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Decomposing Permutation Automata

2021

In this work, we focus on

doi:10.4230/lipics.concur.2021.18
fatcat:uadcdbsgjrc6tkv3jej2akf76u
*permutation*DFAs, i.e., those for which the transition monoid is a group. ... We also consider the variant of the problem that asks whether a DFA is k-factor composite, that is,*decomposable*into k smaller DFAs, for some given integer k ∈ ℕ. ... C O N C U R 2 0 2 1 18:10*Decomposing**Permutation**Automata*▶ Lemma 12. ...##
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A structure theory of automata characterized by groups

1973
*
Journal of computer and system sciences (Print)
*

A characterization theorem for the three classes of

doi:10.1016/s0022-0000(73)80011-x
fatcat:7thx6ukqpzd63edxnwzu4plria
*automata*, a condition for direct product*decomposability*of a strongly connected*permutation*automaton, and some other related results are proposed in ... The structure of a strongly connected*permutation*automaton, a quasiperfect automaton, and a perfect automaton are discussed algebraically using group theory. ... A strongly connected*permutation*automaton is*decomposed*into a direct product of two*automata*if the associated input group is*decomposed*into a direct product of two groups. Proof. ...##
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On permutation-reset automata

1976
*
Information and Control
*

A series decomposition of

doi:10.1016/s0019-9958(76)90466-6
fatcat:uzznllh4djgejpuu3adfcfaji4
*permutation*-reset*automata*is set up. ... It is shown that a realization of this decomposition contains fewer components than in the usual Krohn-Rhodes decomposition of*permutation*-reset*automata*. ... INTRODUCTION In the usual decomposition theories of*automata*,*permutation*-reset*automata*are not*decomposed*directly but are first covered by grouplike and identity-reset*automata*(Ginzburg, 1968; Arbib ...##
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Practical decomposition of automata

1978
*
Information and Control
*

An algebraic criterion is also given for a reduced, strongly connected

doi:10.1016/s0019-9958(78)90320-0
fatcat:zbicjmjdgreqzket36ivehchkq
*permutation*automaton to have a perfect decomposition. ... It will be easily seen that the statesfA(s, ~) andfB(6(s), ~) are equivalent. (2) We say that A is*decomposable*into r*automata*, iff it is simulated by a product M = (Q1 x "' x Q~, x, Y, fM,gM) of r*automata*... If an automaton is not reduced, then it is practically*decomposable*(i.e., it has a practical decomposition.) LEMMA 2. If a reduced automaton is*decomposable*into r*automata*, COROLLARY. ...##
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Yet another proof of the cascade decomposition theorem for finite automata: Correction

1968
*
Mathematical Systems Theory
*

to a cascade of

doi:10.1007/bf01703267
fatcat:7swaj6y44baljnwjja5spnrclm
*permutation*-reset*automata*, Method IIA finishes the job. ... This method (call it IIA) then suffices to*decompose*a permutationreset automaton into a*permutation*automaton followed by a reset automaton; since the methods as originally stated bring an arbitrary automaton ... to a cascade of*permutation*-reset*automata*, Method IIA finishes the job. ...##
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Page 1275 of Mathematical Reviews Vol. 49, Issue 3
[page]

1975
*
Mathematical Reviews
*

Thus strongly connected

*permutation**automata*, quasiperfect*automata*, and perfect*automata*belong to the class of*automata*whose structures are group-characterized. ... The necessary and suffi- cient conditions for a strongly connected*permutation*automaton to be*decomposable*into a direct product is clearly stated using its input ‘group’, from which the necessary and ...##
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Invariance for ordinary differential equations: Correction

1968
*
Mathematical Systems Theory
*

to a cascade of

doi:10.1007/bf01703268
fatcat:2dou3bbcfvbdxeho7x3vqpmwsu
*permutation*-reset*automata*, Method IIA finishes the job. ... This method (call it IIA) then suffices to*decompose*a permutationreset automaton into a*permutation*automaton followed by a reset automaton; since the methods as originally stated bring an arbitrary automaton ... to a cascade of*permutation*-reset*automata*, Method IIA finishes the job. ...##
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Page 1662 of Mathematical Reviews Vol. 55, Issue 5
[page]

