Results 1  10
of
272
Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing b ..."
Abstract

Cited by 114 (22 self)
 Add to MetaCart
In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
Abstract

Cited by 102 (11 self)
 Add to MetaCart
The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
A CurryHoward foundation for functional computation with control
 In Proceedings of ACM SIGPLANSIGACT Symposium on Principle of Programming Languages
, 1997
"... We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatib ..."
Abstract

Cited by 77 (3 self)
 Add to MetaCart
We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant callbyvalue programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and firstclass continuations. Prooftheoretically the dual ¯ v constructs of naming and ¯abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...
Program extraction from classical proofs
 Annals of Pure and Applied Logic
, 1994
"... 1 Introduction It is well known that it is undecidable in general whether a given program meets its specification. In contrast, it can be checked easily by a machine whether a formal proof is correct, and from a constructive proof one can automatically extract a corresponding program, which by its v ..."
Abstract

Cited by 54 (9 self)
 Add to MetaCart
1 Introduction It is well known that it is undecidable in general whether a given program meets its specification. In contrast, it can be checked easily by a machine whether a formal proof is correct, and from a constructive proof one can automatically extract a corresponding program, which by its very construction is correct as well. This at least in principle opens a way to produce correct software, e.g. for safetycritical applications. Moreover, programs obtained from proofs are "commented " in a rather extreme sense. Therefore it is easy to maintain them, and also to adapt them to particular situations. We will concentrate on the question of classical versus constructive proofs. It is known that any classical proof of a specification of the form 8x9yB with B quantifierfree can be transformed into a constructive proof of the same formula. However, when it comes to extraction of a program from a proof obtained in this way, one easily ends up with a mess. Therefore, some refinements of the standard transformation are necessary.
Structural Cut Elimination  I. Intuitionistic and Classical Logic
 Information and Computation
, 2000
"... this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced b ..."
Abstract

Cited by 53 (17 self)
 Add to MetaCart
this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene's sequent system G 3 [Kle52], which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is MartinLo# f 's proof of admissibility [ML68]. In MartinLo# f 's system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this proof directly. Herbelin [Her95] restates this proof and proceeds by assigning proof terms only to restricted sequent calculi LJT and LKT which correspond more immediately to
Classical Logic, Continuation Semantics and Abstract Machines
, 1998
"... Machines Th. STREICHER Fachbereich 4 Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, streiche@mathematik.thdarmstadt.de B. REUS Institut fur Informatik, LudwigMaximiliansUniversitat, Oettingenstr. 67, D80538 Munchen, reus@informatik.unimuenchen.de Abstract One of the ..."
Abstract

Cited by 52 (4 self)
 Add to MetaCart
Machines Th. STREICHER Fachbereich 4 Mathematik, TU Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, streiche@mathematik.thdarmstadt.de B. REUS Institut fur Informatik, LudwigMaximiliansUniversitat, Oettingenstr. 67, D80538 Munchen, reus@informatik.unimuenchen.de Abstract One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped callbyname calculus with Felleisen's control operator C. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ::translation known from logic. The resulting abstract machine appears as an extension of Krivine's Machine implementing head reduction. Though the result, namely Krivine's Machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. callbyvalue variants).
Proofnets and the Hilbert space
 Advances in Linear Logic
, 1995
"... Girard's execution formula (given in [Gir88a]) is a decomposition of usual fireduction (or cutelimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a term or a net, as the sum of maximal paths on the term/net that are not cancelled by th ..."
Abstract

Cited by 50 (3 self)
 Add to MetaCart
Girard's execution formula (given in [Gir88a]) is a decomposition of usual fireduction (or cutelimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a term or a net, as the sum of maximal paths on the term/net that are not cancelled by the algebra L (as was done in [Dan90, Reg92]). It is then natural to ask for a characterization of those paths, that would be only of geometric nature. We prove here that they are exactly those paths that have residuals in any reduct of the term/net. Remarkably, the proof puts to use for the first time the interpretation of terms/nets as operators on the Hilbert space. 1 Presentation Calculus is simple but not completely convincing as a real machinelanguage. Real machine instructions have a fixed runtime; a fireduction step does not. Some implementations do map fireductions into sequences of real elementary steps (as in environment machines for example) but they use a global time t...
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, ext ..."
Abstract

Cited by 49 (27 self)
 Add to MetaCart
We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
Typed lambdacalculus in classical ZermeloFraenkel set theory
 ARCHIVE OF MATHEMATICAL LOGIC
, 2001
"... In this paper, we develop a system of typed lambdacalculus for the ZermeloFraenkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambdacalculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connect ..."
Abstract

Cited by 34 (9 self)
 Add to MetaCart
In this paper, we develop a system of typed lambdacalculus for the ZermeloFraenkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambdacalculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connective #. It is very important, because the well known CurryHoward correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property : every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.Y. Girard[4], under the name of system F, still with the normalization property. More recently, in 1990, the CurryHoward correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming