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Generalised Coinduction
, 2001
"... We introduce the lambdacoiteration schema for a distributive law lambda of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final Fcoalgebra, generalising the basic coiteration schema as given by finality. The duals of ..."
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Cited by 16 (3 self)
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We introduce the lambdacoiteration schema for a distributive law lambda of a functor T over a functor F. Under certain conditions it can be shown to uniquely characterise functions into the carrier of a final Fcoalgebra, generalising the basic coiteration schema as given by finality. The duals of primitive recursion and courseofvalue iteration, which are known extensions of coiteration, arise as instances of our framework. One can furthermore obtain schemata justifying recursive specifications that involve operators such as addition of power series, regular operators on languages, or parallel and sequential composition of processes. Next...
Substitution in nonwellfounded . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the nonwellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.
Abstract From Parity Games to Circular Proofs
"... We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µbicomplete categories. We argue that parity games with a given starting position ..."
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We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µbicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of µbicomplete categories. We show that the interpretation of a parity game in the category of sets and functions is the set of deterministic winning strategies for one player in the game. We discuss bounded memory communication strategies between two parity games and their computational significance. We describe how an attempt to formalize them within the algebra of µbicomplete categories leads to develop a calculus of proofs that are allowed to contain cycles. This paper is a survey on our recent work lifting results on free µlattices [1,2] to a categorical setting. A µlattice is a lattice with enough least and greatest fixed points to interpret formal µterms. A generalization of this notion leads to consider categories with finite products, finite coproducts, and enough initial algebras and final coalgebras of functors. We call these categories µbicomplete. The outcome of this research is so far described in [3,4,5]. A main goal for us is to understand how the algebra of µbicomplete categories describes a computational situation through the combinatorics of games; when attempting to achieve this goal, computational logic and prooftheory become unavoidable ingredients. It is the aim of this note to give insights on how these four worlds – categories, games, computation and logic – relate in this context. As the need of a mathematical formalization has too often hidden these relationships, we shall present here only informal arguments. The reader will find formal proofs of the statements in the references cited above.