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A MultipleConclusion MetaLogic
 In Proceedings of 9th Annual IEEE Symposium On Logic In Computer Science
, 1994
"... The theory of cutfree sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [12], provide data types, higherorder programming) but lack primitives for concurrency. The logic pro ..."
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Cited by 86 (7 self)
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The theory of cutfree sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [12], provide data types, higherorder programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends the languages λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. As a metalanguage, Forum greatly extends the expressiveness of these other logic programming languages. To illustrate its expressive strength, we specify in Forum a sequent calculus proof system and the operational semantics of a functional programming language that incorporates such nonfunctional features as counters and references. 1
A concurrent logical framework I: Judgments and properties
, 2003
"... The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous con ..."
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Cited by 86 (30 self)
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The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives# of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives #, & and #.
Cutelimination for a logic with definitions and induction
 Theoretical Computer Science
, 1997
"... In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The l ..."
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Cited by 72 (22 self)
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In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The literature contains many approaches to formally adding these reasoning principles with logic specifications. We choose an approach based on the sequent calculus and design an intuitionistic logic F Oλ ∆IN that includes natural number induction and a notion of definition. We have detailed elsewhere that this logic has a number of applications. In this paper we prove the cutelimination theorem for F Oλ ∆IN, adapting a technique due to Tait and MartinLöf. This cutelimination proof is technically interesting and significantly extends previous results of this kind. 1
A Uniform ProofTheoretic Investigation Of Linear Logic Programming
, 1994
"... In this paper we consider the problem of identifying logic programming languages for linear logic. Our analysis builds on a notion of goaldirected provability, characterized by the socalled uniform proofs, previously introduced for minimal and intuitionistic logic. A class of uniform proofs in lin ..."
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Cited by 71 (21 self)
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In this paper we consider the problem of identifying logic programming languages for linear logic. Our analysis builds on a notion of goaldirected provability, characterized by the socalled uniform proofs, previously introduced for minimal and intuitionistic logic. A class of uniform proofs in linear logic is identified by an analysis of the permutability of inferences in the linear sequent calculus. We show that this class of proofs is complete (for logical consequence) for a certain (quite large) fragment of linear logic, which thus forms a logic programming language. We obtain a notion of resolution proof, in which only one left rule, of clausedirected resolution, is required. We also consider a translation, resembling those of Girard, of the hereditary Harrop fragment of intuitionistic logic into our framework. We show that goaldirected provability is preserved under this translation.
Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 68 (27 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
Efficient resource management for linear logic proof search
 Proceedings of the 5th International Workshop on Extensions of Logic Programming
, 1996
"... The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses (Prolog) and hereditary Harrop formulas (λProlog and Elf). Among these, the problem of efficiently managing the li ..."
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Cited by 64 (11 self)
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The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses (Prolog) and hereditary Harrop formulas (λProlog and Elf). Among these, the problem of efficiently managing the linear context when solving a goal is of crucial importance for the use of these systems in nontrivial applications. This paper studies this problem in the case of Lolli [HM94], though its results have application to other systems. We first give a prooftheoretic presentation of the operational semantics of this language as a resolution calculus. We then present a series of resource management systems designed to eliminate the nondeterminism in the distribution of linear formulas that undermines the efficiency of a direct implementation of this system. 1
Noncommutativity and MELL in the Calculus of Structures
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2001
"... We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a selfdual noncommutative operator inspired by CCS, that seems not to be expressible in ..."
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Cited by 61 (25 self)
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We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a selfdual noncommutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.
Least and greatest fixed points in linear logic Extended Version
, 2007
"... david.baelde at enslyon.org dale.miller at inria.fr Abstract. The firstorder theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addi ..."
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Cited by 61 (14 self)
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david.baelde at enslyon.org dale.miller at inria.fr Abstract. The firstorder theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and?), we add least and greatest fixed point operators. The resulting logic, which we call µMALL = , satisfies two fundamental proof theoretic properties. In particular, µMALL = satisfies cutelimination, which implies consistency, and has a complete focused proof system. This second result about focused proofs provides a strong normal form for cutfree proof structures that can be used, for example, to help automate proof search. We then consider applying these two results about µMALL = to derive a focused proof system for an intuitionistic logic extended with induction and coinduction. The traditional approach to encoding intuitionistic logic into linear logic relies heavily on using the exponentials, which unfortunately weaken the focusing discipline. We get a better focused proof system by observing that certain fixed points satisfy the structural rules of weakening and contraction (without using exponentials). The resulting focused proof system for intuitionistic logic is closely related to the one implemented in Bedwyr, a recent model checker based on logic programming. We discuss how our proof theory might be used to build a computational system that can partially automate induction and coinduction. 1
A Concurrent Logical Framework II: Examples and Applications
, 2002
"... CLF is a new logical framework with an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives # of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the ..."
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Cited by 58 (36 self)
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CLF is a new logical framework with an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives # of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives #.