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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.
Coalgebras and Monads in the Semantics of Java
 Theoretical Computer Science
, 2002
"... This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at th ..."
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This paper describes the basic structures in the denotational and axiomatic semantics of sequential Java, both from a monadic and a coalgebraic perspective. This semantics is an abstraction of the one used for the verification of (sequential) Java programs using proof tools in the LOOP project at the University of Nijmegen. It is shown how the monadic perspective gives rise to the relevant computational structure in Java (composition, extension and repetition), and how the coalgebraic perspective o#ers an associated program logic (with invariants, bisimulations, and Hoare logics) for reasoning about the computational structure provided by the monad.
Coalgebraic epistemic update without change of model
, 2007
"... We present a coalgebraic semantics for reasoning about information update in multiagent systems. The novelty is that we have one structure for both states and actions and thus our models do not involve the ”changeofmodel” phenomena that arise when using Kripke models. However, we prove that the u ..."
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We present a coalgebraic semantics for reasoning about information update in multiagent systems. The novelty is that we have one structure for both states and actions and thus our models do not involve the ”changeofmodel” phenomena that arise when using Kripke models. However, we prove that the usual models can be constructed from ours by categorical adjunction. The generality and abstraction of our coalgebraic model turns out to be extremely useful in proving preservation properties of update. In particular, we prove that positive knowledge is preserved and acquired as a result of epistemic update. We also prove common and nested knowledge properties of epistemic updates induced by specific epistemic actions such as public and private announcements, lying, and in particular unsafe actions of security protocols. Our model directly gives rise to a coalgebraic logic with both dynamic and epistemic modalities. We prove a soundness and completeness result for this logic, and illustrate the applicability of the logic by deriving knowledge properties of a simple security protocol.
A note on expressive coalgebraic logics for finitary set functors
 J. Log. Comput
"... This paper has two purposes. The first is to present a final coalgebra construction for finitary endofunctors on Set that uses a certain subset L ∗ of the limit L of the first ω terms in the final sequence. L ∗ is the set of points in L which arise from all coalgebras using their canonical morphisms ..."
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This paper has two purposes. The first is to present a final coalgebra construction for finitary endofunctors on Set that uses a certain subset L ∗ of the limit L of the first ω terms in the final sequence. L ∗ is the set of points in L which arise from all coalgebras using their canonical morphisms into L, and it was used earlier for different purposes in Kurz and Pattinson [5]. Viglizzo in [11] showed that the same set L ∗ carried a final coalgebra structure for functors in a certain inductively defined family. Our first goal is to generalize this to all finitary endofunctors; the result is implicit in Worrell [12]. The second goal is to use the final coalgebra construction to propose coalgebraic logics similar to those in [6] but for all finitary endofunctors F on Set. This time one can dispense with all conditions on F, construct a logical language LF directly from it, and prove that two points in a coalgebra satisfy the same sentences of LF iff they are identified by the final coalgebra morphism. The language LF is very spare, having no boolean connectives. This work on LF is thus a reworking of coalgebraic logic for finitary functors on sets. 1
GSOS for probabilistic transition systems (Extended Abstract)
, 2002
"... We introduce probabilistic GSOS, an operator specification format for (reactive) probabilistic transition systems which arises as an adaptation of the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all mode ..."
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We introduce probabilistic GSOS, an operator specification format for (reactive) probabilistic transition systems which arises as an adaptation of the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all models bisimilarity is a congruence and the uptocontext proof principle is valid. Moreover, every specification has a final model which can be shown to offer unique solutions for guarded recursive equations. The format covers operator specifications from the literature, so that the wellbehavedness results given for those arise as instances of our general one.
Uniform Functors on Sets
"... This paper is a contribution to the study of uniformity conditions for endofunctors on sets initiated in Aczel [1] and pursued later in other works such as Turi [17]. The main results have been that the “usual ” functors on sets are uniform in our sense, and that assuming the AntiFoundation Axiom A ..."
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This paper is a contribution to the study of uniformity conditions for endofunctors on sets initiated in Aczel [1] and pursued later in other works such as Turi [17]. The main results have been that the “usual ” functors on sets are uniform in our sense, and that assuming the AntiFoundation Axiom AFA, a uniform functor H has the property that its greatest fixed point H ∗ is a final coalgebra whose structure is the identity map. We propose a notion of uniformity whose definition involves notions from recent work in coalgebraic recursion theory such as completely iterative monads and completely iterative algebras (CIAs), see Adámek et al. [2, 3, 6] and Milius [11]. This simplifies many calculations and makes the definition of uniformity more natural than it had been. We also present several new results, including one which could be called a Paranoia Theorem: For a uniform H, the entire universe is a CIA: the structure is the inclusion of HV into the universe V itself. 1
On compositions and paths for coalgebras
, 2005
"... This report discusses the possibility of defining composed steps and paths in coalgebras. Both of these notions are relevant for defining some semantic relations for coalgebras like weak bisimulation and trace semantics. We present some observations, ideas, and small results on the mentioned topic ..."
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This report discusses the possibility of defining composed steps and paths in coalgebras. Both of these notions are relevant for defining some semantic relations for coalgebras like weak bisimulation and trace semantics. We present some observations, ideas, and small results on the mentioned topics. 1
A coinductive approach to verified exact real number computation. 2009. To appear
 Proceedings of Automated Verification of Critical Systems (AVOCS), Gregynog
"... Abstract: We present an approach to verified programs for exact real number computation that is based on inductive and coinductive definitions and program extraction from proofs. We informally discuss the theoretical background of this method and give examples of extracted programs implementing th ..."
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Abstract: We present an approach to verified programs for exact real number computation that is based on inductive and coinductive definitions and program extraction from proofs. We informally discuss the theoretical background of this method and give examples of extracted programs implementing the translation between the representation by fast converging rational Cauchy sequences and the signed binary digit representations of real numbers.
Automata for Coalgebras: an approach using predicate liftings?
"... Abstract. Universal Coalgebra provides the notion of a coalgebra as the natural mathematical generalization of statebased evolving systems such as (infinite) words, trees, and transition systems. We lift the theory of parity automata to a coalgebraic level of abstraction by introducing, for a set Λ ..."
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Abstract. Universal Coalgebra provides the notion of a coalgebra as the natural mathematical generalization of statebased evolving systems such as (infinite) words, trees, and transition systems. We lift the theory of parity automata to a coalgebraic level of abstraction by introducing, for a set Λ of predicate liftings associated with a set functor T, the notion of a Λautomata operating on coalgebras of type T. In a familiar way these automata correspond to extensions of coalgebraic modal logics with least and greatest fixpoint operators. Our main technical contribution is a general bounded model property result: We provide a construction that transforms an arbitrary Λautomaton A with nonempty language into a small pointed coalgebra (S, s) of type T that is recognized by A, and of size exponential in that of A. S is obtained in a uniform manner, on the basis of the winning strategy in our satisfiability game associated with A. On the basis of our proof we obtain a general upper bound for the complexity of the nonemptiness problem, under some mild conditions on Λ and T. Finally, relating our automatatheoretic approach to the tableauxbased one of Ĉırstea et alii, we indicate how to obtain their results, based on the existence of a complete tableau calculus, in our framework.