Radix enumeration of rational languages is almost co-sequential

Transcription

1 Rdix enumertion of rtionl lnguges is lmost co-sequentil Pierre-Yves Angrnd Jcques Skrovitch July 18, 2008 Abstrct We define nd study here the clss of rtionl functions tht re finite union of sequentil functions. These functions cn be relized by cscdes of sequentil trnsducers. After showing tht cscdes of ny height re equivlent to cscdes of height t most two nd tht this clss strictly contins sequentil functions nd is strictly contined in the clss of rtionl functions, we prove the result whose sttement gives the pper its title. Introduction We define nd study here the clss of rtionl functions tht re finite union of sequentil functions (of pirwise disjoint domins). This clss ppered rther nturlly in the study of the concrete complexity of the successor function in some non stndrd numertion systems (cf. [2, 3]). Without going into detils of tht work, one of the problems which is tckled there is the definition of computtion model tht is powerful enough to describe successor functions nd tht llows the definition of complexity with sufficient degree of bstrction. As we shll see, the successor function is not necessrily sequentil function nd, s we wnt to del with deterministic utomt only, we consider cscdes of sequentil trnsducers, tht is, functions tht compute the vlue of word u by the following procedure : u is red by first sequentil trnsducer τ which outputs word v; then v is red by nother sequentil trnsducer σ q tht depends on the stte q reched by τ t the end of the reding of u, nd so on, for fixed number of steps h. In this pper we first chrcterize these functions: nmely we prove tht they ll re of the kind we just sid: finite union of sequentil functions with pirwise disjoint domins, independently of the prmeter h nd by tht they strictly contin the clss of sequentil functions, nd tht they form proper subfmily of rtionl functions. Then, we prove the min result of this pper, nmely : 1

2 Theorem 1. The rdix enumertion of rtionl lnguge is finite union of co-sequentil functions. This result does not solve the problem of the evlution of the concrete complexity, but t lest tells tht these cscdes of sequentil trnsducers provide sensible computtion model for lrge clss of numertion systems, nd moreover for numertion systems viewed s bstrct ones (cf. [8]). 1 Preliminries We bsiclly follow the definitions nd nottion of [5] (in prticulr for the post-fixed nottion of functions nd reltions), nd of [1] (nd lso of [11]). In the sequel, A is finite lphbet, A the free monoid generted by A, 1 A the empty word, identity of A. For word u in A, u is the length of u. Let A be n lphbet totlly ordered by. The rdix order on A is defined by: u v if either u v or u = v nd there exist w, u, v words nd b letters such tht u = wu nd v = wbv. The rdix order is well order, tht is every non empty subset of A hs smllest element for. For every lnguge L of A, the successor function of L, denoted by Succ L, is the function tht mps every word u in L onto the word v tht is the smllest of ll words of L greter thn u. Rdix enumertion is nother nme, esily understndble, for the successor function. For the generl definitions of rtionl reltions nd finite trnsducers, the reder ie referred to the references quoted bove. In this pper, we shll content ourselves with restricted type of trnsducers. First, nd in order to simplify definitions nd nottion, ll word functions or reltions mp words of A into A. Since we re never interested in the functions or reltions being totl or surjective, this is indeed not restrictive hypothesis. Definition 1. Let us cll trnsducer (from A into itself) n utomton τ = Q, A, A, E, I, T where : Q is finite set of sttes; E is finite set of trnsitions whose lbels re in A A ; I is prtil function, the initil function, from Q into A ; we rther write I q for the vlue I(q), nd q is sid to be initil if I q is defined; T is prtil function, the finl function, from Q into A ; we rther write T q for the vlue T(q), nd q is sid to be finl if T q is defined. The wy reltion ϕ from A into itself is ssocited with trnsducer τ we sy tht τ relizes ϕ is clssicl. A functionl reltion is rtionl (i.e. its grph is rtionl subset of A A ) if, nd only if, it is relized by trnsducer (our definition implies tht trnsducers re finite). 2

