A NOTE ON STAR-FREE EVENTS. Albert R. Meyer. Carnegie-Mellon University Pittsburg _, Pennsylvania February, ]968

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1 A NOTE ON STAR-FREE EVENTS By Albert R. Meyer Carnegie-Mellon University Pittsburg _, Pennsylvania February, ]968 This work was supported by the Advanced Research Projects Agency of the Office of the Secretary of Defense (SD-]46) and is monitored by the Air Force Office of Scientific Research. Distribution of this document is unlimited.

2 ABSTRACT A short proof of the equivalence of star-free and group-free regular events is possible if one is willing to appeal to the Krohn-Rhodes machine decomposition theorem.

3 ]. INTRODUCTION The star-free events are the family of regular events expressible in the extended language of regular expressions (using intersection and complementation, as well as union and concatenation of events) without the use of the Kleene star (closure) operator. The equivalence of the star-free and group-free events was first proved by Sch'dtzenberger []966]. Papert and McNaughton []966] show that the star-free events are precisely the events definable in McNaughton's L-language, and are thereby able to establish the above equivalence without extensive use of the properties of finite semigroups. However, if one is willing to appeal to the machine decomposition theorem of Krohn and Rhodes,the equivalence of star-free, group-free, and also noncounting regular events can be proved more simply. We present such a proof in this note. 2. PRELIMINARIES We assume the reader is already familiar with regular events and finite automata. Our notation follows Yoeli []965] and Ginzburg []968]. In particular, if f and g are functions from a set S into itself, arguments are written on the left (so that sf = f(s)), and the composition f o g means that f is applied first (so that s(f o g) = (sf)g). A semiautomaton is a triple A = <QA, E A, M A> with QA a finite set A A (of states), _ a finite set (of _), and M a set of functions MA: QA_>QAindexe d by o E 7_A. The mapping 5_ is abbreviated,, A,,. The o element q A 6 QA is the next state of q 6 QA under input _ C A. For A A > A A x E (_A), the mapping x" Q- QAis defined inductively: is the

4 -2- identity map on QAwhere A is the null word in (_A),, and if x = y(_ (_A)* A A A oa. xy)a= A A for y C and o _, then x is y o Hence, ( x o y for all x, y E ( A),. For x 6 (EA)* and integers k >- 0, xk is the 0 concatenation of x with itself k times; x = A by convention. Clearly, (xk) A = (xa) k = the composition of xa with itself k times. The (necessarily finite) set of distinct mappings xa for x E (_A) * form a semigroup GA under composition. GA is called the semigrou_ of A. Let A and B be semiat,tomata. B is a subsemiautomaton of A providing _B c A, QB c QA and the mapping o B is the restriction of _ A to QB for each (_ E _B. B is a homomorphic ima$_e of A providing that A QA QB B Ao = _B and there is an onto mapping I]: -> such that I] o o = o I] A for each _ E _ The mapping TI is called a homomorphism of A onto B. A covers B, in symbols "A _ B" if and only if B is a homomorphic image of a subsemiautomaton of A. An automaton is a quintuple i = <QA, A, sa, FA, MA> where A = <QA, _]A, M_ is a semiautomaton, called the semiautomaton of _,s A is an element of QA called the start state, and FA is a subset of QA (A) A A 1 called the final states, The event accepted b i is Ix E * I s x E F A This definition of automaton is merely a notational variant of the usual finite state acceptor (cf. Rabin and Scott []959]), and the events accepted by such automata are precisely the regular events. 3. STAR-F_IEE AND NONCOUNTING EVENTS The star-free events are defined inductively as follows: Definition 1. Let _ be a finite set (of inputs). %he singleton [_ is a star-free event over _. If U, V * are star-free events over _,

5 -3- then U [J V, U (the complement of U relative to _*), and UV (the concatenation of U and V) are star-free events over _. An event is star-free over _ only by implication from the preceding clauses. By DeMorgan's law, U n V = U U _ and so star-free events are also closed under intersection. Since the regular events over _ include the singletons and are closed under union, relative complementation, and concatenation, it follows that every star-free event is regular. Definition 2. (Papert-McNaughton) A regular event U c E* is a noncountin_ regular event over _ if and only if there is an integer ku _. 0 such that for all x, y, z E _* k U ku+1 x y z E U _ x y z 6 U. Intuitively, an automaton accepting a noncounting event U need never count (even modulo any integer greater than one) the number of consecutive occurrences of any word y once ku consecutive y's have occurred in an input word. Lemma I. (Paper-McNaughton) Every star-free event is a noncounting regular event. Proo ff. The singleton [.ul is trivially a noncounting regular event for every a 6 _ (choose k[_ = 2), so it is sufficient to show that if U and V are noncounting regular events over _, then so are U [J V, U, and UV. Let k = max[ku,kvi. Then for any x, y, z 6 _*, xy kz6uuv xy ku (yk-ku z ) 6 U or xy kv(y k-kv z) E V _ xy ku+] (yk-k U z) U or

