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Some Properties and Some Problems on Set Functors

Daniela Cancila, Furio Honsell, Marina Lenisa
2006 Electronical Notes in Theoretical Computer Science  
inclusion preserving, and we discuss simple characterization 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  ... 
doi:10.1016/j.entcs.2006.06.005 fatcat:gcxir234krhvvjuxep62afahpq

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

DANIELE TURI, JAN RUTTEN
1998 Mathematical Structures in Computer Science  
The aim of the present survey is to show that the elementary categorical notion of 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.)  ... 
doi:10.1017/s0960129598002588 fatcat:yszyhcmjsjeqbgzj73exutmbem

On the foundations of final semantics: Non-standard sets, metric spaces, partial orders [chapter]

Jan J. M. M. Rutten, Daniele Turi
1993 Lecture Notes in Computer Science  
Such 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.  ... 
doi:10.1007/3-540-56596-5_45 fatcat:6x2bzicl2va2zhpwqop7eb5obe

Connections of coalgebra and semantic modeling

Lawrence S. Moss
2011 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.  ... 
doi:10.1145/2000378.2000382 dblp:conf/tark/Moss11 fatcat:2a2b5nwcmneyxewpp6giulygsi

Final coalgebras and the Hennessy–Milner property

Robert Goldblatt
2006 Annals of Pure and Applied Logic  
The main 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  ... 
doi:10.1016/j.apal.2005.06.006 fatcat:hftxeswvrbbvpacnjvujft62u4

Coalgebraic Lindströom Theorems

Alexander Kurz, Yde Venema
2010 Advances in Modal Logic  
We study modal Lindström 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).  ... 
dblp:conf/aiml/KurzV10 fatcat:q4tw5yxgindfhiybi4k46iq6m4

Sound and Complete Axiomatizations of Coalgebraic Language Equivalence

Marcello M. Bonsangue, Stefan Milius, Alexandra Silva
2013 ACM Transactions on Computational Logic  
extended to 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  ... 
doi:10.1145/2422085.2422092 fatcat:b6jobbzoinaezn3ujrfzv2svnq

From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems

Marina Lenisa
1999 Electronical Notes in Theoretical Computer Science  
These are morphisms into 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.  ... 
doi:10.1016/s1571-0661(05)80265-8 fatcat:geohj5ixqbbaxoi2jzcda4arua

Models of non-well-founded sets via an indexed final coalgebra theorem

Benno van den Berg, Federico de Marchi
2007 Journal of Symbolic Logic (JSL)  
The paper uses the formalism of indexed categories to recover the proof of 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.  ... 
doi:10.2178/jsl/1191333841 fatcat:k3ptfzcre5btxktbcsae7stmfq

Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem [article]

Benno van den Berg, Federico De Marchi
2006 arXiv   pre-print
The paper uses the formalism of indexed categories to recover the proof of 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.  ... 
arXiv:math/0508531v2 fatcat:efh54o2u7zcl7nit4wdo6zsd2y

A small final coalgebra theorem

Yasuo Kawahara, Masao Mori
2000 Theoretical Computer Science  
This paper presents an elementary and self-contained proof of an existence 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.  ... 
doi:10.1016/s0304-3975(97)00302-2 fatcat:tuyvk2ecwbfntof5knpgethh54

Model Checking : A Co-algebraic Approach [article]

Jianhua Gao, Ying Jiang
2016 arXiv   pre-print
This method is based on the existence of the 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  ... 
arXiv:1606.01014v1 fatcat:y3btdwmswna3piihcgyquk3kty

Well-Pointed Coalgebras (Extended Abstract) [chapter]

Jiří Adámek, Stefan Milius, Lawrence S. Moss, Lurdes Sousa
2012 Lecture Notes in Computer Science  
For set functors preserving intersections, 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.  ... 
doi:10.1007/978-3-642-28729-9_6 fatcat:2hoa7fjhuffdtfam25gr7jxo4i

Recursive coalgebras from comonads

Venanzio Capretta, Tarmo Uustalu, Varmo Vene
2006 Information and Computation  
We discuss Osius's [22] concept of 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.  ... 
doi:10.1016/j.ic.2005.08.005 fatcat:xlxme6iodffxrec4zkjbekczcy

Recursive Coalgebras from Comonads

Venanzio Capretta, Tarmo Uustalu, Varmo Vene
2004 Electronical Notes in Theoretical Computer Science  
We discuss Osius's [22] concept of 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.  ... 
doi:10.1016/j.entcs.2004.02.034 fatcat:nxqxm7l3vvgpjdcielxo2v2nfy
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