Random DFAs over a non-unary alphabet

Transcription

1 Random DFAs over a non-unary alphabet Jean-Marc Champarnaud and Thomas Paranthoën Unversty of Rouen, LIFAR, F Mont-Sant-Agnan Cedex, France {Jean-Marc.Champarnaud, Abstract Ths document gves a generalzaton on the alphabet sze of the method that s descrbed n Ncaud s thess for randomly generatng complete DFAs. Frst we recall some propertes of m-ary trees and we gve a bjecton between the set of m-ary trees and the set (m,n) of generalzed n-tuples. We show that ths bjecton can be bult on any prefx total order on Σ. Then we gve the relatons that exst between the elements of (m,n) and complete DFAs bult on an alphabet of sze greater than 2. We gve algorthms that allow us to randomly generate accessble complete DFAs. Fnally we provde expermental results that show that most of the accessble complete DFAs bult on an alphabet of sze greater than 2 are mnmal. Keywords: Complete determnstc automata; Random generaton; Catalan famles; m-ary trees Introducton The random generaton of DFAs allows us to get some emprcal observatons that lead to theoretcal results n the average case on the classcal algorthms that are appled on DFAs. Although the random generaton of unary DFAs s trval, Ncaud has used ther natural structure to gve the average state complexty of the classcal operatons on unary DFAs (1). Moreover he descrbes n hs thess (2) a method for randomly generatng complete accessble DFAs on an alphabet of sze 2. We show n ths paper how ths method can be extended to the case of DFAs bult on an alphabet of an arbtrary sze. Ncaud s method deals wth bnary trees and an other Catalan famly: the n-tuples. The n-tuples allow one to count the number of determnstc structures that can be produced from a gven bnary tree (a determnstc structure s a DFA wthout fnal states). In ths paper these two Catalan famles are extended to an alphabet of an arbtrary sze. And we thus restate the algorthms presented n (2) n ths case. Wth these algorthms, we carry out some experments that enlght the fact that most of the DFAs are mnmal as far as the sze of the alphabet s greater than 2. 1

2 Let us menton that ths work s a part of a more general study on the random generaton of fnte automata (3). Secton 1 ntroduces defntons and notaton that are necessary to the comprehenson of ths document. Secton 2 gves some propertes of m-ary trees and generalzes the bjecton that exsts between the set of bnary trees, the set of prefx subsets of Σ, wth Σ of sze 2, and the set n of n-tuples to a bjecton between the set of m-ary trees, wth m 2, the set of prefx subsets of Σ, wth Σ 2, and the set (m,n) of generalzed n-tuples. Secton 3 makes explct the relaton between the elements of (m,n) and the determnstc transton structures of sze n on an alphabet of sze m. Fnally Secton 4 descrbes the algorthms for constructng random transton structures, and reports a set of expermental results based on ths random generaton method. 1 Defntons and notaton Readers who are not famlar wth automata theory are referred to (4). A fnte non-determnstc automaton s a 5-tuple A =< Q, Σ, δ, I, F > where Q = {q 1, q 2,..., q n } s the fnte set of states, Σ s the alphabet on whch the automaton s defned, δ s the transton functon (δ : Q Σ 2 Q ) (where 2 Q denotes the set of all subsets of Q) that assocates a subset of Q to each element of Q Σ, I s a non-empty subset of Q whose elements are the ntal states and F s a subset of Q whose elements are the fnal states. In ths paper the sze of an automaton s the number of ts states. An automaton s sad to be accessble f and only f for all states q Q there exsts a path from one of the ntal state to ths state. An automaton s sad to be co-accessble f and only f there exsts a path from ths state to one of the fnal states. An automaton that s both accessble and co-accessble s a trm automaton. An automaton D s determnstc f t has a unque ntal state and f δ(q, x) 1, q Q, x Σ. Moreover D s complete f δ(q, x) = 1, q Q, x Σ. In what follows, (m,n) wll denote the set of accessble complete determnstc automata of sze n on an alphabet of sze m. We wll wrte D =< Q, Σ, δ,, F > for a determnstc automaton (DFA) wth a unque ntal state. A determnstc transton structure s a 4-tuple S =< Q, Σ, δ, >, that s a DFA wthout fnal states. Thus 2 n DFAs can be produced from a transton structure snce there exst 2 n possble sets of fnal states. An m-ary tree s an acyclc drected graph T =< V, E > where V = {v 1, v 2,..., v t } s the set of vertces of the tree and E V V s the set of edges of the tree. We recall that the out-degree (resp. n-degree) of a vertex s the number of edges that are ncdent from (resp. to) ths vertex. We let d + (v) (resp. d (v)) be the out-degree (resp. n-degree) of a vertex v. The n-degree of each vertex of an m-ary tree s equal to 1, except for one vertex called the root and denoted by v 1 that has a zero n-degree. The out-degree of each vertex 2

