D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Proc. of the International Symposium on Logical Foundations of Computer Science, Tver, C.I.S., 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This flowering of developments has been backed by theoretical contributions concerning those aspects of Linear Logic which are essential to proof search. In particular, there have been several investigations of the crucial problem of permutations of inference figures during proof construction [3, 6, 31, 42, 25]. Other contributions have investigated in terms of proof search the complexity of different fragments of Linear Logic [36] In this paper, we contribute to the study of yet another aspect: the abstract interpretation of Linear Logic proofs. The practical outcome that we expect from this ....

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Proc. of the International Symposium on Logical Foundations of Computer Science, Tver, C.I.S., 1992.

....an inference introducing A i , thus eliminating any or branching stemming from rule selection. For connective the applicability of store depends on the context, but, if possible, it is applied immediately. 3.2.4 focusing: Omega , Phi and connectives In papers [1] by J. M. Andreoli and [2] by D.Galmiche and G.Perrier it is shown that the following restriction (called focusing ) can be used immediately after decomposing some formula A with the top connective being either Omega , Phi or . The focusing restriction preserves completeness. Take a sequent Gamma = A 1 ; A 2 ; ....

....presented in our paper is complete when combined with the invertibility and or focusing restrictions for resolution. Proof Any derivation tree violating these restrictions violates the invertibilitycondition (c.f. section 3.2.3) and focusing (c.f. section 3.2. 4) We know (see papers [1] and [2]) that any formula derivable in linear logic has a derivation obeying these restrictions. The completeness proofs of resolution strategies introduced in our paper which rely on transforming derivation trees hold for arbitrary correct derivation trees in linear logic, thus also for derivation trees ....

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D.Galmiche, G.Perrier. Automated deduction in additive and multiplicative linear logic. In LFCS '92: Logic at Tver, LNCS v. 620, Springer-Verlag, pp. 151-162 (1992).

....sequent calculus implementation is mandatory. In Section 3 we remind the reader of the basic tactics and tacticals of Isabelle that we use, in the sequel, to build our dioeerent proof search strategies. Section 4 introduces a general proof search strategy based on the work of Galmiche and Perrier [2, 4]. This strategy, which is complete, appears to be rather ineOEcient. The remainder of the paper is dedicated to the development of heuristics that improve the eOEciency of the general strategy without loosing completeness. Section 5 is concerned with the multiplicative fragment. We take the ....

....using Isabelle s basic tacticals and tactics, is easy. This is because there exist, for classical logic, sequent calculi where all the rules are invertible. Consequently, each single resolution step may be performed deterministically. Unfortunately, this is not so in the case of linear logic. In [2, 4], Galmiche and Perrier study the permutability of the inference rules of linear logic. The result of their work, for the multiplicative additive fragment, is summarised by the following table. R 1 Omega Phi L Phi R R 2 Omega Phi L Phi R A star, in this table, indicates ....

G. Perrier D. Galmiche. Automated deduction in additive and multiplicative linear logic. In A. Nerode and M. Taitslin, editors, Proceedings of the Second International Symposium on Logical Foundations of Computer Science Tver'92, pages 151162. Lecture Notes in Computer Science, 620, Springer Verlag, 1992.

....proof, but no direct algorithm to have directly build a good path. About this notion of canonical proof, we have to mention that we have previously de ned such a notion for di erent fragments of LL as MLL, MALL and also for full linear logic (CLL) but without reference to an initial path [23, 26]. Moreover, this class of proofs is complete w.r.t. the provability and then a simple proof search procedure consists in, from an initial sequent, trying to build such a speci c proof. To summarize, we have presented a characterization of provability based on the notion of path (set of pairs of ....

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Logic at Tver '92, Logical Foundations of Computer Science Symposium, LNCS 620, pages 151162, Tver, Russia, July 1992.

....a large LL fragment and in [12] considering some computational problems, it is to make a suitable logic programming language in a fragment of intuitionistic linear logic. For general theorem proving (not necessarily goal oriented) we also need to de ne complete and tractable subclasses of proofs [3, 5]. Let us mention that in [13] a new linear logic framework, named ACL, is proposed for concurrent computation described in terms of proof construction in linear logic. Thus, eOEcient and adapted proof construction methods are essential for In proceedings of the ICLP 94 Workshop on ....

.... programming frameworks and an important question arises: how to de ne complete (and tractable) proof subclasses and proof search strategies in a linear logical framework Some answers have been partially given in works on proof search and strategies dedicated to theorem proving in LL fragments [3, 5, 14, 17]that can help us to consider LL fragments for (concurrent) logic programming. We propose here a global analysis of this question based on the following procedure. We analyze the logic programming language and then deduce the logical rules involved by the computation process. Thus, we ....

