The authors observe that the theory of prime implicants makes sense for finite join semilattices ... more The authors observe that the theory of prime implicants makes sense for finite join semilattices L having a generating set G. Thus, every element in L can be written as a join of elements in G. An implicant of x in L is defined as an element g in G with g≤x. The maximal implicants of x are called the prime implicants of x. An easy result is that in a finite join semilattice, every element is the join of its prime implicants. A representation of x is a subset A of L such that x is the join of A. If A and B are subsets of L, write A≪B if for each a in A there is some b in B with a≤b. A representation A for x is called syllogistic if for each subset B with join B≤x, it follows that B≪A. It is shown that a representation A of x by a subset of G is syllogistic iff A contains all prime implicants of x. Some other observations for lattices and Boolean functions are included.
A consistent Boolean equation in n unknowns has exactly one triangular reproductive solution sati... more A consistent Boolean equation in n unknowns has exactly one triangular reproductive solution satisfying a certain supplementary condition.
From the authors’ abstract: Given an arbitrary Boolean algebra B, every function f:B n →B that is... more From the authors’ abstract: Given an arbitrary Boolean algebra B, every function f:B n →B that is Boolean (i.e., that has the canonical disjunctive form f(x)=∑ i f(i)m i (x)) can be expressed in the form f(x)=g(x,a), where g:B n+p →B is a simple Boolean function (i.e., g involves no constants from B or g=1 or g=0) and a∈B p . The relationships between f and g, and in particular between the equation f(x)=1 and the equations g(x,j)=1 (j=0,1,...,2 p -1), have been studied in the literature. In the present paper we follow this line of research. The main result provides three necessary and sufficient conditions for the equation g(x,a)=1 to have a unique solution.
The objective of the lecture series is to show the application of 'modern control theory'... more The objective of the lecture series is to show the application of 'modern control theory' to the generation of guidance laws for astronautical as well as atmospheric aerospace vehicles. Time domain analysis is stressed throughout, although frequency domain considerations are used when discussing instrumentation response characteristics. Particular emphasis is placed on the use of digital computers in solving these highly nonlinear problems and techniques are demonstrated for implementing and mechanizing these solutions to airborne and astronautical vehicles. The purpose of the first and second chapters is to review the language of modern control theory as it applies to linear systems. The concepts of input-output, state space representations and matrix analysis are coupled to a discussion of solutions of state space equations and the calculation of state transition matrices for linear systems. Chapters three and four introduce optimization theory and provide various formulat...
In the first part of this paper we determine a necessary and sufficient condition for a subset F ... more In the first part of this paper we determine a necessary and sufficient condition for a subset F 0 of a set F of Boolean functions to be functionally dependent. Then we pass to the general case of a finite set E and a hereditary family of subsets of E which we call independent, while the subsets of the complementary family are said to be dependent. We associate with this framework certain Boolean functions which we call weak dependency functions and the dependency function, respectively. The family of all dependent sets and the complementary family of all independent sets are completely determined by the minimal dependent sets and the maximal independent sets, respectively. The latter sets are shown to be in a one-to-one correspondence with the prime implicants of the dependency function d and the prime implicants of d’, respectively. These results apply in particular to functional dependence of Boolean functions and internal stability of sets of vertices in graphs.
IEEE Transactions on Industrial Electronics and Control Instrumentation, 1975
ABSTRACT This paper describes an approach to the automated programming of microprocessors for app... more ABSTRACT This paper describes an approach to the automated programming of microprocessors for applications in logical control. A procedure is presented for describing control-tasks by state-diagrams. A minicomputer-resident crosscompiler entitled MRS (Microprocessor Realization of State-Diagrams) is discussed; MRS enables a designer with no knowledge of programming to translate a state-diagram into a machine-language microprocessor program. The preset version of MRS runs in 4K of core in the PDP-8 mini-computer and produces programs on paper tape for the Intel 4004 microprocessor.
