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A comonadic account of behavioural covarieties of coalgebras

ROBERT GOLDBLATT
2005 Mathematical Structures in Computer Science  
Then we show that behavioural covarieties K are (isomorphic to) the Eilenberg-Moore categories of coalgebras for certain comonads G K naturally associated with G T .  ...  We show that there is a bijective correspondence between behavioural covarieties of T -coalgebras and isomorphism classes of pure subcomonads of G T .  ...  Thus a categorical account of behavioural covarieties closed under ultrafilter enlargements will need to analyse the relationship between a monad and a comonad, and is a natural topic for further study  ... 
doi:10.1017/s096012950400458x fatcat:g5aflvvy4rbkhoa45l7d3dnina

Covarieties of Coalgebras: Comonads and Coequations [chapter]

Ranald Clouston, Robert Goldblatt
2005 Lecture Notes in Computer Science  
We identify conditions on the underlying category of a comonad G which ensure that there is an exact correspondence between (behavioural/virtual) covarieties of G-coalgebras and subcomonads of G defined  ...  A virtual covariety is a class of coalgebras closed under coproducts, images of coalgebraic morphisms, and subcoalgebras defined by split equalisers.  ...  The authors acknowledge support from a Logic and Computation programme funded by the New Zealand Institute of Mathematics and its Applications.  ... 
doi:10.1007/11560647_19 fatcat:5qu67cnmxva7xfat2d2sb575xi

Modal Predicates and Coequations

Alexander Kurz, Jiří Rosický
2002 Electronical Notes in Theoretical Computer Science  
We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic.  ...  This is a special case of Linton's approach to algebras over a general base category X , namely where X is taken as the dual of sets.  ...  We define two notions, the second one taking 'colourings' into account: Given C, a colouring v for a coalgebra A is an arrow U A → C.  ... 
doi:10.1016/s1571-0661(04)80364-5 fatcat:44m54cet3vaitf3g27szekzvt4

How to write a coequation [article]

Fredrik Dahlqvist, Todd Schmid
2021 arXiv   pre-print
There is a large amount of literature on the topic of covarieties, coequations and coequational specifications, dating back to the early seventies.  ...  We present each of these in a tutorial fashion, relate them to each other, and discuss their respective uses.  ...  By theorem 4.5, the covariety of sequence coalgebras is determined by a set W seq of behaviours.  ... 
arXiv:2109.11967v1 fatcat:gjiohhvtfvhrhnbgwzpqiatvoa

On Specification Logics for Algebra-Coalgebra Structures: Reconciling Reachability and Observability [chapter]

Corina Cîrstea
2002 Lecture Notes in Computer Science  
The two frameworks are then integrated in order to account for structures comprising both a coalgebraic (observational) component and an algebraic (computational) component.  ...  The integration results in logics whose sentences are either coalgebraic (modal) or algebraic (equational) in nature, but whose associated notions of satisfaction take into account both the coalgebraic  ...  Then, a second goal of the paper is to integrate the resulting frameworks in order to account for structures having both a coalgebraic and an algebraic component.  ... 
doi:10.1007/3-540-45931-6_7 fatcat:o4ggwwrhnjdpdoqhgk4xcwyk7i

Algebra, coalgebra, and minimization in polynomial differential equations [article]

Michele Boreale
2019 arXiv   pre-print
We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours.  ...  of a user-specified template.  ...  How to characterise such covariety more explicitly, for example in terms of comonads, is left for future research.  ... 
arXiv:1710.08350v3 fatcat:y3voaiqulrawzoc3lecan7xmyy

Algebra, Coalgebra, and Minimization in Polynomial Differential Equations [chapter]

Michele Boreale
2017 Lecture Notes in Computer Science  
We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours.  ...  of a user-specified template.  ...  How to characterise such covariety more explicitly, for example in terms of comonads, is left for future research.  ... 
doi:10.1007/978-3-662-54458-7_5 fatcat:q4qypnm43zfdzchmpg5yfyzlhq

How to Write a Coequation ((Co)algebraic pearls)

Fredrik Dahlqvist, Todd Schmid, Fabio Gadducci, Alexandra Silva
2021
There is a large amount of literature on the topic of covarieties, coequations and coequational specifications, dating back to the early seventies.  ...  We present each of these in a tutorial fashion, relate them to each other, and discuss their respective uses.  ...  By Theorem 12, the covariety of sequence coalgebras is determined by a set W seq of behaviours.  ... 
doi:10.4230/lipics.calco.2021.13 fatcat:2cih67o7rzgydmv7v3urjsaobm

Interpreted systems and Kripke models for multiagent systems from a categorical perspective

Timothy Porter
2004 Theoretical Computer Science  
This equivalence is exploited in a discussion of colimits of interpreted systems.  ...  This paper enriches previous comparisons of these two forms of semantics by considering categories of models in both cases and then shows that constructions given by Lomuscio and Ryan, extend to give an  ...  The paper has also beneÿtted from the constructive and pertinent comments of two referees for TCS.  ... 
doi:10.1016/j.tcs.2004.04.005 fatcat:wjeigxwv45dbpprbeuert4musa

Interpreted systems and Kripke models for multiagent systems from a categorical perspective

T PORTER
2004 Theoretical Computer Science  
This equivalence is exploited in a discussion of colimits of interpreted systems.  ...  This paper enriches previous comparisons of these two forms of semantics by considering categories of models in both cases and then shows that constructions given by Lomuscio and Ryan, extend to give an  ...  The paper has also beneÿtted from the constructive and pertinent comments of two referees for TCS.  ... 
doi:10.1016/s0304-3975(04)00267-1 fatcat:jsfbwzm3lzdzpp5qtdbaizrwku