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Relating Multifunctions and Predicate Transformers through Closure Operators
 of Lecture Notes in Computer Science
, 1994
"... . We study relations between predicate transformers and multifunctions in a topological setting based on closure operators. We give topological definitions of safety and liveness predicates and using these predicates we define predicate transformers. State transformers are multifunctions with values ..."
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. We study relations between predicate transformers and multifunctions in a topological setting based on closure operators. We give topological definitions of safety and liveness predicates and using these predicates we define predicate transformers. State transformers are multifunctions with values in the collection of fixed points of a closure operator. We derive several isomorphisms between predicate transformers and multifunctions. By choosing different closure operators we obtain multifunctions based on the usual power set construction, on the Hoare, Smyth and Plotkin power domains, and based on the compact and closed metric power constructions. Moreover, they are all related by isomorphisms to the predicate transformers. 1 Introduction There are (at least) two different ways of assigning a denotational semantics to a programming language: forward or backward. A typical forward semantics is a semantics that models a program as a function from initial states to final states. In th...
Towards and infinitary logic of domains: Abramsky logic for transition systems
, 1999
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Duality and the Completeness of the Modal µCalculus
"... We consider the modal µcalculus due to Kozen, which is a finitary modal logic with least and greatest fixed points of monotone operators. We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the ca ..."
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We consider the modal µcalculus due to Kozen, which is a finitary modal logic with least and greatest fixed points of monotone operators. We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the case of having fixed point operators. As a corollary, we obtain completeness results for two proof systems due to Kozen (finitary and infinitary) with respect to certain classes of modal frames. The rules are sound in every model, not only for validity.
Infinitary Domain Logic for Finitary Transition Systems
 Proceedings of TACS'97
, 1997
"... . The Lindenbaum algebra generated by the Abramsky finitary logic is a distributive lattice dual to an SFPdomain obtained as a solution of a recursive domain equation. We extend Abramsky's result by proving that the Lindenbaum algebra generated by the infinitary logic is a completely distribut ..."
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. The Lindenbaum algebra generated by the Abramsky finitary logic is a distributive lattice dual to an SFPdomain obtained as a solution of a recursive domain equation. We extend Abramsky's result by proving that the Lindenbaum algebra generated by the infinitary logic is a completely distributive lattice dual to the same SFPdomain. As a consequence soundness and completeness of the infinitary logic is obtained for the class of finitary transition systems. A corollary of this result is that the same holds for the infinitary HennessyMilner logic. 1 Introduction Complete partial orders were originally introduced as a mathematical structure to model computation [Sco70], in particular as domains for denotational semantics [SS71]. Successively, Scott's presentation of domains as information systems [Sco82] suggested a connection between denotational semantics and logics of programs. Based on the fundamental insight of Smyth [Smy83b] that a topological space may be seen as a `data type' w...
Coalgebras and Their Logics 1
, 2006
"... Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out ..."
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Some comments about the last Logic Column, on nominal logic. Pierre Lescanne points out
Coalgebras, Stone Duality, Modal Logic
, 2006
"... A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand c ..."
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A brief outline of the topics of the course could be as follows. Coalgebras generalise transition systems. Modal logics are the natural logics for coalgebras. Stone duality provides the relationship between coalgebras and modal logic. Furthermore, some basic category theory is needed to understand coalgebras as well as Stone duality. So we
Presenting Functors by Operations and Equations
"... Abstract. 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 Tcoalgebras and investiga ..."
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Abstract. 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 Tcoalgebras 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. 1
ReInterpreting the Modal µCalculus
 MODAL LOGIC AND PROCESS ALGEBRA
, 1995
"... We reexamine the modal µcalculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this in ..."
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We reexamine the modal µcalculus in the light of some classical theory of Boolean algebras and recent results on duality theory for a modal logic with fixed points. We propose interpreting formulas into a field of subsets of states instead of the full power set lattice used by Kozen. Under this interpretation we relate image compact modal frames with Scott continuity of the box modality, msaturated transition systems and descriptive modal frames. Also, it is shown that the class of image compact modal frames satisfies the HennessyMilner property. We conclude by showing that for descriptive modal µframes the standard interpretation coincides with the one we proposed.
Duality for Logics of Transition Systems
"... Abstract. We present a general framework for logics of transition systems based on Stone duality. Transition systems are modelled as coalgebras for a functor T on a category X. The propositional logic used to reason about state spaces from X is modelled by the Stone dual A of X (e.g. if X is Stone s ..."
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Abstract. We present a general framework for logics of transition systems based on Stone duality. Transition systems are modelled as coalgebras for a functor T on a category X. The propositional logic used to reason about state spaces from X is modelled by the Stone dual A of X (e.g. if X is Stone spaces then A is Boolean algebras and the propositional logic is the classical one). In order to obtain a modal logic for transition systems (i.e. for Tcoalgebras) we consider the functor L on A that is dual to T. An adequate modal logic for Tcoalgebras