First-Order Resolution Methods for Modal Logics

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.

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Simulating polyadic modal logics by monadic ones

Journal of Symbolic Logic ◽

10.2178/jsl/1052669058 ◽

2003 ◽

Vol 68 (2) ◽

pp. 419-462 ◽

Cited By ~ 5

Author(s):

George Goguadze ◽

Carla Piazza ◽

Yde Venema

Keyword(s):

Modal Logics ◽

Finite Model ◽

Model Property ◽

Finite Model Property ◽

First Order ◽

First Order Definability

AbstractWe define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic Λsim in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

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Problem Libraries for Non-Classical Logics

10.29007/mkdw ◽

2018 ◽

Author(s):

Jens Otten ◽

Thomas Raths

Keyword(s):

Theorem Proving ◽

General Problem ◽

Crucial Role ◽

Automated Theorem Proving ◽

Modal Logics ◽

First Order ◽

Problem Library ◽

Future Plans

Problem libraries for automated theorem proving (ATP) systems play a crucial role when developing, testing, benchmarking and evaluating ATP systems for classical and non-classical logics. We provide an overview of existing problem libraries for some important non-classical logics, namely first-order intuitionistic and first-order modal logics. We suggest future plans to extend these existing libraries and discuss ideas for a general problem library platform for non-classical logics.

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Recursive enumerability and elementary frame definability in predicate modal logic

Journal of Logic and Computation ◽

10.1093/logcom/exz028 ◽

2019 ◽

Vol 30 (2) ◽

pp. 549-560 ◽

Cited By ~ 1

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.

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Effective Properties of Some First Order Intuitionistic Modal Logics

Automated Deduction in Classical and Non-Classical Logics - Lecture Notes in Computer Science ◽

10.1007/3-540-46508-1_16 ◽

2000 ◽

pp. 236-250

Author(s):

Aida Pliuškevičienė

Keyword(s):

Effective Properties ◽

Modal Logics ◽

First Order ◽

Intuitionistic Modal Logics

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Modal Logics Between Propositional and First-order

Journal of Logic and Computation ◽

10.1093/logcom/12.6.1017 ◽

2002 ◽

Vol 12 (6) ◽

pp. 1017-1026 ◽

Cited By ~ 13

Author(s):

M. Fitting

Keyword(s):

Modal Logics ◽

First Order

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

Journal of Logic and Computation ◽

10.1093/logcom/exaa041 ◽

2020 ◽

Vol 30 (7) ◽

pp. 1305-1329 ◽

Cited By ~ 1

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 $\textbf{S4}$.

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Proof theory for quantified monotone modal logics

Logic Journal of IGPL ◽

10.1093/jigpal/jzz015 ◽

2019 ◽

Vol 27 (4) ◽

pp. 478-506

Author(s):

Sara Negri ◽

Eugenio Orlandelli

Keyword(s):

Proof Theory ◽

Modal Logics ◽

Cut Elimination ◽

Sequent Calculi ◽

Completeness Proof ◽

First Order ◽

Constant Domains ◽

Normal Modal Logics ◽

Order Extension

Abstract This paper provides a proof-theoretic study of quantified non-normal modal logics (NNML). It introduces labelled sequent calculi based on neighbourhood semantics for the first-order extension, with both varying and constant domains, of monotone NNML, and studies the role of the Barcan formulas in these calculi. It will be shown that the calculi introduced have good structural properties: invertibility of the rules, height-preserving admissibility of weakening and contraction and syntactic cut elimination. It will also be shown that each of the calculi introduced is sound and complete with respect to the appropriate class of neighbourhood frames. In particular, the completeness proof constructs a formal derivation for derivable sequents and a countermodel for non-derivable ones, and gives a semantic proof of the admissibility of cut.

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