A JiAx.[(x) U OJ. Therefore, we have: ZIAx.[1 x»)(A ~ A)z((Ax.[(x» U "j l. Jnd since x A x U (j, 01, we get: z(Ax.[(x»)(x) A z((Ax.[(x) U (T,J(x U If, 0)). Now there is an integer Tn E UJ such that: 0» z((Ax.[(x» U (Ti)!x U I f, 0) ~ r(x I U (T';j~ ,)m j(U)x(r~ where we have been able to separate right-hand sidc must contJin ZI(T,)(j, ) ~lf, f and u because j is II). Thus III o~ large. But the and our fixed n are the same, The other element zIAx.!lxl)(x) ~ rlXI for some q E UJ. Thus we have: r(x) Aj"IxlUlf, 111. Again since j is so large, (f, Il)"rlx). Thus by Our choice of A we must havc r(x) = ["Ix). This means then, since n is Iixed, that for all finite f, x: z(Ax/lx))(xl ~ ["(x). But then by continuity this equation holds for all f, x. It follows now that z = Zn, by the proviso of the theorem. These result~ bring up many questions which we leave unanswered here. For example, which combinat r~ (i.e., pure A-terms) ha\'c functionality as in the cxampl ~ above, and can we decide when a term is one ~L1ch? In particular, can the P diagonal comhinator Ax.XlX) be typed" (The argument of Hindley, et al. (I ~72, HI) is purclv formal Jnd does not apPJrent)v apply to thc model.) What ahout lerm~ in LAMBDA beyond the pure ;\ -calculus',' 566 DANA SCOTT Appendix A. Proofs and technical remarks. For Section 1. If we give the two-point space {~, T} the weak T.rtopology with just three open sets: 0, {T}, ,~{ T}, we have what is called the Sierpinski space and its infinite product {~, T}W with the product topology is the same as Pw. The finite sets e" E Pw correspond exactly to the usual basic open sets for the product. For those familiar with such notions, this well-known observation makes many of the facts mentioned in this section fairly obvious. From any point of view, Theorem 1.1 and the remarks in the following paragraph are simple exercises. Proof of Theorem 1.2. Equation (i) as a functional e·quation comes down to which is just another way of writing the definition of continuity. Thus it is indeed true for all x. Next, inclusion (ii) means that if (n, m) E u, then 3e, s; e".(k, m) E u. Clearly all we need to do is take k = n. If we also want the converse inclusion to hold, then what we need is condition (iii). Proof of Theorem 1.3. Substitution is generalized composition of functions of many variables with all possible identifications and permutations of the variables; however, as we are able to define continuity by separating the variables, the argument reduces to a few special cases. The first trick is to take advantage of monotonicity. Thus, suppose fix, y) is continuous in each of its variables. What can we say of fix, x), a very special case of substitution? We calculate fix, x) = U{f(e m x)le" <;: x} =U{f(em em)le" S; x, em <;: x}. Then if we think of ek = en U em and realize that f(e", em) ~ I(ek- ed, we fix, x) = U{f(e" e, )Ie, S; ~e that x}. This means that f(x, x) is continuous in x. This same argument works if other variables are present, as in the passage from f(x, y, z, w) to f(x, x, z, w). When an identification of more than two variables is required, as from fix, y, z, w) to fiX, x, x, x), the principle is just applied several times. Finally to show that f(g(x, y), h(y, x, y)) is continuous, it is sufficient to show that f(g(x, y), h(z, u, v) is continuous in each of its variables separately. By simply overlooking the remaining variables, this comes down to showing that f(g(x)) is continuous if f and g are. But the proof for ordinary composition is very easy with the aid of the characterization theorem (Theorem 1.1). Proof of Theorem IA. This well-known fact holds for continuous functions on many kinds of chain-complete partial orderings; but Pw il'lustrates the idea well enough. Suppose f had a fixed point x = fix). Then since 0 s; x and f is monotonic, we see that f(0) <;: f(x) = x. But then again.