3 Fgure 1: A 3-ary tree and ts assocated complete 3-ary tree. of an m-ary tree s less than or equal to m. A complete m-ary tree of order n s a tree wth a parttonng of ts vertces V = N L, wth N = n, such that v N d + (v) = m and v L d + (v) = 0. The set N = {r 1, r 2,..., r n } s the set of nodes, and L = {l 1, l 2,..., l s } s the set of leaves. There exsts a bjecton between m-ary trees wth n vertces and complete m-ary trees of order n. Indeed t suffces to attach to each vertex v of an m-ary tree m d + (v) leaves n order to obtan a complete m-ary tree of order n (Fgure 1). A set of words X of Σ s prefx f t contans all words u Σ such that there exsts w Σ such that uw X. Let Σ be an alphabet of sze m. A symbol of Σ can be attached to each edge of an m-ary tree such that for all vertces v and all symbols x, there s at most one edge outgong from v that s labeled by x. Thus each vertex of an m-ary tree can be labeled by a word w. The label of each vertex v s the label of the path that leads from the root to ths vertex. The set of these labels s denoted by P(T ). We can show easly that the set P(T ) s prefx. There exsts a bjecton between the set of prefx subsets of Σ of cardnalty n and the set of m-ary trees of order n. In the followng, (m,n) wll denote ether one of these two sets. We assume that Σ s equpped wth a total order <. Let Σ be the free monod over Σ and be a total order on Σ. Let P be a prefx subset of Σ, and T be the m-ary tree assocated wth P. Let P be the lst of words of P ordered by the relaton. Snce the elements of P are n bjecton wth the vertces of T, the order defnes a traversal of the vertces of the tree T. The order n whch the words appear n P corresponds to the order n whch the vertces appear throughout the traversal. We let u = u 1 u 2 u m and w = w 1 w 2 w n be words of Σ and T a Σ -ary tree. We defne: u w for the lexcographc order f one of the two followng condtons holds: () there exsts an nteger 1 k mn(m, n) such that (, 1 < k, u = w ) and u k < w k, () m < n, and (, 1 m, u = w ). The lexcographc order nduces a depth-frst traversal of T. u w for the graded lexcographc order f one of the two followng condtons holds: 3

4 () m < n, () n = m and there exsts k n such that (, 1 < k, u = w ) and u k < w k. The graded lexcographc order nduces a breadth-frst traversal of T. An order on Σ s a prefx order f: ( u Σ )( x Σ ) u ux The lexcographc order and the graded lexcographc order are prefx orders. We call prefx traversal of a tree a traversal nduced by a prefx total order. In what follows, we assume that Σ s an alphabet of sze greater or equal to 2 and that Σ s equpped wth a prefx total order. By conventon a complete m-ary tree of order n s such that m 2 and n 1. 2 Complete m-ary trees and generalzed n- tuples We frst present some propertes of complete m-ary trees. Then wll follow the generalzaton of the classcal n-tuples, that permts us to deduce a bjecton between the set (m,n) of generalzed n-tuples and the set (m,n) of complete m-ary trees. Proposton 1 A complete m-ary tree of order n has (m 1)n + 1 leaves. Proof. In any dgraph, the sum of the n-degrees s equal to the sum of the out-degrees, because they are both equal to the number of edges. Snce n a complete m-ary tree of order n wth L leaves, the sum of the n-degrees s equal to L + n 1, and the sum of the out-degrees s equal to mn, we obtan L = (m 1)n + 1. Lemma 1 We consder a prefx traversal of a complete m-ary tree T of order n. Let k (resp. r) be the number of nodes (resp. leaves) vsted at a step of the prefx traversal. The followng propertes hold: () ( r (m 1)k + 1 ) () ( r = (m 1)k + 1 ) k = n Proof. In the subgraph of T nduced by the prefx traversal the sum of the n-degrees s r + k 1. Moreover the out-degree of each of the k nodes s not greater than m. Thus the sum of the out-degrees s not greater than mk, and we get: r (m 1)k + 1 (1) 4

