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Distributive laws for the coinductive solution of recursive equations
 Information and Computation
"... This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributi ..."
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Cited by 12 (1 self)
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This paper illustrates the relevance of distributive laws for the solution of recursive equations, and shows that one approach for obtaining coinductive solutions of equations via infinite terms is in fact a special case of a more general approach using an extended form of coinduction via distributive laws. 1
Recursion and Corecursion Have the Same Equational Logic
 Comput. Sci
"... This paper is concerned with the equational logic of corecursion, that is of definitions involving final coalgebra maps. The framework for our study is iteration theories (cf. e.g. Bloom and ' Esik [2, 3]), recently reintroduced as models of the FLR 0 fragment of the Formal Language of Recursi ..."
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Cited by 6 (1 self)
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This paper is concerned with the equational logic of corecursion, that is of definitions involving final coalgebra maps. The framework for our study is iteration theories (cf. e.g. Bloom and ' Esik [2, 3]), recently reintroduced as models of the FLR 0 fragment of the Formal Language of Recursion [7, 8, 9]. We present a new class of iteration theories derived from final coalgebras. This allows us to reason with a number of types of fixedpoint equations which heretofore seemed to require metric or ordertheoretic ideas. All of the work can be done using finality properties and equational reasoning. Having a semantics, we obtain the following completeness result: the equations involving fixedpoint terms which are valid for final coalgebra interpretations are exactly those valid in a number of contexts pertaining to recursion. For example, they coincide with the equations valid for leastfixed point recursion on dcpo's. We also present a new version of the proof of the well...
Generalizing Substitution
, 2003
"... It is well known that, given an endofunctor H on a category C, the initial (A + H−)algebras (if existing), i.e., the algebras of (wellfounded) Hterms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Mo ..."
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Cited by 4 (1 self)
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It is well known that, given an endofunctor H on a category C, the initial (A + H−)algebras (if existing), i.e., the algebras of (wellfounded) Hterms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [2] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A + H−)coalgebras (if existing), i.e., the algebras of nonwellfounded Hterms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T ′ (A, −)algebras resp. the inverses of the final T ′ (A, −)coalgebras for any endobifunctor T ′ on any category C such that the functors T ′ (−,X) uniformly carry a monad structure.
COPRODUCTS OF IDEAL MONADS
, 2004
"... The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by ..."
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Cited by 4 (1 self)
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The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by
Relating Two Approaches to Coinductive Solution of Recursive Equations
 Milius (Eds.), Proceedings of the 7th Workshop on Coalgebraic Methods in Computer Science, CMCS’04 (Barcelona, March 2004), Electron. Notes in Theoret. Comput. Sci
, 2004
"... This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws. ..."
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Cited by 3 (2 self)
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This paper shows that the approach of [2,12] for obtaining coinductive solutions of equations on infinite terms is a special case of a more general recent approach of [4] using distributive laws.
The Power of Parameterization in Coinductive Proof
"... Coinduction is one of the most basic concepts in computer science. It is therefore surprising that the commonlyknown latticetheoretic accounts of the principles underlying coinductive proofs are lacking in two key respects: they do not support compositional reasoning (i.e., breaking proofs into se ..."
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Cited by 2 (1 self)
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Coinduction is one of the most basic concepts in computer science. It is therefore surprising that the commonlyknown latticetheoretic accounts of the principles underlying coinductive proofs are lacking in two key respects: they do not support compositional reasoning (i.e., breaking proofs into separate pieces that can be developed in isolation), and they do not support incremental reasoning (i.e., developing proofs interactively by starting from the goal and generalizing the coinduction hypothesis repeatedly as necessary). In this paper, we show how to support coinductive proofs that are both compositional and incremental, using a dead simple construction we call the parameterized greatest fixed point. The basic idea is to parameterize the greatest fixed point of interest over the accumulated knowledge of “the proof so far”. While this idea has been proposed before, by Winskel in 1989 and by Moss in 2001, neither of the previous accounts suggests its general applicability to improving the state of the art in interactive coinductive proof. In addition to presenting the latticetheoretic foundations of parameterized coinduction, demonstrating its utility on representative examples, and studying its composition with “upto ” techniques, we also explore its mechanization in proof assistants like Coq and Isabelle. Unlike traditional approaches to mechanizing coinduction (e.g., Coq’s cofix), which employ syntactic “guardedness checking”, parameterized coinduction offers a semantic account of guardedness. This leads to faster and more robust proof development, as we demonstrate using our new Coq library, Paco.
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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Cited by 1 (0 self)
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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
An AlphaCorecursion Principle for the Infinitary Lambda Calculus
"... Abstract. Gabbay and Pitts proved that lambdaterms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambdacalculus form the final coalgebra for the same functor. This allows us to give a corecur ..."
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Cited by 1 (1 self)
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Abstract. Gabbay and Pitts proved that lambdaterms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambdacalculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alphaequivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Böhm, LévyLongo and Berarducci trees). 1
Coalgebraic Coinduction in (Hyper)settheoretic Categories
, 2000
"... This paper is a contribution to the foundations of coinductive types and coiterative functions, in (Hyper)settheoretical Categories, in terms of coalgebras. We consider atoms as first class citizens. First of all, we give a sharpening, in the way of cardinality, of Aczel's Special Final Coalgebra ..."
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This paper is a contribution to the foundations of coinductive types and coiterative functions, in (Hyper)settheoretical Categories, in terms of coalgebras. We consider atoms as first class citizens. First of all, we give a sharpening, in the way of cardinality, of Aczel's Special Final Coalgebra Theorem, which allows for good estimates of the cardinality of the final coalgebra. To these end, we introduce the notion of Y uniform functor, which subsumes Aczel's original notion. We give also an nary version of it, and we show that the resulting class of functors is closed under many interesting operations used in Final Semantics. We define also canonical wellfounded versions of the final coalgebras of functors uniform on maps. This leads to a reduction of coiteration to ordinal induction, giving a possible answer to a question raised by Moss and Danner. Finally, we introduce a generalization of the notion of F bisimulation inspired by Aczel's notion of precongruence, and we show t...
Nominal Coalgebraic Data Types . . .
"... We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus. ..."
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We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.