TI& letter presents an aIgrit8m for coaPtroctiog stateequations correspoading to a well-formed, b... more TI& letter presents an aIgrit8m for coaPtroctiog stateequations correspoading to a well-formed, bear, time-vlryinp system ut, P)Y = N t , P)U.
The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call prot... more The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call proto-Boolean, and a "general method in Logic" making use of that algebra. Our object is to elucidate these two components of Boole's system, to prove his principal results, and to draw some conclusions not explicit in LT. We also discuss some examples of incoherence in LT; these mask the genius of Boole's design and account for much of the puzzled and disparaging commentary LT has received. Our evaluation of Boole's logical system does not differ substantially from that advanced in Hailperin's exhaustive study, Boole's Logic and Probability. Unlike the latter work, however, we make direct use of the polynomials native to LT rather than appealing to formalisms such as multisets and rings.
A recent paper in this journal, by Abdel-Gawad, Atiya, and Darwish, presents a method of solving ... more A recent paper in this journal, by Abdel-Gawad, Atiya, and Darwish, presents a method of solving a system Boolean equations using the polynomial algebra invented by George Boole in 1854. The authors do not mention Boole, however, or the modern applications of this algebra. Their method entails reduction of the given system to a triangular system, which is solved by back-substitution. We show that the solutions of the triangular system include all those of the given system, but may include others that do not satisfy the given system.
the count to a number such as 64, and to transmit modulo 64 characters. The blockoriented method ... more the count to a number such as 64, and to transmit modulo 64 characters. The blockoriented method permits ASCII symbols to be transmitted regardless of picture complexity, and saves counting stages. SUMMARY The new mode of light-pen operation permits extension of the light-pen to two important areas: 1) time-shared computers, and 2) remote telephone-linked displays. These situations cannot be accommodated by the classical light-pen approach. The implementation described uses the MODEL T display as the viewing device, but the technique can be operated as well with locally refreshed displays (using delay lines, etc.).
The existence problem of the total domination vertex critical graphs has been studied in a series... more The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we settle the existence problem with respect to the parities of the total domination number m and the maximum degree ∆ : for even m except m = 4, there is no m-γ t -critical graph regardless of the parity of ∆; for m = 4 or odd m ≥ 3 and for even ∆, an m-γ t -critical graph exists if and only if ∆ ≥ 2⌊ m−1 2 ⌋; for m = 4 or odd m ≥ 3 and for odd ∆, if ∆ ≥ 2⌊ m−1 2 ⌋ + 7, then m-γ t -critical graphs exist, if ∆ < 2⌊ m−1 2 ⌋, then m-γ t -critical graphs do not exist. The only remaining open cases are ∆ = 2⌊ m−1 2 ⌋ + k, k = 1, 3, 5. Second, we study these remaining open cases when m = 4 or odd m ≥ 9. As the previously known result for m = 3 [1, 2], we also show that for ∆(G) = 3, 5, 7, there is no 4-γ t -critical graph of order ∆(G) + 4. On the contrary, it is shown that for odd m ≥ 9 there exists an m-γ t -critical graph for all ∆ ≥ m − 1.
Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation f(x,, .... more Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation f(x,, . . ..xn) = 0 over a finite Boolean algebra B may be expressed as the union of a collection of subsetsofL+,eachoftheform{(x ,,..., X,)/a,~x,~b,,aiEB,b,EB,i=l ,..., n}. Werefertosuch subsets of B" as segments and to the collection as a segmental cover of K. We show that f(_~,,...,x,,) = 1 is consistent if and only iffcan be expressed by one of a class of sum-of-products
A procedure Is presented for constructing the Blake canonical form for a switching function (the ... more A procedure Is presented for constructing the Blake canonical form for a switching function (the sum of all its prime imphcants) conveniently and rapidly. This procedure combines Blake's method (iterated consensus) with the multiplying method of Samson and Mills
The authors observe that the theory of prime implicants makes sense for finite join semilattices ... more The authors observe that the theory of prime implicants makes sense for finite join semilattices L having a generating set G. Thus, every element in L can be written as a join of elements in G. An implicant of x in L is defined as an element g in G with g≤x. The maximal implicants of x are called the prime implicants of x. An easy result is that in a finite join semilattice, every element is the join of its prime implicants. A representation of x is a subset A of L such that x is the join of A. If A and B are subsets of L, write A≪B if for each a in A there is some b in B with a≤b. A representation A for x is called syllogistic if for each subset B with join B≤x, it follows that B≪A. It is shown that a representation A of x by a subset of G is syllogistic iff A contains all prime implicants of x. Some other observations for lattices and Boolean functions are included.