6 -4- xy kv+l (yk-kv z) E V _ xy k+l z E U U V. Thus, U U V is a noncounting regular event with kuuv= max[ku, kv_. k U k U ku+l ku+l Similarly, xy z 6 U _ xy z _ U _ xy z _ U = xy z 6 D, so that U is a noncounting regular event with k_ = ku. k Finally, let k = 2 max [ku,k V _+ ] and suppose xy z E U V. Then k xy z = uv for some u E U, v E V, and it must be the case that either u = xyk/2w for some w C _*, or that v = w'yk/2z for some w'e _*. In the first case, u = xyk/2w- = xyku(y k/2-ku- w) E U implies that xy ku+1 (yk/2-k U w) = xyk/2 +1 w E U since U is noncounting. In the second case, v w'y k/2z E V similarly implies that w'y k/2 +1 = zev. k+l k+l Hence, in either case xy z C UV. Conversely, if xy z E UV the k arglunent can clearly be reversed to show that xy z E UV. Thus, UV is a noncounting regular event with kuv = 2'ma X[ku,kv!+1. Q.E.D. ku If U is a noncounting regular event over _ and o E _, then o E U implies that U contains all words in o* of length at least k U. Therefore, either U N c_* or U N c_* is a finite event. The regular event (c_c_)*is neither finite nor has finite complement, which proves: Corollary I. The noncounting (and hence the star-free) regular events are a proper subfamily of the regular events. 4. GROUP-FREE EVENTS Associated with any event U c_,* is a congruence relation,--_ (rood U), on E* defined for w, y E E* by: w- y (mod U) _ (Vx, z E _*)[x w z E U _ x y z E U].

7 -5- Noncounting regular events are thus those regular events U such that ku ku+1 y - y (rood U) for all y 6 _*. The relation between this congruence and automata is an immediate consequence of the familiar theorems of Nerode and Myhil] (of. Rabin and Scott []959]): if U is a regular event, then there is an automaton accepting U (viz., the reduced automaton accepting U) such that A A x-= y (rood U) _ x = y. Definition 3. A_ of S is a subsemigroup of S whose elements form an abstract group under multiplication in S. A semigroup is group-free if and only if all its subgroups are isomorphic to the trivial group with one element. A semiautomaton is group-free if and Dnly if the semigroup of the semiautomaton is group-free. A regular set U is group-free if and only if there is an automaton i accepting U such that the semiautomaton A of i is group-free. Lemma 2. Let S be a semigroup. If there is an integer k _ 0 such that k k+] s = s for all s E S, then S is group-free. Proof. Let G be a subgroup of S, and let g be an element of G. Then g k = gk+] implies e = g k (g")] k = gk+] (g-)] k= g where g-i is the inverse of g in G and e is the identity of G. Hence, G = [e I is the trivial group. Q.E.D. C0rol!ary 2. Every noncounting regular event is a group-free regular event. k u ku+1 Proof. If U is a noncounting regular event, then y m y (mod U)

8 -6- implies that (yk U)A = (yku+l) A in the reduced automaton _ accepting U. k U ku+1 Hence, (ya) = (ya) for every element ya E GA, and GA is group-free by lemma 2. Q.E.D. 5. DECOMPOSITION INTO RESETS The machine decomposition theorem of Krohn and Rhodes supplies the key step in the proof that group-free events are star-free. Definition 4. Let A and B be semiautomata and tu: QA _ A B. The cas cade product A _ B of A and B with mapping _0 is the semiautomaton C with QC= QA QB C A oc EC, E = _ and for ct E defined for all sa E QA, sb C QB by: C B <s A, sb> c) = <sac)a, s (<sa,c_>00)b>. A cascade product of three or more automata is defined by association to the left, e.g., a cascade product of semiautomata A, B, and C J is any semiautomaton (A $1B) c02c for any mappings cui and cv2 with appropriate domain and range. Definition 5. A semiautomaton R is a reset providing QR= _1,2_, and R R R _R is the union of three mutually exclusive sets E], E 2, E 1 such that: R R R o E E] = range (c_r) = f1_; c) E E 2 = range (c_r) = _2_; and cy E _I _ a R = the identity on QR. The following weak form of the decomposition theorem is sufficient for our purposes (for a constructive proof of the general theorem see Ginzburg [1968]): Theorem. (Krohn-Rhodes) Every semiautomaton A is covered by a cascade product of seiniautomata AI, A2,..., An such that for 1 i < n, Ai