5 We assume that at the current step we have k < n and r = (m 1)k + 1. Let v be the next vsted vertex. We let k (resp. r ) be the new number of vsted nodes (resp. leaves). We dstngush two cases: v s a leaf: we get k = k and r = r + 1. Snce by hypothess r = (m 1)k +1, we thus have r > (m 1)k + 1, whch s n contradcton wth (1). v s a node: we get k = k + 1 and r = r. Snce the number of edges s less or equal to mk and the sum of the n-degrees s equal to k + r 1, we obtan k + r 1 mk, and r (m 1)k, whch s n contradcton wth the assumptons. Let T be a tree and L be ts set of leaves. Let L be the lst of leaves met durng the prefx traversal of T nduced by. Let the functon φ : L N that assocates wth each leaf of T the number of nodes vsted before t durng ths traversal. We have φ(l +1 ) φ(l ), l L. Proposton 2 Let T be a complete m-ary tree of order n. The number of nodes vsted before the -th leaf (except for the last one) durng a prefx traversal s greater than or equal to m 1. More precsely we have: () ( l L )( 1 < (m 1)n + 1 ) n φ(l ) m 1 ) () φ (l (m 1)n+1 = n Proof. The proof of () s by nducton on the number of nodes vsted before a leaf durng the prefx traversal. Let L = (m 1)n + 1 be the number of leaves. Bass = 1: the number of nodes that are vsted before the frst leaf s strctly postve, otherwse the order of T s zero. Inducton step L 2 1: we assume that the property s true for the -th leaf. We get: φ(l +1 ) φ(l ) (2) m 1 We then dstngush two cases: mod (m 1) 0: we have m 1 = +1 m 1, and the property s true for the ( + 1)-th leaf. mod (m 1) = 0: f at least one of the nequaltes of the assumpton (2) s strct, we get φ(l +1 ) > m 1 and consequently φ(l +1) +1 m 1. Thus the property holds for the ( + 1)-th leaf. Otherwse we get φ(l +1 ) = φ(l ) = m 1. Ths mples + 1 = (m 1)φ(l +1 ) + 1. Accordng to Lemma 1.(), we obtan φ(l +1 ) = n and thus + 1 = L. But by assumpton + 1 < L. Therefore the contradcton. 5

6 Thus the property holds for all leaves except for the last one. On the other hand () s a drect consequence of the Lemma 1.(). The set n of the n-tuples of elements of 1, n s defned as: n = {(k 1,..., k n ) 1, n n 2, n, k k 1 k } Ths set can be generalzed to the set elements of 1, n defned as: { (k 1,..., k s ) (m,n) = (m,n) of the generalzed n-tuples of s 1, n 2, s, k m 1 k k 1 } where s = n(m 1). We consder the functon ϕ : (m,n) (m,n) that assocates wth a complete m-ary tree T of order n the element of (m,n) defned by: ϕ(t ) = ( φ(l 1 ), φ(l 2 ),..., φ(l s ) ) In the followng K wll denote an element of (m,n). Proposton 3 For all n 1, m 2 the functon ϕ s a bjecton from (m,n) to (m,n). Proof. Accordng to Proposton 2 and defnton of (m,n), ϕ has ts values n (m,n). On the other hand let us consder T and T two dstnct trees of (m,n), and L and L the sets of words that label the leaves of these two trees. Let u be the smallest word accordng to such that u L L and u / L L. We assume that u L. We let ϕ(t ) = ( φ(l 1 ), φ(l 2 ),..., φ(l s ) ) and ϕ(t ) = ( φ(l 1 ), φ(l 2 ),..., φ(l s )). By defnton there exsts r such that l r s the leaf labeled by u, and such that for all < r, φ(l ) = φ(l ). Thus φ(l r ) < φ(l r ) and ϕ s njectve. Let us consder a generalzed n-tuple K = (k 1, k 2,..., k s ). We have to show that we can buld an m-ary tree T assocated wth t. We consder a prefx order. We frst gve some general consderaton on a constructon of a tree T accordng to an order then wll follow the constructon of T accordng to K. Let P and L be the sets of words that label respectvely the nodes and the leaves of the tree T durng ts constructon. Moreover let G be the set defne such that G = { ux Σ u Nx Σ }. By the completness property we have G = N m. For more convenence we let C = G\((L N)\{ɛ}). Intutvely C denotes the set of the labels of the paths that are not ended by a leaf. It s clear that f C = { } the tree T s complete. If C { } we can add to t a new vertex accordng to the order. That s, f we add a node, the set N becomes: N = N {mn (C)}, and the sets G and C are redefned from ths new set. And f we add a leaf, the set L becomes L = L {mn (C)}, and the set C s redefned from ths new set. 6

