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Some Properties and Some Problems on Set Functors

2006
*
Electronical Notes in Theoretical Computer Science
*

inclusion preserving, and we discuss simple characterization

doi:10.1016/j.entcs.2006.06.005
fatcat:gcxir234krhvvjuxep62afahpq
*theorems*of*final**coalgebras*as fixpoints. ... Then, starting from Aczel's Special*Final**Coalgebra**Theorem*, we study the class of functors uniform on maps, we present and discuss various examples of functors which are not uniform on maps but still ... Preliminaries Set Theory Preliminaries We will work in*a*universe of sets and classes satisfying the theory NBG of von Neumann-Bernays-Gödel. The theory NBG is closely related to the more familiar ...##
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On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

1998
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Mathematical Structures in Computer Science
*

The aim of the present survey is to show that the elementary categorical notion of

doi:10.1017/s0960129598002588
fatcat:yszyhcmjsjeqbgzj73exutmbem
*a**final**coalgebra*is*a*suitable foundation for such*a*coinduction principle. ... The properties of*coalgebraic*coinduction are studied both at an abstract categorical level and in some specific categories used in semantics, namely categories of non-well-founded sets, partial orders ...*Theorem*5.2. (*Final*P S -*Coalgebra**Theorem*.) ...##
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On the foundations of final semantics: Non-standard sets, metric spaces, partial orders
[chapter]

1993
*
Lecture Notes in Computer Science
*

Such

doi:10.1007/3-540-56596-5_45
fatcat:6x2bzicl2va2zhpwqop7eb5obe
*a**final*semantics enjoys uniformity and*generality*. ... This paper is meant to provide*a*basis to*a*more*general*project aiming at*a*full exploitation of the*finality*of the domains in the semantics of programming languages --concurrent ones among them. ...*final**coalgebra**theorem*. ...##
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Connections of coalgebra and semantic modeling

2011
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Proceedings of hte 13th Conference on Theoretical Aspects of Rationality and Knowledge - TARK XIII
*

*Coalgebra*is

*a*

*general*study of

*a*great many kinds of models, and these include type spaces and Kripke models, and many others. ... But the theory is not overly

*general*, it is not

*a*theory of absolutely everything. ... The surjectivity is tantamount to the completeness

*theorem*for the logic.

*Theorem*3.5. (L * , l * ) is

*a*

*final*

*coalgebra*for F : for all

*coalgebras*

*A*, jA :

*A*→ L * is the unique

*coalgebra*morphism. ...

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Final coalgebras and the Hennessy–Milner property

2006
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Annals of Pure and Applied Logic
*

The main

doi:10.1016/j.apal.2005.06.006
fatcat:hftxeswvrbbvpacnjvujft62u4
*theorem*gives*a*representation of states of the*final**coalgebra*as certain satisfiable sets of formulas. ... The existence of*a**final**coalgebra*is equivalent to the existence of*a*formal logic with*a*set (small class) of formulas that has the Hennessy-Milner property of distinguishing*coalgebraic*states up to ... The main result of this paper (*Theorem*4.3) gives*a*very*general*answer by showing that*a*straightforward logical construction of*final**coalgebras*is always possible whenever there exists*a*formal language ...##
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Coalgebraic Lindströom Theorems

2010
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Advances in Modal Logic
*

We study modal Lindström

dblp:conf/aiml/KurzV10
fatcat:q4tw5yxgindfhiybi4k46iq6m4
*theorems*from*a**coalgebraic*perspective. ... We provide three different Lindström*theorems*for*coalgebraic*logic, one of which is*a*direct generalisation of de Rijke's result for Kripke models. ... Furthermore, when we treat T α 1 (or the carrier Z of the*final**coalgebra*) as*a*topological space in the following, then we do this with respect to the topology*generated*by L α (or L). ...##
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Sound and Complete Axiomatizations of Coalgebraic Language Equivalence

2013
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ACM Transactions on Computational Logic
*

extended to

doi:10.1145/2422085.2422092
fatcat:b6jobbzoinaezn3ujrfzv2svnq
*a*coarser*coalgebraic*language equivalence, which arises from*a*generalised powerset construction that determinises*coalgebras*. ...*Coalgebras*provide*a*uniform framework to study dynamical systems, including several types of automata. ... Soundness, Completeness and Kleene's*theorem*in*general*In this section we obtain*a**generalization*of Kleene's classical*theorem*from automata theory [17] to the setting of*coalgebras*for the liftingF ...##
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From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems

