Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science 232 (2000), no. 1--2, 299--333.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....difficult. Generalizing the right hand side to admit more than one true proposition would violate either linearity or the intuitionistic interpretation in our setting. However, multiple conclusions do not necessarily conflict with natural deduction (see, for example, 20] even for intuitionistic [26] and linear logics [16] Indeed, we can readily incorporate such an approach in our judgmental framework by introducing a new judgment form, C 1 poss C k poss, on the right hand side of a hypothetical judgment. This judgment means that either the hypotheses are contradictory or one of the C i ....

Eike Ritter, David Pym, and Lincoln Wallen. On the intuitionistic force of classical search. Theoretical Computer Science, 232(1--2):299--333, 2000.

....sequent calculus in order to sharpen the distinction between proof search and proof construction. This distinction is based on observations of the differences between the singleconclusioned intuitionistic calculus LJ and the multiple conclusioned calculus, which, following Ritter, Pym and Wallen [10], we refer to as LM [11, 12] In [6] it is noted how LM can be considered as better for search than LJ, due to the greater possibilities for delaying choices (particularly for disjunctions) in LM than LJ. As in [7] for the resource distribution problem of linear logic, we use Boolean constraints ....

E. Ritter, D. Pym and L. Wallen, On the Intuitionistic Force of Classical Search, Theoretical Computer Science 232:299-330, 2000.

....case the constraint is trivial) The shift to the type theoretic point of view has a useful consequence. It can be argued that, from an essentially combinatorial point of view, the search space determined by classical sequent calculus, LK, underlies the search spaces for other logics or systems [91, 141, 154, 24 155]. Given this point of view, a natural question is the following: How can the classical search space be used as a basis for proof search in non classical logics The paper by Ritter, Pym and Wallen, in this volume, addresses this question in detail for propositional intuitionistic logic. The basic ....

E. Ritter D.J. Pym and L.A. Wallen. On the intuitionistic force of classical search. In Proc. Tableaux 96, LNCS 1071, pages 295-31, 1996.

....proof search from classical proof search [13] Then classical logic plays a pivotal role in search calculi and an interesting point consists in understanding how search in a non classical logic can be viewed as a perturbation on search in classical logic. It has been recently studied in [30] by considering a proof theoretic approach, based on terms [29] where the terms permit an axiomatisation of the perturbation which yields the non classical search and also some semantical explanations about embedding in classical logic. In this setting, some questions arise: can we conclude ....

....and semantics analysis. In section 2, we study the intuitionistic provability from classical search by Labelled Proof Systems for Intuitionistic Provability 3 defining a new labelled proof system. This labelled sequent calculus for CL is an alternative to the based on terms approach given in [30], being not type theoretic but proof search oriented. In both cases, there is no completeness result, i.e one cannot conclude the intuitionistic provability from any given classical proof. The main part of this paper deals with characterizations of the intuitionistic provability in the setting of ....

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E. Ritter, D. Pym, and L. Wallen. On the intuitionistic force of classical search. In 5th Int. Workshop TABLEAUX'96, LNAI 1071, pages 294--313, Terrasini, Palermo, Italy, May 1996.

....in a stepwise and incremental manner (Abadi et al. 1991) Among other benefits, explicit substitution makes rewrite steps closer to actual machine operations and allows more laziness in evaluation. Explicit substitution calculi have been used for, among other things, work on proof search (Ritter et al. 1996), unification (Dowek et al. 1995) representation of incomplete proofs (Munoz Hurtado 1996b; Magnusson 1995) proof theory (Herbelin 1995) and functional compiler backends (Hardin et al. 1996) Curry Howard Correspondence for L Systems. While the Curry Howard Correspondence is straightforward ....

Ritter, E., Pym, D., and Wallen, L.A. (1996). On the intuitionistic force of classical search. In Miglioli, P., Moscato, U., Mundici, D., and Ornaghi, M., editors, Proc of the Fifth Workshop on Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of Lecture Notes in Artificial Intelligence, pages 295--311, Springer Verlag.

