Algebra-coalgebra duality in brzozowski's minimization algorithm

Abstract

We give a new presentation of Brzozowski's algorithm to minimize finite automata using elementary facts from universal algebra and coalgebra and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal language equivalent automata from Moore nondeterministic and weighted automata.

Regular Varieties of Automata and Coequations

“…The results of the present paper are an example of that. (Other examples can be found, for instance, in [4,6]). …”

Coalgebraic trace semantics via forgetful logics

“…It turns out that even in the simple case of non-deterministic automata that procedure is not the classical powerset construction; instead, it relies on a double application of contravariant powerset construction. Interestingly, this is what also happens in Brzozowski's algorithm for automata minimization [4], so as a by-product, we get a new perspective on that algorithm which has recently attracted much attention in the coalgebraic community [1,2,3].…”

“…It turns out that even in the simple case of non-deterministic automata that procedure is not the classical powerset construction; instead, it relies on a double application of contravariant powerset construction. Interestingly, this is what also happens in Brzozowski's algorithm for automata minimization [4], so as a by-product, we get a new perspective on that algorithm which has recently attracted much attention in the coalgebraic community [1,2,3].Although we do not assume the branching functor T to be a monad, a forgetful logic for T is equivalent to a transformation from T to a certain monad which, in the case of sets, is the double contravariant powerset monad (a special case of the continuation monad). One might say that the continuation monad is rich enough to handle all types of branching that can be "forgotten" within our framework.This paper is an extended version of [21], adding full proofs and a treatment of the trace semantics of probabilistic systems as a non-trivial instance of the framework.Acknowledgments We thank Marcello Bonsangue, Helle Hvid Hansen, Ichiro Hasuo, Bart Jacobs and Jan Rutten for discussions.…”

Singular value automata and approximate minimization

“…Minimization of automata has been a major subject since the 1950s, starting with the now classical work of the pioneers of automata theory. Recently there has been activity on novel algorithms for minimization based on duality (Bezhanishvili et al, 2012;Bonchi et al, 2014) which are ultimately based on a remarkable algorithm due to Brzozowski from the 1960s (Brzozowski, 1962). The general co-algebraic framework permits one to generalize Brzozowski's algorithm to other classes of automata like weighted automata.…”