7 We can now descrbe how a tree T s bult from the generalzed n-tuple K. We consder that k 0 = 1 and that ntally N = {ɛ}, G = { x x Σ }, and L =. We buld the tree T accordng to K such that at each step t 1, s of the constructon, we add consecutvely, and accordng to the order : k t k t 1 nodes and one leaf. In order to show the correctness of ths constructon, at each step t the set C must be dfferent from { }. It s clear that ntally C = G { }. At the end of each step t, N and L are respectvely equal to k t and t. Moreover from the defnton of the generalzed n-tuple, we have k t t m 1. Thus (m 1)k t (m 1) t m 1, and we have mk t k t t+1 (m 1) t m 1 t+1. t m 1 Snce (m 1) t, we get (m 1) t m 1 t By replacng the value of the cardnals by ther notaton n mk t k t t + 1 > 0, we obtan G L N + {ɛ} > 0, and C > 0. Thus the correctness of the constructon. Fnally f we add to the tree T a leaf after the s-th step, T s complete snce L = (m 1) N +1 (Lemma 1.()). Therefore ϕ s surjectve, and then bjectve. 1 ɛ 1 ɛ a b c a b c 2 a 9 b c 2 a 3 b c a b c a b c 13 a b c a b c 4 3 aa a b ab ac ba bb bc c aa ab ac ba 9 bb bc c 10 a b aaa aab aac a. b. bba bbb bbc Fgure 2: 3-ary trees equvalent to the generalzed n-tuple: (3, 3, 3, 3, 3, 4, 4, 4), accordng to the lexcographc order (a) or to the graded lexcographc one (b). Fgure 2 llustrates the constructon of a complete tree from a generalzed n-tuple. Tree vertces are labeled n the order of ther creaton. We have today a good knowledge of the dfferent objects n bjecton wth m-ary trees, these objects are called Catalan famles. We close ths secton wth some of these famles extended to the case of an alphabet of an arbtrary sze m. We defne for all (m, n) N 2 the generalzed Catalan numbers (5; 6) as: C (m) n = 1 mn + 1 ( ) mn + 1 These numbers descrbe the number of m-ary trees of order n. On the other hand, the bjecton that exsts between bnary trees and Dyck words, can be generalzed to well balanced bracketed words that contan m 1 rght brackets n 7

