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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...
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 distributive l ..."
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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...
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.
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
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