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MATHEMATICS 
 
 Report No. 200 
 
 COO-1018-1073 
 
 ON THE AUTOMORPHISM GROUP OF A REDUCED AUTOMATON 
 
 by 
 Manfred Paul 
 
 1 
 
 
 February 22, 1966 
 
COO -1018- 1073 
 
 Report No. 200 
 
 ON THE AUTOMORPHISM GROUP OF A REDUCED AUTOMATON 
 
 by 
 Manfred Paul 
 
 February 22, 1966 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 61803 
 
 Supported in part jointly by the Atomic Energy Commission and the Advanced 
 Research Projects Agency (ARPA) under AEC Contract AT(ll-l)-10l8 . 
 
In this report we shall investigate the automorphism group G(a/h) 
 of the reduced automaton A/H where A = (S, I, M) is a finite strongly 
 connected automaton and H is a subgroup of the automorphism group G(A) of 
 the automaton A- This problem and other related topics have been dealt 
 with recently by G. P. WEEG, A. C. FLECK, and B. BARNES [1, 2, 3, k, 5], 
 However, the particular problem to give an isomorphic representation of 
 G(a/h) for arbitrary A and H still remained open. Our present purpose is to 
 fill this gap. 
 
 For abbreviation we shall frequently use the following denotations: 
 
 ft S => S 
 
 H < G 
 H <J G 
 fog 
 
 a 
 N 
 
 for 
 for 
 for 
 for 
 
 for 
 for 
 
 a is a unique mapping of S onto S, 
 
 H is a subgroup of or equal to G, 
 
 H is a normal subgroup of or equal to G, 
 
 the function formed by composition of f and g, 
 
 the neutral element of I, 
 
 the order of the group G. 
 
 Furthermore, we shall deal only with finite strongly connected automata 
 A = (S, I, M), i.e. the set S of states of A is not empty and finite, the set I 
 of inputs is a free semigroup over some finite alphabet, and the machine 
 mapping M: S x I -> S has the properties: 
 
 (Vs e S) (y x , yel) M(s, xy) = M(m(s, x), y) [compatibility of M with I], 
 (Vs, teS) (3xel) M(s, x) = t [strong connectedness of A]. 
 
 A mapping g: S =>Sis called an automorphism of A, iff (\/seS) (^xel) g(M(s, x)) = 
 - M(g(s), x). We shall sketch the properties of such an A as far as we shall 
 need them later j 
 
 i) (ysGS) M(s, g) =-s\ 
 ii) G(.A) := (g/g is an automorphism of A) forms a group under 
 composition and /g(a)/ divides /S/. 
 iii) An automorphism of A is completely defined, if its value 
 
 is known for one arbitrary argument seS, i.e. (^g, h£G(A)) 
 ( (C3seS) g (s) = h(s))>((^ses) g(s) = h(s)) ). 
 iv) For an arbitrary subgroup H of G(a) the following reduced 
 automaton A/H can be defined: 
 
 -1- 
 
A/H := (S, I, M) with S being the set of transitivity classes 
 
 in S under H, i.e. 
 
 S := [!/ seS« s = (t/(3heH) t = h(s)}} and M being defined by 
 
 (VseS) (Vxel) M(s", x) := M~(s, x) . This definition of A/H 
 
 is consistent that means it is independent of the choice of 
 
 class representatives. 
 
 v) Be H an arbitrary subgroup of G(a) . Because of iv) we can 
 
 consider the automorphism group G(A/H) of A/H. Furthermore, 
 
 between G(a) and H there is a uniquely determined maximum 
 
 group K which has H as a normal subgroup, i.e. we can 
 GH 
 
 uniquely define K := max {Y/H<J Y < G(A)}. It has been 
 
 shown by A. C FLECK [k] for a special case and by R. BAYER 
 
 [6] in general that the factor group K /H is isomorphic 
 
 GH 
 
 to a subgroup of G(A/H) . 
 
 In this last paragraph v) we made reference to a group K which 
 
 GH 
 
 was defined purely with the help of the subgroup lattice of G(A) for an 
 
 arbitrary pair of groups G, H with H < G. Since v) also suggests a 
 
 generalization of K in order to find an isomorphic representation of 
 
 GH 
 
 G(A/H) we shall establish a characterization of K in terms of the automat 
 involved by means of the following 
 
 on 
 
 THEOREM 1: Let A = (S, I, M) be a finite strongly connected automaton, 
 
 H be a subgroup of G(a), and K be the max {Y/H^d Y < G(a)}. 
 