8 for one left bracket (Fgure 3.d). The grammar of these words for an alphabet of sze m s: S a } SbSb {{ Sb } S ɛ m 1 terms Sb These words can also be vewed as sequences u = x 1 x 2 x n(m 1)+n of 0s and 1s called well m-balanced sequences that satsfy the followng propertes (5): () u contans n(m 1) 1s for n 0s, () for all, such that 1 n(m 1) + n we have : {j 1 j, x j = 0} {j 1 j, x j = 1} m 1 These sequences have been studed n probablstc mathematcs n the general case, and n combnatorcs n the case of bnary trees ( ballot problem (7), Dyck word (8)). They are n bjecton wth the walks above the sea level that have an ncreasng slope m 1 tmes greater than the decreasng one (Fgure 3.b). Computer scentsts also call them Dyck paths. Fnally the graphcal representaton of the n-tuples gves rse to the player sequence whch s a set of blocks that are contaned n a rectangle and that contans the negatve slope dagonal of ths rectangle (Fgure 3.c). a. ((()))))())) (d) b. (3,3,3,3,3,4,4,4) (e) c. Fgure 3: Illustraton of the dfferent objects n bjecton: (a) complete m- ary tree, (b) path above the sea level, (c) player sequence, (d) well balanced sequence, (e) generalzed n-tuple. 3 Relaton between complete determnstc automata and complete m-ary trees Ncaud s study shows that the classcal n-tuples allow us to buld and to count the DFAs on an alphabet of sze 2. We show that the noton of canoncal labelng extends naturally to the case of an alphabet of sze m 2. Ths permts us to 8

9 establsh the relatons that exst between the elements of (m,n) and those of (m,n), and to gve some bounds of (m,n). Let D =< Q, Σ, δ,, F >, D (m,n) be an accessble complete determnstc automaton. We recall that Σ s equpped wth a prefx total order. We assocate wth each state q of ths automaton the word: w(q) = mn { u Σ δ(, u) = q and u s the label of a smple path } Snce the automaton s accessble ths word exsts. Snce the automaton s determnstc and the order s total, ths word s unque. The labelng nduced by the applcaton w s canoncal. Two dstnct complete accessble determnstc automata that are canoncally labeled cannot be somorphc (f the labellngs of ther states are dentcal, ther transton tables are necessarly dfferent). We denote by P(D) the set of labels of the states of D by w: P(D) = { w(q) q Q } Proposton 4 For all automata D of (m,n) the set P(D) s prefx. Proof. We assume that there exsts a word uv P(D) such that u / P(D). Snce the automaton s complete, w(δ(q 0, u)) exsts, and w(δ(q 0, u)) u. Snce the order s prefx w(δ(q 0, u))v uv. Ths leads to a contradcton. Prefx sets are n bjecton wth complete m-ary trees. Thus the transton structures reduced to the set of the smallest paths from the ntal state to each one of the DFA states are n bjecton wth complete m-ary trees. Proposton 5 The set of the accessble complete determnstc transton structures of sze n on an alphabet of sze m can be generated wth the elements of (m,n). Each element K of (m,n) can generate a number of structures equal to: n(m 1) n K = n (k 1,..., k n(m 1) ) = n Thus, we have: (m,n) = 2 n K (m,n) n K Proof. Let K be an element of (m,n), and T = (V, E) be ts unque assocated complete tree. We denote by N and L respectvely the sets of nodes and of leaves of T. The transton structure defned by S =< N, Σ, E (N N), v 1 > contans n 1 transtons and s accessble. In order to obtan a complete determnstc transton structure, we add to ths structure the (m 1)n + 1 transtons correspondng to the edges that lead from a node to a leaf. Let l be a leaf of the tree, and u be ts label. Let p be the parent of l. We consder the edge (p, l) that s labeled by x. The addton of the edge =1 k 9

