Decidable Elementary Modal Logics

Michaliszyn, Jakub; Otop, Jan

doi:10.1109/lics.2012.59

Cited by 5 publications

(14 citation statements)

References 15 publications

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“…The authors of [10] conjectured that the problem is decidable in PSPACE for all universal Horn formulas. This conjecture was confirmed in [17].…”

Section: Introductionmentioning

confidence: 57%

“…Finite case is similar -we show that D tiles Z m × Z m for some m if and only if there exists a finite K Γ -based model of τ ∧ λ D (see [17]).…”

Section: Logic With Undecidable Sat and Finsatmentioning

confidence: 86%

“…As a warm-up exercise, we recall the undecidability result from [17], as it will be a part of our later proofs. We define Γ as…”

Section: Logic With Undecidable Sat and Finsatmentioning

confidence: 99%

“…In case of some universal Horn formulas, decidability of corresponding modal logics is shown in [17] by demonstrating the finite model property, i.e. by proving that every modal formula satisfiable over a class of frame K has also a finite model in K .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

Michaliszyn¹,

Otop²,

Witkowski³

2012

Electron. Proc. Theor. Comput. Sci.

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We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our attention to finite structures. Many decidability and undecidability results for the elementary modal logics were proved separately for general satisfiability and for finite satisfiability [11, 12, 16, 17]. In this paper, we show that there is a reason why we must deal with both kinds of satisfiability separately – we prove that there is a universal first-order formula that defines an elementary modal logic with decidable (global) satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal formula that defines an elementary modal logic with decidable finite satisfiability problem, but undecidable general satisfiability problem

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“…The authors of [10] conjectured that the problem is decidable in PSPACE for all universal Horn formulas. This conjecture was confirmed in [17].…”

Section: Introductionmentioning

confidence: 57%

“…Finite case is similar -we show that D tiles Z m × Z m for some m if and only if there exists a finite K Γ -based model of τ ∧ λ D (see [17]).…”

Section: Logic With Undecidable Sat and Finsatmentioning

confidence: 86%

“…As a warm-up exercise, we recall the undecidability result from [17], as it will be a part of our later proofs. We define Γ as…”

Section: Logic With Undecidable Sat and Finsatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

Michaliszyn¹,

Otop²,

Witkowski³

2012

Electron. Proc. Theor. Comput. Sci.

Self Cite

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“…Indeed, for m = n = 0, (iii)-(v ) follow from our assumption and (60). We have d n,0 for all n > 0 satisfying (i), (iii) and (iv ) by (59) and (60). Now suppose inductively that we have d n,m for some m < H and all n < ω.…”

Section: Proof It Is Easy To See That {✸mentioning

confidence: 98%

Kripke Completeness of Strictly Positive Modal Logics Over Meet-Semilattices With Operators

et al. 2019

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Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi, and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic.

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A Dichotomy for Some Elementarily Generated Modal Logics

Kikot

2015

Stud Logica

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In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form ∀x0∃x1 . . . ∃xn xiR λ xj. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold. arXiv:1406.5700v2 [math.LO] 27 Feb 2015Briefly, we prove that for any class C in question, conditions (I-i) -(I-x) either simultaneously hold, or simultaneously do not hold, and this is determined by the existence in the corresponding diagram of an undirected cycle not passing through the universally quantified point, provided that the diagram is "minimal", i.e., none of its edges may be removed without affecting the corresponding formula, and "rooted", i.e., each of its points is reachable from x 0 via a directed path.We exclude from our list such algorithmical properties as decidability, finite model property and complexity, and do not deal with them in this paper, since an easy (but seemingly unpublished) argument shows that all logics in our class have f.m.p. and are PSPACE-complete regardless of the mentioned cycle. But we cannot help mentioning that the dichotomies in the complexity-theoretic setting have recently become known to the logical community. For example, in [13] the modal logics given by universal Horn sentences are classified into those that are in NP and those that are PSPACE-hard and this classification was further refined in [27]. The authors of [25] classified universal relational constraints with respect to the complexity of reasoning in the description logic EL.This work is in line with current research in theoretical modal logic. First, this result can be considered as a straighforward generalisation of Hughes' paper [16] about the reflexive-successor logic. The axiomatics of [16] was generalised in [1] to the case of first-order conditions of the form ∀x∃y(xR λ y ∧ φ(y)) where φ(y) is a generalised Kracht formula [18], and for some particular logics of this form finite axiomatisability, the finite model property and elementarity are studied there. The authors of [1] also conjectured that within their class there is a coincidence between finite axiomatisability and elementarity, and between ∆-elementarity and elementarity (cf. [2]).Another central problem of modal logic is: given an elementary class, i.e., a first-order formula, provide an explicit axiomatisation of the corresponding modal logic (this was done in [14]), and describe its properties, for example, in terms of (I-i)-(I-x) (cf. problems 6.6 and 6.8 ibid.) Since the product of two elementary classes is elementary [9], the school of many dimensional modal logic deals mainly with such problems (e.g., [23], [24], and [8] for older results). In general, the algorithmic problem 'given a firstorder formula, decide whether each of (I-i)-(I-x) holds' should be undecidable due to the undecidability ...

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Decidable Elementary Modal Logics

Cited by 5 publications

References 15 publications

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

Kripke Completeness of Strictly Positive Modal Logics Over Meet-Semilattices With Operators

A Dichotomy for Some Elementarily Generated Modal Logics

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