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Coalgebraic Logic
 Annals of Pure and Applied Logic
, 1999
"... We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The ..."
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We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the final coalgebra. Keywords: infinitary modal logic, characterization theorem, functor on sets, coalgebra, greatest fixed point. 1 Intr...
Characteristic Formulae for FixedPoint Semantics: A General Framework
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... The literature on concurrency theory offers a wealth of examples of characteristicformula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proof ..."
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The literature on concurrency theory offers a wealth of examples of characteristicformula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proofs of correctness have been developed independently, but have a common underlying structure. This study provides a general view of characteristic formulae that are expressed in terms of logics with a facility for the recursive definition of formulae. It is shown how several examples of characteristicformula constructions from the literature can be recovered as instances of the proposed general framework, and how the framework can be used to yield novel constructions. The paper also offers general results pertaining to the definition of cocharacteristic formulae and of characteristic formulae expressed in terms of infinitary modal logics.
Global Definability in Basic Modal Logic
"... We present results on global definability in basic modal logic, and contrast our modeltheoretic results and proof techniques with known results about local definability. ..."
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We present results on global definability in basic modal logic, and contrast our modeltheoretic results and proof techniques with known results about local definability.
Bounded Variable Logics: Two, Three, and More
 In preparation
, 1997
"... Consider the bounded variable logics L k 1! (with k variable symbols), and C k 1! (with k variables in the presence of counting quantifiers 9 ?m ). These fragments of infinitary logic L1! are well known to provide an adequate logical framework for some important issues in finite model theory. ..."
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Consider the bounded variable logics L k 1! (with k variable symbols), and C k 1! (with k variables in the presence of counting quantifiers 9 ?m ). These fragments of infinitary logic L1! are well known to provide an adequate logical framework for some important issues in finite model theory. This paper deals with a translation that associates equivalence of structures in the k variable fragments with bisimulation equivalence between derived structures. Apart from a uniform and intuitively appealing treatment of these equivalences, this approach relates some interesting issues for the case of an arbitrary number of variables to the case of just three variables. Invertibility of the invariants for jC 3 , in particular, would imply a positive answer to the tempting conjecture that fixedpoint logic with counting captures S k Ptime " C k 1! . 1 Introduction The bounded variable fragments of infinitary logic play an important role in finite model theory. In contrast to firs...
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... Circularity and the axioms of set theory In recent years, various kinds of circular phenomena have been studied by researchers from different disciplines. Examples are the study of automata in the science of computation, which generally contain cycles of state transitions; recursive domain equations ..."
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Circularity and the axioms of set theory In recent years, various kinds of circular phenomena have been studied by researchers from different disciplines. Examples are the study of automata in the science of computation, which generally contain cycles of state transitions; recursive domain equations in computer science, which play a role in the semantics of concurrent programming languages with recursion; theories of truth and (self)reference in philosophy; and terminological cycles in artificial intelligence. Let us look at an example, taken from computation science, and treated in Vicious circles in all detail. Streams are infinite sequences of elements over an (alphabet) set A. Let a and b be in A and let s be the stream that informally is defined as consisting of an a followed by a b, again followed by an a and a b, and so on. Because of the fact that when the first two elements of s are removed, the same stream s is obtained again, it seems natural to model it by using the ordinary pairing operator: s = 〈a, 〈b, s〉〉, or, using t to denote the stream obtained by removing the first a from s, by the following system of equations: s = 〈a, t 〉 and