10 (p, r), r N labeled by x to the transton structure S does not change the labelng of the states of S f w(r) u. The number of dfferent edges (p, r) that can be added s thus equal to the number of nodes r whose labels are smaller than u. Ths number s equal to k for the leaf l, 1, (m 1)n. Hence the expresson of the number of transton structures that can be bult from a generalzed n-tuple. Fnally there exst 2 n dfferent sets of fnal states, hence the number of complete determnstc automata of sze n on an alphabet of sze m. Ths result permts to defne some bounds on the number of automata of a gven sze. Proposton 6 We have the followng nequaltes: () (2π) m 2 2 e 1 s m n+α 1 n s 1+α+m /2 (m,n) 2 n e 2π m n+α 1 n s 1 /2+α wth s = (m 1)n α = ( s )( 1 log(s + 1) ) log(mn) () (2) 24 n e n n n (1 + o(1)) (2,n) 2 n 1 π 4 n n n 1 /2 (1 + o(1)) () (9) (m,n) 2 n nmn (n 1)! Proof. The product of the elements of an element K of (m,n) s bounded by: (n!) (m 1) K n n(m 1) Thus, by usng the fact that the generalzed Catalan numbers descrbe the number of elements of (m,n), we get the followng nequaltes: ( ) n (n!)(m 1) mn + 1 ( ) (m,n) mn + 1 n 2 n n nn(m 1) mn + 1 mn + 1 n Thanks to some smplfcatons and usng Strlng approxmaton we get the bounds () by usng the followng approxmaton of the generalzed Catalan numbers: ( ) mn n mn + 1 = e m α+n 1 n α 3 /2 2π In the case of a bnary alphabet, the above expresson can be approxmated and we get the bounds () gven by Ncaud. Fnally, () can be mproved, snce the number of accessble transton structures s smaller than the number n nm of sets of m determnstc but not necessarly accessble transton functons. And snce there exst (n 1)! dfferent ways to label these structures, we deduce the nequalty (). Notce that a better upper bound, based on accessble DFAs, s presented n (10; 11; 9). 10

11 4 Algorthms for the constructon of transton structures We gve frst a recurrence relaton that expresses the number of determnstc complete transton structures of sze n on an alphabet of sze m. We deduce from ths relaton an algorthm that computes ths class of numbers; ths allows us to gve an algorthm that randomly generates a generalzed n-tuple accordng to the number of dfferent transton structures that can be deduced from ths n-tuple. 4.1 Constructon of the elements of (m,n) In (2), t s shown that n-tuples can be computed va recursve formulae. Followng ths approach, we defne the followng generalzaton of (m,n): (m,t,p) = { (k 1, k 2,...,k t ) 1, p } t 2, t, k k k 1 m 1 Notce that for all m and n, an element of (m,n) s an element of (m,n(m 1),n). In the followng K wll denote an element of (m,t,p). We let for all m, t and p: = K (m,t,p) K (m,t,p) (m,t,p) Proposton 7 For all t, p 1 and m 2, the followng relatons hold: = 0 f p < t m 1, c (m,t,p) = 1 2p(p + 1) f t = 1, = c (m,t,p 1) + p c (m,t 1,p) otherwse. t Proof. If p < m 1 then k p < m 1, and the condton k m 1 cannot be satsfed. If t = 1 then c (m,1,p) = p =1 = 1 2p(p + 1). For the recurrence relaton, t s suffcent to remark that an element of (m,t,p) not endng wth p s n (m,t,p 1). If t ends wth p then t has the form (k 1, k 2,..., k t 1, p), wth (k 1, k 2,..., k t 1 ) (m,t 1,p). Thus (k 1, k 2,..., k t 1, p) = p (k 1, k 2,...,k t 1 ). t The elements allow us to compute the number of complete accessble determnstc transton structures on an alphabet of sze m and to generate these structures. The algorthm that bulds the elements s descrbed n Fgure 4. The array bult by ths algorthm can be vewed as a Pascal-lke trangle. It avods computng the same values many tmes, due to the recursve defnton of. Fgure 5 represents for m = 3, 1 t 16 and 1 p 8. 11

12 1 Functon arrayofthec(m : nteger, t : nteger, p : nteger) array 2 var 3 T : array [1, t] [0, p] of nteger 4 Begn 5 for j 1 to p do 6 T[1][j] 1 j(j + 1) 2 7 od 8 for 2 to t do 9 for j 0 l to p m do 10 f j < m 1 11 then T[][j] 0 12 else T[][j] T[][j 1] + jt[ 1][j] 13 f 14 od 15 od 16 return T 17 End Fgure 4: Algorthm that bulds the elements. It shows for example that there exst c (3,4,2) 2 = 28 2 = 56 complete determnstc structures of transton of sze 2 on an alphabet of sze 3. From the bounds gven n Proposton 6 the growth of the numbers c(m, (m (m 1)n+ n 1 1)n, n) s n the worst case of order n log(n), thus ther sze s of order ((m 1)n + n 1 log(n) )log(n). The sze of these numbers gves rse to some mplementaton problems, snce the memory space used to buld the table becomes quckly huge; for example the table necessary to randomly generate automata of sze 1000 on an alphabet of sze 2 needs around 250 MB wth the GMP mathematcs lbrary (12). The algorthm that generates a random generalzed n-tuple K uses the array bult by the prevous algorthm, and produces a random element of (m,n) t\p Fgure 5: Table of the c (3,t,p) elements for t from 1 to 16 and p from 1 to 8. 12

