G. Bellin and J. Ketonen. A Decision Procedure Revisited: Notes on Direct Logic, Linear Logic and its Implementation, Theoretical Computer Science 95, 1992 n. 1, pp. 115-142.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one: x:nat) P x) B (P O) and then uses Auto which completes the proof. 4.10.7 Linear. The tactic Linear, due to Jean Christophe Filliatre [38] implements a decision procedure for Direct Predicate Calculus, that is first order Gentzen s Sequent Calculus without contraction rules [57, 10]. Intuitively, a first order goal is provable in Direct Predicate Calculus if it can be proved using each hypothesis at most once. Unlike the previous tactics, the Linear tactic does not belong to the initial state of the system, and it must be loaded explicitly with the command Coq Cd ....

G. Bellin and J. Ketonen. A decision procedure revisited : Notes on direct logic, linear logic and its implementation. Theoretical Computer Science, 95:115--142, 1992.

....subtle way, it prevents proofs by case, like for instance the drinkers theorem 9y:8x: P (y) oe P (x) which is provable in Gentzen sequent calculus but not in Direct Predicate Calculus. In [6] a decision procedure for Direct Predicate Calculus is explicitly given. It has been studied again in [1], which mentions a mistake in the original paper, carries out relations with linear logic and gives details about implementation of the decision procedure. The basic idea is simple: each atomic subformula This research was partly supported by ESPRIT Basic Research Action Types and by the GDR ....

....see a derivation as the set of its axioms. The decision procedure consists of looking for such sets (called paths) which are finite and in finite number, then to construct derivation from paths. Quantification, in the case of prenex formulas, in handled through Herbrand functions and unification [1, 9]. The result is no longer true for non prenex formulas. The skolemization does not assure the eigenvariable condition: it can now depend on the order of the quantifier rules, which was obvious and fixed in the prenex case. We extend the decision procedure to handle the case of non prenex formulas; ....

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G. Bellin and J. Ketonen. A decision procedure revisited : Notes on direct logic, linear logic and its implementation. Theoretical Computer Science, 95:115--142, 1992.

....clear that these proof methods deal with concepts that are implicitly related to concepts of LL s proof theory. Let us notice that in 1984, 33] describes a decision procedure for direct logic, where contraction is eliminated, such that a formula can be used at most once in a proof. More recently, [8] has analysed its relationships with linear logic and in fact some concepts used in such procedures are implicitly related to concepts de ned, or used, by computer scientists and logicians working on proof theory applications. In this paper, we propose a new approach to de ne connection ....

.... of proof search (such as path, formula tree or connection) and logical concepts related to the proof net notion (as axiom link, decomposition tree, proof structure) Such correspondences were often considered implicitly in works devoted to proof construction in direct logic [33] or linear logic [7, 8, 24] and the presentation of the relationships between connections and proof nets provides an opportunity to make them more explicit, from both the automated proof search and the logical points of view. 2 Linear logic and proofs The linear logic (LL) introduced by J.Y. Girard [27] is a logic of ....

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G. Bellin and J. Ketonen. A decision procedure revisited: Notes on direct logic, linear logic and its implementation. Theoretical Computer Science, 95:115142, 1992.

....one: x:nat) P x) B (P O) and then uses Auto which completes the proof. 4.9.6 Linear. The tactic Linear, due to Jean Christophe Filliatre [34] implements a decision procedure for Direct Predicate Calculus, that is first order Gentzen s Sequent Calculus without contraction rules [49, 8]. Intuitively, a first order goal is provable in Direct Predicate Calculus if it can be proved using each hypothesis at most once. Unlike the previous tactics, the Linear tactic does not belong to the initial state of the system, and it must be loaded explicitly with the command Coq Cd ....

G. Bellin and J. Ketonen. A decision procedure revisited : Notes on direct logic, linear logic and its implementation. Theoretical Computer Science, 95:115--142, 1992.

....specific subclass of formulas of a logic in rough analogy like horn formulas being part of the first order formulas. The aim of this paper is to present this logic and identify the appropriate subclass of formulas. The central idea is simple: Combine Direct Logic (i.e. LK without contraction) [1, 14] with the exponentials of Linear Logic, to reintroduce contraction. Because contraction is not available in general, no formula (resource) can be reused in a proof unless it is marked as reusable. In contrast to Linear Logic weakening is still present and therefore formulas need not necessarily be ....

....the simultaneous substitution of the variables x 1 ; xn by the terms t 1 ; t n . For arbitrary formulas OE; 2 F we introduce the implication OE as an abbreviation for :OE . The meaning of the connectives are defined by a sequent calculus which is a combination of Direct Logic [1, 14] (LK without the contraction rule) and the exponentials of Linear Logic [12] Because weakening is already present, we can drop the explicit weakening rules for the exponentials. In the following possible indexed occurrences of Gamma and Delta denote finite (possible empty) multi sets of ....

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G. Bellin, J. Ketonen. A decision procedure revisited: Notes on Direct Logic, Linear Logic and its implementation. Theoretical Computer Science, 95:115--142, 1992.

....arbitrary choices to be made, so these fragments are somewhat speculative. Variants of linear logic with unrestricted weakening (sometimes called Affine Logic) have also been studied [32, 7] Here again the logics are somewhat speculative, although there is a close relationship with direct logic [27, 11]. Some fragments of linear logic with weakening have the same complexity as the same fragment without weakening. For example, just as for linear logic, full first order affine logic is undecidable, as can be seen by the fact that Girard s encoding of classical logic into linear logic [17] is also ....

.... area of study is directly relevant to the logic programming use of linear logic, where linear logic sequents are taken to be logic programs which execute by performing proof search [20, 5, 4, 19] This area of research is also directly relevant to the construction of linear logic theorem provers [40, 44, 35, 6, 11, 10, 41]. The results here also lead into the study of semantics of linear logic, pointing to deep connections between various fragments of linear logic and familiar structures from computer science [12, 29, 30] In particular, work has progressed in attempting to find viewpoints where the proof theory of ....

G. Bellin and J. Ketonen. A decision procedure revisited: Notes on direct logic, linear logic, and its implementation. Theoretical Computer Science, 95:115--142, 1992.

.... cut static and dynamic elimination of irrelevance coincide. iii) The property of being in weakening normal form is not preserved by cutelimination: in MAL a times cut reduction may introduce new irrelevance. iv) A representation of proofs in MAL is available in terms of proof nets in [13, 3] (the system of first order multiplicative affine logic is called there Direct Logic; those papers belong in the tradition of automated proof searching, and do not give a treatment of cut elimination) The above definition of irrelevance may be rephrased for proof structures, and a proof net is ....

....computation with garbage collection for intuitionistic systems is best understood as a sequential process 2 Proof nets in MAL with Mix We define proof nets for MAL MIX and prove that Cut elimination in this system has the Church Rosser property. For full definitions, we refer the reader to [2, 3]. 2.1 Language, Sequent Calculus The propositional language of Multiplicative Affine Logic MAL has the connectives Omega (times) and (par) of Multiplicative Linear Logic with the propositional constants 1 and . Definition 1. The sequent calculus for propositional MAL MIX is as follows: ....

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G. Bellin and J. Ketonen. A Decision Procedure Revisited: Notes on Direct Logic, Linear Logic and its Implementation, Theoretical Computer Science 95, 1992 n. 1, pp. 115-142.

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G. Bellin and J. Ketonen. A decision procedure revisited: Notes on direct logic, linear logic, and its implementation. Theoretical Computer Science, 95:115--142, 1992.

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