1999
*
Electronical Notes in Theoretical Computer Science
*

These are morphisms into

doi:10.1016/s1571-0661(05)80265-8
fatcat:geohj5ixqbbaxoi2jzcda4arua
*final**coalgebras*, satisfying the T -coiteration scheme, which is*a**generalization*of the corecursion scheme. ... We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint*Theorem*, and the categorical description of coinduction based on*coalgebras*. ...*Theorem*2.8 (*Coalgebraic*Coinduction) Suppose that F : C S → C S has*final*F -*coalgebra*(νF, α νF ). Let (X, α X ) be an F -*coalgebra*. ...##
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Models of non-well-founded sets via an indexed final coalgebra theorem

2007
*
Journal of Symbolic Logic (JSL)
*

The paper uses the formalism of indexed categories to recover the proof of

doi:10.2178/jsl/1191333841
fatcat:k3ptfzcre5btxktbcsae7stmfq
*a*standard*final**coalgebra**theorem*, thus showing existence of*final**coalgebras*for*a*special class of functors on finitely complete ... As an instance of this result, we build the*final**coalgebra*for the powerclass functor, in the context of*a*Heyting pretopos with*a*class of small maps. ...*Final**coalgebra**theorems*In this section, we are going to use the machinery of Section 2 in order to prove an indexed*final**coalgebra**theorem*. ...##
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Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem
[article]

2006
*
arXiv
*
pre-print

The paper uses the formalism of indexed categories to recover the proof of

arXiv:math/0508531v2
fatcat:efh54o2u7zcl7nit4wdo6zsd2y
*a*standard*final**coalgebra**theorem*, thus showing existence of*final**coalgebras*for*a*special class of functors on categories with ... As an instance of this result, we build the*final**coalgebra*for the powerclass functor, in the context of*a*Heyting pretopos with*a*class of small maps. ...*Final**coalgebra**theorems*In this section, we are going to use the machinery of Section 2 in order to prove an indexed*final**coalgebra**theorem*. ...##
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A small final coalgebra theorem

2000
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Theoretical Computer Science
*

This paper presents an elementary and self-contained proof of an existence

doi:10.1016/s0304-3975(97)00302-2
fatcat:tuyvk2ecwbfntof5knpgethh54
*theorem*of ÿnal*coalgebras*for endofunctors on the category of sets and functions. ...*Finally*it is easy to see that fit = git, where i : H →*A*denotes the inclusion. This proves that Set M ( ) is*a**generating*set of Set( ). Conversely assume that G is*a**generating*set of Set( ). ...*Theorem*4 . 6 . 46 Every -*coalgebra*(*A*;*a*) has the maximum congruence*A*:Proof. ...##
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Model Checking : A Co-algebraic Approach
[article]

2016
*
arXiv
*
pre-print

This method is based on the existence of the

arXiv:1606.01014v1
fatcat:y3btdwmswna3piihcgyquk3kty
*final**coalgebra*of*a*suitable endofunctor and can be*generalized*smoothly to other*coalgebraic*structures. ... In this paper, we try to solve this problem from*a**coalgebraic*approach. ... The next step is using first algorithm to act on the Kripke structure K, and we will get the concrete smallest Kripke structure of the infinite Kripke structure with (G 0 ,*A*) as its finite representation ...##
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Well-Pointed Coalgebras (Extended Abstract)
[chapter]

2012
*
Lecture Notes in Computer Science
*

For set functors preserving intersections,

doi:10.1007/978-3-642-28729-9_6
fatcat:2hoa7fjhuffdtfam25gr7jxo4i
*a*new description of the*final**coalgebra*and the initial algebra is presented: the former consists of all well-pointed*coalgebras*. ...*Finally*, the initial iterative algebra consists of all finite well-pointed*coalgebras*. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems. ... Moreover, H k 0 is by (*a*)*a**final*well-founded*coalgebra*, thus, isomorphic to ψ : I / / HI. Thus (I, ψ −1 ) is isomorphic to the initial algebra.*Theorem*2.14. ...##
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Recursive coalgebras from comonads

2006
*
Information and Computation
*

We discuss Osius's [22] concept of

doi:10.1016/j.ic.2005.08.005
fatcat:xlxme6iodffxrec4zkjbekczcy
*a*recursive*coalgebra*of*a*functor from the perspective of programming semantics and give some new sufficient conditions for the recursiveness of*a*functor-*coalgebra*... that are based on comonads, comonad-*coalgebras*and distributive laws. ... We begin by the main*theorem*, which uses*a**general*comonad. ...##
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Recursive Coalgebras from Comonads

2004
*
Electronical Notes in Theoretical Computer Science
*

We discuss Osius's [22] concept of

doi:10.1016/j.entcs.2004.02.034
fatcat:nxqxm7l3vvgpjdcielxo2v2nfy
*a*recursive*coalgebra*of*a*functor from the perspective of programming semantics and give some new sufficient conditions for the recursiveness of*a*functor-*coalgebra*... that are based on comonads, comonad-*coalgebras*and distributive laws. ... We begin by the main*theorem*, which uses*a**general*comonad. ...
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