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Hierarchies of Modal and Temporal Logics with Reference Pointers
 Journal of Logic, Language and Information
, 1995
"... . We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: "point of reference  reference pointer" which enable semantic references to be made within a formula. We propose three different but equivalent sema ..."
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Cited by 43 (2 self)
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. We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: "point of reference  reference pointer" which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. The universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and strong completeness theorem is proved for them and extended to some classes of their extensions. Key words: Modal and Temporal Logics, Reference Pointers, Expressi...
Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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Cited by 29 (16 self)
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We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Combining Temporal Logic Systems
 Notre Dame Journal of Formal Logic
, 1994
"... This paper is a continuation of the work started in [FG92] on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component onedimensional temporal logics to the resulting twodimensional tempora ..."
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Cited by 29 (2 self)
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This paper is a continuation of the work started in [FG92] on combining temporal logics. In this work, four combination methods are described and studied with respect to the transference of logical properties from the component onedimensional temporal logics to the resulting twodimensional temporal logic. Three basic logical properties are analysed, namely soundness, completeness and decidability. Each combination method is composed of three submethods that combine the languages, the inference systems and the semantics of two onedimensional temporal logic systems, generating families of twodimensional temporal languages with varying expressivity and varying degree of transference of logical properties. The temporalisation method and the independent combination method are shown to transfer all three basic logical properties. The method of full interlacing of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several...
Temporal Logic with Reference Pointers
 Proceedings of the 1st International Conference on Temporal Logic, volume 827 of LNAI
"... An extension of the propositional temporal language is introduced with a simple syntactic device, called "reference pointer" which provides for making references within a formula to "instants of reference" specified in the formula. The language with reference pointers L_trp has a ..."
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Cited by 14 (1 self)
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An extension of the propositional temporal language is introduced with a simple syntactic device, called "reference pointer" which provides for making references within a formula to "instants of reference" specified in the formula. The language with reference pointers L_trp has a great expressive power (e.g. Kamp's and Stavi's operators as well as Prior's clock variables are definable in it), especially compared to its frugal syntax, perspicuous semantics and simple deductive system. The minimal temporal logic K_trp of this language is axiomatized and strong completeness theorem is proved for it and extended to an important class of extensions of K_trp . The validity in L_trp is proved undecidable.
Nondeterministic temporal logics for general Flow systems
, 2004
"... In this paper, we use the constructs of branching temporal logic to formalize reasoning about a class of general flow systems, including discretetime transition systems, continuoustime differential inclusions, and hybridtime systems such as hybrid automata. We introduce Full General Flow Logic, ..."
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Cited by 13 (5 self)
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In this paper, we use the constructs of branching temporal logic to formalize reasoning about a class of general flow systems, including discretetime transition systems, continuoustime differential inclusions, and hybridtime systems such as hybrid automata. We introduce Full General Flow Logic, GFL which has essentially the same syntax as the wellknown Full Computation Tree Logic, CTL , but generalizes the semantics to general flow systems over arbitrary timelines. We propose an axiomatic proof system for GFL and establish its soundness w.r.t. the general flow semantics.
Verification in Continuous Time by Discrete Reasoning
 Proceedings of the 4th International Conference on Algebraic Methodology and Software Technology AMAST'95, Lecture Notes in Computer Science 936
, 1995
"... This paper shows how the advantages of both semantics can be combined by adapting the simple verification rules of the discrete semantics to the continuous one. Specifically, we show that if a temporal logic formula has the property of ..."
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Cited by 10 (0 self)
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This paper shows how the advantages of both semantics can be combined by adapting the simple verification rules of the discrete semantics to the continuous one. Specifically, we show that if a temporal logic formula has the property of
Continuous Verification by Discrete Reasoning
, 1994
"... Two semantics are commonly used for the behavior of realtime and hybrid systems: a discrete semantics, in which the temporal evolution is represented as a sequence of snapshots describing the state of the system at certain times, and a continuous semantics, in which the temporal evolution is repres ..."
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Two semantics are commonly used for the behavior of realtime and hybrid systems: a discrete semantics, in which the temporal evolution is represented as a sequence of snapshots describing the state of the system at certain times, and a continuous semantics, in which the temporal evolution is represented by a series of time intervals, and therefore corresponds more closely to the physical reality. Powerful verification rules are known for temporal logic formulas based on the discrete semantics. This paper shows how to transfer the verification techniques of the discrete semantics to the continuous one. We show that if a temporal logic formula has the property of finite variability, its validity in the discrete semantics implies its validity in the continuous one. This leads to a verification method based on three components: verification rules for the discrete semantics, axioms about time, and some temporal reasoning to bring the results together. This approach enables the verification...
Composing and Refining Dense Temporal Logic Specifications
 Formal Aspects of Computing
, 1999
"... . A dense temporal logic development method for the specification, refinement, composition and verification of reactive systems is introduced. A reactive system is specified by a pair consisting of a machine and a condition that indicate the valid computations of this machine. Compositionality is ac ..."
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. A dense temporal logic development method for the specification, refinement, composition and verification of reactive systems is introduced. A reactive system is specified by a pair consisting of a machine and a condition that indicate the valid computations of this machine. Compositionality is achieved by adding to each machine step whether it is a environment, system or communication step. Refinement can be expressed straightforward in the logic because the stutter problem is elegantly solved by using the dense structure of the logic. Compositionality enables us to break refinement between complex systems into refinement between small and simple systems. The latter can then be verified by existing proof rules for refinement which are reformulated in our formalism. 1. Introduction We present a compositional refinement method for reactive systems. A system is called reactive if it maintains some ongoing interaction with its environment, for example an operating system. This contrast...
Synthesising Axioms By Games
"... Contents 1 Introduction 2 2 Story 2 3 To the games 4 4 Generalisations 6 5 Discussion 9 Research partially supported by UK EPSRC grants GR/K54946, GR/L85978, GR/L85961. Thanks to ' Agnes Kurucz and Szabolcs Mikul'as for comments. 1 1 Introduction We would like to begin by hoping that y ..."
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Contents 1 Introduction 2 2 Story 2 3 To the games 4 4 Generalisations 6 5 Discussion 9 Research partially supported by UK EPSRC grants GR/K54946, GR/L85978, GR/L85961. Thanks to ' Agnes Kurucz and Szabolcs Mikul'as for comments. 1 1 Introduction We would like to begin by hoping that you had a very happy birthday (we assume you found more enjoyable ways to spend it than reading this), and we wish you many more years of fruitful research. In this short article we would like to discuss from our current perspective the problem of providing axioms for classes of algebras, and the way in which games can contribute to solving it. Towards the end, we will describe general settings in which this can be done. We will be rather discursive and opinionated  you may well disagree with what we say (and we'd like to hear from you about it), but we are certainly not trying to provoke you or implying that you do disagree with us. We will use natural langua