[(f(0) <;:f(x) = x; and so by induction, ["(0) <;: x. This proves that m(f) S; x; and thus if m(f) is a fixed point, it must be the least one. To prove that it is a fixed point, we need a fact that will often be useful: . LEMMA. [f x" S; X"+I for all n, and iff is continuous, then f(U{x"ln E w}) = U{f(x")ln E wi· DATA TYPES AS LATTICeS 567 Proof. By monolOnicity, the inclusion holds in one direction. Suppose eM s; f(U{x.ln E wi)· 1ben by Theorem 1.1 we have eM S;f(ed for some e. ~ U{x.ln E wi. Because e. is finite and the sequence is increasing, we can argue that e. S; x. for 80me n. But then f(e.) s;f(x.). This shows that eM s; U{f(x.)ln E w) and proves the inclusion in the other direction. (Exercise: Does this property characterize continuous functions?) Proof of Theorem 1.4 concluded. Noting that ["(0)s; ["+1(0) holds for all n, we can calculate: [(fix,(m = U{f(f"(0))!n E w) = U{r'(0)/n EwI. But this is just fix(n), since the only term left out is /,(0) = 0. Proof of Theorem 1.5. The function [is clearly well-defined even when y E Y has a neighborhood V where xn u= 0: in that case [(y) = w by convention on in Pw. In case XEX, it is obvious that [(xls;f(x). For the the meaning of opposite inclusion. suppose that m E f(x). Because f is continuousand {zlm E z} is open in Pw, there is an open subset Vof X such that x' E V always implies that m Ef(x'). But X is a subspace of Y, so V= xn V for some open subset Uof Y. Thus we can see why mE [(x). It remains to show that [is itself continuous. We must show that the inverse image under [ of every open subset of Pw is open in Y. But the open subsets of Pw are unions of finite intersections of sets of the form {z ImE z}. Thus it is enough to show that {yj m E[(y)} is always open in Y. But this set equals U{UjmEn{f(x)lxEXn U)), which being a union of open sets is open. Note that what we have proved is that [is continuous no matter what function f is given; however, if f is not continuous, then [ cannot be an extension of f. For readers not as familiar with general topology we note that the idea of Theorem 1.5 can be turned into a definition. Suppose X s; Pw is a subset of P w. It n becomes a subspace with the relative topology. What are the continuous functions f: X -> Pw? From Theorem 1.5 we see that a necessary and sufficient condition is that the T: Pw -> Pw be an extension of f. Thus for x E X we can write the equation f(x) = [(xl as a biconditional: mEf(x) if! 3e.s;xVx'EX[e.s;x'impliesmEf(x')], which is to hold for all mEw. This form of the definition of continuity on a subspace is more complicated than the original definition. because in general e.~ X and we cannot write f(e.). Proof of Theorem 1.6. What To means is that every point of X is uniquely determined by its neighborhoods. Now e(x) just tells you the set of indices of the (basic) neighborhoods of x. Thus it is clear that, is one-to-one. To prove that it is continuous, we need only note: (xln E e(x)) = V., which is always open. To show that, is an embedding, we must finally check that the images of the open sets U. are open in e(X). This comes down to showing: ,(V.) = ,(X) n {zln which is clear. E Z}, 5hX DANA ~COTr For Section 2. Equation (2.1) defines a continuous lunction because it is a special case of Theorem 1.5. where we have been able to simplify the definition into cases because (I) is a very elementary subset of Pw. Equation (2.2) gives a continuous function since the definition makes pdistributive, as remarked in the text for finite unions, byt it is just as easy to show that p distributes over arbitrary unions. The ditterence between a continuous f and a distributive p is this: to find mE f(x) we need a finite subset e" c; x with m E fie"); howpver, to find m E p(x) we need only one element n E x with mE p((n}) = p(n) = p( n). Continuous functions are generalizations of distributive functions. The generality is necessary. For example, in (2.3) we see another function x; y distributive in each of its variables; but take care: the function x; x is not distributive in x-there is no closure under substitution. This is just one reason why continuous functions are better. Another good example comes from (2.4) if you compare the functions x, x+x, x+x+x, etc. Equations (2.5H2. 7) are very elementary. Note that z => x, y is distributive in each of its variables. We could write: z => x, Y = P(z), where piO) = x and PIn + \ 1= y, to show that it is distributive in z. Proof of Theorem 2.1. If we did not usc the A-notation, then all LAMBDAdefinable runctions would be obtained by substitution from the first five (ef Table 2). Since they are all seen to be continuous, the result would then follow by tbe substitution theorem (Theorem 1.3). Bringing in A-abstraction means tbat we have to combine Theorem 1.3 with this fact: LEMMA. If fix. y, z, ... ) is a continuous function of all its variables, then Ax.