A consistent Boolean equation in n unknowns has exactly one triangular reproductive solution sati... more A consistent Boolean equation in n unknowns has exactly one triangular reproductive solution satisfying a certain supplementary condition.
From the authors’ abstract: Given an arbitrary Boolean algebra B, every function f:B n →B that is... more From the authors’ abstract: Given an arbitrary Boolean algebra B, every function f:B n →B that is Boolean (i.e., that has the canonical disjunctive form f(x)=∑ i f(i)m i (x)) can be expressed in the form f(x)=g(x,a), where g:B n+p →B is a simple Boolean function (i.e., g involves no constants from B or g=1 or g=0) and a∈B p . The relationships between f and g, and in particular between the equation f(x)=1 and the equations g(x,j)=1 (j=0,1,...,2 p -1), have been studied in the literature. In the present paper we follow this line of research. The main result provides three necessary and sufficient conditions for the equation g(x,a)=1 to have a unique solution.
The objective of the lecture series is to show the application of 'modern control theory'... more The objective of the lecture series is to show the application of 'modern control theory' to the generation of guidance laws for astronautical as well as atmospheric aerospace vehicles. Time domain analysis is stressed throughout, although frequency domain considerations are used when discussing instrumentation response characteristics. Particular emphasis is placed on the use of digital computers in solving these highly nonlinear problems and techniques are demonstrated for implementing and mechanizing these solutions to airborne and astronautical vehicles. The purpose of the first and second chapters is to review the language of modern control theory as it applies to linear systems. The concepts of input-output, state space representations and matrix analysis are coupled to a discussion of solutions of state space equations and the calculation of state transition matrices for linear systems. Chapters three and four introduce optimization theory and provide various formulat...
In the first part of this paper we determine a necessary and sufficient condition for a subset F ... more In the first part of this paper we determine a necessary and sufficient condition for a subset F 0 of a set F of Boolean functions to be functionally dependent. Then we pass to the general case of a finite set E and a hereditary family of subsets of E which we call independent, while the subsets of the complementary family are said to be dependent. We associate with this framework certain Boolean functions which we call weak dependency functions and the dependency function, respectively. The family of all dependent sets and the complementary family of all independent sets are completely determined by the minimal dependent sets and the maximal independent sets, respectively. The latter sets are shown to be in a one-to-one correspondence with the prime implicants of the dependency function d and the prime implicants of d’, respectively. These results apply in particular to functional dependence of Boolean functions and internal stability of sets of vertices in graphs.
IEEE Transactions on Industrial Electronics and Control Instrumentation, 1975
ABSTRACT This paper describes an approach to the automated programming of microprocessors for app... more ABSTRACT This paper describes an approach to the automated programming of microprocessors for applications in logical control. A procedure is presented for describing control-tasks by state-diagrams. A minicomputer-resident crosscompiler entitled MRS (Microprocessor Realization of State-Diagrams) is discussed; MRS enables a designer with no knowledge of programming to translate a state-diagram into a machine-language microprocessor program. The preset version of MRS runs in 4K of core in the PDP-8 mini-computer and produces programs on paper tape for the Intel 4004 microprocessor.