3 The underlying input utomton of trnsducer τ = Q, A, A, E, I, T is the finite utomton obtined from τ by keeping the first component of the lbel of every trnsition nd by replcing the initil nd finl functions by their domins. Definition 2. A trnsducer is sid to be sequentil (resp. co-sequentil) trnsducer if the underlying input utomton is deterministic (resp. co-deterministic). A rtionl function is sequentil (resp. co-sequentil) if it is relized by sequentil (resp. co-sequentil) trnsducer. We denote by Seq nd coseq the sets of sequentil nd co-sequentil functions. These sets re closed under composition. Remrk 1. A co-sequentil trnsducer is lso often seen s trnsducer with deterministic input utomton but which reds nd writes words from the right to the left, which is then right sequentil trnsducer. Sequentil functions were introduced by Schützenberger in [13] under the nme of subsequentil functions (cf. lso [1]). It will be no surprise tht we follow the terminology of [11]. Moreover, nd s in this lst reference, we reserve the qulifiers right nd left for utomt (nd hence for trnsducers) which model physicl mchines, nd ccording to s they red words from right to left or from left to right respectively. Functions re neither left or right but they re relized by trnsducers which cn be right or left. Sequentil functions re chrcterized within rtionl functions by topologicl criterion in the following wy : the prefix distnce d of two words u nd v is defined s d(u, v) = u + v u v, where u v is the longest prefix common to u nd v. Definition 3. A function ϕ is sid k-lipschitz (for the prefix distnce) if : f, g Dom(ϕ), d(fϕ, gϕ) k d(f, g). The function ϕ is Lipschitz if there exists k such tht ϕ is k-lipschitz. Theorem 2 ([4]). A rtionl function is sequentil if, nd only if, it is Lipschitz. Remrk 2. By the left right dulity we define the suffix distnce. A rtionl function is co-sequentil if nd only if it is Lipschitz for the suffix distnce. Exmple 1. Let us consider the function ψ 0 = Succ A, the successor function on A itself, with A = {0, 1}. ψ 0 is co-sequentil but not sequentil. The Figure?? shows co-sequentil trnsducer relizing this function. Let u = (1) k 0 nd v = (1) k it holds then : uψ 0 = (1) k 1 nd vψ 0 = (0) k. Let us consider the prefix distnce : since d(u, v) = 1 nd d(uψ 0, vψ 0 ) = 2k, ψ 0 is not Lipschitz. Exmple 2. Let U τ = {1, 3, 8, 21,...} be the numertion system defined by the reccurence u n+2 = 3u n+1 u n. It is known tht the integers re ll represented by the rtionl lnguge L 1 = A \ {A (21 2)A 0A } where A = {0, 1, 2} (cf. [7]). 3

4 1 0, Figure 1: A co-sequentil trnsducer relizing ψ 0 Let u = 2(1) k nd v = (1) k. It holds then: Succ L1 (u) = 1(0) k+1 nd Succ L1 (v) = (1) k 1 2, nd s d(u, v) = 1 nd d(succ L1 (u), Succ L1 (v)) = 2k + 1, the function Succ L1 is not k-lipschitz for the suffix distnce, for ny k, nd hence not Lipschitz nd not co-sequentil. 2 Cscdes of Sequentil Trnsducers 2.1 Definition nd Chrcteriztion We busively denote by the sme letter τ the function nd the trnsducer tht relizes it, the context mking cler which object we re referring to. Definition 4. (i) A sequentil trnsducer τ with set of finl sttes T is cscde of height 1. With τ, we ssocite the (prtil) stte mp τ from A onto T such tht f τ is the stte reched t the end of the (unique) computtion of τ tht reds f. (ii) Let θ be cscde of height h with set of finl sttes T, θ : A T the ssocited stte mp, nd, for every q in T, let σ q be sequentil trnsducer with set of finl sttes S q. The function ω, denoted ω = (θ, {σ q }), nd defined by: f A fω = (fθ)σ q if q = f θ is cscde of height h + 1, whose set of finl sttes is S = q T S q nd whose ssocited stte mp ω is defined by : f A f ω = (fθ)σ q if q = f θ. Tht is, fω is the result of composition of two functions θ nd σ q where the second one, σ q, depends upon the stte q reched by θ t the end of its computtion on f. By the left right dulity we define cscde of co-sequentil trnsducers. Exmple 3 (Ex. 2 cont.). The function Succ L1 is recognized by cscde of height 2 of right sequentil trnsducers (reding from right to left), nmely the cscde (τ, {σ p, σ q }) where τ, σ p nd σ q re described in Figure 2. 4

5 0 1, ,, 0 1 p q,, ,, 1 0 1, 1 2,, () τ (b) σ p nd σ q Figure 2: A cscde of height 2 From this exmple, it follows tht cscdes of sequentil trnsducers llow to describe more functions thn sequentil trnsducers. We shll not elborte much on cscde of rbitrry height becuse of the following theorem. Theorem 3. Let θ: A A be function. The following re equivlent: (i) θ is relized by cscde of height 1 or 2; (ii) θ is relized by cscde of height h, h 1; (iii) θ is finite union of sequentil functions whose domins re pirwise disjoint. Proof. (ii) (iii). Let θ be cscde of height h, with set of finl sttes T nd for ech q T, let K q = (q)[θ ] 1 the set of words tht re mpped by θ on the stte q. We shll prove the more precise properties: ) The K q re pirwise disjoint rtionl sets. b) The restriction of θ to K q is sequentil function denoted θ q. The impliction, nd the two properties, obviously hold if h = 1. Let {σ q } q T be fmily of sequentil functions with sttes S q nd ω = (θ, {σ q }) the cscde of height h + 1 it llows to define. Let r in S = q T S q, the set of finl sttes of ω; there exists unique q such tht r is in S q nd we hve f K q fω = (fθ)σ q = (f)[θ q σ q ] tht is, the restriction of ω to K q is sequentil function nd the lnguge ) L r = r[ω] 1 = ((r)[σ ] 1 θ 1 q q is rtionl set (contined in K q ). Since ω is function, the L r re pirwise disjoint. The properties () nd (b) hold for ω, which is then the union of its restrictions to the L r. (iii) (i). Let θ be the function relized by finite union of sequentil trnsducers {σ 1,...,σ k } with pirwise disjoint domins L 1,..., L k. It is esy to build deterministic finite utomton A whose set of finl sttes T is prtitionned 5

