A resolution calculus for modal logics

Ohlbach, Hans Jürgen

doi:10.1007/bfb0012852

Cited by 66 publications

(36 citation statements)

References 9 publications

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“…Rather than by giving a recursive algorithm we consider the process of unification as a sequence of transformations on systems of equations. This concept is similar to the ideas of Martelli & Montanari ( [11], see also [12,18]). …”

Section: Transforming Equationssupporting

confidence: 76%

“…In his matrix characterization of non-classical validity Wallen [19] has shown that in addition the prefixes of atomic formulae need to be unified in order to make them complementary where a prefix of an atom essentially describes its position in the formula tree. Similarly Ohlbach's resolution calculi for modal logics [12,13] require a unification of world-paths which characterize the modal context of an atom in a formula. Both conditions are nearly identical 1 and express the peculiarities and restrictions of these logics.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

T-string unification: Unifying prefixes in non-classical proof methods

Otten

Kreitz

1996

Theorem Proving With Analytic Tableaux and Related Methods

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Abstract. For an efficient proof search in non-classical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallen's matrix characterization and Ohlbach's resolution calculus. Beside the usual term-unification both methods require a specialized string-unification to unify the so-called prefixes of atomic formulae (in Wallen's notation) or world-paths (in Ohlbach's notation). For this purpose we present an efficient algorithm, called T-String-Unification, which computes a minimal set of most general unifiers. By transforming systems of equations we obtain an elegant unification procedure, which is applicable to the intuitionistic logic J and the modal logic S4. With some modifications we are able to treat the modal logics D, K, D4, K4, S5, and T. We explain our method by an intuitive graphical presentation, prove correctness, completeness, minimality, and termination and investigate its complexity.

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Section: Transforming Equationssupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

T-string unification: Unifying prefixes in non-classical proof methods

Otten

Kreitz

1996

Theorem Proving With Analytic Tableaux and Related Methods

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“…In symmetric logics we can return to an earlier world, and we need a mechanism for remembering what we knew when we were there last. For this purpose prefixes (as in [7] and [9]) or world paths (as in [21]) are used.…”

Section: Example By This Rule the Clause List [P ] [ Q R] [ P ¬♦mentioning

confidence: 99%

“…So far several distinct versions of resolution for modal logics have been proposed. [21] presents a system in which explicit notation is introduced to denote possible worlds. This is related to the tableau systems of [7] and some of those in [9] (see also [24]).…”

Section: Introductionmentioning

confidence: 99%

Destructive Modal Resolution

Fitting¹

1990

J Logic Computation

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We present non-clausal resolution systems for propositional modal logics whose Kripke models do not involve symmetry, and for first order versions whose Kripke models do not involve constant domains. We give systems for K, T , K4 and S4; other logics are also possible. Our systems do not require preliminary reduction to a normal form and, in the first order case, intermingle resolution steps with Skolemization steps.

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“…Equational unification also plays a rôle in the context of modal logics in a quite different setting. In the so-called (optimized) functional translation of modal logics into first-order logic, frame properties are translated into equational axioms, which can be dealt with by equational unification [51,26,37,52,2,53,66]. This kind of unification for modal logics is very different from the one mentioned before, and will not be treated in the present paper.…”

Section: Introductionmentioning

confidence: 98%