[(x, y, z, ... ) is a continuous function of the remaining variables. Proof It is enough to consider one extra variable. We compute from the definition of A in Table 2 as follows: Ax.[lx, y) = (In, m)lm Efle", ~ y)} ((n, m)13e, c; y.m Ef(e", e,)) = U{{(n, mllm Ef(e", e,J}Ie, c; y} = U{Ax.[lx, e,!Ie, c; y}. Thus Ax.[lx, y) is continuous in y. Proof of Theorem 2.2. The reason behind this result is the restriction to continuous functions. Theorem 2.1 shows that we cannot violate the restriction by giving definitions in LAMBDA, and the graph theorem (Theorem 1.2) shows that continuous functions correspond perfectly with their graphs. The verific"tion of (a) of Table 1 is obvious as the 'x' in ·AX.T' is a bound variable. (Care should be taken in making the proviso that' y' is not otherwise free in T.) The same would of course hold for any other pair of variables. We do not bother very much about alphabetic questions. The verification of (f3) is just a restatement of Theorem 1.21i I. Let T define a Also funl u)( x) function f (of x). Then by definition f( x) = T and Ax. T ~ graphl n. in the notation of Theorem 1.2 is the same as the binary operation Id.n in the notation of LAMBDA. Thus in Theorem 1.2(i) if we apply both sides to v we get nothing else than (f3I. 569 DATA TYPES AS LAITICES Half of property «() is already implied by (/3): tbe implication from left to rigbt. (Just apply botb sides to x.) In tbe olher direction, 'rt x. T ~ IT means thaI T and IT define the same function of x; thus, the two graphs must be equal. Remarks on other laws. The failure of (TI) simply means that not every set in Pw is the graph of a function. Condition (iii) of Theorem 1.2 is equivalent to saying that U ~ Ax.u(x), in other words, U is the graph of some function if and only if it is the graph of the function determined by u. Law (iL) is the monotone property of application (in both variables); therefore, (iLl and W together imply «(*J from left to right. Suppose that 'rt x. T <;; IT; then clearly: {{n, m)1 mE T[ eo / x]} <;; {In, m)[m E e x]}, IT[ o / which gives «(*) from right to left. There are, by the way, other laws valid in the model, as explained in the later resullS. Proof of Theorem 2.3. This is a standard result combinatory logic. We have only to put: That is, we use the iteration of the process of forming the graph of a continuous function. As each step (from the inside out) keeps everything continuous, we are sure that the equation of Theorem 2.3 will hold for iterated applicalion. Proof of Theorem 2.4. This can be found in almost any reference on combinatory logic or A -conversion. The main idea is to eliminate the A in favor of the combinators. The fact thaI we have a few other kinds of terms causes no problem if wc introduce the corresponding combinators. The method of proof is to show, for any LAMBDA-term T with free variables among x o. Xl. . . , , X.. _ h that there is a combination y of combinators such that: T= 1'(x,,)(xt!··· (X'_l)' Thi . . can bc done by induction On the complexity ut T. Proof of Theorem 2.5. The well-known calculation shows that we have from (2.H): Y( u) ~ (Ax.u(x(x)))(Ax.u(x(x))) ~ u( YI u)). Thus Yi u) is a fixed point of thefunction u(x). What is needed is the proof to show that it is the least one. Let d ~ Ax.u(x(xJ) and let a be any other fixed point of u(x). To show, as we must, that dId) ~ a. it is enough to show that e, ~ d always implies f,(e,) <;; a; because by continuity we have: ~Ld( U{eJf,)lf, <;; d) By way of i"duct(on. supp"," that Ihis implication hold, for all n < l. Assume that d and that mE e,(ed. We will want to use the induction hypothesis to show that mEa. By the definition of application. there exists an integer n such that tn. m) r:: 1:>/ and en S; e{. But n ;? (n, m) < I. and en s;: d. By the hypothesis, we have eol f,,) s: a. Note Ihat (n, m I € d also, and that d is defined hy A -abstraction; thus, <'I <;; 570 DANA scurr mE d(e,) by definition. By monotonicity u(e,(e,») c: u(a) = a; therefore mEa. This shows that e,(e,) c: a, and the inductive proof is complete. Remark. Note that we did not actually use the fixed-point theorem in the proof, but we did use rather special properties of the pairing (n, m) and the finite sets elEquation (2.9) is proved easily from the definition of application; indeed u(x) is distributive in u. Equation (2.10) is proved even more easily from the definition of A-abstraction. For (2.11), we see that the inclusion from left to right would hold in general by