13 accordng to the number of transton structures t can generate (Fgure 6). It assumes that we have a functon append whch concatenates an nteger e to the end of an nteger lst l and returns the new lst: append(l : lst, e : nteger) lst. 1 Functon randomelementofk(m : nteger, t : nteger, p : nteger) nteger lst 2 Begn 3 f p < t then return m 1 4 f 5 f t = 1 6 then 7 De Random( 1, T[1][p] ) 8 De De 9 x 1 10 whle De > x do 11 De De x 12 x x od 14 return (x) 15 else 16 De Random( 1, T[t][p] ) 17 f (De T[t][p 1]) and (p > 1) 18 then return randomelementofk(m, t, p 1) 19 else return append(randomelementofk(m, t 1, p), p) 20 f 21 f 22 End Fgure 6: Algorthm that randomly generates a generalzed n-tuple accordng to the number of determnstc structures assocated wth. Proposton 8 The algorthm of Fgure 6 randomly bulds an element K of K (m,t,p) such that each K has a probablty equal to to be generated. Proof. (Lnes 3-4) Snce there s no element K such that p < m 1 the functon returns an empty lst f t s called wth such parameters. The proof of the sequel of the algorthm s by nducton snce the functon s recursve. We assume that = T[t][p]. (Lnes 7-14) Bass, the elements K of (m,1,p): the elements K of (m,1,p) are the lsts wth a unque element: (1), (2),..., (p). In order to satsfy the property, the algorthm has to determne an nteger x 1, p such that the probablty that x s equal to r (r 1, p ) s equal to t r c (m,1,p) = r p(p+1) 2 (second equalty of Proposton 7). Thus n Lne 7 we choose a random nteger De 1, c (m,1,p), and at the end of the loop (Lnes 10-13), x s the value such that x(x 1) 2 < De x(x+1) 2. Hence the property holds. 13

14 (Lnes 16-20) Inducton step: we assume that the algorthm returns the elements of (m,t 1,p) and the elements of (m,t,p 1) wth the property. That s each element I of (m,t,p 1) s randomly choosen wth a probablty equal to I c (m,t,p 1), and each element J of (m,t 1,p) s randomly choosen wth a probablty equal to J c (m,t 1,p). We thus have to show that each element K of (m,t,p) s randomly choosen wth a probablty equal to K. We let K = (k 1, k 2,...,k t ) be any element of K (m,t,p). We dstngush three cases: (Lnes 17,19) Case K (m,t,1) : necessarly K ends wth 1, thus the condton p > 1. Snce c (m,t,1) = 1, t, m, the property holds. (Lnes 16-18) Case K (m,t,p 1) : the probablty that the algorthm returns a K that belongs to (m,t,p 1) s equal to c (m,t,p 1). Moreover by assumpton each element I of (m,t,p 1) s randomly choosen wth a probablty equal to I c (m,t,p 1). Thus each element K of (m,t,p) that belongs to (m,t,p 1) s randomly choosen wth a probablty equal to c (m,t,p 1) K c (m,t,p 1), and the property holds. (Lnes 16-17,19) Case K = (k 1,...,k t 1,p) and (k 1,...,k t 1 ) (m,t 1,p) : the probablty that the algorthm returns a K that ends wth p and such that (k 1, k 2,...,k (t 1) ) (m,t 1,p) s equal to c (m,t,p 1) = p c (m,t 1,p) (thrd equalty of Proposton 7). Moreover by assumpton each element J J of (m,t 1,p) s randomly choosen wth a probablty equal to c (m,t 1,p). Thus each element K that has such a form s randomly choosen wth a probablty equal to p c (m,t 1,p) (k1,k2,...,kt 1) c (m,t 1,p) = (k1,k2,...,kt 1) p = K (m,t,p), and the property holds. In order to generate a random K of (m,n) we call ths recursve functon as follows: randomelementofk(m, n(m 1), n). Once an element K s randomly generated and ts assocated tree s bult accordng to Proposton 3, one of the automata assocated wth ths tree can be bult accordng to Proposton Expermental results The tests have been performed wth a program wrtten n C++ that uses the lbrary GMP. The generated DFAs are of sze 100, and for each test and each possble number of fnal states, DFAs have been randomly generated. For an alphabet of sze 2, t appears (Fgure 7) that accessble complete DFAs are mnmal wth a probablty of 0.8. That s consonant wth Ncaud s results. For an alphabet of sze greater than 2, we have observed that almost all accessble complete DFAs are mnmal (except for those whose fnal state set s empty or contans all states). Ths observaton s llustrated by Fgure 8, for DFAs of sze 100; notce that t s stll vald for DFAs of smaller sze. 14

