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285
Implementing Tactics and Tacticals in a HigherOrder Logic Programming Language
 Journal of Automated Reasoning
, 1993
"... We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement tactic style theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped terms ..."
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Cited by 69 (15 self)
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We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement tactic style theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped terms, replacing firstorder unification with higherorder unification, and allowing implication and universal quantification in queries and the bodies of clauses. Inference rules for a variety of inference systems can be naturally specified in this language. The higherorder features of the language contribute to a concise specification of provisos concerning variable occurrences in formulas and the discharge of assumptions present in many inference systems. Tactics and tacticals, which provide a framework for highlevel control over search for proofs, can be directly and naturally implemented in the extended language. This framework serves as a starting point for implementing theorem provers an...
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 66 (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.
From operational semantics to abstract machines
 Mathematical Structures in Computer Science
, 1992
"... We consider the problem of mechanically constructing abstract machines from operational semantics, producing intermediatelevel specifications of evaluators guaranteed to be correct with respect to the operational semantics. We construct these machines by repeatedly applying correctnesspreserving t ..."
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Cited by 65 (6 self)
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We consider the problem of mechanically constructing abstract machines from operational semantics, producing intermediatelevel specifications of evaluators guaranteed to be correct with respect to the operational semantics. We construct these machines by repeatedly applying correctnesspreserving transformations to operational semantics until the resulting specifications have the form of abstract machines. Though not automatable in general, this approach to constructing machine implementations can be mechanized, providing machineverified correctness proofs. As examples we present the transformation of specifications for both callbyname and callbyvalue evaluation of the untyped λcalculus into abstract machines that implement such evaluation strategies. We also present extensions to the callbyvalue machine for a language containing constructs for recursion, conditionals, concrete data types, and builtin functions. In all cases, the correctness of the derived abstract machines follows from the (generally transparent) correctness of the initial operational semantic specification and the correctness of the transformations applied. 1.
Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λCalculi
, 1992
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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 60 (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 56 (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 proof theory for generic judgments: An extended abstract
 In LICS 2003
, 2003
"... A powerful and declarative means of specifying computations containing abstractions involves metalevel, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and t ..."
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Cited by 55 (20 self)
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A powerful and declarative means of specifying computations containing abstractions involves metalevel, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cutelimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the πcalculus and the encoding of objectlevel provability.
Specifying and Implementing Theorem Provers in a HigherOrder Logic Programming Language
, 1989
"... We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped λterms, replacing ..."
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Cited by 50 (8 self)
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We argue that a logic programming language with a higherorder intuitionistic logic as its foundation can be used both to naturally specify and implement theorem provers. The language extends traditional logic programming languages by replacing firstorder terms with simplytyped λterms, replacing firstorder unification with higherorder unification, and allowing implication and universal quantification in queries and the bodies of clauses. Inference rules for a variety of proof systems can be naturally specified in this language. The higherorder features of the language contribute to a concise specification of provisos concerning variable occurrences in formulas and the discharge of assumptions present in many proof systems. In addition, abstraction in metaterms allows the construction of terms representing object level proofs which capture the notions of abstractions found in many proof systems. The operational interpretations of the connectives of the language provide a set of basic search operations which describe goaldirected search for proofs. To emphasize the generality of the metalanguage, we compare it to another general specification language: the Logical Framework (LF). We describe a translation which compiles a specification of a logic in LF to a set of formulas of our metalanguage, and
Two party immediate response disputes: properties and efficiency
 Artificial Intelligence
, 2001
"... Abstract. Two Party Immediate Response Disputes (TPIdisputes) are one class of dialogue or argument game in which the protagonists take turns producing counter arguments to the ‘most recent ’ argument advanced by their opponent. Argument games have been found useful as a means of modelling dialecti ..."
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Cited by 47 (18 self)
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Abstract. Two Party Immediate Response Disputes (TPIdisputes) are one class of dialogue or argument game in which the protagonists take turns producing counter arguments to the ‘most recent ’ argument advanced by their opponent. Argument games have been found useful as a means of modelling dialectical discourse and in providing semantic bases for proof theoretic aspects of reasoning. In this article we consider a formalisation of TPIdisputes in the context of finite Argument Systems. Our principal concern may, informally, be phrased as follows: given a specific argument system, À and argument, x within À, what can be stated concerning the number of rounds a dispute might take for one of its protagonists to accept that x has some defence respectively cannot be defended?
Minimal Classical Logic and Control Operators
 In ICALP: Annual International Colloquium on Automata, Languages and Programming, volume 2719 of LNCS
, 2003
"... We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a \natural" implementation of this logic is Parigot's classical natural deduction. ..."
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Cited by 40 (4 self)
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We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a \natural" implementation of this logic is Parigot's classical natural deduction.