monotonicity. In the other direction, suppose m E/(x) n g(x). Then for suitablc k and I we have (k, m) E I and e, c: x, also (I, m) E g and e, c: x. Lct e, = e, U e, c: x. Because I and g are graphs, we can say (n, m) E In g; and thus mE g)(x). This is the only point where we require the assumption on graphs. Equation (2.12) follows directly from the definition of abstraction. For (2.13), which generalizes (2.9), we can also argue directly from the definition of application. In the case of intersection it is easy to find U" such that 0 E u,(T) for all n, but n{u,lnEw}=L Equation (2.14) is obvious because the least fixed point of the identity function must be L A less mysterious definition would be ~ = 0-1, but the chosen one is more "logical". For (2.15) we note that by definition: un Az.O = {(n, m)lm E O) = {(n, O)ln E w). Because 0 = (0, 0) and I = (I, 0), we get the hinl. Equation (2.16) makes use of U for iteration. If x ~ 0 U (x + I), then x must contain all integers; hence x = T. The iteration for n in (2.17) is more complex. The fundamental equation we need is: x ny~ x => (y => O. lJ, «x - 1) n (y - 1) + I. This says to compute the intersection of two scts x and y, we first test whether o EX. If so, then test whether 0 E y. If so, then we know 0 EX n y. In the meantime we begin testing x for posilive elements. If we could compute (by the ,ame program) the intersection (x - I) n (y - I), then we would gct the positive eleby adding one. This is a very slow program, but we ments of the intersection x can argue by induction that it gives us all the desired elements. Of course, n is the least function satisfying this equation. In the case of (2.18) it is clear that we have: ny Ax.~ = {(n, m)lm E~) =~; Ax.T={(n, m)lmET}=T; because in the last every integer is a (number of a) pair. Suppose now that a ~ Ax. a and a '" T. Let k be the least integer where k E a. Now k ~ (n, m) for some nand m. If mEa, then (n, m) E a = Ax.a; hence mE a. But m ~ k and k is minimal; therefore, m = k. But this is only possible if k = m = n = O. Suppose further a " land that I is the least integer where lEa. Now I = (i, j) with j E a and j ~ I. So j = I and 1= j = i = O. This contradiction proves that a = ~ or a = T. Equations (2.19)-(2.22) are definitions, and (2.23) is proved easily by induction on i. Equation (2.24) is also a definition. To prove (2.25) we note thaI 571 'ES AS LADICES {u,li E x} is continuous (even: distributive) in u and x. Thus, there is a continuous function seq(u)(x) giving this value. What is required is to prove that it is LAMBDA-definable. We see: seq(u)(x) = {n E uolO EX}U{m E U{ui+di + 1 E x}13k.k + 1 EX} = X :0 Uo, seq(AI.u,+t!(x -1): that is, seq satisfies the fixed-point equation for S. Thus S ~seq. other inclusion we argue by induction on i for: To establish the 'rIx, uri EX=? u, l)x( u S~ This is easy by cases using what we are given about $ in (2.24); it implies that seq ~ $. Note that: AnEw.T=AnEW.u iff 'Vn EW.T=: U. For primitive recursive functions, even of several variables, there is no trouble in transcribing into LAMBDA-notation any standard defin tion~­ especially as we can use the abstraction operator An E w. If we recall that every Le. set a has the form: a = {mI3n.p(n) = m +1}, where p is primitive recursive, we then see that a = piT) -1. This means that every Le. set is LAMBDA-definable. Proof of Theorem 2.6. In case of a function of several variables, we remark: hoAx, ... AXk_'.f(xo)(x,)· .. (Xk_l) ~{(no, · ,(n, "m)" ')))!mE!(e",)(e",)'" (e",_,)}. This makes the implication from (i) to Iii) obvious. Conversely, if a graph u is r.e., then from the definition of application we have: mE u(e",)(e",)' .. te,,) iff Ino, In" (... , (n,_" m)" ')))E u, which is Le. in m, no, nl,"': nk-!' Therefore (Sand (ii) are equivalent. We have already proved that (ii) implies (iii). For the converse we have only to show that all LAMBDA-definable sets are Le. For this argument we could take advantage of the combinator theorem, (Theorem 2.4). Each of the six COmbinators are Le., and there is no problem of showing that if u and x are Le., then so is u(x); because it is defined in such an elementary way with existential and bounded universal number quantifiers and with membership in u and in x occurring positively. Explicitly we have: mE u(x) ~ iff 3e" iff 3n 'rim < n[[m E e" implies m E xl and (n, m) E u]. x.