15 100% 80% 60% 40% 20% 0% Number of fnal states Fgure 7: Percentage of complete mnmal DFAs of sze 100 on an alphabet of sze 2, accordng to the number of fnal states. 100% 80% 60% 40% 20% 0% % 80% 60% 40% 20% 0% Number of fnal states (a) Number of fnal states (b) Fgure 8: Percentage of complete mnmal DFAs of sze 100 on an alphabet of sze 3 (a) and 5 (b) accordng to the number of fnal states. 5 Concluson The extenson from bnary trees to m-ary trees gves rse to a natural generalzaton of the Catalan famles. Ths generalzaton allows us to gve an algorthm that bulds random DFAs on an alphabet of an arbtrary sze. Expermental results show that the use of such a generaton method allows us to buld random mnmal complete automata. Indeed, as observed by Ncaud, automata generated on a bnary alphabet are mnmal wth an emprcal probablty of 0.8. Moreover, as ponted out by our experments, almost all automata generated on an alphabet of a larger sze are mnmal. Thus a random generaton method wth rejecton can be used to randomly generate mnmal DFAs. The two emprcal observatons on the mnmalty of DFAs are gven as conjectures. Acknowledgements: We want to thank P. Gastn and our anonymous referees of an earler verson of ths paper. The frst one for GasTeX, a useful 15

16 L A TEX package for drawng graphs and automata, and the second ones for ther helpful advce. References [1] C. Ncaud, Average state complexty of operatons on unary automata, MFCS 1999, Lecture Notes n Computer Scence 1672 (1999) [2] C. Ncaud, Etude du comportement en moyenne des automates fns et des langages ratonnels, Ph.D. thess, Unversté Pars 7 (2000). [3] J.-M. Champarnaud, G. Hansel, T. Paranthoën, D. Zad, Nfas btstreambased random generaton, n: J. Dassow, M. Hoeberechts, H. Jürgensen, D. Wotschke (Eds.), Proceedngs of DCFS Descrptonal Complexty of Formal Systems, London Ontaro Canada, 2002, pp [4] S. Yu, Regular languages, n: G. Rozenberg, A. Salomaa (Eds.), Handbook of Formal Languages, Volume I, Word, Language, Grammar, Sprnger, Berln, 1997, pp [5] U. Tamm, Lattce paths not touchng a gven boundary, Journal of Statstcal Plannng and Inference 105 (2) (2002) [6] P. Hlton, J. Pedersen, Catalan numbers, ther generalzaton, and ther uses, Math. Intellgencer 13 (2) (1991) [7] W. Feller, An Introducton to Probablty Theory and ts Applcaton, Wley, [8] M. Lothare, Combnatorcs on Words, Addson-Wesley, [9] M. Domaratzk, D. Ksman, J. Shallt, On the number of dstnct languages accepted by fnte automata wth n states, n: Proceedngs, Descrptonal Complexty of Automata, Grammars and Related Structures (DCAGRS), 2001, pp [10] V. A. Lskovets, The number of connected ntal automata, Kbernetka 3 (5) (1969) [11] R. W. Robnson, Countng strongly connected fnte automata, Graph Theory wth Applcatons to Algorthms and Computer Scence (1985) [12] GMP, gnu multple precson lbrary, 16