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215
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 59 (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.
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 58 (22 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.
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 46 (7 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
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 43 (18 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.
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 41 (15 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.
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 37 (17 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?
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 35 (12 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
Practical RefinementType Checking
, 1997
"... Refinement types allow many more properties of programs to be expressed and statically checked than conventional type systems. We present a practical algorithm for refinementtype checking in a calculus enriched with refinementtype annotations. We prove that our basic algorithm is sound and comple ..."
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Cited by 34 (1 self)
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Refinement types allow many more properties of programs to be expressed and statically checked than conventional type systems. We present a practical algorithm for refinementtype checking in a calculus enriched with refinementtype annotations. We prove that our basic algorithm is sound and complete, and show that every term which has a refinement type can be annotated as required by our algorithm. Our positive experience with an implementation of an extension of this algorithm to the full core language of Standard ML demonstrates that refinement types can be a practical program development tool in a realistic programming language. The required refinement type definitions and annotations are not much of a burden and serve as formal, machinechecked explanations of code invariants which otherwise would remain implicit. 1 Introduction The advantages of staticallytyped programming languages are well known, and have been described many times (e.g. see [Car97]). However, conventional ty...
Encoding Transition Systems in Sequent Calculus
 Theoretical Computer Science
, 1996
"... Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. I ..."
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Cited by 33 (10 self)
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Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. In both cases, it is possible to relate transition paths to proofs in sequent calculus. In neither encoding, however, does it seem possible to capture properties, such as simulation and bisimulation, that need to consider all possible transitions or all possible computation paths. We consider augmenting both intuitionistic and linear logics with a proof theoretical treatment of definitions. In both cases, this addition allows proving various judgments concerning simulation and bisimulation (especially for noetherian transition systems). We also explore the use of infinite proofs to reason about infinite sequences of transitions. Finally, combining definitions and induction into sequent calculus proofs makes it possible to reason more richly about properties of transition systems completely within the formal setting of sequent calculus.
A Proof Procedure for the Logic of Hereditary Harrop Formulas
 JOURNAL OF AUTOMATED REASONING
, 1993
"... A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the socalled goal formulas in the theory of hereditary Harrop formulas. Proof search inintuitionistic logic is complicated by the nonexistence of a Herbrandlike theorem for this logic: formulas cann ..."
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Cited by 30 (12 self)
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A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the socalled goal formulas in the theory of hereditary Harrop formulas. Proof search inintuitionistic logic is complicated by the nonexistence of a Herbrandlike theorem for this logic: formulas cannot in general be preprocessed into a form such as the clausal form and the construction of a proof is often sensitive to the order in which the connectives and quantifiers are analyzed. An interesting aspect of the formulas we consider here is that this analysis can be carried out in a relatively controlled manner in their context. In particular, the task of finding a proof can be reduced to one of demonstrating that a formula follows from a set of assumptions with the next step in this process being determined by the structure of the conclusion formula. An acceptable implementation of this observation must utilize unification. However, since our formulas may contain universal and existential quantifiers in mixed order, care must be exercised to ensure the correctness of unification. One way of realizing this requirement involves labelling constants and variables and then using these labels to constrain unification. This form of unification is presented and used in a proof procedure for goal formulas in a firstorder version of hereditary Harrop formulas. Modifications to this procedure for the relevant formulas in a higherorder logic are also described. The proof procedure that we present has a practical value in that it provides the basis for an implementation of the logic programming language lambdaProlog.