Residuated Algebraic Structures in the Vicinity of Pre-rough Algebra and Decidability

Varieties of topological quasi-Boolean algebras in the vicinity of pre-rough algebras [28, 29] are expanded to residuated algebraic structures by introducing a new implication operation and its residual in these structures. Sequent calculi for some classes of residuated algebraic structures are established. These sequent calculi have the strong finite model property which yields the decidability of the word problem for corresponding classes of algebraic structures.

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Semantical Proof of Subformula Property for the Modal Logics K 4.3, KD 4.3, and S4.3

Bulletin of the Section of Logic ◽

10.18778/0138-0680.48.4.01 ◽

2019 ◽

Vol 48 (4) ◽

Author(s):

Daishi Yazaki

Keyword(s):

Sequent Calculus ◽

Modal Logics ◽

Finite Model ◽

Inference Rules ◽

Sequent Calculi ◽

Model Property ◽

Finite Model Property ◽

Subformula Property

The main purpose of this paper is to give alternative proofs of syntactical and semantical properties, i.e. the subformula property and the nite model property, of the sequent calculi for the modal logics K4.3, KD4.3, and S4.3. The application of the inference rules is said to be acceptable, if all the formulas in the upper sequents are subformula of the formulas in lower sequent. For some modal logics, Takano analyzed the relationships between the acceptable inference rules and semantical properties by constructing models. By using these relationships, he showed Kripke completeness and subformula property. However, his method is difficult to apply to inference rules for the sequent calculi for K4.3, KD4.3, and S4.3. Lookinglosely at Takano's proof, we nd that his method can be modied to construct nite models based on the sequent calculus for K4.3, if the calculus has (cut) and all the applications of the inference rules are acceptable. Similarly, we can apply our results to the calculi for KD4.3 and S4.3. This leads not only to Kripke completeness and subformula property, but also to finite model property of these logics simultaneously.

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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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On the Modal μ-Calculus Over Finite Symmetric Graphs

Mathematica Slovaca ◽

10.1515/ms-2015-0052 ◽

2015 ◽

Vol 65 (4) ◽

Author(s):

Giovanna D’Agostino ◽

Giacomo Lenzi

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Satisfiability Problem ◽

Finite Model ◽

Model Property ◽

Finite Model Property ◽

Symmetric Graphs ◽

Trivial Consequence

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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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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