This paper illustrates a simple idea on how to interpret undecompsable generalized multiplicative... more This paper illustrates a simple idea on how to interpret undecompsable generalized multiplicative modules of linear logic as probabilistic methods of a logic prgramming language. These new modules/methods allow to express kinds of propagation of probability distribution inside Bayesian Networks that cannot be expressed by standard Prolog-like programming languages. 1 Logic programming with multiplicative modules In the logic programming paradigm, a program execution is interpreted by the process of proof construction. A state of a computation is represented by a (cut-free) proof which may be partial, i.e. with open branches, or, equivalently, containing non-logical axioms. A program is defined by means of a set of rules (or methods) which can be applied to an open branch of a proof to close it (by an axiom) or to expand it with new branches. The standard methods of an abstract logic programming language (typically, Prolog-like) are represented as H : −B1, . . . ,Bn≥0 where the head ...
Linear Logic [4] has raised a lot of interest in computer research, especially because of its res... more Linear Logic [4] has raised a lot of interest in computer research, especially because of its resource sensitive nature. One line of research studies proof construction procedures and their interpretation as computational models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Focusing", has been described in [2]. Focusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Focusing can also be interpreted in the proof-net formalism, where it appears, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the case of the Non-Commutative logic of [1], and all the computational e...
All in-text references underlined in blue are linked to publications on ResearchGate, letting you... more All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
In this paper we investigate the notion of generalized connective for multiplicative linear logic... more In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ⊗ and disjunction`, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms.
This paper studies the so-called generalized multiplicative connectives of linear logic, focusing... more This paper studies the so-called generalized multiplicative connectives of linear logic, focusing on the question of finding the "non-decomposable" ones, i.e., those that may not be expressed as combinations of the default binary connectives of multiplicative linear logic, ⊗ (tensor) and (par). In particular, we concentrate on generalized connectives of a surprisingly simple form, called "entangled connectives", and prove a characterization theorem giving a criterion for identifying the undecomposable ones.
This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e... more This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e. de-sequentialized proofs, of linear logic (LL). In particular, it presents a syntax for PNs of the cyclic multiplicative and additive fragment of linear logic (CyMALL). Any proof structure (PS), in Girards style, is weighted by boolean monomial weights, moreover, its conclusions Γ (a sequence of formulas occurrences) are endowed with a cyclic order σ, i.e., σ(Γ). Naively, a CyMALL PS π with conclusions σ(Γ) is correct if, for any slice ϕ(π) (obtained by a boolean valuation ϕ of π) there exists an additive resolution (i.e. a multiplicative refinement of ϕ(π)) that is a CyMLL PN with conclusions σ(Γr), where Γr is an additive resolution of Γ (i.e. a choice of an additive subformula for each formula of Γ). In its turn, the correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of LL. The main intuition relies on the fact that any DR-switching (i.e. any correction or test graph for a given PN) can be naturally viewed as a seaweed, i.e. a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Dislike the most part of current syntaxes for non-commutative PNs our syntax allows a sequentialization for the full class of CyMLL PNs, without requiring these latter must be cut-free. Moreover, we give a characterization of CyMALL PNs for the extended (MALL) Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences. In particular additive Lambek PNs allow to parse phrases containing words with syntactical ambiguity (i.e. words with polymorphic type).
This work presents a computational interpretation of the construction process for cyclic (CyLL) a... more This work presents a computational interpretation of the construction process for cyclic (CyLL) and noncommutative (NL) sequential proofs. We assume a proof construction paradigm, based on a normalization procedure, known as focussing which manages eÆciently the non-determinism of the construction. Similarly to the linear case, a new formulation of focussing for NL is used to introduce a general constraintbased technique in order to deal with partial information during proof construction. In particular, the procedure develops through construction steps building \intermediate objects" called abstract proofs, similar to the designs of ludics.
We present a simple cut-elimination procedure for MALL proof nets with monomial weights ( a la Gi... more We present a simple cut-elimination procedure for MALL proof nets with monomial weights ( a la Girard) and explicit contraction links, based on an almost local cut reduction steps. This procedure preserves correctness of proof nets and it is strong normalizing and conuen t.
Linear Logic [4] has raised a lot of interest in computer research, especially because of its res... more Linear Logic [4] has raised a lot of interest in computer research, especially because of its resource sensitive nature. One line of research studies proof construction procedures and their interpretation as computational models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Focusing", has been described in [2]. Focusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Focusing can also be interpreted in the proof-net formalism, where it appears, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the case of the Non-Commutative logic of [1], and all the computational e...
