P. D. Lincoln and N. Shankar. Proof search in rst order linear logic and other cut-free sequent calculi. In S. Abramsky, editor, Proc. of the Ninth Annual IEEE Symposium on Logic In Computer Science (LICS), pages 282-291. IEE Computer Societ Press, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a freevariable sequent system which is sound and complete for linear logic is proposed, it is shown that an analogous liberalization of the 8R and the 9L rules is unsound: in that case, too, restricting the set of arguments of Skolem functions results in a shift to a di erent logic . 20] and [18] make a ne analysis of the relationship between the arguments of Skolem functions in run time Skolemization and rule permutability in cut free sequent calculi for intuitionist and linear logic, respectively. 2 Although complete proofs have been carried out only for logics whose propositional ....

P. D. Lincoln and N. Shankar. Proof search in rst order linear logic and other cut-free sequent calculi. In S. Abramsky, editor, Proc. of the Ninth Annual IEEE Symposium on Logic In Computer Science (LICS), pages 282-291. IEE Computer Societ Press, 1994.

....results of substructural logics. Horn programming in linear logic is NP complete [20] Provability in the non modal fragment of Propositional Linear Logic (MALL) is PSPACE complete ( 21] Provability in the non modal fragment of (predicative) Linear Logic (MALL1) is NEXPTIME complete ([22], 23] We note that one still needs to be careful. If one re introduces the structural rules in a controlled manner in a substructural logic, as is done in Linear logic with the bang and of course modality, one again looses decidability Propositional Linear Logic (with full modalities) ....

P. Lincoln, N. Shankar, Proof search in rst order linear logic and other cutfree sequent calculi, Proc. of the ninth (IEEE) symposium on Logic in Computer Science, pp. 282-291, 1994.

....proceeds by induction on a sequent calculus proof of A z 1 ; A z n . We shall consider, without loss of generality, that in the left introduction of 8 and of ( are always consecutive (if it is not the case, the rules can be permuted to obtain such a proof, see for instance [20], noting that , that is the only case of unpermutability with 8, appears only in the constraint part and thus not below a ( we will thus group them as a single rule, so that each logical rule simulates an LCC transition rule. is an axiom: one uses the re exivity of in the case of a ....

P. Lincoln and N. Shankar. Proof search in rst-order linear logic and other cut-free sequent calculi. In Proc. 9th Annual IEEE Symposium on Logic in Computer Science, Paris, 1994.

....rule of column R 1 can always permute below the rule of row R 2 . The cross means that R 1 and R 2 are not in a situation of permutability. A numeral in the table means impermutability of R 1 below R 2 , and we exhibit some counter examples in table 2; most of them are taken or adapted from [5, 11]. For impermutability 7 (10 is similar) A = B C) D E) and the correct proof is: 26 R 2 nR 1 O P abs entropy O 1 P 1 1 2 2 3 4 5 6 ....

P. Lincoln and N. Shankar. Proof search in rst-order linear logic and other cut-free sequent calculi. In Proc. IEEE Int. Conf. on Logic in Computer Science, pages 282-291. IEEE, 1994.

....D of the sequent pSq; 0 p[ 0 = n] 2 ; 2 ) 0 q where = p( 1 ; 1 ) q; p( 0 ; 0 ) 0 q, that does use neither dl nor any of the quanti er rules 8l, 9l, 9r. Proof: We exploit the relative permutability of these inference rules, as described in [25]. More precisely, we apply the following four steps: 1. We permute rule dl below every other rule. The resulting derivation consists then of a sequence of applications of dl followed by a subderivation D 0 that does not use this rule. The applications of dl correspond to committing to the ....

....] 2 ) q that uses only rules from the left half of Figure 3, then there exists a cut free derivation D of this sequent such that Every use of r appear just below id, 1r or r; Rules 1l and l are applied eagerly. Proof: Again, we take advantage of the permutability results in [25]. Rule r can be pushed up past any other rule (except id and 1r) On the other hand, l and 1l can always be permuted down, as long as the nesting of subformulas is respected (clearly if the left hand side contains a formula (A B) C, a proof fragment that dismantles this formula must apply l ....

Lincoln, P., and Shankar, N. Proof search in rstorder linear logic and other cut-free sequent calculi. In Ninth Annual Symposium on Logic in Computer Science (Paris, France, 1994), S. Abramsky, Ed., IEEE Computer Society Press, pp. 282-291.

....D of the sequent pSq; 9 0 : p( 2 ; 2 ) 0 q where = p( 1 ; 1 ) q; p( 0 ; 0 ) 0 q, that does use neither dl nor any of the quanti er rules 8l, 9l, 9r. Proof: We exploit the relative permutability of these inference rules, as described in [22]. More precisely, we apply the following four steps: 1. We permute rule dl below every other rule. The resulting derivation consists then of a sequence of applications of dl followed by a subderivation D 0 that does not use this rule. The applications of dl correspond to committing to the ....

.... p( 2 ; 2 ) q that uses only rules from the left half of Figure 3, then there exists a cut free derivation D of this sequent such that Every use of r appear just below id or r; Rule l is applied eagerly. Proof: Again, we take advantage of the permutability results in [22]. Rule r can be pushed up past any other rule (except id) On the other hand, l can always be permuted down, as long as the nesting of subformulas is respected (clearly if the left hand side contains a formula (A B) C, a proof fragment that dismantles this formula must apply l above l) 2 At ....

P. Lincoln and N. Shankar. Proof search in rstorder linear logic and other cut-free sequent calculi. In S. Abramsky, editor, Ninth Annual Symposium on Logic in Computer Science, pages 282-291, Paris, France, 1994. IEEE Computer Society Press.

....the non determinism in term instantiation for an existential quanti er using Herbrand s theorem and uni cation. Even if Herbrand s theorem cannot be applied directly to most logics, such as intuitionistic, linear or modal logics, this non determinism can often be eliminated via other procedures [94]. Another level of non determinism has to do with the order in which rules are applied and in many sequent calculi, order of rules is crucial for the success of the proof search process. The permutability results of a sequent calculus indicate when the order of two inference rules can be permuted ....

....when the order of two inference rules can be permuted without invalidating a proof and are used to reduce the non determinism on the ordering of rules applications. Moreover, optimizations can be based on the reduction of the amount of backtracking in the proof search as well illustrated in [94]. It is important to notice that a way to solve proof search problems, such as the occurrence of loops in the search procedure, consists in proposing new sequent calculi, the rules of which integrate the solution mechanisms [42] In this section, we brie y review some of the key aspects of some ....

P. Lincoln and N. Shankar. Proof search in rst-order linear logic and other cut-free sequent calculi. In 9th IEEE Symposium on Logic in Computer Science, pages 282-291, Paris, France, July 1994.

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