9 -7- Ai is a reset or else G is a non-trivial homomorphic image of a subgroup of GA. Since the trivial group has only itself as a homomorphic image, the following lemma is inmlediate: Lemma 3. Every group-free semiautomaton is covered by a cascade product of resets. Corollary 3. Every group-free regular event is accepted by an automaton whose semiautomaton is a cascade product of resets. Proof. Let i, with group-free semiautomaton A, be an automaton accepting a group-free regular event U. By lemma 3 and the definition of covering, A is the image under a homomorphism ]] of a subsemiautomaton of a cascade product C of resets. There is no loss of generality in assuming that _A = C, since the subsemiautomaton of C obtained by restricting _C to _A is also a cascade product of resets which covers A. Choose any sc E QC such that sc]] = s A (the start state of i) and define F C = {q E QC I q _] E FAI. Then for any x E ( A),, x E U saxa E FA _ scq] xa E F A _ scxc]] E FA _ scx C E FC. Hence, the automaton C with semiautomaton C, start state s, and final states F C is the required automaton accepting U. Q.E.D. 6. THE MAIN THEOREM. The behavior of cascades of resets can be described in terms of star-free events using

10 -8- Definition 6. For a semiautomaton A and states p, q E QA, the set A Pq of p-q-inputs is Ix E (_A)* I PxA = q l. Lemma 4. Let C = B _ R with B a semiautomaton, R a reset, and _: QB _B ER X -0. If B is a star-free event (over E B) for all p, q E QB, Pq then Cab is a star-free event (over _C = EB) for all a, b E QC. Proof. Write '_" for the (equal) sets E B and C. By the definition of cascade product, the first cc_nponent of <p, 1> yc is simply py B for any p E QB, Y E _". "" Since R is a reset, in order for the second component of <p, I> yc to be 2, R must receive an input <r, o> _ E _2 for some r EQB, oe_. Suppose x E C<p,]Xq,2 >. Then px = q and so x E Bpq, but also * B x must equal y o z for some y, z E _, o E E such that: py = r for some r E QB and <r, _> m E E2" R Choose tile shortest z such that x = y _ z for y and _ satisfying the preceding conditions. Then no prefix of z R (where s _ QB, 8 C _), i.e., causes R to receive an input <s, 6> _0 E _] B z _ BraB,s6 _. B R * Conversely, if py = r for <r ' o> w E _2 and z _ Bro B,S 6 _ for any <s,6> _ E _IR' then yc_z E C<p,1_q,2 > providing yoz E Bpq. Altogether, one has : C<p,iXq,2>= Bpq fl [U Bpr _ (U BroB,s 6 _*)] QB the lefthand union being over all r C, _ E _,such that <r,g_"ioe _2' and the righthand union being over all s E QB R, 6 E _ such that <s,6_ E _I" The unions in the expression for C<p,]>_q,2> are finite, and _ is a star-free event (_ = _ and _ = [o_n _-_), so that C<p,iXq,2> is a star-free event. The set of x E C<p,1>_q,1> is precisely the set of

11 -9- x 6 _ such that px B = q and x _ C<p,iXq,_, i.e., C<p,1><q,1> = Bpq rl C<p, IXq, 2>' and so C<p,l><q,l> is also a star-free event. Since the argument is symmetric in states 1 and 2 of QR, Ca b is a star-free event for all a, b E QC. Q.E.D. Lemma 5. If C is a cascade product of resets, then Cab is a star-free event for all a,b E QC. Proof. Let R be a reset and B a semiautomaton such that QB = [p_ and _B = 7R. For _ E _B, define 0J: QB X _B - R by the condition <p,_>0_ = o. In this trivial case of cascade product, Rij = (B R)<p,i><p, j> for QR. (_B * all i, j E Since B = ) is star-free, lemma 4 implies that PP Rij is star-free. The rest of the proof follows immediately by lemma 4 and induction on the number of resets in Co Q.E.D. Corollary 4. Every event accepted by an automaton i, whose semiautomaton A is a cascade product of resets, is a star-free event. Proof. Let a C QA be the start state of i, and FA the final states. The event accepted by i is U A Aab which is a star-free event since b6f the union is finite and Aab is star-free by lemma 5. Q.E.D. This completes the proof of the following Theorem. (Sch'dtzenberger, Papert-McNaughton) l_e following are equivalent for events U c _ : 1) U is a star-free event. 2) U is a noncounting regular event. 3) U is a group-free event. 4) U is accepted by a cascade product of resets.

12 -10- References I. Ginzburg, A., A!gebraic ]heoliy- o_fi Automata, t_ooappe#qr (1968). 2. Papert, S. and R. McNaughton, "On topological events", _ l_eory of Autc_nata, University of Michigan Engineering Summer Conferences (1966). 3. Rabin, M. O. and D. Scott, "Finite automata and their decision problems", IBM J. Re ss. and Dev., 3,2 (April, 1959),! Sch'dtzenberger, M. P_, "On a family of sets related to McNaughto_s L-language", Automata The0rx, Caianiello, E. R. (ed.), Academic Press, N. Y. (1966), _ 5. Yoeli, M., "Generalized Cascade Decompositions of Automata", JAC MM, 12, 3 (July, 1965), _