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16
NonDeterministic Kleene Coalgebras
"... In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Miln ..."
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In this paper, we present a systematic way of deriving (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and deterministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.
A.: Coalgebraic Logic and Synthesis of Mealy Machines
"... Abstract. We present a novel coalgebraic logic for deterministic Mealy machines that is sound, complete and expressive w.r.t. bisimulation. Every finite Mealy machine corresponds to a finite formula in the language. For the converse, we give a compositional synthesis algorithm which transforms every ..."
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Cited by 10 (7 self)
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Abstract. We present a novel coalgebraic logic for deterministic Mealy machines that is sound, complete and expressive w.r.t. bisimulation. Every finite Mealy machine corresponds to a finite formula in the language. For the converse, we give a compositional synthesis algorithm which transforms every formula into a finite Mealy machine whose behaviour is exactly the set of causal functions satisfying the formula. 1
An algebra for Kripke polynomial coalgebras
 24TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 2009
"... Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of socalled Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be charact ..."
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Cited by 7 (7 self)
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Several dynamical systems, such as deterministic automata and labelled transition systems, can be described as coalgebras of socalled Kripke polynomial functors, built up from constants and identities, using product, coproduct and powerset. Locally finite Kripke polynomial coalgebras can be characterized up to bisimulation by a specification language that generalizes Kleene’s regular expressions for finite automata. In this paper, we equip this specification language with an axiomatization and prove it sound and complete with respect to bisimulation, using a purely coalgebraic argument. We demonstrate the usefulness of our framework by providing a finite equational system for (non)deterministic finite automata, labelled transition systems with explicit termination, and automata on guarded strings.
Modal Logics are Coalgebraic
, 2008
"... Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large vari ..."
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Cited by 4 (0 self)
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Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pickandchoose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors ’ firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility.
Completeness for flat modal fixpoint logics
 Annals of Pure and Applied Logic, 162(1):55 – 82
, 2010
"... This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for ..."
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This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for each γ ∈ Γ. The term ♯γ(ϕ1,..., ϕn) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x, ϕ1,..., ϕn). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L♯(Γ) on Kripke frames. We prove two results that solve this problem. First, let K♯(Γ) be the logic obtained from the basic polymodal K by adding a KozenPark style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ♯γ. Provided that each indexing formula γ satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K + ♯ (Γ) of K♯(Γ). This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ♯γ, of size bounded by the length of γ. Thus the axiom system K + (Γ) is finite whenever Γ is finite.
Completeness of the finitary Moss logic
 In Areces and Goldblatt [3
"... abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type. ..."
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abstract. We give a sound and complete derivation system for the valid formulas in the finitary version of Moss ’ coalgebraic logic, for coalgebras of arbitrary type.
Equational Coalgebraic Logic
 MFPS
, 2009
"... Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible onestep behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for Tcoalgebras. We compare two existing proposals, ..."
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Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible onestep behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for Tcoalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing onestep translations between them, extending the results in [21] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss’s logic. The three logics are equivalent for a natural but restricted class of functors. We give examples showing that the logics fall apart in general. Finally, we argue that the quest for a generic logic for Tcoalgebras is still open in the general case.
Stream Differential Equations: concrete formats for coinductive definitions
, 2011
"... In this article we give an accessible introduction to stream differential equations, ie., equations that take the shape of differential equations from analysis and that are used to define infinite streams. Furthermore we discuss a syntactic format for stream differential equations that ensures that ..."
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In this article we give an accessible introduction to stream differential equations, ie., equations that take the shape of differential equations from analysis and that are used to define infinite streams. Furthermore we discuss a syntactic format for stream differential equations that ensures that any system of equations that fits into the format has a unique solution. It turns out that the stream functions that can be defined using our format are precisely the causal stream functions. Finally, we are going to discuss nonstandard stream calculus that uses basic (co)operations different from the usual head and tail operations in order to define and to reason about streams and stream functions. 1
Kleene Coalgebra  an overview
"... Coalgebras provide a uniform framework for the study of dynamical systems, including several types of automata. The coalgebraic view on systems has recently been proved relevant by the development of a number of expression calculi which generalize classical results by Kleene, on regular expressions, ..."
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Coalgebras provide a uniform framework for the study of dynamical systems, including several types of automata. The coalgebraic view on systems has recently been proved relevant by the development of a number of expression calculi which generalize classical results by Kleene, on regular expressions, and by Kozen, on Kleene algebra. This note contains an overview of the motivation and results of the generic framework we developed – Kleene Coalgebra – to uniformly derive the aforementioned calculi. We present an historical overview of work on regular expressions and axiomatizations, as well a discussion of related work. We show applications of the framework to three types of probabilistic systems: simple Segala, stratified and PnueliZuck.
Contents
, 1981
"... under the supervision of Dr. Yde Venema and Dr. Clemens Kupke, and submitted to ..."
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under the supervision of Dr. Yde Venema and Dr. Clemens Kupke, and submitted to