scholarly journals Axiomatizability of non-normal and quasi-normal modal predicate logics of first-order definable classes of Kripke frames

2018 ◽  
pp. 81-94
Author(s):  
Mikhail Nikolayevich Rybakov ◽  
◽  
◽  
Keyword(s):  
First Order ◽  
Kripke Frames ◽  
Modal Predicate Logics

scholarly journals KRIPKE COMPLETENESS OF STRICTLY POSITIVE MODAL LOGICS OVER MEET-SEMILATTICES WITH OPERATORS

2019 ◽  
Vol 84 (02) ◽  
pp. 533-588 ◽  
Author(s):  
STANISLAV KIKOT ◽  
AGI KURUCZ ◽  
YOSHIHITO TANAKA ◽  
FRANK WOLTER ◽  
MICHAEL ZAKHARYASCHEV
Keyword(s):  
Computer Science ◽  
Monotone Operators ◽  
Modal Logics ◽  
Consequence Relations ◽  
First Order ◽  
Completeness Problem ◽  
Relational Structures ◽  
Kripke Frames

AbstractOur 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.

scholarly journals Neighbourhood semantics for graded modal logic

2021 ◽  
Author(s):  
Jinsheng Chen ◽  
Hans Van Ditmarsch ◽  
Giuseppe Greco ◽  
Apostolos Tzimoulis
Keyword(s):  
Modal Logic ◽  
First Order ◽  
Kripke Frames ◽  
Definition Of

We introduce a class of neighbourhood frames for graded modal logic using an operation from Kripke frames to neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation by modifying the definition of monotonic bisimulation.  

scholarly journals Recursive enumerability and elementary frame definability in predicate modal logic

2019 ◽  
Vol 30 (2) ◽  
pp. 549-560 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
The Other ◽  
Kripke Semantics ◽  
First Order ◽  
Elementary Class ◽  
Recursive Enumerability ◽  
Kripke Frames ◽  
Recursively Enumerable ◽  
The Relationship

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

2020 ◽  
Vol 30 (7) ◽  
pp. 1305-1329 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Possible Worlds ◽  
Modal Logics ◽  
Finite Frames ◽  
First Order ◽  
Kripke Frames ◽  
Finite Frame ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Individual Variables ◽  
Natural Classes

Abstract We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

scholarly journals Kripke Incompleteness of First-order Calculi with Temporal Modalities of CTL and Near Logics

2015 ◽  
Vol 21 (1) ◽  
pp. 86-99
Author(s):  
Е. А. Котикова ◽  
М. Н. Рыбаков
Keyword(s):  
Temporal Logic ◽  
Expressive Power ◽  
Branching Time ◽  
First Order ◽  
Kripke Frames ◽  
Order Language ◽  
Constant Domains ◽  
Kripke Incompleteness ◽  
Computational Tree Logic

We study an expressive power of temporal operators used in such logics of branching time as computational tree logic or alternating-time temporal logic. To do this we investigate calculi in the first-order language enriched with the temporal operators used in such logics. We show that the resulting languages are so powerful that many ‘natural’ calculi in the languages are not Kripke complete; for example, if a calculus in such language is correct with respect to the class of all serial linear Kripke frames (even just with constant domains) then it is not Kripke complete. Some near questions are discussed.

On axiomatising products of Kripke frames

2000 ◽  
Vol 65 (2) ◽  
pp. 923-945 ◽  
Author(s):  
Ágnes Kurucz
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Modal Language ◽  
First Order ◽  
Kripke Frames ◽  
Other Hand ◽  
The Many ◽  
By Products

AbstractIt is shown that the many-dimensional modal logic Kn, determined by products of n-many Kripke frames, is not finitely axiomatisable in the n-modal language, for any n > 2. On the other hand, Kn is determined by a class of frames satisfying a single first-order sentence.

scholarly journals A Finite Model Property for Gödel Modal Logics

10.29007/vgh2 ◽  
2018 ◽  
Author(s):  
Xavier Caicedo ◽  
George Metcalfe ◽  
Ricardo Rodriguez ◽  
Jonas Rogger
Keyword(s):  
Modal Logics ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
First Order ◽  
Gödel Logic ◽  
Np Completeness ◽  
Kripke Frames ◽  
The One

A new semantics with the finite model property is provided and used to establish decidability for Gödel modal logics based on (crisp or fuzzy) Kripke frames combined locally with Gödel logic. A similar methodology is also used to establish decidability, indeed co-NP-completeness, for a Gödel S5 logic that coincides with the one-variable fragment of first-order Gödel logic.

scholarly journals Fine’s Theorem on First-Order Complete Modal Logics

2020 ◽  
pp. 316-334
Author(s):  
Robert Goldblatt
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Modal Algebra ◽  
First Order ◽  
Kripke Frames ◽  
Canonical Frame ◽  
History Of ◽  
Order Completeness

Fine’s influential Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This article reviews the background and context of this result, and the history of its impact on further research. It then develops a new characterization of when a logic is canonically valid, providing a precise point of distinction with the property of first-order completeness. The ultimate point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction.

scholarly journals First-order logics of branching time

2013 ◽  
Vol 19 ◽  
pp. 68-99
Author(s):  
Е.А. Котикова ◽  
М.Н. Рыбаков
Keyword(s):  
Branching Time ◽  
First Order ◽  
Recursive Enumerability ◽  
Kripke Frames

We consider the logic QCTL, a first-order exten- sion of CTL defined as a logic of Kripke frames for CTL. We study the question about recursive enumerability of its fragments specified by a set of temporal modalities we use. Then we discuss some questions concerned axiomatizability and Kripke completeness.

scholarly journals Decidable fragments of first-order modal logics

2001 ◽  
Vol 66 (3) ◽  
pp. 1415-1438 ◽  
Author(s):  
Frank Wolter ◽  
Michael Zakharyaschev
Keyword(s):  
Free Variable ◽  
Modal Logics ◽  
First Order ◽  
Modal Operators ◽  
Modal Predicate Logics

AbstractThe paper considers the setof first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in, which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.

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