In this work we present a computation paradigm based on a concurrent and incremental construction... more In this work we present a computation paradigm based on a concurrent and incremental construction of proof nets (de-sequentialized or graphical proofs) of the pure multiplicative and additive fragment of Linear Logic, a resources conscious refinement of Classical Logic. Moreover, we set a correspon- dence between this paradigm and those more pragmatic ones inspired to transactional or distributed systems. In particular we show that the construction of additive proof nets can be interpreted as a model for super-ACID (or co-operative) transactions over distributed transactional systems (typi- cally, multi-databases).
Linesir Logic [4] has radsed a lot of interest in computer research, especially because of its re... more Linesir Logic [4] has radsed a lot of interest in computer research, especially because of its resource sensitive nature. One hne of research studies proof construction procedures and their interpretation as computation£il models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Pocusing", has been described in [2]. Pocusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Pocusing Ccin also be interpreted in the proof-net formalism, where it appecirs, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the CEise of the Non-Commutative logic of [1], and all the computational exploitation of Pocusing which has been performed in the commutative case can thus be revised and adapted to the non commutative case. * This work was performed while the second author was visiting XRCE; this visit was supported by the European TMR [Training and Mobility for Researchers) Network "Linear Logic in Computer Science" (esp. the Rome and Marseille sites, XRCE being attEiched to the latter).
Proof nets are a parallel syntax for sequential proofs of linear logic, firstly introduced by Gir... more Proof nets are a parallel syntax for sequential proofs of linear logic, firstly introduced by Girard in 1987. Here we present and intrinsic (geometrical) characterization of proof nets, that is a correctness criterion (an algorithm) for checking those proof structures which correspond to proofs of the purely multiplicative and additive fragment of linear logic. This criterion is formulated in terms of simple graph rewriting rules and it extends an initial idea of a retraction correctness criterion for proof nets of the purely multiplicative fragment of linear logic presented by Danos in his Thesis in 1990.
2008 23rd Annual IEEE Symposium on Logic in Computer Science, 2008
We present a syntax for MALL (multiplicative additive linear logic without units) proof nets whic... more We present a syntax for MALL (multiplicative additive linear logic without units) proof nets which refines Girard's one. It is also based on the use of monomial weights for identifying additive components (slices). Our generalization gives the possibility of representing a kind of sharing of nodes which does not exist in Girard's nets. This sharing leads to the definition of a strong cut elimination procedure for MALL. We give a correctness criterion which is proved to be stable by reduction and to give a sequentialization theorem with respect to the sequent calculus. Sequentialization is proved by showing that an expansion procedure allows us to unfold any of our proof nets into a Girard proof net.
In this work we present a paradigm of focusing proof search based on an incremental construction ... more In this work we present a paradigm of focusing proof search based on an incremental construction of retractile (i.e, correct or sequentializable) proof structures of the pure (units free) multiplicative and additive fragment of linear logic. The correctness of proof construction steps (or expansion steps) is ensured by means of a system of graph retraction rules; this graph rewriting system is shown to be convergent, that is, terminating and confluent. Moreover, the proposed proof construction follows an optimal (parsimonious, indeed) retraction strategy that, at each expansion step, allows to take into account (abstract) graphs that are "smaller" (w.r.t. the size) than the starting proof structures.
We introduce a new correctness criterion for multiplicative non commutative proof nets which can ... more We introduce a new correctness criterion for multiplicative non commutative proof nets which can be considered as the non-commutative counterpart to the Danos-Regnier criterion for proof nets of linear logic. The main intuition relies on the fact that any switching for a proof net (obtained by mutilating one premise of each disjunction link) can be naturally viewed as a series-parallel order variety (a cyclic relation) on the conclusions of the proof net.
When we cut a multiplicative proof-net of linear logic in two parts we get two modules with a cer... more When we cut a multiplicative proof-net of linear logic in two parts we get two modules with a certain border. We call pretype of a module the set of partitions over its border induced by Danos-Regnier switchings. The type of a module is then defined as the double orthogonal of its pretype. This is an optimal notion describing the behaviour of a module: two modules behave in the same way precisely if they have the same type. In this paper we define a procedure which allows to characterize (and calculate) the type of a module only exploiting its intrinsic geometrical properties and without any explicit mention to the notion of orthogonality. This procedure is simply based on elementary graph rewriting steps, corresponding to the associativity, commutativity and weak-distributivity of the multiplicative connectives of linear logic.
