Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193-215, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are introduced. The minimum system of Positive Modal Logic is the (#, #, #, #, #, #) fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed in [12] The frames of this semantics consists of a set of indexes, a quasi order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasiorder ....

....paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d persistency results as well as a Sahlqvist Theorem for sequents and the semantics proposed in [6]. Also a GoldblattThomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved. Keywords: positive modal logic, distributive lattices with operators, duality theory, correspondence theory, d persistency. 1 Introduction The ....

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Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

....c #Oxford University Press 780 An Expectation Transformer Model for Probabilistic Temporal Logic probabilities (neither 0 nor 1) or other quantitative aspects (expected e#ciency) requires di#erent techniques. Other approaches address explicit probabilities, but then forgo demonic nondeterminism [6]; yet nondeterminism is not only an unavoidable aspect of some problems but as abstraction is essential for forming layered descriptions of complex systems. Finally, recent work by Bianco and de Alfaro [3] based on the pCTL of Aziz [1] and ultimately ideas of Hansson and Jonsson [8] ....

....from the latter we have that X: #A satisfies X = #A # #X, of which however ##A is the # least solution. 3 Probabilistic temporal logic 3. 1 Expectations and their transformers Standard predicate transformer semantics can be generalised to replace demonic choice by probabilistic choice [6] but more interesting is to add probabilistic choice while retaining demonic nondeterminism [10, 20] By doing the latter, and generalising the definitions of the previous section, we obtain a model for probabilistic branching time temporal logic. Again take state space S; but define now the ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

....c Oxford University Press 780 An Expectation Transformer Model for Probabilistic Temporal Logic probabilities (neither 0 nor 1) or other quantitative aspects (expected eciency) requires di erent techniques. Other approaches address explicit probabilities, but then forgo demonic nondeterminism [6]; yet nondeterminism is not only an unavoidable aspect of some problems but as abstraction is essential for forming layered descriptions of complex systems. Finally, recent work by Bianco and de Alfaro [3] based on the pCTL of Aziz [1] and ultimately ideas of Hansson and Jonsson [8] allows ....

....from the latter we have that X : 3A satis es X = 3A [ X ; of which however 33A is the least solution. 3 Probabilistic temporal logic 3. 1 Expectations and their transformers Standard predicate transformer semantics can be generalised to replace demonic choice by probabilistic choice [6] but more interesting is to add probabilistic choice while retaining demonic nondeterminism [10, 20] By doing the latter, and generalising the de nitions of the previous section, we obtain a model for probabilistic branching time temporal logic. Again take state space S; but de ne now the ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193-215, 1984.

....Pnueli and Hart [26] shows that there are many systems in which such validity depends only on the transition probabilities being nonzero, and not on their precise values. Other approaches address explicit probabilities, but until recently only at the cost of disallowing demonic nondeterminism [7, 6]. Both authors are members of the Programming Research Group at Oxford University: Wolfson Building, Parks Road, Oxford OX1 3QD, fcarroll,anabelg comlab.ox.ac.uk. McIver is supported by the EPSRC. Recent work by Bianco and de Alfaro [3] based on the pCTL of Aziz et al. 1] and ultimately ....

....the latter we have that X : 3A satisfies X = 3A [ ffiX ; of which however 33A is the least solution. Xi 3 Probabilistic temporal logic 3. 1 Expectations and their transformers Standard predicate transformer semantics can be generalised to replace demonic choice by probabilistic choice [7] or to add probabilistic choice while retaining demonic nondeterminism [11, 20] By doing the latter, and generalising the definitions of the previous section, we obtain a model for probabilistic branching time temporal logic. Again take state space S; but define now the expectations ES over S as ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

....and divergence (catastrophic nontermination, or abort) are included naturally. The use of [0; 1] means that numbers do occur explicitly in the calculations (in contrast to [9] however the use of expectations rather than explicit probabilities may mean that the expressiveness is not as great as [4]. To recover explicit probabilities, when desired, we rely on this observation: If a random variable A is f0; 1g valued over some state space S, selecting as a characteristic function some sub space S 0 of it, then the expected value of A over a probability distribution Pr on S is in fact the ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

....with either this or that ; and the implementor may choose between them according to his own concerns. Early research on probabilistic semantics took a different route: demonic choice was not regarded as fundamental rather it was abandoned altogether, being replaced by probabilistic choice [9, 4, 3, 8, 7]. Thus probabilistic semantics was divorced from the contemporaneous work on specification and refinement, because without demonic choice there is no means of abstraction. More recently however it has been discovered [6, 15] how to bring the two topics back together, taking the more natural ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

....and divergence (catastrophic non termination, or abort) are included naturally. The use of [0; 1] means that numbers do occur explicitly in the calculations (in contrast to [9] however the use of expectations rather than explicit probabilities may mean that the expressiveness is not as great as [4]. To recover explicit probabilities, when desired, we rely on this observation: If a random variable A is f0; 1g valued over some state space S, selecting as a characteristic function some sub space S 0 of it, then the expected value of A over a probability distribution Pr on S is in fact the ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

....in GSL whenever one writes test this [ test that (2) instead of the more conventional (1) above. Early research on probabilistic semantics took a different route: demonic choice was not regarded as fundamental rather it was abandoned altogether, being replaced by probabilistic choice [8, 4, 3, 7, 6]. Thus probabilistic semantics was divorced from the contemporaneous work on specification and refinement, because without demonic choice there is no means of abstraction. More recently however it has been discovered [5, 15] how to bring the two topics back together, taking the more natural ....

Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193--215, 1984.

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Yishai A. Feldman and David Harel. A probabilistic dynamic logic. J. Computing and System Sciences, 28:193-215, 1984.

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