1978
*
Mathematical Reviews
*

.@) is a

*permutation*auto- maton if and only if for all s, s’ e S and ce X, &(s, c)=M(s', c) implies that s=s’. The following assertion is valid, in particular, for*permutation**automata*. ... on APN A; if the PA B is arbitrary*decomposable*on the APN A and there are imposed certain restrictions on the interconnections in A, then B is said to be*decomposable*into APN A. ...##
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A decomposition theorem for probabilistic transition systems

1995
*
Theoretical Computer Science
*

In this note we prove that every finite Markov chain can be

doi:10.1016/0304-3975(95)00004-g
fatcat:vrbavctx5jaw3akgseamto2ezm
*decomposed*into a cascade product of a Bernoulli process and several simple*permutation*-reset deterministic*automata*. ... By doing so we give a positive answer to an open question stated in (Paz, 1971) concerning the*decomposability*of probabilistic systems. ... The*permutation*groups of the components divide the subgroup of the transformation semigroup of d (which implies that counter-free*automata*can be*decomposed*into a cascade of reset*automata*). 2. ...##
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Page 131 of Automation and Remote Control Vol. 36, Issue 1
[page]

1975
*
Automation and Remote Control
*

+le; =1, The least common multiple is denoted by (i, j), while the largest common divisor of i and j is denoted by <i, j>,
Proof, The isomorphism of the

*automata*is induced by the group of*permutations*... @ G x H of the set S x X and the group of*permutations*§G x K of the set S x Y, since the class of all*automata*A <S, X, Y, 6, A> is a class of mappings SS**XYS** == (SX Y) SXxz Let there exist a substitution ...##
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On the Krohn-Rhodes Cascaded Decomposition Theorem
[chapter]

2010
*
Lecture Notes in Computer Science
*

The Krohn-Rhodes theorem states that any deterministic automaton is a homomorphic image of a cascade of very simple

doi:10.1007/978-3-642-13754-9_12
fatcat:ei4c25zbcnhmveg4liv7wdn5ye
*automata*which realize either resets or*permutations*. ... Moreover, if the automaton is counter-free, only reset*automata*are needed. In this paper we give a very constructive proof of a variant of this theorem due to Eilenberg. ... Every automaton A can be*decomposed*into a cascade of*permutation*-reset*automata*, satisfying the conditions of Theorem 3, whose size is at most exponential in the size of A. ...##
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Page 4560 of Mathematical Reviews Vol. , Issue 92h
[page]

1992
*
Mathematical Reviews
*

Summary: “The algebraic properties of an extended function of

*permutable*finite-state deterministic*automata*(FDA) are investi- gated in this article. ... In the present paper a stochastic Petri net is analysed by*decomposing*the transition rate matrix. ...##
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Induced permutation automata and coverings of strongly connected automata

1999
*
Discrete Applied Mathematics
*

Using these two

doi:10.1016/s0166-218x(98)00150-4
fatcat:rakooocupfbbhpwkuhtnjfl7da
*automata*, we construct a cascade product covering of A, if e%(A)c is not {id} (identity*permutation*group). ... First, we introduce the induced*permutation*automaton whose characteristic semigroup is isomorphic to e%(A)e, and the generalized factor automaton. ... Some structures of*permutation**automata*are simple. but those of elemental admissible subset systems are not so simple. ...##
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A census of finite automata

1965
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

Clearly, isomorphism of

doi:10.4153/cjm-1965-010-9
fatcat:6zvbd5qawzgb7p6bm27f25uyra
*automata*is an equivalence relation and hence*decomposes*the family of all machines into equivalence classes. ... TABLE I THE I NUMBER OF NON-ISOMORPHIC BINARY*AUTOMATA*n &n,2,l Cn.2,1 1 1 1 2 10 9 3 129 119 TABLE II THE II NUMBER OF NON-ISOMORPHIC BINARY*AUTOMATA*UNDER INPUT*PERMUTATIONS*n #n, ... Another unsolved problem is the determination of the number of minimal*automata*with n states. ...##
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On the decomposition of generalized semiautomata
[article]

2020
*
arXiv
*
pre-print

Semi-

arXiv:2004.08805v1
fatcat:ix25nctgqbfwzjlqmhv3zvymd4
*automata*are abstractions of electronic devices that are deterministic finite-state machines having inputs but no outputs. ... It is well-known that each stochastic semiautomaton can be*decomposed*into a sequential product of a dependent source and a deterministic semiautomaton making partly use of the celebrated theorem of Birkhoff-von ... In particular, each*permutation*matrix is deterministic. ...
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