(n, m) E u For Section 3. For the proof of (3.1) we distinguish cases. In case x = y = 1-, we note that condl 1-)( 1- I = 1- and 1- (1-) = 1-, so the equation checks in this case. Recall: cond(x) y ~Az. :ox, y={ln, m)!mEle" :ox, y)}. 572 DANA ,')COTT We can show 0 e' eond(x)(y), Note first 0 = (n, m) iff n = 0 = m; furthermore, eo ~ ~ and 1 =0 x, Y = 1; but 0" 1. Also we have: eond(x)(y)(O)=x and eond(r)(y)(J)= y; so if either xi' 1 or y i' 1, then eond(x)(y) 'i' 1. In this case, eond(x)(y) must contain positive elements. The result now follows. Theorem 3.1 is obvious from the construction of G, because G(G) = 0 and G(O)(O) = sue, and so the G(O)(i) give us all the other eombinators. The primitive recursive funetions needed for (3.4)-(3.6) are standard. Equation (3.7) is a definto~ we rewrote it using the Y-()perato ~and the proof of Theorem 3.2 is easy by induction. There is also no difficulty with (3.8)-(3.12), The idea of the proof of Theorem 3.3 is eontained in the statement of the theorem itself. The proof of Theorem 3.4 is already outlined in the text. Proof of Theorem 3.5. The argument is essentially the original one of Myhill-Shepherdson, Suppose p is computable, total and extensional. Define: q = {(j, mJlm E val(p(fin(j)))), where fin is primitive recursivc l and for all JEW: val(fin(})) = er Certainly q ERE, and we will establish the theorem if we can prove "continuity": val( p(n)) = U{val( p(fin(j) ))Ie, <; val( n)). We proceed by contradiction. Suppose first we have a kE val(p(n)), where ke'val(p(fin(j)) whenever e, <;val(n). Pick r to be a primitive recursive function whose range is not recursive. Define s. primitive recursive, so that for all mEw: val( s( m)) ~ (j E val(n Jim" Irt ill i '" ill. The set val(n) must be infinite, because p is extensional, and if val(n) ~ e, ~ val(fin(j)), then ke' val( p(fin(j))) = val( p(n»). Note that val(s(m)), as a suhset of the infinite set, is finite if m ;s in the;: range ()f r: otherwise it is equal to val(n). Again by the extensionality of p we see that k E val!pls(m)) if and only if m IS not in the range of r. But this puts an Le. condition on m equivalent to a non-r.e. condition. which shows there is no such k. For the second case suppose we have a ke'val(p(n), where for a suitahle ej <; val(n) it is the case that k E val(p(fin(j))). Define: t = Am E w.e, U Ival(mJ =0 val(n), val(n»). We have: t(mJ= { e, if val(m') val(n) if not. = 1; We choose u primnivL: fL:cursive, where: val(u(m)) = tim). By the chOice llf k, and by the extensionality of p, and by the fact that valin) i' e" DATA TYPES AS LATTICES 573 we have: k EVal(p(u(m))) iff tim) = e} iff val(m)=.L But this is impossible, since one side is Le. in m and the other is not by Theorem 3.4. As both cases lead to contradiction, continuity is established and the proof is complete. Proof of Theorem 3.6. Consider.a degree Deg(a). This set is closed under application, because: u(a)( v(a)) = S(u)(v )(a), and S(u)(v) is Le. if both u and v are. Note that it also contains the element G; hence, as a subalgebra, it is generated by a and G. Let A be any finitely generated subalgebra with generators ab, ~a • ... , a~-l' Consider the element a = cond«ab, a~,· .. , a~-l»(G). As in the proof of Theorem 3.1, a generates A under application. It is then easy to see why A = Deg(a). Proof of Theorem 3.7. We first establish (3.16) and (3.17): Lou 0 R = Ax.L(u«O, x»)) = Ax.L«I, u, x» = Ax.u(x). u 0 v 0 R = Ax.u(v«O, x))) = Ax.u((J, v, x») = Ax.u(v)(x) = u(v). Now starting with any u ERE, we write u = T, where T is formed from G by application alone. By (3.17), we can write u in terms of G and R using only o. That is, u belongs to a special subsemigroup. [n view of (3.16), we find that Ax.u(x) belongs to that generated by R, Land G. But RE n FUN ~ {Ax.u(x)!u ERE}, and so the theorem is proved. For Section 4. The notion of a continuous latrice is due to Scott (1970/71) and we shall not review all the facts here. One special feature of these lattices is that the lattice operations of meet and join (n and U) are continuous (that is. commute with directed sups). As topological spaces, they can be cbaracterized as those To-spaces satisfying the extension theorem (which we proved for Pw in Theorem 1.5). Proof of Theorem 4.1. Consider a continuous function a. and let A = {xix = a(x)}. By the fixed-point theorem (Theorem 1.4) we know that A is nonempty and that it has a Ie "51 element under <;;. Certainly A is