 GH 
 
 Then for a mapping cP". S => S the following three propositions 
 are equivalent: 
 
 (a) <J>eK GH . 
 
 (b) (Vh€H) GkeH) (VseS) (^xel) ho^ok (m(s, x)) = M(^(s), x). 
 
 (c) (Vh'eH) (3k'eH) (VseS) (Vxel) k*c^oh'(M(s, x)) = M(f(s), x) 
 
 Proof: First we shall show that (b) implies (c). 
 
 Proposition (b) states that there is a function k which maps H 
 into H such that for all heH, seS, and xelsho^okfh] (m(s, x)) = 
 
 = M(^(s), x) . We used here and shall use in the sequel brackets 
 for arguments of functions the value of which is a function. We 
 shall see that the function k is a one-to-one mapping of H onto H 
 and has, therefore, an inverse k " which also maps H one-to-one 
 onto H. 
 
 -2- 
 
i) Be k and k two automorphisms corresponding to a certain h 
 
 according to (b). Then we have for this particular h 
 
 (^seS) (Vxel) h o f o k (m(s, x)) = h o cf o k p (m(s, x)). 
 
 Now, since <0 : S => S and S is finite, & has an inverse 
 
 (C : S => S and we can, therefore, apply d> "oh" which 
 
 leads to (VseS) (Vxel) k (M(s, x)) = k 2 (M(s, x)). 
 
 This means that k n and k^ are two automorphisms which 
 12 
 
 coincide for at least one argument, since neither S nor I 
 is empty. According to iii) we have, therefore, k = k r 
 
 1 
 
 2 
 
 li 
 
 using this as the normal abbreviation for (t^s) k (s) = k p (s). 
 Be k[h J = k[h p ] for two elements h„ and h p of He Then we 
 have for these particular h and h„ 
 
 (V'seS) (tfxel) h^cfo k[h ] (m(s, x)) = h p o cf o k[h ] (m(s, x)) 
 'This means that h and h p coincide for at least one argument 
 and by the same reasoning as before in i) we find h = h . 
 
 Together, i) and ii) show that k is a one-to-one mapping of H into 
 and, hence onto H, since H is finite. 
 
 Having this we see immediately that (b) implies (c). We 
 have only to take k' - k (h'J for any h'eH in (c). The proof 
 that (c) implies (b) can be omitted. It runs analogously 
 mutatis mutandis. 
 
 Next we shall show that (b) implies (a). 
 
 i) Choosing the identity e as a particular heH and £ as a 
 particular xel we get from (b) 
 
 (VseS) ef o k[e] (s) =£f(s). Applying^" 1 we find (t^seS) k[e] (s) 
 and this means that k[e] = e. This result leads to a special 
 case of (b) for h = e: (VseS) (Vx£l) dj (m(s, x)) = M(f(s), x). 
 Therefore, tf is an automorphism. 
 ii) From (b) we deduct: 
 
 GfaeH)ft(seS) 
 
 f" 1 oh ocj(s) = df^o h o if (M(s, € ) ) = 
 = f" 1 o h o f o k[h] (MCk^O] (s), B )) = 
 = ^' 1 (M(«f(k" 1 [h](s)),€)). 
 
 Since we know already that eP is an automorphism, the last expression 
 
 -3- 
 
 = s 
 
becomes M(k [h](s),£) = k [h](s). Therefore we have: 
 
 (Vheli) (^seS) df o h o d{ (s) = k~ [h](s), which means that 
 
 (VheH) <p o h o <f e H. From the theory on the subgroup 
 
 lattice of a given group we know that K = (y/yeG(A); 
 -, uH 
 
 (/heH) y "ohoyeH} and, since i) and ii) just showed 
 
 that <8 meets both conditions for a y to be in K , we find 
 ' GH 
 
 rfeK . Therefore, (b) implies (a). Finally we shall show 
 
 * GH 
 
 that (a) implies (b). 
 
 Since <PeK and H<J K , we have 
 
 ' GH GH , -, 
 
 (^heH; QkGH) (VseS) (*xel) f " o h" o ^ (m(s, x)) = k(M(s, x)). 
 
 Applying h o <0 we get 
 
 (VheH) (^keH) 0/seS) (/xel) h o <£ o k (m(s, x)) = <£ (m(s, x)). 
 
