scholarly journals Coalescing: Syntactic Abstraction for Reasoning in First-Order Modal Logics

10.29007/cpbz ◽  
2018 ◽  
Author(s):  
Damien Doligez ◽  
Jael Kriener ◽  
Leslie Lamport ◽  
Tomer Libal ◽  
Stephan Merz
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Propositional Modal Logic ◽  
Theorem Provers ◽  
Syntactic Abstraction

We present a syntactic abstraction method to reason about first-order modal logics by using theorem provers for standard first-order logic and for propositional modal logic.

On formalizing the distinction between logical and factual truth

1966 ◽  
Vol 31 (3) ◽  
pp. 460-477
Author(s):  
William H. Hanson
Keyword(s):  
Modal Logic ◽  
Propositional Logic ◽  
Modal Logics ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Feature Section

Semantical systems that distinguish between logically true and factually true formulas are well-known from the work of Carnap. The present paper is concerned with extending the formalization of this distinction in two ways. First, we show how to construct syntactical (i.e., logistic) systems that correspond to semantical systems of the type just mentioned. Such a syntactical system for propositional logic is developed in section 3. Similar systems for first-order logic are sketched in section 5. Second, we show how to extend semantical systems that make the logical-factual distinction to languages containing modal connectives. Carnap's work on modal logic conspicuously lacks this feature. Section 4 contains such semantical systems for four well-known modal logics. It also contains a syntactical equivalent of one of these modal semantical systems.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

Logical Tools

2021 ◽  
pp. 14-52
Author(s):  
Cian Dorr ◽  
John Hawthorne ◽  
Juhani Yli-Vakkuri
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
System P

This chapter presents the system of classical higher-order modal logic which will be employed throughout this book. Nothing more than a passing familiarity with classical first-order logic and standard systems of modal logic is presupposed. We offer some general remarks about the kind of commitment involved in endorsing this logic, and motivate some of its more non-standard features. We also discuss how talk about possible worlds can be represented within the system.

Lifting propositional proof compression algorithms to first-order logic

2020 ◽  
Author(s):  
Jan Gorzny ◽  
Ezequiel Postan ◽  
Bruno Woltzenlogel Paleo
Keyword(s):  
Propositional Logic ◽  
Empirical Evaluation ◽  
Order Logic ◽  
Automated Deduction ◽  
First Order Logic ◽  
Compression Algorithms ◽  
International Conference ◽  
First Order ◽  
Theorem Provers ◽  
Resolution Proofs

Abstract Proofs are a key feature of modern propositional and first-order theorem provers. Proofs generated by such tools serve as explanations for unsatisfiability of statements. However, these explanations are complicated by proofs which are not necessarily as concise as possible. There are a wide variety of compression techniques for propositional resolution proofs but fewer compression techniques for first-order resolution proofs generated by automated theorem provers. This paper describes an approach to compressing first-order logic proofs based on lifting proof compression ideas used in propositional logic to first-order logic. The first approach lifted from propositional logic delays resolution with unit clauses, which are clauses that have a single literal. The second approach is partial regularization, which removes an inference $\eta $ when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference in every path from $\eta $ to the root of the proof. This paper describes the generalization of the algorithms LowerUnits and RecyclePivotsWithIntersection (Fontaine et al.. Compression of propositional resolution proofs via partial regularization. In Automated Deduction—CADE-23—23rd International Conference on Automated Deduction, Wroclaw, Poland, July 31–August 5, 2011, p. 237--251. Springer, 2011) from propositional logic to first-order logic. The generalized algorithms compresses resolution proofs containing resolution and factoring inferences with unification. An empirical evaluation of these approaches is included.

scholarly journals Checkable Proofs for First-Order Theorem Proving

10.29007/s6d1 ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Martin Suda
Keyword(s):  
Theorem Proving ◽  
Critical Mass ◽  
Current Situation ◽  
Boolean Satisfiability ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Theorem Provers ◽  
Interactive Theorem Provers ◽  
Theory Reasoning

Inspired by the success of the DRAT proof format for certification of boolean satisfiability (SAT),we argue that a similar goal of having unified automatically checkable proofs should be soughtby the developers of automated first-order theorem provers (ATPs). This would not onlyhelp to further increase assurance about the correctness of prover results,but would also be indispensable for tools which rely on ATPs,such as ``hammers'' employed within interactive theorem provers.The current situation, represented by the TSTP format is unsatisfactory,because this format does not have a standardised semantics and thus cannot be checked automatically.Providing such semantics, however, is a challenging endeavour. One would ideallylike to have a proof format which covers only-satisfiability-preserving operations such as Skolemisationand is versatile enough to encompass various proving methods (i.e. not just superposition)or is perhaps even open ended towards yet to be conceived methods or at least easily extendable in principle.Going beyond pure first-order logic to theory reasoning in the style of SMT orbeyond proofs to certification of satisfiability are further interesting challenges.Although several projects have already provided partial solutions in this direction,we would like to use the opportunity of ARCADE to further promote the idea andgather critical mass needed for its satisfactory realisation.

scholarly journals Tableau-based translation from first-order logic to modal logic

2021 ◽  
Vol 56 ◽  
pp. 57-74
Author(s):  
Tin Perkov ◽  
Luka Mikec
Keyword(s):  
Modal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Modal Formula ◽  
First Order ◽  
Starting Point ◽  
Invariance Test

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

scholarly journals On the multi-dimensional modal logic of substitutions

2019 ◽  
Vol 56 (4) ◽  
pp. 454-481
Author(s):  
Tarek Sayed Ahmed ◽  
Mohammad Assem Mahmoud
Keyword(s):  
Amalgamation Property ◽  
Modal Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Semantics ◽  
Infinite Dimensions ◽  
Finite Axiomatizability ◽  
First Order ◽  
Abstract Entities ◽  
Omitting Types

Abstract We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an N P complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

scholarly journals A Quantified Coalgebraic van Benthem Theorem

2021 ◽  
pp. 551-571
Author(s):  
Paul Wild ◽  
Lutz Schröder
Keyword(s):  
Modal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Transition Systems ◽  
First Order ◽  
Wide Range ◽  
Symmetry Assumption ◽  
Invariant Properties ◽  
Bounded Rank

AbstractThe classical van Benthem theorem characterizes modal logic as the bisimulation-invariant fragment of first-order logic; put differently, modal logic is as expressive as full first-order logic on bisimulation-invariant properties. This result has recently been extended to two flavours of quantitative modal logic, viz. fuzzy modal logic and probabilistic modal logic. In both cases, the quantitative van Benthem theorem states that every formula in the respective quantitative variant of first-order logic that is bisimulation-invariant, in the sense of being nonexpansive w.r.t. behavioural distance, can be approximated by quantitative modal formulae of bounded rank. In the present paper, we unify and generalize these results in three directions: We lift them to full coalgebraic generality, thus covering a wide range of system types including, besides fuzzy and probabilistic transition systems as in the existing examples, e.g. also metric transition systems; and we generalize from real-valued to quantale-valued behavioural distances, e.g. nondeterministic behavioural distances on metric transition systems; and we remove the symmetry assumption on behavioural distances, thus covering also quantitative notions of simulation.

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