....is a restriction of LK, the latter is complete for intuitionistic logic, but not sound. By studying the structure of LK derivations under permutation of inference rules, it is possible to assess their intuitionistic force and hence use classical search to determine intuitionistic provability. In [16], we used terms of the ffl calculus, a variant of Parigot s calculus, as a system of realizers for sequent derivations to present such an analysis, and hence give a characterization of the search space for intuitionistic logic in terms of that for classical logic. Operations on the realizers ....

....intuitionistically sound, but to express the intuitionistic force of standard (classical) resolution, and thereby give a characterization of the search space for intuitionistic logic in terms of that for classical resolution. 1. 2 Overview of technical results In x 2, we summarize the results of [16]. A translation of sequent derivations into ffl terms is given for the disjunction free fragment of classical logic. The terms are seen as realizers for the derivations. A realizer is said to be intuitionistic if it satisfies a certain structural condition related to weakening and rule ....

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E. Ritter, D. Pym and L. Wallen. On the intuitionistic force of classical search. In: P. Miglioli, U. Moscato, D. Mundici and M. Ornaghi (eds.), Proc. Tableaux '96, LNCS 1071, 295--311, 1996.

....a full account of the computations involved in computing a term, one must describe a method to compute substitutions. Since the seminal work of Abadi, Cardelli, Curien and L evy, explicit substitutions have developed into a subject of their own, finding further applications e.g. in proof search [29], unification [11] representation of incomplete proofs [23, 21] and proof theory [16] In this paper, we generalise some of the results on explicit substitutions for calculi to classical calculi, i.e. calculi with control like structures. More precisely, we consider a specific calculus with ....

.... of explicit substitutions for another classical calculus, namely Parigot s calculus [25] Audebaud s calculus [3] of explicit substitutions is an explicit substitution calculus with de Bruijn indices and composition of substitutions in the spirit of oe whereas Pym, Ritter and Wallen s ffl [29] calculus is a named explicit substitution calculus without composition of substitutions in the spirit of exp. In [3] the system presented is shown to be confluent on open terms. Confluence on open terms is not however a question that is usually studied in calculi written with named variables ....

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E. Ritter, D. Pym, and L. A. Wallen. On the intuitionistic force of classical search. In P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, editors, Procedings of TABLEAU'96, volume 1071 of Lecture Notes in Artificial Intelligence, pages 295--311. Springer Verlag, 1996.

....extensively characterized by Orevkov [7] More recently, people are interested in the question how and under which condition classical proofs of a formula can be transformed into an intuitionistic proof of the same formula, 1 and how the sizes of the corresponding search spaces are related. In [10, 11], translations are considered based on the idea to permute inferences in classical proofs until an intuitionistic proof is obtained or all classical proofs and all permutations have been tried. The motivation for investigating such a translation is at least twofold. Firstly, the complexity of the ....

....then analytic cuts can be simulated even in cut free calculi [2, 3] Now, the procedure is obvious. Translate the formula into definitional form and prove it classically. Then try to get an intuitionistic proof of the intuitionistic definitional form of the same formula. Roughly speaking, in [10], classical cut free proofs in LK are translated to intuitionistic cut free proofs in LJ, whereas, in a second paper [11] a definitional translation is used resulting in a more powerful translation, mainly because analytic cuts can then be simulated in LJ. The structure of the paper is as ....

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D. Pym, E. Ritter, and L. Wallen. On the Intuitionistic Force of Classical Search. In P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, editors, Theorem Proving with Analytic Tableaux and Related Methods, pages 295--311. Springer Verlag, 1996.