partially ordered by £; further, A is closed under directed unions but not under arbitrary unions. DANA 5H scorr That is, A is not a complete sublattice of Pw with regard to the lattice operations of Pw, but it could be a complete lattice on its own-if we can show sups extst. Thus, let S s:: A be an arbitrary subset of A By the fixed-point theorem, find the least solution to the equation: y= U{X\XE S}U aryl. Clearly x s y for all XES; and so x = a(x) s:: aryl, for all XES. This means that y = a(y), and thus yEA. By construction, then, y is an upper bound to the elements of S. Suppose Z E A is another upper bound for S. It will also satisfy the above equation; thus y S z, and so y is the least upper bound. A partially ordered set with sups also has infs, as is well known, and is a complete lattice. Suppose that a is a retract. We can easily show that the fixed-point set A (with the relative topology from Pw) satisfies the extension theorem. For assume f: X -> A is continuous, and X S Y as a subspace. Now we can also regard f: X -> Pw as continuous because A is a subspace of Pw. By Theorem 1.5 there is an extension to a continuous {: Y -+ Pw. But then a 0 f: Y -+ A is the continuous function we want for A, and the proof is complete. The space of retracts. Let us define: RET={a\ ~ oa), the set of all retracts, which is a complete lattice in view of Theorem 4.1. It will be proved to be not a retract itself by showing it is not a continuous lattice; in fact, the meet operation on RET is not continuous on RET. The proof was kindly communicated by Y. L. Ershov and rests on distinguishingsome extreme cases of retracts. Call a retract a nonexrensive if for all nonempty finite sets x we have x~ a(x). Call a retract b finite if all its values are finite (i.e., beT) is finite). If a is nonextcnsive and b is finite. then Ershov notes that they are "orthogonal" in RET in the sense that c = a n b = ~. The reason is that, since c c;; b, it is finite; but c ~ a, too, so c(x) s; a(x) for all x. Because c is a retract. we have c(x) = c(c(x» s a(c(x). As c(x) is finite and a is nonextensive, it follows for all x. that c(x) ~.L This orthogonality :s unfortunate, because consider the finite retracts bn = Ax. e•. We have here a directed set of retracts where U{b. In E w} = Ax. T = T. If n were continuous, it would follow that for nonextensive a: a =a n T~ a nUlb.ln Ew}=U{a n b.]nEw}= 1, showing that there are no nontrivial such a. But this is not so. Let « be a strict linear ordering of w in the order type of the rational numbers. Define: a(x) = {ml3n E X.m« n}. We see at once that a is continuous; and, because« is transitive and dense, a is a retract. Since « is irreflexive, it is the case for finite nonempty sets x that max~ (x)'" a(x); hence, a is nonextensive. As afT) = T, we find a "'.L The proof is complete. Note that there are many transitive, dense, irreftexive relations on w, so there are many nonextensive retracts. These retracts, like a above, are distributive. A 7 DATA TYPES AS LATTICES 575 nondistributive example is: a':; {mI3n, n' E x.n« m« n'l. Many other examples are possible. Proof of Theorem 4.2. The relation (lE; is by tbe definition of retract reflexive on RET; it is also obviously antisymmetric. To prove transitivity, suppose a (lE; b (lE; c, then a :; a 0 b == a 0 b 0 c = a 0 c. Similarly, a = c 0 a. Note. by the way, that a ()<; b implies that the range of a is included in that of b; but that the relationship a,; b does not imply this faCL The relationship a ()<; b, however, is stronger than inclusion of ranges. Proofs of Theorems 4.3-4.5. We will not give full details as all the parts of these theorems are direct calculations. Consider by way of example Theorem 4.3(i). We find: (a .... b) 0 (a .... b) = Au.b 0 (b = Au.b 0 0 u 0 a) 0 a u0 a =QG--Job, provided that a and b are retracts. A very similar computation would verify part (iv), if one writes out the composition: (a .... b') 0 (f .... f) 0 (b .... a') and uses the equations: f =b f a 0 0 and r = b' r 0 0 a'. The main point of the proof of Theorem 4.6 has already been given in the texL For Sections 5-7. Sufficient hints for proofs have been given in the texL Appendix B. Acknowledgments and references. My greatest overall debt is to the late Christopher Strachey, who provided not only the initial stimulus and continuing encouragement, but also what may be termed the necessary irritation. Not