Introduction to Coalgebra

Jacobs, B.

doi:10.1017/cbo9781316823187

Cited by 92 publications

(40 citation statements)

References 126 publications

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“…From Theorem 6.1 we also derive the soundness of up-to Bhv for unary predicates: the monotone predicate liftings used in coalgebraic modal logic [17] are fibration maps [27], so they satisfy the hypothesis of Theorem 6.1.…”

Section: Corollary 62 If F Is a Set-functor Preserving Weak Pullbackmentioning

confidence: 99%

“…If F and G preserve weak pullbacks and F and G are the canonical Pre-liftings Pre(F) and Pre(G), then is obtained via the restriction of the natural transformation Rel(ρ) between the corresponding canonical relation liftings (Rel(−) is functorial, see [27]). The situation is slightly more complex for non-canonical liftings, such as the lax lifting of the LTS functor.…”

Section: Example 102mentioning

confidence: 99%

“…Proof Recall that the canonical relation lifting Rel(G)(R) of a relation i : R → X × X is obtained via the (epi,mono)-factorisation in (27). We assume further that all the monos in the diagrams below are inclusions.…”

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confidence: 99%

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A general account of coinduction up-to

et al. 2016

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Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras. The fact that bisimulations up to context can be safely used in any language specified by GSOS rules can also be seen as an instance of our framework, using the well-known observation by Turi and Plotkin that such languages form bialgebras. In the second part of the paper, we provide a new categorical treatment of weak bisimilarity on labeled transition systems and we prove the soundness of up-to context for weak bisimulations of systems specified by cool rule formats, as defined by Bloom to ensure congruence of weak bisimilarity. The weak transition systems obtained from such cool rules give rise to lax bialgebras, rather than to bialgebras. Hence, to reach our goal, we extend the categorical framework developed in the first part to an ordered setting. B Daniela Petrişan

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Section: Corollary 62 If F Is a Set-functor Preserving Weak Pullbackmentioning

confidence: 99%

Section: Example 102mentioning

confidence: 99%

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A general account of coinduction up-to

et al. 2016

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“…e.g. [1]). Intriguingly little, however, has been done to formalise the process of making observations and of using these observations to learn how a system works.…”

Section: Introductionmentioning

confidence: 99%

Angluin Learning via Logic

Barlocco

Kupke

2017

Logical Foundations of Computer Science

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Abstract. In this paper we will provide a fresh take on Dana Angluin's algorithm for learning using ideas from coalgebraic modal logic. Our work opens up possibilities for applications of tools & techniques from modal logic to automata learning and vice versa. As main technical result we obtain a generalisation of Angluin's original algorithm from DFAs to coalgebras for an arbitrary finitary set functor T in the following sense: given a (possibly infinite) pointed T -coalgebra that we assume to be regular (i.e. having an equivalent finite representation) we can learn its finite representation by asking (i) "logical queries" (corresponding to membership queries) and (ii) making conjectures to which the teacher has to reply with a counterexample. This covers (a known variant) of the original L* algorithm and the learning of Mealy/Moore machines. Other examples are bisimulation quotients of (probabilistic) transition systems.

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“…It is a study of states and their operations and properties. The set of states is usually seen as a black box, to which one has limited access [6]. A relation between what is actually inside and what can be observed externally is the foundation of the theory of coalgebras [3].…”

Section: Introductionmentioning

confidence: 99%

Some properties of coalgebras and their rôle in computer science

Steingartner

Radaković

Valkošák

et al. 2016

J. Appl. Math. Comput. Mech.

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Abstract. This paper introduces basic theoretical knowledge of coalgebras in computer science. Coalgebras are, specifically in category theory, structures defined according to an endofunctor. For both algebra and coalgebra, a functor is a convenient and general way of defining a signature. We present practical usage of the coalgebras in an example. We observe a behavior of a simple Sequencer developed in SLGeometry framework. We model its behavior with the simple program written in Python, and we describe its behavior within coalgebra of endofunctor. The computation of the values stored in internal states is performed coinductively. Our approach can be used in the teaching process of formal methods for young software engineers.

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Introduction to Coalgebra

Cited by 92 publications

References 126 publications

A general account of coinduction up-to

A general account of coinduction up-to

Angluin Learning via Logic

Some properties of coalgebras and their rôle in computer science

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