Resolution Proof Systems

Stachniak, Zbigniew

doi:10.1007/978-94-009-1677-7

Cited by 14 publications

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“…[8,9,11,12]). With every resolution proof system Rs we associate three types of inference rules in terms of which the deductive process is carried out.…”

Section: Resolution Logicsmentioning

confidence: 96%

“…These rules are: the resolution rule, the transformation rules, and the 0-rules. Since these rules are discussed in details in [8,9,11,12]), below, we provide only short descriptions of the rules. where Y E F and 0 is a designated constant symbol (not in L ) denoting falsehood.…”

Section: Resolution Logicsmentioning

confidence: 99%

“…Let C be a propositional language, Rs =< d,F > be a [8,9] In other words, two formulas are Os-congruent if and only if one can replace the other without affecting the consistency status of any finite context (cf. (r2)).…”

Section: Resolution Approximation Of Propositional Logicsmentioning

confidence: 99%

“…The formal association between a propositional logic and a resolution proof system is captured by the notion of a resolution counterpart introduced in [8] (see also [3,9]). If P =< C, C > is a propositional logic and Rs is a resolution proof system on .C, then we call Rs a resolution counterpart of P , and P a resolution logic, if the following three conditions are satisfied:…”

Section: Resolution Logicsmentioning

confidence: 99%

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Note on resolution approximation of many-valued logics

Stachniak

Proceedings of the Twentieth International Symposium on Multiple-Valued Logic

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This paper discusses the problem of resolution approximation of many-valued logics, i.e. the problem of defining propositional logics in terms of finite sets of resolution proof systems. Resolution proof systems discussed in this paper are based on the weak form of transformation rules. It is shown that for manyvalued logics (and more generally, for resolution logics) minimal resolution approximations can be effectively constructed.

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“…[8,9,11,12]). With every resolution proof system Rs we associate three types of inference rules in terms of which the deductive process is carried out.…”

Section: Resolution Logicsmentioning

confidence: 96%

Section: Resolution Logicsmentioning

confidence: 99%

Section: Resolution Approximation Of Propositional Logicsmentioning

confidence: 99%

Section: Resolution Logicsmentioning

confidence: 99%

See 2 more Smart Citations

Note on resolution approximation of many-valued logics

Stachniak

Proceedings of the Twentieth International Symposium on Multiple-Valued Logic

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“…When such a connective is absent, the resolution rule can be written as a multiple-conclusion rule (cf. [ 7 ] ) . ) The verifiers as well as their order in the resolution rule are fixed, however, the selection of the set of verifiers is by no means unique.…”

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confidence: 92%

Minimal resolution proof systems for finitely-valued Lukasiewicz logics

Harley

Stachniak

[1993] Proceedings of the Twenty-Third International Symposium on Multiple-Valued Logic

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Ab st rac tIn lhis paper we describe an eflective method for, constructing minimal (in the sense described in thi1, paper) non-clausal resolution proof systems for tudosiewicz logics. W e show that every minimal resolution counterpart of the n-valued tukasiewicz logic, n > 2 , has 2n verifiers, and we provide a polynomial time algorithm to generate them. "has paper extends the resinlts reported in [9]. Introduction:The results reported in [l, 2, 5, 81 suggcsst that a number of well-known automated theorcm I'roving techniques, originally developed for the clas4cal locic, can also be successfully applied to many-valiicd log i t -3 and to finitely-valued calculi in particular. In [5, r j ] it is shown that for every finitely-valiwtl logic P one can effectively construct a resolution based automated proof system Rs refutationally equivalent to ' P. The main rule of inference of Rs, the so-called resolution rule, is defined on a propositional level by the following scheme: a o ( p ) , ,am(p) a o ( p l u 0 ) V V a m ( P l f " ) ' In this rule, CQ, ...,a, are arbitrary (not necessarily different) formulas with a common variable p , and V O , ..., vm are preselected formulas called verzfiers. The conclusion of this rule is obtained by replacing every occurrence of p in a l (~) with the it h verifier o, and taking the disjunction of the results. Loosely speaking, this rule says that if a w t A' = { a o ( p ) , .,., a , ( p ) } of formulas is consistent, then so is X U {ao(p/vo) V ... v a,(p/vnz)}. (This formalization of the resolution rule assumes the existence of a disjunction connective in a given language. When such a connective is absent, the resolution rule can be written as a multiple-conclusion rule (cf . [ 7 ] ) . ) The verifiers as well as their order in the resolution rule are fixed, however, the selection of the set of verifiers is by no means unique. Since the computational complexity of the deductive process based on (lie resolution rule depends, among other factors, on tlie the nuniber of verifiers occurring in the rule, i he scvirch for the smallest set of verifiers for a. givrii logic is of a primary concern. In [9] it is shown that the smalkst number of verifiers required by a resolut,ion rule of the n-valued Lukasiewicz logic is 271, provided tha.t n -1 is prime and n > 2. In this paper we show that thc assumption that, n-1 is prime can be dropped. Formally, we show that jor every n > 2, the minimal namber of verifiers required b y resolution rule of the n-valued Lukasiewicz logic is 2n.In fact, by slight modification of the notjion of a verifier, we can prove a sharper result, that only n verifiers are required. This result, together with the fact that the two-valued Lukasiewicz logic (i.e., the classical propositional logic) requires only two verifiers (cf.[4]), closes the search for the lower bound of the nrimber of verifiers for finikly-valued tukasiewicz logics. The solution of this problem constitutes the first part of this paper. In the second part, we present a polynomial time algorithm for g...

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Finite Algebras and AI: From Matrix Semantics to Stochastic Local Search

Stachniak

2004

Artificial Intelligence and Symbolic Computation

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Resolution Proof Systems

Cited by 14 publications

References 0 publications

Note on resolution approximation of many-valued logics

Note on resolution approximation of many-valued logics

Minimal resolution proof systems for finitely-valued Lukasiewicz logics

Finite Algebras and AI: From Matrix Semantics to Stochastic Local Search

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