Search results for "coalgebra"
showing 5 items of 5 documents
The dual equivalence of equations and coequations for automata
2015
The transition structure α : X ? X A of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. We use this algebra-coalgebra duality as a common perspective for the study of equations and coequations. For every automaton ( X , α ) , we define two new automata: free ( X , α ) and cofree ( X , α ) representing, respectively, the greatest set of equations and the smallest set of coequations satisfied by ( X , α ) . Both constructions are shown to be functorial. Our main result is that the restrictions of free and cofree to, respectively, preformations of languages and to quotients A * / C of A * with respect to a congr…
Varieties and Covarieties of Languages (Extended Abstract)
2013
AbstractBecause of the isomorphism (X×A)→X≅X→(A→X), the transition structure of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. This algebra-coalgebra duality goes back to Arbib and Manes, who formulated it as a duality between reachability and observability, and is ultimately based on Kalmanʼs duality in systems theory between controllability and observability. Recently, it was used to give a new proof of Brzozowskiʼs minimization algorithm for deterministic automata. Here we will use the algebra-coalgebra duality of automata as a common perspective for the study of both varieties and covarieties, which are …
Algebra Structures on Hom(C,L)
1999
info:eu-repo/semantics/published
The Bianchi variety
2010
The totality Lie(V) of all Lie algebra structures on a vector space V over a field F is an algebraic variety over F on which the group GL(V) acts naturally. We give an explicit description of Lie(V) for dim V=3 which is based on the notion of compatibility of Lie algebra structures.
A description based on languages of the final non-deterministic automaton
2014
The study of the behaviour of non-deterministic automata has traditionally focused on the languages which can be associated to the different states. Under this interpretation, the different branches that can be taken at every step are ignored. However, we can also take into account the different decisions which can be made at every state, that is, the branches that can be taken, and these decisions might change the possible future behaviour. In this case, the behaviour of the automata can be described with the help of the concept of bisimilarity. This is the kind of description that is usually obtained when the automata are regarded as labelled transition systems or coalgebras. Contrarily t…