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Decomposing Permutation Automata

Ismaël Jecker, Nicolas Mazzocchi, Petra Wolf, Serge Haddad, Daniele Varacca
2021
In this work, we focus on 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.  ... 
doi:10.4230/lipics.concur.2021.18 fatcat:uadcdbsgjrc6tkv3jej2akf76u

A structure theory of automata characterized by groups

Yoshifumi Masunaga, Shoichi Noguchi, Juro Oizumi
1973 Journal of computer and system sciences (Print)  
A characterization theorem for the three classes of 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.  ... 
doi:10.1016/s0022-0000(73)80011-x fatcat:7thx6ukqpzd63edxnwzu4plria

On permutation-reset automata

Elmar Dilger
1976 Information and Control  
A series decomposition of 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  ... 
doi:10.1016/s0019-9958(76)90466-6 fatcat:uzznllh4djgejpuu3adfcfaji4

Practical decomposition of automata

A. Nozaki
1978 Information and Control  
An algebraic criterion is also given for a reduced, strongly connected 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.  ... 
doi:10.1016/s0019-9958(78)90320-0 fatcat:zbicjmjdgreqzket36ivehchkq

Yet another proof of the cascade decomposition theorem for finite automata: Correction

Paul Zeiger
1968 Mathematical Systems Theory  
to a cascade of 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.  ... 
doi:10.1007/bf01703267 fatcat:7swaj6y44baljnwjja5spnrclm

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  ... 

Invariance for ordinary differential equations: Correction

James A. Yorke
1968 Mathematical Systems Theory  
to a cascade of 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.  ... 
doi:10.1007/bf01703268 fatcat:2dou3bbcfvbdxeho7x3vqpmwsu

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.  ... 

A decomposition theorem for probabilistic transition systems

Oded Maler
1995 Theoretical Computer Science  
In this note we prove that every finite Markov chain can be 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.  ... 
doi:10.1016/0304-3975(95)00004-g fatcat:vrbavctx5jaw3akgseamto2ezm

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  ... 

On the Krohn-Rhodes Cascaded Decomposition Theorem [chapter]

Oded Maler
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 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.  ... 
doi:10.1007/978-3-642-13754-9_12 fatcat:ei4c25zbcnhmveg4liv7wdn5ye

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.  ... 

Induced permutation automata and coverings of strongly connected automata

Kenji Uemura, Takeo Yaku, Kimio Sugita
1999 Discrete Applied Mathematics  
Using these two 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.  ... 
doi:10.1016/s0166-218x(98)00150-4 fatcat:rakooocupfbbhpwkuhtnjfl7da

A census of finite automata

Michael A. Harrison
1965 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Clearly, isomorphism of 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.  ... 
doi:10.4153/cjm-1965-010-9 fatcat:6zvbd5qawzgb7p6bm27f25uyra

On the decomposition of generalized semiautomata [article]

Merve Nur Cakir, Karl-Heinz Zimmermann
2020 arXiv   pre-print
Semi-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.  ... 
arXiv:2004.08805v1 fatcat:ix25nctgqbfwzjlqmhv3zvymd4
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