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 pro… Show more
“…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.…”
Section: Similar Dfasmentioning
confidence: 94%
“…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.…”
Section: ⊓ ⊔mentioning
confidence: 99%
“…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). )…”
Section: Monoids Groups and Actionsmentioning
confidence: 99%
“…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.…”
A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. We prove that a permutation DFA is uniformly minimal if and only if its transition monoid is a primitive group. We use this to study boolean operations on group languages, which are recognized by direct products of permutation DFAs. A direct product cannot be uniformly minimal, except in the trivial case where one of the DFAs in the product is a one-state DFA. However, nontrivial direct products can satisfy a weaker condition we call uniform boolean minimality, where only final state sets used to recognize boolean operations are considered. We give sufficient conditions for a direct product of two DFAs to be uniformly boolean minimal, which in turn gives sufficient conditions for pairs of group languages to have maximal state complexity under all binary boolean operations ("maximal boolean complexity"). In the case of permutation DFAs with one final state, we give necessary and sufficient conditions for pairs of group languages to have maximal boolean complexity. Our results demonstrate a connection between primitive groups and automata with strong minimality properties.
“…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.…”
Section: Similar Dfasmentioning
confidence: 94%
“…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.…”
Section: ⊓ ⊔mentioning
confidence: 99%
“…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). )…”
Section: Monoids Groups and Actionsmentioning
confidence: 99%
“…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.…”
A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. We prove that a permutation DFA is uniformly minimal if and only if its transition monoid is a primitive group. We use this to study boolean operations on group languages, which are recognized by direct products of permutation DFAs. A direct product cannot be uniformly minimal, except in the trivial case where one of the DFAs in the product is a one-state DFA. However, nontrivial direct products can satisfy a weaker condition we call uniform boolean minimality, where only final state sets used to recognize boolean operations are considered. We give sufficient conditions for a direct product of two DFAs to be uniformly boolean minimal, which in turn gives sufficient conditions for pairs of group languages to have maximal state complexity under all binary boolean operations ("maximal boolean complexity"). In the case of permutation DFAs with one final state, we give necessary and sufficient conditions for pairs of group languages to have maximal boolean complexity. Our results demonstrate a connection between primitive groups and automata with strong minimality properties.
We show that if a semisimple synchronizing automaton with n states has a minimal reachable non-unary subset of cardinality r ≥ 2, then there is a reset word of length at most (n − 1)D(2, r, n), where D(2, r, n) is the 2-packing number for families of r-subsets of [1, n].
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