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Logic at Tver '92, Logical Foundations of Computer Science Symposium, LNCS 620, pages 151162, Tver, Russia, July 1992.

....we have two possibilities: the first one is to extend the proof net notion to AMLL [4] and to consider its automated construction by a similar approach. A second one is to consider the problem of proof construction directly in AMLL with a specific, and completely different, decision procedure [8] and to study its relationship with extension of proof nets notion. Even so, this mechanization of proof net construction in MLL is a step for the use of proof net, having in mind its computational interpretation. ....

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Logic at Tver '92, Logical Foundations of Computer Science Symposium, Tver, Russia, July 1992.

....will not consist only in connecting open edges and considering the additive connectors terminal branches are not given a priori with the conclusion Gamma. A second one is to consider the problem of proof construction directly in AMLL with a specific, and completely different, decision procedure [8] and to study its relationship with extension of proof nets notion. Even so, the mechanization of proof net construction in MLL, presented here, is a first attempt for the use of proof net and linear logic, having in mind the computational interpretation and its applications. ....

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. To appear in Logic at Tver '92, Symposium on Logical Foundations of Computer Science, Tver, Russia, July 1992.

....(CLL) it is important to propose efficient proof search procedures knowing that theorem proving is significantly more difficult for CLL than for classical logic since there is no convenient normal forms. Some works have been devoted to theorem proving and decision procedures in such fragments [4, 5, 11, 14] and to linear logic programming [2, 8, 9] including various notions or proposals, mainly with a bottom up approach. Considering proof construction in linear sequent calculus, regardless of proof direction, we point out the necessity to systematically study permutability properties because they ....

....and support efficient proof search strategies proposals by defining, for example, complete and tractable subclasses of normal proofs. Thus, it appears that notions as inference movement, proof transformation and normal proof are logical foundations for the conception of proof search strategies [5, 6]. In previous works, we have worked on an automated deduction procedure in the MALL fragment with a bottom up approach by defining complete and tractable subclasses of proofs [5] But what about logical bases for bottom up, top down or mixed proof directions in full linear logic In this paper, ....

[Article contains additional citation context not shown here]

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Logic at Tver '92, Logical Foundations of Computer Science Symposium, LNCS 620, pages 151--162, Tver, Russia, July 1992.

....Logic (CLL) it is important to propose eOEcient proof search procedures knowing that theorem proving is signi cantly more diOEcult for CLL than for classical logic since there is no convenient normal forms. Some works have been devoted to theorem proving and decision procedures in such fragments [4, 5, 11, 14] and to linear logic programming [2, 8, 9] including various notions or proposals, mainly with a bottom up approach. Considering proof construction in linear sequent calculus, regardless of proof direction, we point out the necessity to systematically study permutability properties because they ....

....and support eOEcient proof search strategies proposals by de ning, for example, complete and tractable subclasses of normal proofs. Thus, it appears that notions as inference movement, proof transformation and normal proof are logical foundations for the conception of proof search strategies [5, 6]. In previous works, we have worked on an automated deduction procedure in the MALL fragment with a bottom up approach by de ning complete and tractable subclasses of proofs [5] But what about logical bases for bottom up, top down or mixed proof directions in full linear logic In this paper, we ....

[Article contains additional citation context not shown here]

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In A. Nerode and M. Taitslin, editors, Proceedings of Logical Foundations of Computer Science, Tver, Russia, july 1992, volume 620 of Lecture Notes in Computer Science, pages 151162, 1992.

....of efficient proof search methods in non classical logics [26, 29] Thus, our approach begins with a systematical study of the inference permutability possibilities in full linear logic aiming to efficient proof construction mechanization. A first attempt in this direction has been developed in [10] where we have focused on the additive and multiplicative fragment of CLL and proposed an algorithm for automated deduction in this fragment. In this paper we completely refine the permutability notion and we extend the approach to full linear logic. Thus, after having systematically studied, the ....

....2 ) Intuitively, the notion of permutability is easy to understand. It means the possibility to invert two inferences in a proof without disturbing the rest of the proof (the parts below and above the inferences) In a previous work on deduction in the additive and multiplicative fragment of CLL [10], we have called it perfect permutability (pp) Should one inference be of type , there is some difficulty because of the duplication of the other inference (due to the duplication in the context) In [10] we have translated this variant by the notion of quasi permutability (qp) But such ....

[Article contains additional citation context not shown here]

D. Galmiche and G. Perrier. Automated deduction in additive and multiplicative linear logic. In Logic at Tver '92, Logical Foundations of Computer Science Symposium, LNCS 620, pages 151--162, Tver, Russia, July 1992.

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