 Now, since tfeK < G(A), we also have 
 * GH 
 
 (VseS) (^xel)^(M(s, x)) = M(«p(s), x) and, therefore, 
 (VheH) (3keH) (/seS) (Kxel) h o <J> o k(M(s, x)) = M(f(s), x) . 
 This concludes the proof of theorem 1. 
 
 As we shall see later the generalization of K which we are looking for will 
 
 GH 
 
 be to allow in proposition (b) of theorem 1 the function k: H => H to 
 depend on seS and xel, i.e. we then will have for any seS and xel a function 
 H => Ho On the way towards our aim we shall need the following 
 
 Let A = (S, I, M) be a finite strongly connected automaton and 
 H be a subgroup of G(a) . Then for a mapping tf\ S => S the 
 following two propositions are equivalent: 
 
 (d) (tfieH) (\/seS) (Vxel) QkeH) h o^o k (m(s, x)) = M(f(s), x). 
 
 (e) (tfti'eH) (^seS) (/xel)' (3k' eH) k< o«^o h' (m(s, x)) = M(f(s), x) 
 
 The proof can be omitted since it is essentially analogous to the equivalence 
 proof for propositions (b) and (c) in theorem 1. Only now we have throughout 
 
 sx 
 
 LEMMA: 
 
 the proof to consider the function k 
 
 sx 
 
 H => H for a given pair seS and xel 
 
 instead of the function k: H => H which was independent of s and x. 
 
 The similarity of propositions (d) and (e) to propositions (b) 
 and ^c) suggests and our main result later will justify the following 
 
 DEFINITION 1: Let A = (S, I, M) be a finite strongly connected automaton 
 and H be a subgroup of G(A) . Then a mapping <0: S => S is 
 
 •if- 
 
called compatible with H in A , iff 
 
 (YheH) (^seS) (Vxel) (ikeH) h o c% o k(M(s, x)) - M(«f(s), x) . 
 
 This definition together with the lemma gives us immediately the following 
 
 COROLLARY: Let A = (S, I, M) be a finite strongly connected automaton 
 and H be a subgroup of G(A) . Then a mapping <0 : S => S is 
 compatible with H in A, iff 
 (Vh'eH) (^seS) (Vxel) (3k'eH) k' o <£ o h f (m(s, x)) = M(«p(s), x) 
 
 The mappings which are compatible with a subgroup of G(a) in A have an 
 important property which we shall state in the following 
 
 THEOREM 2: Let A = (S, I, M) be a finite strongly connected automaton, 
 H be a subgroup of G(A), and <J> be the set of all mappings 
 
 An 
 
 df which are compatible with H in A. 
 
 Then $ ATJ forms a group with composition as its group operation. 
 An 
 
 Proof: Since $ obviously contains K (compare definition 1 and 
 An On 
 
 theorem l), the identity e is an element of <l> ° Furthermore, 
 the function composition is an associative operation. Therefore, 
 
 we can confine the proof to showing that c|> is closed under 
 
 AH 
 
 composition and inversion. 
 
 H 
 
 m 
 
 ■ ■ < . ; *» 
 
 ■ 
 
 v. k .tv 
 
 i) Be <f,e$ Arr and CP o €0 AU - Then, using the corollary we have for 
 ' 1 AH * d. AH 
 
 ^••(yh'eH) (tfseS) flfrel) fik^H) ^ o <f ± o h ! (m(s, x)) = Mty^s), x) 
 
 From the corollary in a slightly modified Aversion we find for 
 
 -1 
 
 •H) 
 
 fseS) (Vxel) (3k ; eH) k' of 2 o k (M^'s), x)) 
 
 f 2 and f ± l (tfk^ 
 
 = M(f 2 0^(5), x). 
 
 Stringed together these propositions yield 
 (tfh'eH) -(^seS) (Vxel) (ik^H) (3k *eH) 
 
 k"o# o tt oh 1 (M(s, x)) - k ' oif p ok o k o <P o h' (m(s, x) ) = 
 
 = h'o£ o k^M^s), x)) = M(tf 2 of^s), x) 
 
 and, therefore, we have finally 
 
 (Kh'eH) (Vs e S) (tfxel) (3k ! eH) k'o^ o^ o h'<M(s, x)) = 
 
 = M(f 2 0^(5), x) 
 
 which according to the corollary means that 
 
 •5- 
 
f 2 °fl £ *AH' ' 
 
 ii) Be df £<£„„ • Then., using definition 1 slightly modified we 
 
 have for^heH) faseS) (Vxel) (3keH) h o & o k (m(s, x)) = 
 
 = M(f(s), x). 
 
