[...]... and we give their complexities in time (O(n 3)) The first test of the section 4 is based on the interlaced product of automata, the second one' and the third one are based on the computation of the left quotients of automata 2 Preliminaries Let C be a finite alphabet The set C* is the set of all finite words over C, Cw is the set of all infinite words, and C" is the union of C* and C" The empty word... Section 3, we show the existence of an infinite hierarchy of Petri net languages on the number of transitions and places of their recognizing nets For any finite alphabet C, we denote C*, (resp C', El') the set of all words (resp of all words of length r , of all words of length at most r ) on the alphabet C, A denotes the empty word For any word w E C*, Z(w) denotes the length of w Every subset L... 2 The purpose of this paper is study of growth function of Petri nets and its applications The definitions of Petri net and of Petri net language are recalled in Section 2 The Section 3 deals with the notion of growth function of a Petri net The main result of this part is the growth speed theorem which shows that the growth function of any Petri net is bounded by a certain polynomial The Section 4... the word u is a prefix of the word u It follows that v = uu‘ for some word u’in C*, the word u’stands for u-lv and is a suffix of the word v We denote by pref(v) (respectively suff(v)) the set of all words that are prefixes (resp suffixes) of the word w , pref(L) (resp suff(L)) is the set of prefixes (resp suffixes) of words of L For two languages L and L‘, the left quotient of L’ by L is L-lL‘ = {w... Words of C+ are obtained by finite concatenation of letters of C: u = u u .u,E 9, n > i > 0, lz for u E C Words of C" are obtained by infinite concatenation of letters of C: i u = ~ 1 ~ 2 E C", for n > i > 0, u E C .u, i Let L c C* be a language over C,then L* is the set of finite words obtained by finite concatenation of words of L , that is the submonoid generated by L , and L" is the set of infinite... decide whether a given rational language is an w-code Those algorithms have a complexity in the worst case in O ( n 3 ) ,n being the number of states of the automaton representing the language 1 Introduction Let C be a finite alphabet We denote by C* (respectively C") the set of words (resp infinite words) over C Given L of C*, the language L* is the submonoid generated by L and it denotes the set of words... function of T However, it almost always happens that for large value of T , gN(r) or hN(r) can be closely approximated by a much simpler function which will provide us about state growth speed of the net N 4 3.2 In the sequel, we use the notations and definitions of the theory of computational complexity Definition 6 If f and g are functions defined on the positive integers, then (1) f = O(g)if there... transition t of N is said to be firable at the marking p if VP E P : 11.03) 2 I(P,t) If t is firable at p then when t fires, the Petri net N will go into a new marking p' given by VP E p : P Y P ) = P(P) - I(P,t) O(t,P) + We write then 6 ( p ,t ) = p' and call 6 the state changing function of the net N A firing sequence of N can be defined as a sequence of transitions such that the firing of each of its... net Proof The proof is analogous to the one of corollary 1 By the Corollaries 1 and 2, we can show a lot of rather simple languages not being acceptable by either any Petri net or a Petri net whose number of transitions and number of places are less than a given contant Example 1 Let 11 = k x 2 2, c @ C and : L = { x c x / x E C+} It can verify that if q , x 2 E X' 2 1 # 2 2 then x1&,.x2(modL) Therefore... factorization of a word u in L+ (respectively in Lw)is a finite sequence (resp an infinite sequence) fiL (211, u2, ,u , .) of words of L such that = , u = u1u2 u, We call sequence of left factors of u for the factorization f u , the sequence ( p i ) t l , where n is the number of factors in the factorization fu (infinite for infinite words), where for i 2 1, pi is the concatenation of the i first . this book is available from the British Library. THE MATHEMATICAL FOUNDATION OF INFORMATICS Proceedings of the Conference Copyright Q 2005. see in the sequel, in the theory of growth function, only the state growth speed of the system matters, no attention is paid to the states themselves.