Results 1 - 10
of
42
Explicit Provability And Constructive Semantics
- Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
Abstract
-
Cited by 139 (25 self)
- Add to MetaCart
(Show Context)
In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.
A Sahlqvist theorem for distributive modal logic
- Annals of Pure and Applied Logic 131, Issues
, 2002
"... Dedicated to Bjarni Jónsson In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For ..."
Abstract
-
Cited by 43 (13 self)
- Add to MetaCart
Dedicated to Bjarni Jónsson In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.
Presenting Functors by Operations and Equations
, 2006
"... We take the point of view that, if transition systems are coalgebras for a functor T, then an adequate logic for these transition systems should arise from the ‘Stone dual ’ L of T. We show that such a functor always gives rise to an ‘abstract’ adequate logic for T-coalgebras and investigate under ..."
Abstract
-
Cited by 33 (17 self)
- Add to MetaCart
We take the point of view that, if transition systems are coalgebras for a functor T, then an adequate logic for these transition systems should arise from the ‘Stone dual ’ L of T. We show that such a functor always gives rise to an ‘abstract’ adequate logic for T-coalgebras and investigate under which circumstances it gives rise to a ‘concrete ’ such logic, that is, a logic with an inductively defined syntax and proof system. We obtain a result that allows us to prove adequateness of logics uniformly for a large number of different types of transition systems and give some examples of its usefulness.
A Principle for Incorporating Axioms into the First-Order Translation of Modal Formulae
- Automated Deduction—CADE-19, volume 2741 of Lecture Notes in Artificial Intelligence
, 2003
"... In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable logics. The goal of the axiomatic translation ..."
Abstract
-
Cited by 20 (7 self)
- Add to MetaCart
(Show Context)
In this paper we present a translation principle, called the axiomatic translation, for reducing propositional modal logics with background theories, including triangular properties such as transitivity, Euclideanness and functionality, to decidable logics. The goal of the axiomatic translation principle is to find simplified theories, which capture the inference problems in the original theory, but in a way that is more amenable to automation and easier to deal with by existing theorem provers. The principle of the axiomatic translation is conceptually very simple and can be largely automated. Soundness is automatic under reasonable assumptions, and termination of ordered resolution is easily achieved, but the non-trivial part of the approach is proving completeness.
Canonical Varieties with No Canonical Axiomatisation
- Trans. Amer. Math. Soc
, 2003
"... We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every ..."
Abstract
-
Cited by 20 (8 self)
- Add to MetaCart
(Show Context)
We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabilistic methods...
Erdős Graphs Resolve Fine's Canonicity Problem
- THE BULLETIN OF SYMBOLIC LOGIC
, 2003
"... We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting ..."
Abstract
-
Cited by 16 (8 self)
- Add to MetaCart
We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames. The constructions use the result of Erdős that there are finite graphs with arbitrarily large chromatic number and girth.
On Fibring Semantics for BDI Logics
- Logics in computer science
, 2002
"... This study examines BDI logics in the context of Gabbay's fibring semantics. We show that dovetailing (a special form of fibring) can be adopted as a semantic methodology to combine BDI logics. We develop a set of interaction axioms that can capture static as well as dynamic aspects of the ment ..."
Abstract
-
Cited by 10 (4 self)
- Add to MetaCart
This study examines BDI logics in the context of Gabbay's fibring semantics. We show that dovetailing (a special form of fibring) can be adopted as a semantic methodology to combine BDI logics. We develop a set of interaction axioms that can capture static as well as dynamic aspects of the mental states in BDI systems, using Catach's incestual schema $G^{a, b, c, d}$. Further we exemplify the constraints required on fibring function to capture the semantics of interactions among modalities. The advantages of having a fibred approach is discussed in the final section.
Explaining Subsumption by Optimal Interpolation
- In Proc. of the European Conf. of Logics in Artificial Intelligence
"... Abstract. We describe ongoing research to support the construction of terminologies with Description Logics. For the explanation of subsump-tion we search for particular concepts because of their syntactic and se-mantic properties. More precisely, the set of explanations for a subsump-tion P N is t ..."
Abstract
-
Cited by 10 (0 self)
- Add to MetaCart
(Show Context)
Abstract. We describe ongoing research to support the construction of terminologies with Description Logics. For the explanation of subsump-tion we search for particular concepts because of their syntactic and se-mantic properties. More precisely, the set of explanations for a subsump-tion P N is the set of optimal interpolants for P and N. We provide definitions for optimal interpolation and an algorithm based on Boolean minimisation of concept-names in a tableau proof for ALC-satisfiability. Finally, we describe our implementation and some experiments to assess the computational scalability of our proposal. 1
First-order resolution methods for modal logics
- In Volume in Memoriam of Harald Ganzinger, LNCS
, 2006
"... Abstract. In this paper we give an overview of results for modal logic which can be shown using techniques and methods from first-order logic and resolution. Because of the breadth of the area and the many applications we focus on the use of first-order resolution methods for modal logics. In additi ..."
Abstract
-
Cited by 7 (4 self)
- Add to MetaCart
(Show Context)
Abstract. In this paper we give an overview of results for modal logic which can be shown using techniques and methods from first-order logic and resolution. Because of the breadth of the area and the many applications we focus on the use of first-order resolution methods for modal logics. In addition to traditional propositional modal logics we consider more expressive PDL-like dynamic modal logics which are closely related to description logics. Without going into too much detail, we survey different ways of translating modal logics into first-order logic, we explore different ways of using first-order resolution theorem provers, and we discuss a variety of results which have been obtained in the setting of first-order resolution. 1