 Applying k o ^ o h we get 
 
 (Kh€H) (VseS) (*xel) (3keH) 
 
 k o/ 1 o h(M(«f(s), x)) = M(s, x) = M(<f _1 (<y(s)), x). 
 
 -1 
 Hence <f e$ according to the corollary, since <P maps S onto 
 
 S. This concludes the proof of theorem 2. 
 
 DEFINITION 2: Let A = (S, I, M) be a finite strongly connected automaton 
 and H be a subgroup of G(A) . Then the set 
 
 Y 
 
 AH 
 
 := [y/y- s => S; (VseS) (3heH)-y(s) = h(s)} 
 
 Remark: 
 
 is called the extension of H in A . 
 
 The extension of a subgroup of G(a) in A forms a group under 
 composition. 
 
 The proof for this remark is part of the proof for the main result 
 of this report which will now be established in the following 
 
 THEOREM 3- Let A = (S, I, M) be a finite strongly connected automaton, H 
 
 be a subgroup of G(A), <J> be the set of all mappings if which 
 
 are compatible with H in A, and T ATJ be the extension of 
 
 An 
 
 Proof: 
 
 H in A. 
 
 Then 
 
 (1) $. TT is a group under composition which contains ¥ 
 
 AH AH 
 
 as a normal subgroup, and 
 
 (2) the factor group <£ /Y is isomorphic to the automorphism 
 group G(A/H) of the reduced automaton A/H. 
 
 We shall begin with (l). Since we know already that <J> is a 
 
 An 
 
 group under composition we need only to show that ¥ -*S $.„• 
 
 AH AH 
 
 First we shall show that ¥ forms a group under composition. 
 
 AH 
 
 From definition 2 it is obvious that H is contained in ¥ and 
 
 An 
 
 that, therefore, the identity e is an element of ¥ • Accordingly 
 
 AH 
 
 it suffices to show that ¥. is closed under composition and inversion. 
 
 AH 
 
 -6- 
 
i) Be Y e¥ and f2 eY AH' Then We haVe ^ seS ) ( 5h ! eH ) VA S ) = h n ( s ) 
 and 
 frseS) (3h 2 eH)y 2 (y i (s)) - ^(^(s)). 
 
 Together these propositions yield 
 
 (VseS) (3h 1 , h 2 eH)^ 2 o-y/^s) = h 2 o h^s) 
 
 and, since h„ oh eH, we find 
 
 (teeS) (3heH)^ 2 oy^s) - h(s), 
 
 which means that y/„ oltt £ ^au° 
 
 ii) Be 1pe¥ . . Then we have 
 
 -(VseS) (3h eH) ^(^(s)) = h 1 (^ 1 (s)). 
 
 ■1 
 
 Applying h , which is an element h of H, we get immediately 
 
 (VseS) (^heH) y 1 (s) - h(s) 
 and, therefore, yi e^^- 
 
 Next we shall show that ¥ Arr < <I> ° So be^ef . Using an obvious modification 
 
 An An ' ^n 
 
 of definition 2 we have (^heH) (l/seS) (Vxel) (3k eH) ^(h(M(s, x))) = 
 = k (h(M(s, x)))o This proposition can be transformed as follows 
 (VheH) (\/seS) (Vxel) ^k n eH) (Vk eH) 'U/(h(M(s, x))) = k n oho k ~ 1 (M(k (s), x)). 
 
 _ 1 d ' _L d c 
 
 We only inserted k o k„ exploiting that k„ is an automorphism., Now, since 
 
 d d d 
 
 ^eY , we have of course (VseS) (3k p eH) ^(s) - k p (s). Inserting this properly 
 into our previous proposition we find 
 (VheH) (VseS) (Vxel) (3k , k 2 eH) y o h (m(s, x)) = k o h o k 2 ~ 1 (M(y<s), x)). 
 
 Applying k p o h ok , which is an element k of H, we get 
 
 (VheH) (VseS) (Vxel) (3keH) k oy o h(M(s, x)) =-• M(y(s), x)). 
 
 y, 
 
 Therefore, according to the corollary, ye$ . Now we shall prove that . 
 
