Coalgebraic constructions of canonical nondeterministic automata
For each regular language L we describe a family of canonical nondeterministic acceptors (nfas). Their construction follows a uniform recipe: build the minimal dfa for L  in a locally finite variety V V , and apply an equivalence between the category of finite V V -algebras and a suitable category of finite structured sets and relations. By instantiating this to different varieties, we recover three well-studied canonical nfas: V V = boolean algebras yields the tomaton of Brzozowski and Tamm, V V = semilattices yields the jiromaton of Denis, Lemay and Terlutte, and V=Z2 V = Z 2 -vector spaces yields the minimal xor automaton of Vuillemin and Gama. Moreover, we obtain a new canonical nfa called the distromaton by taking V V = distributive lattices. Each of these nfas is shown to be minimal relative to a suitable measure, and we derive sufficient conditions for their state-minimality. Our approach is coalgebraic, exhibiting additional structure and universal properties of the canonical nfas.
coalgebras minimization non-deterministic automata join-semilattices