....from [10] or [19] and this is probably the only sensible way of doing it for nonclassical first order logics. Techniques based on encoding semantical Kripke accessibility relationships as path expressions [17, 23] seem to pose similar problems and to offer similar solutions. A recent technique [21] is to look for intuitionistic proofs as (hidden) subproofs of classical proofs, by a clever deciphering on proof terms for classical logic expressed in Parigot s calculus. Unfortunately, in the latter technique we need to decorate formulas on paths by terms; if done naively, this will require ....

E. Ritter, D. Pym, and L. Wallen. On the intuitionistic force of classical search. In Tableaux'96, 1996. Lecture Notes in Artificial Intelligence, Springer Verlag.

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E. Ritter, D. Pym, and L. A. Wallen. On the intuitionistic force of classical search. Journal of Logic and Computation, 10(2):173--207, 2000.

....a dependently typed calculus which provides a partial analysis, being both in the spirit of BI and yet somewhat reliant on the presence of a form of Dereliction. Nevertheless, can be interpreted in the general bred framework sketched in Figure 2; Classical logic: the and calculi [22,26,27,29,30] allow the presentation of a semantics of the proofs of propositional classical logic within the bred framework [29] In ( sequents are structured so as to have a chosen active position on the right (the left most position) t : where the term t inhabits (in the sense of the ....

....presence of a form of Dereliction. Nevertheless, can be interpreted in the general bred framework sketched in Figure 2; Classical logic: the and calculi [22,26,27,29,30] allow the presentation of a semantics of the proofs of propositional classical logic within the bred framework [29]. In ( sequents are structured so as to have a chosen active position on the right (the left most position) t : where the term t inhabits (in the sense of the propositions as types correspondence) the formula, in the context of the formul in . A bred semantics is obtained by ....

[Article contains additional citation context not shown here]

Ritter, E. and D. Pym and L. Wallen, On the intuitionistic force of classical search, Theoretical Computer Science 232 (2000) 299-333.

....latter is complete for intuitionistic logic, but not sound. By studying the structure of LK derivations under permutation of inference rules, it is possible to assess their intuitionistic force and hence use classical search with its simple combinatorics to determine intuitionistic provability. In [25, 26], we developed a system of terms based on Parigot s calculus to serve as a system of realizers for sequent derivations in a disjunction free fragment of classical and intuitionistic logic. Combinatorial operations on the realizers were used to enlarge the set of classical derivations that could ....

....translations between resolution systems and the sequent calculus, to extend the analysis to resolution yielding a characterization of the search space for that fragment of intuitionistic logic in terms of the search space generated by classical resolution. In this paper we extend the results of [25, 26] and [27] to include disjunction thereby simplifying and completing the picture of the intuitionistic search space in terms of classical resolution. This involves two steps. The first is the formulation of a suitable system of realizers for full classical logic and the characterization of the ....

[Article contains additional citation context not shown here]

E. Ritter, D. Pym and L. Wallen. On the intuitionistic force of classical search. To appear, Theoretical Computer Science, 1999.

....latter is complete for intuitionistic logic, but not sound. By studying the structure of LK derivations under permutation of inference rules, it is possible to assess their intuitionistic force and hence use classical search with its simple combinatorics to determine intuitionistic provability. In [25, 26], we developed a system of terms based on Parigot s calculus to serve as a system of realizers for sequent derivations in a disjunction free fragment of classical and intuitionistic logic. Combinatorial operations on the realizers were used to enlarge the set of classical derivations that could ....

....translations between resolution systems and the sequent calculus, to extend the analysis to resolution yielding a characterization of the search space for that fragment of intuitionistic logic in terms of the search space generated by classical resolution. In this paper we extend the results of [25, 26] and [27] to include disjunction thereby simplifying and completing the picture of the intuitionistic search space in terms of classical resolution. This involves two steps. The first is the formulation of a suitable system of realizers for full classical logic and the characterization of the ....

[Article contains additional citation context not shown here]

E. Ritter, D. Pym and L. Wallen. On the intuitionistic force of classical search (extended abstract). In: P. Miglioli, U. Moscato, D. Mundici and M. Ornaghi (eds.), Proc. Tableaux '96, LNCS 1071, 295--311, 1996.