 ' AH -. AH 
 
 normal in <£.„« So be ye^ and^e$ . Then we have (VseS) a? o y/ o (0 (s) 
 = ^oy(M.(^(s),£)). SinceyeY^, we get 
 (VseS) (ah^H) <f X oyo<f{s) = f 1 o h 1 (M(^(s),5 )). 
 Using the corollary we find 
 (VseS) (3h x eH) (3h p eH) f' 1 oyo<f{s) = h 2 _1 o h g o <f> " " o h ± (yi(f(s) , e )) 
 
 = h "VC? 1 ^))^)) 
 
 'T- 
 
 IS 
 
 ii 
 
 m* 
 
 ■ 
 
and this means that 
 
 (VseS) (3heH) <9~ oyjo(P{s) = h(s). Therefore, cp~ o-y/oj? g¥ and Y is 
 
 indeed a normal subgroup of $,,„■> This concludes the proof for statement (l) 
 of theorem 3- 
 
 The proof for statement (2) comprises quite a few single steps. 
 For the sake of clarity we shall, therefore outline briefly which path we 
 are going to follow. 
 
 i) A mapping f will be defined with f: /¥ -* {f/f: S => S } • 
 
 ii) It will be shown that f indeed maps $ /l into G(A/H). 
 
 iii) The mapping f will be proved to be a one-to-one mapping. 
 
 iv) We shall see that f is a mapping of <S> /¥ onto G(A/H) • 
 
 An AH 
 
 v) It will be shown finally that f is a homomorphism. 
 Together i) up to v) prove statement (2). 
 
 i) 'The mapping f is defined as follows: 
 
 fmay denotate the class of $»„/¥.„ which contains 0e<& . Then 
 An An ' An 
 
 we define f[J] : = fy with (\fseS) Jf_(s) := *(s) . 
 
 For this definition we have of course to prove that it is 
 consistent, i.e. independent of the choice of class representatives, 
 and that each y- is indeed a mapping of S onto S. 
 
 Regarding the consistency we consider for arbitrary 
 
 y e¥ , <?£<§ , heH, and seS the following sequence of propositi 
 
 AH 
 
 ons; 
 
 ffyorf] (h(s)) = yo^oh(s) by definition of f ; 
 (3h eH) yo^oh ( s ) = h otfoh(s) by definition 2; 
 
 (3k gH) h o^oh(s) = h ok ok ctfoh(s) = h ok o(p(s) by the 
 corollary; 
 
 (Vh.,k gh) h ok X o^(s) = ^(s) by definition of S of A/H. 
 
 Hereby we have shown that 
 
 OtyG^) (Vfe$ AH ) VheH) (^sgS) f $Zf] $^s)) = f{7j and, 
 
 therefore, the definition of f is consistent. 
 
 -8- 
 
Next we shall show that 
 (Vfe$ m ) ftyJ: S => S. So be f ty] (1^ = f[f](ip) 
 for arbitrary ^e$. ; s eS, and Sp£S. Then we have by 
 definition of f : <tf(s ) = <^(sp) which means 
 (3heH) tf(s ) = h ocp(s p ). Using definition 1 we get 
 (3h, keH)f ( Sl ) = h o f ( j^-J 
 
 -1 
 
 # 
 
 -1 
 
 <^(k (s 2 )) and, applying 
 
 this means that s = k (s_) for some keH. Therefore, 
 
 '1 " v 2' 
 s = s and f[cp] is indeed for each 4 a one-to-one mapping 
 
 of S into, hence onto S, since S is finite. 
 
 ii) Since we have already shown that (fape$ /¥ ) f [a] : S => S, 
 
 An AH ' 
 
 it suffices to show that f[<f] is a homomorphism of A/H for 
 
 each ^. So be $£<§.„/¥ , seS, and xel. Then we have f [<y] (m(s, x)) 
 
 = f[<?] (M(s, x/) by definition of M of A/H; fty] (M(s, x)) = 
 
 = #(M(s, x))" by definition of f ; 
 
 (3keH) ^"(M(s", xJT = k o <y (M(s, x)) = M(cp(s), x)' by definition 
 of S of A/H and using the corollary for h' = e; 
 M(<f(s), x) = M(^7J, x) = M(f[j](s"), x) by definition of M of 
 A/H and by definition of f. So finally we have found that 
 f[tf] is indeed a homomorphism of A/H for each tf. 
 iii) Be f[^j = f typ] for arbitrary ^ ± ^ AE /^ m and <P 2 g<|) AeAaH° 
 
 Then we have (>'s e S) ff^Ks) - f[<£p] (s) ; (Vs€S) ^(s) = f 2 (s) 
 
 by definition of f; 
 
 (^s e S) (3h e H) <f,(s) = h o <f 2 (s) by definition of S. 
 