being a trained mathematician, he often assumed that various operations made sense without looking too closely or rigorously at the details. This was particularly the case with the A-calculus, which he used as freely as everyday algebra. Having repeatedly and outspokenly condemned the A-calculus as a formal system without clear mathematical foundations, it was up to me to provide some alternative. The first suggestion was a typed system put forward in Scott (1969) (unpUblished, but later developed as LCF by Robin Milner and his collaborators). Experience with the type structure of function spaces, which had come to my attention from work in recursion theory by Nerode, Platek and others. soon convinced me that there were many more similar structures than might at first be imagined. In particular. a vague idea about a space with a "dense" basis led quickly to the more direct construction, by inverse limits, of function spaces of "infinite" type that were very reasonable models of the classical "type-free" 576 DANA SCOTT A-calculus (as well as many other calculi for other "reflexive domains). The details Can be found in Scott (1971) and Scott (1973b). Algebra was justified, but the work in doing so was tiring and the exact connections with computability were not all that easy to describe. In the meantime Plotkin (1972) contained suggestions for a "set-theoretical" construction of models, but not much notice was taken of the plan at the time it was cirCUlated-perhaps owing to a fairly sketchy presentation of the precise semantics of the A-calculus. The present paper evolved particularly from the project of making the connections with ordinary recursion theory easier to comprehend, since a satisfactory theory of computability and programming language semantics had to face this problem. The idea of using sets of integers for a model was first put forward by the author at a meeting at Oberwolfach at Easter in 1973 and in a more definitive form at the Third Scandinavian Logic Symposium shortly thereafter (see Scott (1975a) which is a preliminary and shorter version of this paper). The author gave a report on the model at the Bristol Logic Colloquium in July 1973, but did not submit a paper for the proceedings. A series of several lectures was presented at the Kiel Logic Colloquium in July 1974. covering most of the present paper which was distributed as a pre print at the meeting. The text (but unfortunately neither acknowledgments nor references) was printed in the proceedings (Springer Lecture Notes in Mathematics, vol. 499). In 1973 after experimentation with definitions somehow forced him into the definition of the model, the author realized that it was essentially the same as Plotkin's idea and, even more surprising, it was already implicit in a very precise form in much earlier work by Myhill-ShepherdSllO (1955) and Friedberg-Rogers (1959) (see also Rogers (1967)) on enumeration operators. What had happened was that Plotkin had not made enough tie-up with recursion theory, and the recursive people had nol seen 'he tie-up with A-calculus, even though they knew that one could do a lot with such operators. Actually, if the author had taken his own advice in 1971 (see Continuous lattices, Scott (1972a, end of § 2)), he would have seen that many spaces have their own continuous-function spaces as computable retracts, a fact which is just exactly the basis for the present construc- tion; but instead he said: "it [the representation as a retract] does not seem to be of too much help in proving theorems." Over the years in work on A­calculus and programming language semantics. personal contact and correspondence with a large number of people has been very stimulating and helpful. I must mention particularly de Bakker. Barendregt, Bekic, Blikle, Bohm, Curry, Egli, Engeler, Ershov, Goodman, Hyland, Kreisel, Landin, Milne, Milner, Mosses, Nivat, Park, Plotkin, Reynolds, de Roever, Smyth, Stay, Tang, Tennent, Wadsworth. (I apologize to those 1 have inadvertently left out of this lis!.) In the reference list a very imperfect attempt has been made to collect references directly relevant to the topics of this paper as well as pointers to related areas that may be of inspiration for future work. The list of papers is undoubtedly incomplete, and inaccurate as well. but the author hopes it may be of some use for those seeking orientation. It is a very vexing problem