This paper illustrates a simple idea on how to interpret undecompsable generalized multiplicative... more This paper illustrates a simple idea on how to interpret undecompsable generalized multiplicative modules of linear logic as probabilistic methods of a logic prgramming language. These new modules/methods allow to express kinds of propagation of probability distribution inside Bayesian Networks that cannot be expressed by standard Prolog-like programming languages. 1 Logic programming with multiplicative modules In the logic programming paradigm, a program execution is interpreted by the process of proof construction. A state of a computation is represented by a (cut-free) proof which may be partial, i.e. with open branches, or, equivalently, containing non-logical axioms. A program is defined by means of a set of rules (or methods) which can be applied to an open branch of a proof to close it (by an axiom) or to expand it with new branches. The standard methods of an abstract logic programming language (typically, Prolog-like) are represented as H : −B1, . . . ,Bn≥0 where the head ...
Linear Logic [4] has raised a lot of interest in computer research, especially because of its res... more Linear Logic [4] has raised a lot of interest in computer research, especially because of its resource sensitive nature. One line of research studies proof construction procedures and their interpretation as computational models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Focusing", has been described in [2]. Focusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Focusing can also be interpreted in the proof-net formalism, where it appears, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the case of the Non-Commutative logic of [1], and all the computational e...
All in-text references underlined in blue are linked to publications on ResearchGate, letting you... more All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
In this paper we investigate the notion of generalized connective for multiplicative linear logic... more In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ⊗ and disjunction`, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms.
This paper studies the so-called generalized multiplicative connectives of linear logic, focusing... more This paper studies the so-called generalized multiplicative connectives of linear logic, focusing on the question of finding the "non-decomposable" ones, i.e., those that may not be expressed as combinations of the default binary connectives of multiplicative linear logic, ⊗ (tensor) and (par). In particular, we concentrate on generalized connectives of a surprisingly simple form, called "entangled connectives", and prove a characterization theorem giving a criterion for identifying the undecomposable ones.
This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e... more This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e. de-sequentialized proofs, of linear logic (LL). In particular, it presents a syntax for PNs of the cyclic multiplicative and additive fragment of linear logic (CyMALL). Any proof structure (PS), in Girards style, is weighted by boolean monomial weights, moreover, its conclusions Γ (a sequence of formulas occurrences) are endowed with a cyclic order σ, i.e., σ(Γ). Naively, a CyMALL PS π with conclusions σ(Γ) is correct if, for any slice ϕ(π) (obtained by a boolean valuation ϕ of π) there exists an additive resolution (i.e. a multiplicative refinement of ϕ(π)) that is a CyMLL PN with conclusions σ(Γr), where Γr is an additive resolution of Γ (i.e. a choice of an additive subformula for each formula of Γ). In its turn, the correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of LL. The main intuition relies on the fact that any DR-switching (i.e. any correction or test graph for a given PN) can be naturally viewed as a seaweed, i.e. a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Dislike the most part of current syntaxes for non-commutative PNs our syntax allows a sequentialization for the full class of CyMLL PNs, without requiring these latter must be cut-free. Moreover, we give a characterization of CyMALL PNs for the extended (MALL) Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences. In particular additive Lambek PNs allow to parse phrases containing words with syntactical ambiguity (i.e. words with polymorphic type).
This work presents a computational interpretation of the construction process for cyclic (CyLL) a... more This work presents a computational interpretation of the construction process for cyclic (CyLL) and noncommutative (NL) sequential proofs. We assume a proof construction paradigm, based on a normalization procedure, known as focussing which manages eÆciently the non-determinism of the construction. Similarly to the linear case, a new formulation of focussing for NL is used to introduce a general constraintbased technique in order to deal with partial information during proof construction. In particular, the procedure develops through construction steps building \intermediate objects" called abstract proofs, similar to the designs of ludics.
We present a simple cut-elimination procedure for MALL proof nets with monomial weights ( a la Gi... more We present a simple cut-elimination procedure for MALL proof nets with monomial weights ( a la Girard) and explicit contraction links, based on an almost local cut reduction steps. This procedure preserves correctness of proof nets and it is strong normalizing and conuen t.
Linear Logic [4] has raised a lot of interest in computer research, especially because of its res... more Linear Logic [4] has raised a lot of interest in computer research, especially because of its resource sensitive nature. One line of research studies proof construction procedures and their interpretation as computational models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Focusing", has been described in [2]. Focusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Focusing can also be interpreted in the proof-net formalism, where it appears, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the case of the Non-Commutative logic of [1], and all the computational e...
