Results 11  20
of
100
A Computational Interpretation of Modal Proofs
 Proof Theory of Modal Logics
, 1994
"... The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, exten ..."
Abstract

Cited by 28 (2 self)
 Add to MetaCart
The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, extending previous work by Masini on a twodimensional generalization of Gentzen's sequents (2sequents). The modal rules closely match the standard rules for an universal quantifier and different logics are obtained with simple conditions on the elimination rule for 2. We give an explicit term calculus corresponding to proofs in these systems and, after defining a notion of reduction on terms, we prove its confluence and strong normalization. 1. Introduction Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent ex...
Decidable Fragments of FirstOrder Modal Logics
 JOURNAL OF SYMBOLIC LOGIC
, 1999
"... The paper considers the set ML1 of firstorder 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 ML1, which reduces the modal satisfiability to the classica ..."
Abstract

Cited by 28 (8 self)
 Add to MetaCart
The paper considers the set ML1 of firstorder 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 ML1, 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.
Modal Logic, Transition Systems and Processes
, 1994
"... Transition systems can be viewed either as process diagrams or as Kripke structures. The first perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be mo ..."
Abstract

Cited by 24 (3 self)
 Add to MetaCart
Transition systems can be viewed either as process diagrams or as Kripke structures. The first perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be motivated by operations on transition systems, but they can also be suggested by investigations of modal formalisms. To show that the equational view of processes from process algebra is closely related to modal logic, we consider various ways of looking at the relation between the calculus of basic process algebra and propositional dynamic logic. More concretely, the paper contains preservation results for various bisimulation notions, a result on the expressive power of propositional dynamic logic, and a definition of bisimulation which is the proper notion of invariance for concurrent propositional dynamic logic. Keywords: modal logic, transition systems, bisimulation, process algebra 1 In...
On the Products of Linear Modal Logics
 JOURNAL OF LOGIC AND COMPUTATION
, 2001
"... We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz: ..."
Abstract

Cited by 24 (9 self)
 Add to MetaCart
We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using nonstructural Gabbaytype inference rules.
Reasoning about Action and Change  A Dynamic Logic Approach
 Journal of Logic, Language, and Information
, 1996
"... this paper, we pursue a monotonic approach to the frame problem and concentrate on the combinatorial problem and the overcommitment problem. We will propose a solution within the framework of propositional dynamic logic (PDL)the modal logic of actions and of computer programs (see Pratt, 1976, 19 ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
this paper, we pursue a monotonic approach to the frame problem and concentrate on the combinatorial problem and the overcommitment problem. We will propose a solution within the framework of propositional dynamic logic (PDL)the modal logic of actions and of computer programs (see Pratt, 1976, 1980; Segerberg 1980; Harel, 1984). It is based on the idea of associating an operator [ff] with each action ff, the brackets being reminiscient of the box operator 2 of ordinary modal logic (see Hughes & Cresswell, 1984). The reading of a formula [ff]A is "after every terminating (halting) execution of ff, A is true." PDL provides a powerful language for describing compound actions such as sequential composition of actions ff and fi, written ff; fi, (nondeterministic) choice between ff and fi, written ff + fi, and (nondeterministic) iteration of ff, written ff
A systematic proof theory for several modal logics
 Advances in Modal Logic, volume 5 of King’s College Publications
, 2005
"... abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence betw ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a HilbertLewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is adhoc. While we can formulate several modal logics in the sequent calculus that enjoy cutelimination, their formalisation arises through systembysystem fine tuning to ensure that the cutelimination holds, and the correspondence to the axioms of the HilbertLewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the HilbertLewis axiomatisation. We show that the calculus possesses a cutelimination property directly analogous to cutelimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1
Intuitionistic Necessity Revisited
 PROCEEDINGS OF THE LOGIC AT WORK CONFERENCE
, 1996
"... In this paper we consider an intuitionistic modal logic, which we call IS42 . Our approach is different to others in that we favour the natural deduction and sequent calculus proof systems rather than the axiomatic, or Hilbertstyle, system. Our natural deduction formulation is simpler than other pr ..."
Abstract

Cited by 23 (7 self)
 Add to MetaCart
In this paper we consider an intuitionistic modal logic, which we call IS42 . Our approach is different to others in that we favour the natural deduction and sequent calculus proof systems rather than the axiomatic, or Hilbertstyle, system. Our natural deduction formulation is simpler than other proposals. The traditional means of devising a modal logic is with reference to a model, and almost always, in terms of a Kripke model. Again our approach is different in that we favour categorical models. This facilitates not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.
Cutfree Sequent and Tableau Systems for Propositional Diodorean Modal Logics
"... We present sound, (weakly) complete and cutfree tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of po ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
We present sound, (weakly) complete and cutfree tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points. Although cutfree, the last two systems do not possess the subformula property. But for any given finite set of formulae X the "superformulae" involved are always bounded by a finite set of formulae X L depending only on X and the logic L. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. Each tableau system has a cutfree sequent analogue proving that Gentzen's cutelimination theorem holds for these latter systems. The techniques are due to Hi...