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Decision Problems for Propositional Linear Logic
, 1990
"... Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, ..."
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Cited by 107 (19 self)
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Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspacecomplete. We also establish membership in np for the multiplicative fragment, npcompleteness for the multiplicative fragment extended with unrestricted weakening, and undecidability for certain fragments of noncommutative propositional linear logic. 1 Introduction Linear logic, introduced by Girard [14, 18, 17], is a refinement of classical logic which may be derived from a Gentzenstyle sequent calculus axiomatization of classical logic in three steps. The resulting sequent system Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Labo...
Linear Logic
, 1992
"... this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may ..."
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Cited by 24 (1 self)
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this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics
Rewriting for Fitch style natural deductions
 RTA, Lecture Notes in Computer Science 3091 (2004
"... Abstract. Logical systems in natural deduction style are usually presented in the Gentzen style. A different definition of natural deduction, that corresponds more closely to proofs in ordinary mathematical practice, is given in [Fitch 1952]. We define precisely a CurryHoward interpretation that ma ..."
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Cited by 7 (0 self)
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Abstract. Logical systems in natural deduction style are usually presented in the Gentzen style. A different definition of natural deduction, that corresponds more closely to proofs in ordinary mathematical practice, is given in [Fitch 1952]. We define precisely a CurryHoward interpretation that maps Fitch style deductions to simply typed terms, and we analyze why it is not an isomorphism. We then describe three reduction relations on Fitch style natural deductions: one that removes garbage (subproofs that are not needed for the conclusion), one that removes repeats and one that unshares shared subproofs. We also define an equivalence relation that allows to interchange independent steps. We prove that two Fitch deductions are mapped to the same λterm if and only if they are equal via the congruence closure of the aforementioned relations (the reduction relations plus the equivalence relation). This gives a CurryHoward isomorphism between equivalence classes of Fitch deductions and simply typed λterms. Then we define the notion of cutelimination on Fitch deductions, which is only possible for deductions that are completely unshared (normal forms of the unsharing reduction). For conciseness, we restrict in this paper to the implicational fragment of propositional logic, but we believe that our results extend to full first order predicate logic. 1
Sequents, Frames, and Completeness
"... . Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the ..."
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Cited by 2 (1 self)
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. Entailment relations, originated from Scott, have been used for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the unifying principle of the spatiality of coherence logic. In particular, the domain of disjunctive states, derived from the hyperresolution rule as used in disjunctive logic programs, can be seen as the frame freely generated from the opposite of a sequent structure. At the categorical level, we present equivalences among the categories of sequent structures, distributive lattices, and spectral locales using appropriate morphisms. Key words: sequent structures, lattices, frames, domain theory, resolution, category. Introduction Entailment relations were introduced by Scott as an abstract description of Gentzen's sequent calculus [1315]. It can be seen as a generalisation of the ear...
Distributed Reasoning in a PeertoPeer Setting: Application to the Semantic Web
"... In a peertopeer inference system, each peer can reason locally but can also solicit some of its acquaintances, which are peers sharing part of its vocabulary. In this paper, we consider peertopeer inference systems in which the local theory of each peer is a set of propositional clauses defined ..."
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Cited by 1 (0 self)
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In a peertopeer inference system, each peer can reason locally but can also solicit some of its acquaintances, which are peers sharing part of its vocabulary. In this paper, we consider peertopeer inference systems in which the local theory of each peer is a set of propositional clauses defined upon a local vocabulary. An important characteristic of peertopeer inference systems is that the global theory (the union of all peer theories) is not known (as opposed to partitionbased reasoning systems). The main contribution of this paper is to provide the first consequence finding algorithm in a peertopeer setting: DeCA. It is anytime and computes consequences gradually from the solicited peer to peers that are more and more distant. We exhibit a sufficient condition on the acquaintance graph of the peertopeer inference system for guaranteeing the completeness of this algorithm. Another important contribution is to apply this general distributed reasoning setting to the setting of the Semantic Web through the Somewhere semantic peertopeer data management system. The last contribution of this paper is to provide an experimental analysis of the scalability of the peertopeer infrastructure that we propose, on large networks of 1000 peers. 1.
unknown title
, 1994
"... Linear logic, introduced by Girard, is a renement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the \structural " rules of contraction and weakening; adding a modal operator; and adding ner versions of the propositional connec ..."
