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15
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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Cited by 89 (0 self)
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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...
Set Theory for Verification: II  Induction and Recursion
 Journal of Automated Reasoning
, 2000
"... A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. ..."
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Cited by 42 (20 self)
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A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.
Mechanizing Coinduction and Corecursion in Higherorder Logic
 Journal of Logic and Computation
, 1997
"... A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresse ..."
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Cited by 41 (5 self)
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A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresses recursive functions over inductive data types; corecursion expresses functions that yield elements of coinductive data types. The theory rests on a traditional formalization of infinite trees. The theory is intended for use in specification and verification. It supports reasoning about a wide range of computable functions, but it does not formalize their operational semantics and can express noncomputable functions also. The theory is illustrated using finite and infinite lists. Corecursion expresses functions over infinite lists; coinduction reasons about such functions. Key words. Isabelle, higherorder logic, coinduction, corecursion Copyright c fl 1996 by Lawrence C. Paulson Content...
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
"... ..."
A Fixedpoint Approach to (Co)Inductive and (Co)Datatype Definitions
, 1997
"... This paper presents a fixedpoint approach to inductive definitions. Instead of using a syntactic test such as "strictly positive," the approach lets definitions involve any operators that have been proved monotone. It is conceptually simple, which has allowed the easy implementation of mutual re ..."
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Cited by 20 (2 self)
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This paper presents a fixedpoint approach to inductive definitions. Instead of using a syntactic test such as "strictly positive," the approach lets definitions involve any operators that have been proved monotone. It is conceptually simple, which has allowed the easy implementation of mutual recursion and iterated definitions. It also handles coinductive definitions: simply replace the least fixedpoint by a greatest fixedpoint. The method
A Case Study of Coinduction in Isabelle
, 1995
"... The consistency of the dynamic and static semantics for a small functional programming language was informally proved by R.Milner and M.Tofte. The notions of coinductive definitions and the associated principle of coinduction played a pivotal role in the proof. With emphasis on coinduction, the w ..."
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Cited by 7 (0 self)
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The consistency of the dynamic and static semantics for a small functional programming language was informally proved by R.Milner and M.Tofte. The notions of coinductive definitions and the associated principle of coinduction played a pivotal role in the proof. With emphasis on coinduction, the work presented here deals with the formalisation of this result in the generic theorem prover Isabelle. Contents 1 Introduction 1 2 Coinduction in Relation Semantics 2 2.1 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.2 The Language : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.3 Dynamic Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.4 Static Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.5 Consistency : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3 Isabelle 7 3.1 Documentation : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 3.2 Notation : : : : : : : : : : : : : : : : : : : : : ...
Tool Support for Logics of Programs
 Mathematical Methods in Program Development: Summer School Marktoberdorf 1996, NATO ASI Series F
, 1996
"... Proof tools must be well designed if they... ..."
Final Coalgebras as Greatest Fixed Points in ZF Set Theory
, 1999
"... this paper is not to change the axiom system but to adopt new definitions of ordered pairs, functions, and derived concepts such as Cartesian products. Under the new definitions, the stream functor's final coalgebra is indeed its (exact) greatest fixedpoint and each stream is an infinite nest of pai ..."
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Cited by 2 (2 self)
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this paper is not to change the axiom system but to adopt new definitions of ordered pairs, functions, and derived concepts such as Cartesian products. Under the new definitions, the stream functor's final coalgebra is indeed its (exact) greatest fixedpoint and each stream is an infinite nest of pairs. Recursion equations are solved up to equality
Declarative Combinatorics: Isomorphisms, Hylomorphisms and Hereditarily Finite Data Types in Haskell – unpublished draft –
, 808
"... This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals etc.) and their extension to hereditarily fin ..."
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Cited by 1 (1 self)
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This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals etc.) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides anytoany encodings automatically. A few examples of “free algorithms ” obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to succinct data representations and generation of random instances. The paper is part of a larger effort to cover in a declarative programming paradigm some fundamental combinatorial generation algorithms along the lines of Knuth’s recent work (Knuth 2006). In 440 lines of Haskell code we cover 20 data types and, through the use of the embedded combinator language, provide 380 distinct bijective encodings between them. The selfcontained source code of the paper, as generated from a literate Haskell program, is available at