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 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.

An interpretation of “finite” modal first-order languages in classical second-order languages

1976 ◽  
Vol 41 (2) ◽  
pp. 337-340
Author(s):  
Scott K. Lehmann
Keyword(s):  
Possible Worlds ◽  
Second Order ◽  
Universal Quantifier ◽  
Possible World ◽  
Simple Interpretation ◽  
First Order ◽  
Order Language ◽  
Finite Set ◽  
Modal Semantics ◽  
Individual Variables

This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.

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.

Bimodal logics for extensions of arithmetical theories

1996 ◽  
Vol 61 (1) ◽  
pp. 91-124 ◽  
Author(s):  
Lev D. Beklemishev
Keyword(s):  
Decision Procedures ◽  
Modal Logics ◽  
Kripke Semantics ◽  
Recursively Enumerable ◽  
Propositional Constants ◽  
Bimodal Logics ◽  
Natural Classes

AbstractWe characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ0 + EXP, PRA); (PRA, IΣn); (IΣm, IΣn) for 1 ≤ m < n; (PA, ACA0); (ZFC, ZFC + CH); (ZFC, ZFC + ¬CH) etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

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 Finite representability of semigroups with demonic refinement

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Robin Hirsch ◽  
Jaš Šemrl
Keyword(s):  
Partial Order ◽  
Turing Machines ◽  
Binary Relations ◽  
Finite Representation ◽  
First Order ◽  
Finite Structures ◽  
Finite Base ◽  
Finite Set ◽  
Finite Representability ◽  
Representation Class

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

A weak completeness theorem for infinite valued first-order logic

1963 ◽  
Vol 28 (1) ◽  
pp. 43-50 ◽  
Author(s):  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Unit Interval ◽  
Order Logic ◽  
First Order Logic ◽  
Order System ◽  
First Order ◽  
Recursively Enumerable ◽  
Weak Completeness ◽  
Truth Value

This paper contains some results concerning the completeness of a first-order system of infinite valued logicThere are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

On completely recursively enumerable classes and their key arrays

1956 ◽  
Vol 21 (3) ◽  
pp. 304-308 ◽  
Author(s):  
H. G. Rice
Keyword(s):  
Decision Procedure ◽  
Cardinal Number ◽  
Maximum Amount ◽  
Infinite Class ◽  
Finite Sets ◽  
Amount Of Information ◽  
Recursively Enumerable ◽  
Finite Set ◽  
Partial Recursive Functions ◽  
Set Up

The two results of this paper (a theorem and an example) are applications of a device described in section 1. Our notation is that of [4], with which we assume familiarity. It may be worth while to mention in particular the function Φ(n, x) which recursively enumerates the partial recursive functions of one variable, the Cantor enumerating functions J(x, y), K(x), L(x), and the classes F and Q of r.e. (recursively enumerable) and finite sets respectively.It is possible to “give” a finite set in a way which conveys the maximum amount of information; this may be called “giving explicitly”, and it requires that in addition to an effective enumeration or decision procedure for the set we give its cardinal number. It is sometimes desired to enumerate effectively an infinite class of finite sets, each given explicitly (e.g., [4] p. 360, or Dekker [1] p. 497), and we suggest here a device for doing this.We set up an effective one-to-one correspondence between the finite sets of non-negative integers and these integers themselves: the integer , corresponds to the set αi, = {a1, a2, …, an} and inversely. α0 is the empty set. Clearly i can be effectively computed from the elements of αi and its cardinal number.

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