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Logic Programming in a Fragment of Intuitionistic Linear Logic
"... When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ..."
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Cited by 302 (39 self)
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When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottomup search for a cutfree proof contexts, represented as the lefthand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goaldirected interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model.
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
Deciding Provability of Linear Logic Formulas
 Advances in Linear Logic
, 1994
"... Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, ..."
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Cited by 21 (0 self)
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Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, and the multiplicatives\Omega and . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . SRI International Computer Science Laboratory, Menlo Park CA 94025 USA. Work supported under NSF Grant CCR9224858. lincoln@csl.sri.com http://www.csl.sri.com/lincoln/lincoln.html Patrick Lincoln For the most part we will consider fragments of linear logic built up using these connectives in any combination. For example, full linear logic formulas may employ any connective, while multiplic
Constructive Logics. Part II: Linear Logic and Proof Nets
 Proceedings of the International Joint Conference and Symposium on Logic Programming
, 1997
"... . The purpose of this paper is to give an exposition of material dealing with constructive logics, typed calculi, and linear logic. The first part of this paper gives an exposition of background material (with a few exceptions). This second part is devoted to linear logic and proof nets. Particular ..."
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Cited by 14 (0 self)
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. The purpose of this paper is to give an exposition of material dealing with constructive logics, typed calculi, and linear logic. The first part of this paper gives an exposition of background material (with a few exceptions). This second part is devoted to linear logic and proof nets. Particular attention is given to the algebraic semantics (in Girard's terminology, phase semantics) of linear logic. We show how phase spaces arise as an instance of a Galois connection. We also give a direct proof of the correctness of the DanosRegnier criterion for proof nets. This proof is based on a purely graphtheoretic decomposition lemma. As a corollary, we give an O(n 2 )time algorithm for testing whether a proof net is correct. Although the existence of such an algorithm has been announced by Girard, our algorithm appears to be original. This research was partially supported by ONR Grant NOOO1488K0593. Contents 1 Core Linear Logic and Propositional Linear Logic 3 2 Representing I...
On the Linear Decoration of Intuitionistic Derivations
, 1993
"... We define an optimal proofbyproof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained. 1 Introduction Uniform translations of intuitionistic into linear logic, with their plethoric use of exponentials, are bound to gi ..."
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Cited by 13 (1 self)
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We define an optimal proofbyproof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained. 1 Introduction Uniform translations of intuitionistic into linear logic, with their plethoric use of exponentials, are bound to give only `universal linearity information' about proofs. This paper aims at displaying the structure of `specific linearity information ' hidden in a given derivation. How can we apply this to intuitionistic proofs? We have to build a translation into linear logic such that reductions of the intuitionistic proof can be simulated by reductions of its linear image. A necessary condition for this to hold, is that the `skeleton' of the original proof is preserved by the translation. We call translations with this property `decorations '. Specifically, we construct a proofbyproof embedding of IL into LL (formulated as sequent calculi) such that: 1/ the skeleton of the original proof is preserve...
The Undecidability Of Second Order Linear Logic Without Exponentials
 Journal of Symbolic Logic
, 1995
"... . Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicativeadditive fragment, but it does not work in the classical c ..."
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Cited by 12 (3 self)
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. Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicativeadditive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicativeadditive fragment of second order classical linear logic is also undecidable, using an encoding of twocounter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics. In this paper, we write LL for the full propositional fragment of linear logic, MLL for the multiplicative fragment, MALL for the multiplicativeadditive fragment, and MELL for the multiplicativeexponential fragment. Similarly, we write ILL, IMLL, etc. for the fragments of intuitionistic linear logic, LL2, MLL2, etc. for the second order fragments of linear logic, and ILL2, IML...
Connection Methods in Linear Logic and Proof Nets Construction
 Theoretical Computer Science
, 1999
"... Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. A ..."
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Cited by 12 (2 self)
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Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. Aiming at using proof nets as a tool for automated deduction in linear logic, we define a connectionbased characterization of provability in Multiplicative Linear Logic (MLL). We show that an algorithm for proof net construction can be seen as a proofsearch connection method. This central result is illustrated with a specific algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof. From these results we expect to extend to other LL fragments, we analyse what happens with the additive connectives of LL by tackling the additive fragment in a similar way.
Classical BI (A Logic for Reasoning about Dualising Resources)
"... We show how to extend O’Hearn and Pym’s logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a nonconservative extension of (propositional) Boolean BI that includes multiplicative versions of ..."
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Cited by 9 (6 self)
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We show how to extend O’Hearn and Pym’s logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a nonconservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cuteliminating proof system for CBI, based on Belnap’s display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.