Terminal coalgebras in well-founded set theory

Barr, Michael

doi:10.1016/0304-3975(93)90076-6

Cited by 221 publications

(161 citation statements)

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“…As proved in [21], it is enough to show that F W is finitary, i.e. that for any set X and any x ∈ F W X there is a finite subset Y ⊆ X such that x arises as an element of F W Y .…”

Section: Proof Any W-ltsmentioning

confidence: 99%

Structural Operational Semantics for Weighted Transition Systems

Klin

2009

Semantics and Algebraic Specification

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Abstract. Weighted transition systems are defined, parametrized by a commutative monoid of weights. These systems are further understood as coalgebras for functors of a specific form. A general rule format for the SOS specification of weighted systems is obtained via the coalgebraic approach of Turi and Plotkin. Previously known formats for labelled transition systems (GSOS) and stochastic systems (SGSOS) appear as special cases.

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“…As proved in [21], it is enough to show that F W is finitary, i.e. that for any set X and any x ∈ F W X there is a finite subset Y ⊆ X such that x arises as an element of F W Y .…”

Section: Proof Any W-ltsmentioning

confidence: 99%

Structural Operational Semantics for Weighted Transition Systems

Klin

2009

Semantics and Algebraic Specification

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“…From this point of view, the general theorems of Aczel and Mendler [3] and Barr [4] yield final coalgebras for a great many functors.…”

Section: Discussionmentioning

confidence: 96%

“…This category-theoretic notion relates to the methods of bisimulation and coinduction, which are heavily used in concurrency theory [6], functional programming [1] and operational semantics [7]. Aczel and Mendler [3] and also Barr [4] have proved that final coalgebras exist in set theory for large classes of naturally occurring functors. This might be supposed to satisfy most people's requirements.…”

Section: Introductionmentioning

confidence: 99%

A concrete final coalgebra theorem for ZF set theory

Paulson

1995

Lecture Notes in Computer Science

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Abstract.A special final coalgebra theorem, in the style of Aczel's [2], is proved within standard Zermelo-Fraenkel set theory. Aczel's AntiFoundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's Solution and Substitution Lemmas are proved in the style of Rutten and Turi [12]. The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint. The theory is intended for machine implementation and a simple case of it is already implemented using the theorem prover Isabelle [10].

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“…In order to carry this development through, Rutten, building on results and notions from Aczel and Mendler [AM89], Barr [Bar93,Bar94] and Lambek [Lam68], needed to assume two properties of the type functor F : that it should preserve weak pullbacks, and, somehow implicitly, that it should also preserve intersections. These assumptions, which we shall explain below, seem to be satisfied in all standard examples.…”

Section: Introductionmentioning

confidence: 99%