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Some results on the relation between automata and their automorphism groups

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Summary

In this paper the structure of automata is investigated using the concept of the automorphism group. The investigations about strongly connected automata are extended to cyclic (Oehmke) and normal automata. The set of states is divided into equivalence classes of strongly connected subsets (SCEC). In the set of all SCEC we explain a partial ordering whose minimal elements are called sourceclasses. If there is only one source-classe, the automaton is called cyclic. If each automorphism maps every SCEC onto itself, then the automaton is said to be normal. We generalize some results ofA. Fleck [1]. In some cases we restrict ourselves to Abelian automata.

Zusammenfassung

In dieser Arbeit wird die algebraische Struktur von abstrakten Automaten untersucht. Als Hilfsmittel dazu dient die Automorphismengruppe von Automaten, das ist die Gruppe aller Zustandspermutationen, bei welchen die Übergangsfunktion erhalten bleibt.

Die Zustandsmenge eines Automaten wird in Äquivalenzklassen von “eng verbundenen” (strongly connected) Teilmengen zerlegt. In der Menge dieser Äquivalenzklassen erklären wir eine Teilordnung, deren minimale Elemente “Quellklassen” (source-classes) genannt werden. Wenn nur eine Quellklasse existiert, dann heißt der Automat zyklisch (Oehmke [3]); wenn jeder Automorphismus alle eng verbundenen Äquivalenzklassen auf sich selbst abbildet, dann nennen wir den Automaten normal. In den bisherigen Arbeiten wurden fast ausschließlich nur eng verbundene Automaten behandelt. Hier hingegen erstrecken sich die Untersuchungen auf zyklische und normale Automaten. Dabei werden auch einige Resultate vonA. Fleck [1] über eng verbundene Automaten verallgemeinert. Gelegentlich beschränken wir uns auf abelsche Automaten.

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Abbreviations

A :

automaton

S :

set of states ofA

I :

semigroup of input strings

M :

transition function

h :

automorphism ofA

G (A) :

automorphism group ofA

s∼t :

the two statess andt are strongly connected

\(\bar s\) :

strongly connected equivalence class (SCEC) ofS withs as representative

≃:

homomorphism

≊:

isomorphism

N :

normal subgroup of all normal permutations inG (A)

G/N :

factor group ofG with respect toN

\(\bar S\) :

set of all\(\bar s\)

\(\bar h\) :

permutation on\(\bar S\) defined by\(\bar h{\mathbf{ }}(\bar s){\mathbf{ }} = {\mathbf{ }}\overline {h{\mathbf{ }}(s)} ,{\mathbf{ }}s{\mathbf{ }} \in {\mathbf{ }}\bar s\)

\(\bar G\) :

group of all\(\bar h\)

≤:

partial ordering in\(\bar S\)

\(\overline{\overline s} \) :

source-class

\(\overline T \) :

basis set

E :

source-class of a cyclic automaton

|S|:

cardinality of the setS

\(\overline{\overline S} \) :

set of all source-classes

I T :

set of all strings inI which transform an arbitrary state inT in a state inT (T⊂S)

э:

notation for an identity

Ф:

homomorphism fromI E→G (A) defined by Φ(x)=h iffh (s)=M (s, x), x∈I E

x ϱy :

congruence relation inI E induced by Φ

\(\hat I_E \) :

factor-semigroup modulo ϱ

\(\hat x\) :

congruence class modulo ϱ

x ϱy :

congruence relation inI defined byM (s, x)M (s, y) for any sourcesE (for cyclic automata)

\(\tilde I\) :

set of equivalence classes mod σ inI

\(\tilde x\) :

congruence class modulo σ

\(\tilde A{\mathbf{ }} = {\mathbf{ }}\tilde I,{\mathbf{ }}I,{\mathbf{ }}\tilde M\) :

automaton, where\(\tilde M\) is defined by\(\tilde M{\mathbf{ }}(\tilde x;{\mathbf{ }}y){\mathbf{ }} = {\mathbf{ }}\widetilde{xy}\)

\(\overline A {\mathbf{ }} = {\mathbf{ }}(\overline S ,{\mathbf{ }}I,{\mathbf{ }}\overline M )\) :

automaton, where\(\overline M \) is defined by\(\overline M {\mathbf{ }}(\bar s,{\mathbf{ }}x){\mathbf{ }} = {\mathbf{ }}\overline {M{\mathbf{ }}(s,x)} \) (we assume thatA=(S, I, M) is Abelian)

s τt :

equivalence relation inS defined byh (s)=h(t), whereh is a homomorphism fromA=(S, I, M) andB=(T, I, N)

s * :

equivalence class modulo τ

S * :

set of all equivalence classess *

A *=(S *, I, M*):

quotient automaton, whereM * is defined byM * (s*, x)=M (s, x)*

ϰ:

LetA be cyclic andsεE. Letf be the homomorphism fromA ontoà given by\(f{\mathbf{ }}(M(s,{\mathbf{ }}x)){\mathbf{ }} = {\mathbf{ }}\tilde x.\). Now ϰ is defined byM (s, x) ϰM (s, y) ifff (M (s, x))=f (M (s, y))

References

  1. Fleck, A. C.: Isomorphism Groups of Automata. J. ACM, Vol. 9, Oct. 1962, No. 4, pp. 469–476.

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  3. Oehmke, R. H.: On the Structures of an Automaton and its Input Semigroup. J. ACM, Vol. 10, Oct. 1963, No. 4, pp. 521–525.

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  4. Clifford-Preston: The Algebraic Theory of Semigroups. Math. Surveys, Vol. 1, No. 7, Am. Math. Soc., 1961.

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Feichtinger, G. Some results on the relation between automata and their automorphism groups. Computing 1, 327–340 (1966). https://doi.org/10.1007/BF02345486

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