Lifting Adjunctions to Coalgebras to (Re)Discover Automata Constructions

  • Henning Kerstan
  • Barbara König
  • Bram Westerbaan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8446)


It is a well-known fact that a nondeterministic automaton can be transformed into an equivalent deterministic automaton via the powerset construction. From a categorical perspective this construction is the right adjoint to the inclusion functor from the category of deterministic automata to the category of nondeterministic automata. This is in fact an adjunction between two categories of coalgebras: deterministic automata are coalgebras over \(\mathbf {Set}\) and nondeterministic automata are coalgebras over \(\mathbf {Rel}\). We will argue that this adjunction between coalgebras originates from a canonical adjunction between \(\mathbf {Set}\) and \(\mathbf {Rel}\).

In this paper we describe how, in a quite generic setting, an adjunction can be lifted to coalgebras, and we compare some sufficient conditions. Then we illustrate this technique in length: we recover several constructions on automata as liftings of basic adjunctions including determinization of nondeterministic and join automata, codeterminization, and the dualization of linear weighted automata. Finally, we show how to use the lifted adjunction to check behavioral equivalence.



We would like to thank Marcello Bonsangue, Alexandra Silva and Filippo Bonchi for raising and discussing the problem with us. Furthermore we would like to acknowledge Ana Sokolova for interesting discussions on this topic.


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Copyright information

© IFIP International Federation for Information Processing 2014

Authors and Affiliations

  • Henning Kerstan
    • 1
  • Barbara König
    • 1
  • Bram Westerbaan
    • 2
  1. 1.Universität Duisburg-EssenEssenGermany
  2. 2.Radboud Universiteit NijmegenNijmegenThe Netherlands

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