Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Abstract

Abstract. We present a ring theoretic approach toČerný's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those inČerný's series. Semisimplicity gives also the advantage of "factorizing" the problem of finding a synchronizing word into the sub-problems of finding "short" words that are zeros into the projection of the simple components in the WedderburnArtin decomposition. In the general case this last problem is related to the search of radical words of length at most (n − 1) 2 where n is the number of states of the automaton. We show that the solution of this "Radical Conjecture" would give an upper bound 2(n−1) 2 for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases andČerný's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

Primitivity, Uniform Minimality, and State Complexity of Boolean Operations

“…Then we computed the setwise stabilizer R 0,1 of {0, 1} × F 8 (the rows indexed by 0 and 1), and the setwise stabilizer C 0,1 of F 8 × {0, 1} (the columns indexed by 0 and 1). Next, we computed R 0,1 π ′ ≤ G ′ and C 0,1 π ≤ G. These groups turned out to both be equal to T , the subgroup of translations in AGL (1,8). We saw earlier than T is imprimitive.…”

“…Primitive groups have seen increasing application in automata theory over the past decade, particularly in connection with the classical synchronization problem for DFAs; for a survey of such work see [2]. The connection between simple DFAs and primitive groups was recently noted by Almeida and Rodaro [1]. However, primitive groups are not mentioned in Restivo and Vaglica's work on uniformly minimal DFAs, nor in any other work on DFA minimality that we are aware of.…”

“…Cyclic groups of composite order give examples of transitive imprimitive groups, while cyclic groups of prime order p ≥ 5 give examples of primitive, non-2-transitive groups. (For example, the group (1, 2, 3, 4, 5) ≤ S 5 is not 2-transitive on {1, 2, 3, 4, 5} since nothing maps the pair (1, 2) to the pair (1,3). )…”

“…Let A ′ have states {1, 2} and transformations a ′ = () and b ′ = (1, 2). Notice that in the transition group of A×A ′ , we have (b, b ′ ) = ((), (1,2)) and (ε, ε ′ ) = ((), ()). Thus (b, b ′ )π = (ε, ε ′ )π = () and it follows that π is not injective.…”

“…gap> IsomorphismGroups(SymmetricGroup(5),PrimitiveGroup(6,2)); [ (3,4), (1,2,3)(4,5) ] -> [ (1,2)(3,4) (5,6), (1,2,3,5,4,6) ]…”