 This last proposition means that there is a mapping h* : S -* H 
 
 such that 0/seS) f (s) = h*[ s] (dp ? (s)) . The mapping y* , 
 
 -1 
 defined by ()/seS)"w* (s) := h*[tf p " L (s)](s), is clearly an 
 
 element of Y. TT . With that we can continue from our last 
 AH 
 
 proposition (^seS) tf (s) = 'y* (tfL ( s ) } and this means ^ = <9 . 
 iv) In order to prove this point we take an arbitrary g£G(A/H) 
 and shall define a mapping <P: S => S with the help of g 
 as fellows. Since each class seS contains exactly /H/ 
 elements, we can find an index mapping 
 [1, 2, ..., /S/} X (1, 2, . .., /E/)=> S such that 
 (Vi = 1, 2, 
 
 ., /S/) (VJ - 1, 2, ., /H/) s ±J == Bil . 
 
 Having chosen one such indexing we define (Vi, j)<^(s. .) 
 
 -*- J 
 
 'kj' 
 
iff g(s ) - s i n order to show that <o maps S onto S it 
 suffices to prove that k covers its full range, if i does 
 so. But this is indeed an immediate consequence of g 
 mapping S onto S. Similarly we find from the uniqueness of 
 g that (O is unique, since s 
 
 . = s. . implies k n 
 k^ k 2 J F 1 
 
 V 
 
 Next we shall show that <9 e$ . First we find immediately 
 
 ' AH 
 
 from the definition of (0: (VseS) <o(s) - g(s). Therefore, 
 
 we have 
 
 (VheH) (tfseS) (Vxel) dpVh (M(s, x)) = g(h(M(s, x))) = 
 
 = g(M(s, x)) = g(M(s", x)) = M(g(s~), x) = M(^T, x) = 
 
 = M^s), x). 
 This means that 
 
 (tfheH) (^s e S) (Vxel) (3k e H) k odj o h(M(s, x)) = M(<f(s), x), 
 and according to the corollary tf e< ^Au° 
 
 It remains to "be shown that f [^p] = g, and in fact we have 
 for all i and j»f[<f] (s. .) = (f(s . .) = g(s. .) which means 
 that fty] = g. 
 
 v) Be ^ 6 ^/^ and ^^AH^AH* Then W6 haV6 
 
 (^s £ S) fCf x • ^ 2 ](F) - f[f|°5£] (s) = f^^lsj = f^J^FT) = 
 
 = ft^jCft^jCs)) = f[fj_] o f[f 2 ](s), which means that 
 
 f tyl " ^ = f ffl^° f tf2^ " 
 
 This concludes the proof of theorem 3* 
 
 Considering the results of A. C FLECK and R. BAYER it would be interesting 
 
 to know in which cases K /H is isomorphic to the full automorphism group 
 
 GH 
 
 G(A/H), even if /K / < /S/. We can give an answer to this question through 
 GH 
 
 the following investigation. 
 
 Let us introduce an equivalence relation in $ such that the 
 
 An 
 
 equivalence classes will cover the classes of the factor group <J> /f in a 
 
 An An 
 
 certain manner. 
 
 DEFINITION 3: Let A = (S, I, M) be a finite strongly connected automaton, 
 
 H be a subgroup of G(a), <t> be the set of all mappings which 
 
 AH 
 are compatible with H, and r be an element of S. 
 
 -10- 
 
Then two elements dj and ($ of $ are called 
 r-equivalent (denotated by $ =&) > i^ tfi ( r ) - ^>( r ) • 
 
 Obviously the relation just introduced is an equivalence relation. 
 Its main properties with respect to our purpose will be stated in the 
 following 
 
 THEOREM h: Let A = (S, I, M) be a finite strongly connected automaton, 
 
 H be a subgroup of G(A), K.„„ be the max [Y/H<i Y < G(a)}, 
 
 Gxi 
 
 <J> be the set of all mappings which are compatible with H, 
 AH 
 
 ^ be the extension of H in A, and r be an element of S. 
 
 AH 
 
 Then we have: 
 
 i) Each element k of K is contained in an r-equivalence 
 
 GH 
 
 class of $.„» 
 AH 
 
 ii) 'The elements of K are pairwise r-unequivalent . 
 