TI& letter presents an aIgrit8m for coaPtroctiog stateequations correspoading to a well-formed, b... more TI& letter presents an aIgrit8m for coaPtroctiog stateequations correspoading to a well-formed, bear, time-vlryinp system ut, P)Y = N t , P)U.
The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call prot... more The deductive system in Boole's Laws of Thought (LT) involves both an algebra, which we call proto-Boolean, and a "general method in Logic" making use of that algebra. Our object is to elucidate these two components of Boole's system, to prove his principal results, and to draw some conclusions not explicit in LT. We also discuss some examples of incoherence in LT; these mask the genius of Boole's design and account for much of the puzzled and disparaging commentary LT has received. Our evaluation of Boole's logical system does not differ substantially from that advanced in Hailperin's exhaustive study, Boole's Logic and Probability. Unlike the latter work, however, we make direct use of the polynomials native to LT rather than appealing to formalisms such as multisets and rings.
A recent paper in this journal, by Abdel-Gawad, Atiya, and Darwish, presents a method of solving ... more A recent paper in this journal, by Abdel-Gawad, Atiya, and Darwish, presents a method of solving a system Boolean equations using the polynomial algebra invented by George Boole in 1854. The authors do not mention Boole, however, or the modern applications of this algebra. Their method entails reduction of the given system to a triangular system, which is solved by back-substitution. We show that the solutions of the triangular system include all those of the given system, but may include others that do not satisfy the given system.
the count to a number such as 64, and to transmit modulo 64 characters. The blockoriented method ... more the count to a number such as 64, and to transmit modulo 64 characters. The blockoriented method permits ASCII symbols to be transmitted regardless of picture complexity, and saves counting stages. SUMMARY The new mode of light-pen operation permits extension of the light-pen to two important areas: 1) time-shared computers, and 2) remote telephone-linked displays. These situations cannot be accommodated by the classical light-pen approach. The implementation described uses the MODEL T display as the viewing device, but the technique can be operated as well with locally refreshed displays (using delay lines, etc.).
The existence problem of the total domination vertex critical graphs has been studied in a series... more The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we settle the existence problem with respect to the parities of the total domination number m and the maximum degree ∆ : for even m except m = 4, there is no m-γ t -critical graph regardless of the parity of ∆; for m = 4 or odd m ≥ 3 and for even ∆, an m-γ t -critical graph exists if and only if ∆ ≥ 2⌊ m−1 2 ⌋; for m = 4 or odd m ≥ 3 and for odd ∆, if ∆ ≥ 2⌊ m−1 2 ⌋ + 7, then m-γ t -critical graphs exist, if ∆ < 2⌊ m−1 2 ⌋, then m-γ t -critical graphs do not exist. The only remaining open cases are ∆ = 2⌊ m−1 2 ⌋ + k, k = 1, 3, 5. Second, we study these remaining open cases when m = 4 or odd m ≥ 9. As the previously known result for m = 3 [1, 2], we also show that for ∆(G) = 3, 5, 7, there is no 4-γ t -critical graph of order ∆(G) + 4. On the contrary, it is shown that for odd m ≥ 9 there exists an m-γ t -critical graph for all ∆ ≥ m − 1.
Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation f(x,, .... more Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation f(x,, . . ..xn) = 0 over a finite Boolean algebra B may be expressed as the union of a collection of subsetsofL+,eachoftheform{(x ,,..., X,)/a,~x,~b,,aiEB,b,EB,i=l ,..., n}. Werefertosuch subsets of B" as segments and to the collection as a segmental cover of K. We show that f(_~,,...,x,,) = 1 is consistent if and only iffcan be expressed by one of a class of sum-of-products
A procedure Is presented for constructing the Blake canonical form for a switching function (the ... more A procedure Is presented for constructing the Blake canonical form for a switching function (the sum of all its prime imphcants) conveniently and rapidly. This procedure combines Blake's method (iterated consensus) with the multiplying method of Samson and Mills
Uploads
Papers by Frank Brown