to keep such references up to date. Some remarks toward references and acknow- ledgments on the specific results in the various sections follow. Felipe Bracho deserves special thanks for help in the preparation of the final manuscript and with the reference lis!. DATA TYPES AS LAITJCES 577 Section 1. The relevance of the "positive" or "weak" topology first came to the author's attention through the work of Nerode (1959). Continuous functionals were studied by Kleene and Platek and many other researchers in recursion theory. Monotonicity was particularly stressed by Platek (1964). The graphs and the definition of application are used in the same way by Plotkin (1972) and Rogers (1967, see p. 147). The fixed-point theorem is very well-known. See, e.g., Tarski (1955). The extension theorem was formulated by the author, but it is very similar to many results in point-set topology; it was used in a prominent way in Scott (1972a) to characterize continuous lattices. The embedding theorem is well-known; see, e.g., tech (1966). Section 2. The language LAMBDA is due to the author. Note in particular that Plotkin and Rogers do not define A-abstraction, even though they know of the existence of many combinators and could have defined abstraction if anyone had ever asked them. In particular, they understood about conversion in many instances. The reduction and combinator theorems are well known from combinatory logic and can be found in any reference. The first recursion theorem is basic to all of elementary recursion theory; what is new here is the adaptation of David Park's proof (Park (l970c), unpublished) to the present model to show that Curry's "paradoxical" combinator actually does the recursion. The definition of computability and the definability theorem tie up the present theory with the older theory of enumeration operators. Section 3. The idea of reduction to a few combinators is an old one in combinatory logic; the author only needed to find a small trick (formula (3.1» which would take care of the arithmetical combinators. The ideas for the Godel numbering and the proof of the second recursion theorem are standard, as is rhe proof of the incompleteness theorem. It only looks a little different since we combine arithmetic with the "type-free" combinators. The proof of the completeness theorem for definability (Theorem 3.5) is taken directly from MyhillShepherdson (1955). The author is indebted to Hyland for pointing this out. The subalgebra theorem is an easy reformulation of talk about enumeration degrees: for more information on such degrees consult Rogers (1967), Sasso (1975), and also Gutteridge (1971). The area is underdeveloped as compared to Turing degrees. Semigroups of combinators have been discussed by Church and Bohm. Section 4. The notion of a ferracr is common in topology, butthe idea of using them to define data types and of having a calculus of computable retracts is original with the author. Of course the connection between lattices and fixed points was known; more about lattices is to be found in Scott (1972a). The various operations on retracts and the idea of using fixed-point equations to define retracts recursively are due to the author. Applications to semantics were given in Scott (1971) for flow diagrams, and this has been followed up by many people. in particular Goguen, et al. (1975) and Reynolds (1974b). Section 5. Algebraic lattices have been known for a long time (see Gratzer (\968) and also closure operations (sec, e.g., Tarski (1930)). It was Per MartinLOf and Peter Hancock who suggested that they might form a "universe"; in particular the construction of V is essentially due to them. The limit theorem is due to the author. Section 6. More information on the classification by notions in descriptive set theory of various subsets can be found in the work of Tang who also makes 578 DANA SCOTT connections with the work of Wadge. The various normal forms for the classes of sets (e.g., Table 3) are due to the author. Section 7. Functionality has been studied for some time in combinatory logic (see, e.g., Hindley, et al. (1972) for an introduction). The author had the idea to see what it all means in the models; there are, of course, connections going back to Curry and Kleene, with functional interpretations of intuition istic logic (cf. Theorem 7.3, which is well-known). 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