 GH 
 
 iii) Each class 
 
 of the factor group $. /f consists of 
 
 AH AH 
 
 exactly /H/ r-equivalence classes of c£> 
 Proof: Statement i) is obviously true, since K C $ 
 
 AH 
 
 Consequently, 
 
 definition 3 applies to the elements of K , and statement ii) 
 
 GH - AH 
 
 W 
 
 holds, since two arbitrary elements k and k of K are automorphisms 
 
 -L o. On 
 
 of a strongly connected automaton for which k (r) = kp(r) implies 
 
 k l :; k 2 ■ 
 
 In order to prove iii) we shall first show that 
 
 frfl'f2 e< W(3l ?^2.)r ^W #1 y°f2j Which means that 
 
 the r -equivalence containing an arbitrary fl^eO falls completely 
 
 J d AH 
 
 into the cP n containing class & of the factor group ^.tt/^.tt- 
 J c- * c. AH AH 
 
 So, be (t and <D arbitrary elements of <£ with <f (r) = ^ 3 ( r )° 
 Then, exploiting the strong connectedness of A and the corollary 
 for h ; - e we have (VseS) (3xel) (3h , h 2 eH) 
 
 h 1 o£f(s) = h l of(M(r, x)) = M^Cr), x) = M(f 2 (r), x) - 
 
 = ku o 
 
 f 2 (M(r, x)) = h ocf 2 (s) and, 
 
 since h o bu is an element of H, this means 
 
 (^seS) (3heH) ^, (s) = h o ^ (s)° Defining a mapping y/ by 
 
 -1 
 (^seS)'Jp(s) := h[<p (s)](s), we see thaty/e^ and that for this 
 
 fwe have indeed (^s £ S) (0 (s) =-yo«(s), i.e.^ =y Q d>. 
 
 -11- 
 
Next we shall show that (VfeQ ) (VyeY ) {3heH)fof = h of 
 
 which means that <2 consists of at most /h/ r-equivalence classes. 
 
 So, be <P£<$ and^e^ arbitrarily. Then we have by definition 2 
 ' AH An 
 
 (3heH) yo^(r) = h o^?(r), i.e. ip of '= h o^for some heH. 
 Finally we shall prove that for any pair h and h p of 
 
 different elements of H and an arbitrary 4 €.<$>.„ we have always 
 
 An 
 
 }i ocP^f- h ? od?- This means that ^ consists of at least /H/ 
 
 r-equivalence classes, since obviously H C Y and, therefore, 
 
 — An 
 
 (VheH) h o q e <f° So, be h and h„ two elements of H with 
 
 h ^ h p and be dpe<£ arbitrarily. Then we have h otfi(r) =ft h p o^(r), 
 
 since h and h„ as different automorphisms in a strongly connected 
 
 automaton cannot coincide for any argument. This shows that 
 
 h o^ihp o^ and we have concluded the proof of theorem k. 
 
 If we denotate by n the number of r-equivalence classes in <l> , then 
 we have shown in theorem k that /$.„/ = ; AH 7 and also that 
 
 m ~7h7 
 
 n > /K /, the equalsign holding, if and only if each r-equivalence class 
 = GH 
 
 of $ contains an element k of K . 
 An GH 
 
 So, if in a finite strongly connected 
 
 automaton for one of its states r each r-equivalence class of $ contains 
 
 AH 
 
 an e 
 
 /K / . /¥ / 
 
 lement k of K , then /<!> / becomes ; GH ' AH 7 , i.e., using theorem 3, 
 GH AH 7557 
 
 /g(VH)/ - /V*«/ " Affi/ 1 /- before, K GH /H 
 
 is in this case 
 
 isomorphic to G(A/H), since we know from [6] that K /H is in general 
 
 GH 
 
 isomorphic to a subgroup of G(A/H). 
 
 On the other hand, if in such an automaton A for one of its states 
 r there is an r-equivalence class that does not contain any keK , then we 
 find /g(A/H)/ > /k /h/. Therefore, in this case K /H is not isomorphic 
 to G(A/H). 
 
 By this discussion of theorem h we have proved in fact the 
 following theorem 5 which gives a necessary and sufficient condition for 
 K /H to be isomorphic to G(A/h) . 
 
 THEOREM 5: Let A = (S, I, M) be a finite strongly connected automaton, 
 
 H be a subgroup of G(A), K„ be the max {Y/H<4 Y < OCA)}, 
 
 GH 
 
 and r be an element of S. 
 