In this work we present a computation paradigm based on a concurrent and incremental construction... more In this work we present a computation paradigm based on a concurrent and incremental construction of proof nets (de-sequentialized or graphical proofs) of the pure multiplicative and additive fragment of Linear Logic, a resources conscious refinement of Classical Logic. Moreover, we set a correspon- dence between this paradigm and those more pragmatic ones inspired to transactional or distributed systems. In particular we show that the construction of additive proof nets can be interpreted as a model for super-ACID (or co-operative) transactions over distributed transactional systems (typi- cally, multi-databases).
Linesir Logic [4] has radsed a lot of interest in computer research, especially because of its re... more Linesir Logic [4] has radsed a lot of interest in computer research, especially because of its resource sensitive nature. One hne of research studies proof construction procedures and their interpretation as computation£il models, in the "Logic Programming" tradition. An efficient proof search procedure, based on a proof normalization result called "Pocusing", has been described in [2]. Pocusing is described in terms of the sequent system of commutative Linear Logic, which it refines in two steps. It is shown here that Pocusing Ccin also be interpreted in the proof-net formalism, where it appecirs, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets. This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds. This is, in particular, the CEise of the Non-Commutative logic of [1], and all the computational exploitation of Pocusing which has been performed in the commutative case can thus be revised and adapted to the non commutative case. * This work was performed while the second author was visiting XRCE; this visit was supported by the European TMR [Training and Mobility for Researchers) Network "Linear Logic in Computer Science" (esp. the Rome and Marseille sites, XRCE being attEiched to the latter).
Proof nets are a parallel syntax for sequential proofs of linear logic, firstly introduced by Gir... more Proof nets are a parallel syntax for sequential proofs of linear logic, firstly introduced by Girard in 1987. Here we present and intrinsic (geometrical) characterization of proof nets, that is a correctness criterion (an algorithm) for checking those proof structures which correspond to proofs of the purely multiplicative and additive fragment of linear logic. This criterion is formulated in terms of simple graph rewriting rules and it extends an initial idea of a retraction correctness criterion for proof nets of the purely multiplicative fragment of linear logic presented by Danos in his Thesis in 1990.
2008 23rd Annual IEEE Symposium on Logic in Computer Science, 2008
We present a syntax for MALL (multiplicative additive linear logic without units) proof nets whic... more We present a syntax for MALL (multiplicative additive linear logic without units) proof nets which refines Girard's one. It is also based on the use of monomial weights for identifying additive components (slices). Our generalization gives the possibility of representing a kind of sharing of nodes which does not exist in Girard's nets. This sharing leads to the definition of a strong cut elimination procedure for MALL. We give a correctness criterion which is proved to be stable by reduction and to give a sequentialization theorem with respect to the sequent calculus. Sequentialization is proved by showing that an expansion procedure allows us to unfold any of our proof nets into a Girard proof net.
In this work we present a paradigm of focusing proof search based on an incremental construction ... more In this work we present a paradigm of focusing proof search based on an incremental construction of retractile (i.e, correct or sequentializable) proof structures of the pure (units free) multiplicative and additive fragment of linear logic. The correctness of proof construction steps (or expansion steps) is ensured by means of a system of graph retraction rules; this graph rewriting system is shown to be convergent, that is, terminating and confluent. Moreover, the proposed proof construction follows an optimal (parsimonious, indeed) retraction strategy that, at each expansion step, allows to take into account (abstract) graphs that are "smaller" (w.r.t. the size) than the starting proof structures.
We introduce a new correctness criterion for multiplicative non commutative proof nets which can ... more We introduce a new correctness criterion for multiplicative non commutative proof nets which can be considered as the non-commutative counterpart to the Danos-Regnier criterion for proof nets of linear logic. The main intuition relies on the fact that any switching for a proof net (obtained by mutilating one premise of each disjunction link) can be naturally viewed as a series-parallel order variety (a cyclic relation) on the conclusions of the proof net.
When we cut a multiplicative proof-net of linear logic in two parts we get two modules with a cer... more When we cut a multiplicative proof-net of linear logic in two parts we get two modules with a certain border. We call pretype of a module the set of partitions over its border induced by Danos-Regnier switchings. The type of a module is then defined as the double orthogonal of its pretype. This is an optimal notion describing the behaviour of a module: two modules behave in the same way precisely if they have the same type. In this paper we define a procedure which allows to characterize (and calculate) the type of a module only exploiting its intrinsic geometrical properties and without any explicit mention to the notion of orthogonality. This procedure is simply based on elementary graph rewriting steps, corresponding to the associativity, commutativity and weak-distributivity of the multiplicative connectives of linear logic.
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