....latter is complete for intuitionistic logic, but not sound. By studying the structure of LK derivations under permutation of inference rules, it is possible to assess their intuitionistic force and hence use classical search with its simple combinatorics to determine intuitionistic provability. In [22], we developed a system of terms based on Parigot s calculus to serve as a system of realizers for sequent derivations in a disjunction free fragment of classical and intuitionistic logic. Combinatorial operations on the realizers were used to enlarge the set of classical derivations that could ....

....translations between resolution systems and the sequent calculus, to extend the analysis to resolution yielding a characterization of the search space for that fragment of intuitionistic logic in terms of the search space generated by classical resolution. In this paper we extend the results of [22] and [23] to include disjunction thereby simplifying and completing the picture of the intuitionistic search space in terms of classical resolution. This involves two steps. The first is the formulation of a suitable system of realizers for full classical logic and the characterisation of the ....

[Article contains additional citation context not shown here]

E. Ritter, D. Pym and L. Wallen. On the intuitionistic force of classical search (extended abstract). In: P. Miglioli, U. Moscato, D. Mundici and M. Ornaghi (eds.), Proc. Tableaux '96, LNCS 1071, 295--311, 1996.

....case the constraint is trivial) The shift to the type theoretic point of view has a useful consequence. It can be argued that, from an essentially combinatorial point of view, the search space determined by classical sequent calculus, LK, underlies the search spaces for other logics or systems [91, 141, 154, 155]. Given this point of view, a natural question is the following: How can the classical search space be used as a basis for proof search in non classical logics The paper by Ritter, Pym and Wallen, in this volume, addresses this question in detail for propositional intuitionistic logic. The ....

E. Ritter D.J. Pym and L.A. Wallen. On the intuitionistic force of classical search. In Proc. Tableaux 96, LNCS 1071, pages 295--31, 1996.

....by some of us (Pym and Ritter) elsewhere. An application of the results to intuitionistic and classical resolution has been developed in [29] A preliminary version of the work presented herein for the fragment of intuitionistic propositional logic without negation or disjunction can be found in [28]. 2 Proof objects and calculus 2.1 Proof objects and realizers For propositional intuitionistic (oe; logic, proofs of a sequent Gamma A, within a single conclusioned calculus of sequents LJ, can be interpreted as constructions of natural deductions of the succedent formula A from ....

E. Ritter, D.J. Pym and L.A. Wallen. On the intuitionistic force of classical search (extended abstract). In: P. Miglioli, U. Moscato, D. Mundici and M. Ornaghi (eds.), Proc. Tableaux '96, LNCS 1071, 295-311, 1996.

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Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science 232 (2000), no. 1--2, 299--333.

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Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science 232 (2000), no. 1--2, 299--333.

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Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science, vol. 232 (2000), no. 1--2, pp. 299--333.

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Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science 232 (2000), no. 1--2, 299--333.

No context found.

E. Ritter, D. Pym, and L. A. Wallen. On the intuitionistic force of classical search. In P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi, editors, Proc of the Fifth Workshop on Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of Lecture Notes in Arti cial Intelligence, pages 295-311, Berlin, Heidelberg, New York, 1996. Springer Verlag.

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Ritter, E., Pym, D., and Wallen, L.A. (1996). On the intuitionistic force of classical search. In Miglioli, P., Moscato, U., Mundici, D., and Ornaghi, M., editors, Proc of the Fifth Workshop on Theorem Proving with Analytic Tableaux and Related Methods, volume 1071 of Lecture Notes in Artificial Intelligence, pages 295--311, Springer Verlag.

No context found.

Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science 232 (2000), no. 1--2, 299--333.

No context found.

Eike Ritter, David Pym, and Lincoln Wallen, On the intuitionistic force of classical search, Theoretical Computer Science, vol. 232 (2000), no. 1--2, pp. 299--333.

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