 -12- 
 
Then the following two propositions are equivalent: 
 
 (a) The factor group K_ TT /H is isomorphic to the 
 
 GH 
 
 automorphism group G(a/h) of the reduced automaton 
 A/H. 
 
 (b) For each mapping <J>: S => S which is compatible with 
 
 H there is a mapping k in K such that k(r) = eP(r) . 
 
 uH * 
 
 The fact that K /H is isomorphic to G(a/h), if /K / = /s/, 
 
 appears now as a special instance of theorem 5° Namely, in this case the 
 
 elements of K comprising /s/ automorphisms in a strongly connected 
 GH 
 
 automaton are forced to meet the condition (^r, seS) (iJkeK ) k(r) = s 
 
 GH 
 
 which means that proposition (b) of theorem 5 is implied by /K / = /S/. 
 
 GH 
 
 We can obtain a stronger result through the following 
 
 THEOREM 6: Let A = (S, I, M) be a finite strongly connected automaton 
 
 and let H, K nzJ , $ , and r be defined as before. Then 
 Gn AH 
 
 Proof: 
 
 we have (Vf& M ) (VgeG(A)) (<f( T ) = g(r)^ g eK GH ) 
 
 Be <ye$ and g£G(A) arbitrarily such that tf'(r) = g(r). Then, 
 
 using the strong connectedness of A and the main properties 
 
 of G(A) and <£ we get the following straight sequence of 
 AH 
 
 equations: 
 
 (VheH) (VseS) (3xei) (^en) 
 
 h o g(s) -ho g(M(r, x)) = h(M(g(r), x)) - 
 
 = h(M(ep(r), x)) =■ h o h~" o cf o h (M(r, x)) = cPo h (s). 
 
 Together with definition 2 this yields: 
 
 (Vh€H) (3y^ m ) hog^oy 
 
 Choosing the identity e as a particular h, we find g = ys o<& 
 
 -I e 
 
 and, therefore, (/heH) (jye^.-j) g " o h o g =y/, The mapping 
 
 g o h o g is apparently an automorphism and, therefore, the 
 
 corresponding *to too is an automorphism. Now, the only automorphisms 
 
 in ¥ are the elements of H and this s?iows that 
 AH , 
 
 (\^h£H) g o h o g e H and, therefore, g is indeed an element of 
 
 K 
 
 GH" 
 
 Theorem 6 permits us to supplement theorem 5 by a proposition that is slightly 
 weaker than proposition (b): 
 
 -13- 
 
COROLLARY: Let A, H, K , and r be defined as in theorem 5. Then the 
 
 GH 
 
 following three propositions are equivalent: 
 
 (a) as in theorem 5. 
 
 (b) as in theorem 5* 
 
 (c) For each mapping^: S => S which is compatible with 
 
 H there is an automorphism geG(A) such that g(r) =^(r). 
 
 This corollary contains the special case [6] that K /H is isomorphic to 
 
 Gn 
 
 G(A/H) ; if the automaton A is strongly connected and total, i.e. /g(a)/ = /S/ 
 
 -ll+- 
 
REFERENCES 
 
 *h I i 
 
 [1] Weeg, G. P., "The Structure of an Automaton and its Operati 
 frSr 1 ^ Transformation Group", J. ACM 9, pages 3U5-3I19, 
 
 on - 
 
 [2] 
 
 j "The Group and Semigroup Associated with Automata", 
 
 Proc. Symp. Ma th. Th eor y o f Automata , pages 257-266, Polytechnic 
 ±Tess, (1962). 
 
 [3] 469-^6^' (l^62)? SOra ° rPhlSm Gr ° UPS ° f Automata "> J - ACM 9, pages 
 
 W 
 
 , ."On the Automorphism Group of an Automaton", J. ACM 12 
 
 pages 566-569, (1965) 
 
 sses 
 
 [5] Barnes, B., "Groups of Automorphisms and Sets of Equivalence Cla 
 oi Input for Automata", J. ACM 12, pages 56I-565, (1965). 
 
 [6] lZ e V, \' ^"/f ° mor P h i sm Grou P of a Strongly Connected Automaton 
 and its Quotient Automata", Department of Computer Science 
 
 1 t^fiC^?° ±B > ^^ Illin ° is 6l8 ° 3 - Re ^t *■ 199, 
 
 C?] (1966)! ^'^ CH ' A " " GrOUp " :i ^ ,e Automata", J. ACM 13. pages I7O-I75, 
 
 
 ■15- 
 


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