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A Principle for Incorporating Axioms into the FirstOrder Translation of Modal Formulae
 Automated Deduction—CADE19, volume 2741 of Lecture Notes in Artificial Intelligence
, 2003
"... In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable logics. The goal of the axiomatic translation ..."
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In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable logics. The goal of the axiomatic translation principle is to find simplified theories, which capture the inference problems in the original theory, but in a way that is more amenable to automation and easier to deal with by existing theorem provers. The principle of the axiomatic translation is conceptually very simple and can be largely automated. Soundness is automatic under reasonable assumptions, and termination of ordered resolution is easily achieved, but the nontrivial part of the approach is proving completeness.
Splittings and the finite model property
 Journal of Symbolic Logic
, 1993
"... An old and conjecture of modal logics states that every splitting of the major systems K4, S4 and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof tec ..."
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An old and conjecture of modal logics states that every splitting of the major systems K4, S4 and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ / f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive. 1 Splittings and the finite model property 2
The Axiomatic Translation Principle for Modal Logic
, 2007
"... In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable fragments of firstorder logic. The goal of the ax ..."
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In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable fragments of firstorder logic. The goal of the axiomatic translation principle is to find simplified theories, which capture the inference problems in the original theory, but in a way that can be readily automated and is easier to deal with by existing (firstorder) theorem provers than the standard translation. The principle of the axiomatic translation is conceptually very simple and can be almost completely automated. Soundness is automatic under reasonable assumptions, general decidability results can be stated and termination of ordered resolution is easily achieved. The nontrivial part of the approach is proving completeness. We prove results of completeness, decidability, model generation, the small model property and the interpolation property for a number of common and less common modal logics. We also present results of experiments with a number of firstorder logic theorem provers which are very encouraging.
Conservative extensions in modal logics
 In Proceedings of AiML6
, 2006
"... Every normal modal logic L gives rise to the consequence relation ϕ =L ψ which holds iff ψ is true in a world of an Lmodel whenever ϕ is true in that world. We consider the following algorithmic problem for L. Given two modal formulas ϕ1 and ϕ2, decide whether ϕ1∧ϕ2 is a conservative extension of ..."
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Every normal modal logic L gives rise to the consequence relation ϕ =L ψ which holds iff ψ is true in a world of an Lmodel whenever ϕ is true in that world. We consider the following algorithmic problem for L. Given two modal formulas ϕ1 and ϕ2, decide whether ϕ1∧ϕ2 is a conservative extension of ϕ1 in the sense that whenever ϕ1 ∧ ϕ2 =L ψ and ψ does not contain propositional variables not occurring in ϕ1, then ϕ1 =L ψ. We first prove that the conservativeness problem is coNExpTimehard for all modal logics of unbounded width (which have rooted frames with more than N successors of the root, for any N < ω). Then we show that this problem is (i) coNExpTimecomplete for S5 and K, (ii) in ExpSpace for S4 and (iii) ExpSpacecomplete for GL.3 (the logic of finite strict linear orders). The proofs for S5 and K use the fact that these logics have uniform interpolation. 1
Notes on the Space Requirements for Checking Satisfiability in Modal Logics
, 2002
"... . Recently, there has been growing attention on the space requirements of tableau methods (see for example [6], [2], [10]). We have proposed in [9] a method of reducing modal consequence relations to the global and local consequence relation of (polymodal) K. The reductions used there did however ..."
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. Recently, there has been growing attention on the space requirements of tableau methods (see for example [6], [2], [10]). We have proposed in [9] a method of reducing modal consequence relations to the global and local consequence relation of (polymodal) K. The reductions used there did however not give rise to ecient time complexity bounds. In this note we shall use reduction functions to obtain rather sharp space bounds. These bounds can be applied to ordinary tableau systems, and do not make use of the Mints transform. It has been shown by Hudelmaier ([6]) that satisability in S4 is O(n 2 log n){space computable, while satisability in K and T are O(n log n){space computable. A O(n log n){space bound for K.D has been obtained by Basin, Matthews and Vigano ([2]). Vigano ([14]) has shown that satisability in K4, KD4 and S4 is in O(n 2 log n){ space. Nguyen has reduced these bounds to O(n log n) for K4, K4D and S4 in [11]. Additionally, O(n log n){bounds are shown for ...
Computing Minimal ELunifiers is Hard
"... Unification has been investigated both in modal logics and in description logics, albeit with different motivations. In description logics, unification can be used to detect redundancies in ontologies. In this context, it is not sufficient to decide unifiability, one must also compute appropriate un ..."
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Unification has been investigated both in modal logics and in description logics, albeit with different motivations. In description logics, unification can be used to detect redundancies in ontologies. In this context, it is not sufficient to decide unifiability, one must also compute appropriate unifiers and present them to the user. For the description logic EL, which is used to define several large biomedical ontologies, deciding unifiability is an NPcomplete problem. It is known that every solvable ELunification problem has a minimal unifier, and that every minimal unifier is a local unifier. Existing unification algorithms for EL compute all minimal unifiers, but additionally (all or some) nonminimal local unifiers. Computing only the minimal unifiers would be better since there are considerably less minimal unifiers than local ones, and their size is usually also quite small. In this paper we investigate the question whether the known algorithms for ELunification can be modified such that they compute exactly the minimal unifiers without changing the complexity and the basic nature of the algorithms. Basically, the answer we give to this question is negative. Keywords:
Beth Definability in Expressive Description Logics
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2011
"... The Beth definability property, a wellknown property from classical logic, is investigated in the context of description logics (DLs): if a general LTBox implicitly defines an Lconcept in terms of a given signature, where L is a DL, then does there always exist over this signature an explicit def ..."
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The Beth definability property, a wellknown property from classical logic, is investigated in the context of description logics (DLs): if a general LTBox implicitly defines an Lconcept in terms of a given signature, where L is a DL, then does there always exist over this signature an explicit definition in L for the concept? This property has been studied before and used to optimize reasoning in DLs. In this paper a complete classification of Beth definability is provided for extensions of the basic DL ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures. Moreover, we present a tableaubased algorithm which computes explicit definitions of at most double exponential size. This algorithm is optimal because it is also shown that the smallest explicit definition of an implicitly defined concept may be double exponentially long in the size of the input TBox. Finally, if explicit definitions are allowed to be expressed in firstorder logic then we show how to compute them in EXPTIME.
A Tableau Calculus with AutomatonLabelled Formulae for Regular Grammar Logics
"... Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automatonlabelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cutfree and has the analytic superf ..."
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Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automatonlabelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cutfree and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar logics using our calculus. 1
DOI: 10.12775/LLP.2015.007
"... Abstract. This work was intended to be an attempt to introduce the metalanguage for working with multipleconclusion inference rules that admit asserted propositions along with the rejected propositions. The presence of rejected propositions, and especially the presence of the rule of reverse subst ..."
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Abstract. This work was intended to be an attempt to introduce the metalanguage for working with multipleconclusion inference rules that admit asserted propositions along with the rejected propositions. The presence of rejected propositions, and especially the presence of the rule of reverse substitution, requires certain change the definition of structurality.