On the Modal μ-Calculus Over Finite Symmetric Graphs

AbstractIn this paper we consider the alternation hierarchy of the modal μ-calculus over finite symmetric graphs and show that in this class the hierarchy is infinite. The μ-calculus over the symmetric class does not enjoy the finite model property, hence this result is not a trivial consequence of the strictness of the hierarchy over symmetric graphs. We also find a lower bound and an upper bound for the satisfiability problem of the μ-calculus over finite symmetric graphs.

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On the Decision Problem for Two-Variable First-Order Logic

Bulletin of Symbolic Logic ◽

10.2307/421196 ◽

1997 ◽

Vol 3 (1) ◽

pp. 53-69 ◽

Cited By ~ 137

Author(s):

Erich Grädel ◽

Phokion G. Kolaitis ◽

Moshe Y. Vardi

Keyword(s):

Computational Complexity ◽

Decision Problem ◽

Order Logic ◽

First Order Logic ◽

Satisfiability Problem ◽

Finite Model ◽

Model Property ◽

Finite Model Property ◽

First Order ◽

Long Time

AbstractWe identify the computational complexity of the satisfiability problem for FO2, the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO2 has the finite-model property, which means that if an FO2-sentence is satisiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO2-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO2-sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO2 is NEXPTIME-complete.

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On the satisfiability problem for fragments of two-variable logic with one transitive relation

Journal of Logic and Computation ◽

10.1093/logcom/exz012 ◽

2019 ◽

Vol 29 (6) ◽

pp. 881-911

Author(s):

Wiesław Szwast ◽

Lidia Tendera

Keyword(s):

Satisfiability Problem ◽

Finite Model ◽

Model Property ◽

Transitive Relation ◽

Tree Model ◽

Construction Technique ◽

Ramsey Theorem ◽

Finite Model Property ◽

The Status ◽

Variable Logic

Abstract We study the satisfiability problem for two-variable first-order logic over structures with one transitive relation. We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model property nor the tree model property, to show decidability we introduce a novel model construction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decidability for the full two-variable logic with one transitive relation; hence, contrary to our previous claim, [FO$^2$ with one transitive relation is decidable, STACS 2013: 317-328], the status of the latter problem remains open.

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Canonical rules

Journal of Symbolic Logic ◽

10.2178/jsl/1254748686 ◽

2009 ◽

Vol 74 (4) ◽

pp. 1171-1205 ◽

Cited By ~ 16

Author(s):

Emil Jeřábek

Keyword(s):

Alternative Proof ◽

Finite Model ◽

Consequence Relations ◽

Model Property ◽

Finite Model Property ◽

Admissible Rules ◽

Superintuitionistic Logics ◽

Canonical Formulas ◽

Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

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Finite model property for five modal calculi in the neighbourhood of $S3$.

Notre Dame Journal of Formal Logic ◽

10.1305/ndjfl/1093894152 ◽

1971 ◽

Vol 12 (1) ◽

pp. 69-74 ◽

Cited By ~ 1

Author(s):

Anjan Shukla

Keyword(s):

Finite Model ◽

Model Property ◽

Finite Model Property

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On simple, weak and strong models of propositional calculi

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410004771x ◽

1973 ◽

Vol 74 (1) ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Ronald Harrop

Keyword(s):

Simple Model ◽

Propositional Calculus ◽

Finite Model ◽

Model Property ◽

Finite Model Property ◽

Finite Structure ◽

Propositional Calculi ◽

Simple Models

In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

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On the Decidability of Intuitionistic Tense Logic without Disjunction

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2020/249 ◽

2020 ◽

Author(s):

Fei Liang ◽

Zhe Lin

Keyword(s):

Proof Theory ◽

Tense Logic ◽

Heyting Algebras ◽

Finite Model ◽

Algebraic Proof ◽

Model Property ◽

Algebraic Models ◽

Finite Axiomatizability ◽

Finite Model Property ◽

Modal Operators

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

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Small substructures and decidability issues for first-order logic with two variables

Journal of Symbolic Logic ◽

10.2178/jsl/1344862160 ◽

2012 ◽

Vol 77 (3) ◽

pp. 729-765 ◽

Cited By ~ 11

Author(s):

Emanuel Kieroński ◽

Martin Otto

Keyword(s):

Order Logic ◽

First Order Logic ◽

Equivalence Relations ◽

Finite Model ◽

Model Property ◽

Finite Model Property ◽

First Order ◽

Surrounding Structure ◽

Small Model ◽

Exponential Size

AbstractWe study first-order logic with two variables FO2 and establish a small substructure property. Similar to the small model property for FO2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO2 has the finite model property and is complete for non-deterministic exponential time, just as for plain FO2. With two equivalence relations, FO2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO2 is undecidable.

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