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Linear logic, introduced by Girard, is a renement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the \structural " rules of contraction and weakening; adding a modal operator; and adding ner versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection, and the control structure of logic programs. In addition, there is a direct correspondence between polynomialtime computation and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional (quantierfree) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes pspacecomplete. We also establish membership in np for the multiplicative fragment, npcompleteness for the multiplicative fragment extended with unrestricted weakening, and undecidability for fragments of noncommutative
1 Linear Logic Appeared in SIGACT 1992 Linear Logic
"... Girard’s claims such as “Linear logic is a resource conscious logic”. Increasingly, computer scientists have recognized linear logic as an expressive and powerful logic, with deep connections to concepts from computer science. The expressive power of linear logic is evidenced by some very natural en ..."
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Girard’s claims such as “Linear logic is a resource conscious logic”. Increasingly, computer scientists have recognized linear logic as an expressive and powerful logic, with deep connections to concepts from computer science. The expressive power of linear logic is evidenced by some very natural encodings of computational models such as Petri nets, counter machines, Turing machines, and others. This note presents an intuitive overview of linear logic, some recent theoretical results, and some interesting applications of linear logic to computer science. Other introductions to linear logic may be found in [12, 36]. 2 Linear Logic vs Classical and Intuitionistic Logics Linear logic differs from classical and intuitionistic logic in several fundamental ways. Classical logic may be viewed as if it deals with static propositions about the world where each proposition is either true or false. Because of the static nature of propositions in classical logic, one may “duplicate ” propositions: P implies (P and P). Implicitly we learn that “one P is as good as two”. Also, one may discard propositions: (P and Q) implies P. Here the proposition Q has been “thrown away”. Both of these sentences are valid in classical logic
Callbyvalue isn’t dual to callbyname, callbyname is dual to callbyvalue!
, 2004
"... Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ⊢ ∆ is a theorem o ..."
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Gentzen’s sequent calculus for classical logic shows great symmetry: for example, the rule introducing ∧ on the left of a sequent is mirror symmetric to the introduction rule for the dual operator ∨ on the right of a sequent. A consequence of this casual observation is that when Γ ⊢ ∆ is a theorem over operators {∨,∧,¬}, then so is ∆◦ ⊢ Γ◦, where Σ◦ reverses the order of formulas in Σ, and exchanges each instance of A ∧ B (C ∨ D) in a formula of Σ with B ∨ A (D ∧ C). This symmetry of rules means that the duality of theorems is also a duality of proof structures.
Appeared in SIGACT 1992 Linear Logic
"... Linear logic was introduced by Girard in 1987 [11]. Since then many results have supported Girard's claims such as \Linear logic is a resource conscious logic". Increasingly, computer scientists have recognized linear logic as an expressive and powerful logic, with deep connections to con ..."
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Linear logic was introduced by Girard in 1987 [11]. Since then many results have supported Girard's claims such as \Linear logic is a resource conscious logic". Increasingly, computer scientists have recognized linear logic as an expressive and powerful logic, with deep connections to concepts from computer science. The expressive power of linear logic is evidenced by some very natural encodings of computational models such as Petri nets, counter machines, Turing machines, and others. This note presents an intuitive overview of linear logic, some recent theoretical results, and some interesting applications of linear logic to computer science. Other introductions to linear logic may be found in [12, 36]. 2 Linear Logic vs Classical and Intuitionistic Logics Linear logic diers from classical and intuitionistic logic in several fundamental ways. Classical logic may be viewed as if it deals with static propositions about the world where each proposition is either true or false. Because of the static nature of propositions in classical logic, one may \duplicate " propositions: P implies (P and P). Implicitly we learn that \one P is as good as two". Also, one may discard propositions: (P and Q) implies P. Here the proposition Q has been \thrown away". Both of these sentences are valid in classical logic for any P and Q. In linear logic, these sentences are not valid. A linear logician might ask \Where did the second P come from? " and \Where did the Q go?". Of course these questions are nonsequiturs in the classical setting, since propositions are assumed to be static, unchanging facts about the world. On the other hand, the rules of linear logic imply that linear propositions stand for dynamic properties or nite resources. For example, consider the propositions D, M, and C, conceived as resources: 1 D