6 in T 1,..., T k nd such tht the (unique) computtion of A on ny word u ends in T i if, nd only if, u belongs to L i. Let τ be the identity trnsducer built on the underlying input utomton A. For every t in T, let ψ t = σ i if t is in T i. The cscde of height 2, (τ, {ψ t } t T ) obviously relizes θ. By the left right dulity, it lso holds: Corollry 1. Let θ : A A be function. The following re equivlent: (i) θ is relized by cscde of right sequentil trnsducers of height 1 or 2; (ii) θ is relized by cscde of right sequentil trnsducers of height h, h 1; (iii) θ is finite union of co-sequentil functions whose domins re pirwise disjoint. Corollry 2. Cscdes of sequentil (resp. co-sequentil) functions re rtionl functions. 2.2 Cscdes nd rtionl functions Let us now denote CSeq nd CcoSeq the sets of cscdes of respectively sequentil nd co-sequentil trnsducers. From the previous subsection we know tht CSeq RtF, CcoSeq RtF nd Seq CSeq nd coseq CcoSeq. We now prove tht cscdes form strict subfmily of rtionl functions. Let A = {, b} nd let ϕ: A A be the function tht mps every word w to the unique word wϕ obtined from w by the rewriting rule bb b nd tht contins no fctor bb nymore. The function ϕ, often clled Fiboncci reduction, is rtionl function (see for instnce [1, Exer. III.5.5] or [9]), tht is neither sequentil nor co-sequentil (cf. [6]). Let us consider for instnce the two words u k = (b) k nd v k = (b) k b ; we hve u k ϕ = u k nd v k ϕ = b() k, thus d(u k, v k ) = 2 nd d(u k ϕ, v k ϕ) = 4k + 2, hence ϕ is not Lipschitz. Proposition 1. The Fiboncci reduction is not relized by cscde of either sequentil or co-sequentil trnsducers. Proof. Let us first remrk tht, even if (uv)ϕ (u)ϕ(v)ϕ, the following identity holds for ny two words u nd v of A : (uv)ϕ = (u)ϕ(v)ϕ. (1) Consider now the fmily of n+1 words X i, 1 i n+1 defined in the following wy : X i = w i,n w i,n 1... w i,1 where w i,k = v li if k = i nd w i,k = u li if k i nd where the u li = (b) l i nd v li = (b) l i b re s bove nd where the sequence of the n integers l i will be define inductively below. From (1) follows tht: X i ϕ = u ln u ln 1...u li+1 b() l i u li 1...u l1 6

7 for 1 i n nd: X n+1 ϕ = X n+1 = u ln u ln 1... u l1. From which we deduce tht for ll i, j, 1 i j n + 1 : ( ) d(x i, X j ) = 2 (2l m + 3), d(x i ϕ, X j ϕ) = 2 1 m i 1 ( 1 m i 1 (2l m + 3) + 2l i + 1 ). If we inductively choose the integers l 1,...,l n such tht: l 1 > 1 (K 1) 2 ( l i > 1 2 (K 1) then for ll i, j, 1 i j n + 1: 1 m i 1 (2l m + 3) d(x i ϕ, X j ϕ) > Kd(X i, X j ) nd ϕ cnnot be the union of only n K-Lipschitz functions. Choosing u k = (bb) k nd v k = b(bb) k, we prove similrly tht ϕ cnnot be the union of n co-sequentil functions. A similr proof gives the following proposition (referring to exmple 1): ) itself is co- Proposition 2. The successor function ψ 0 on the whole set A sequentil function tht is not finite union of sequentil functions. Remrk 3. Let A = {, b, c} be n lphbet. Let us consider the following functions : Dom(χ 1 ) = {() k k N} χ 1 (() k ) = () k, Dom(χ 2 ) = {() k k N} χ 2 (() k ) = b(bb) k, Dom(χ 3 ) = {c() k k N} χ 3 (c() k ) = (bb) k+1. ψ 1 = χ 1 χ 2 ψ 2 = χ 1 χ 3 The rtionl functions χ 1, χ 2 nd χ 3 re obviously sequentil nd co-sequentil nd hve pirwise disjoint domins: ψ 1 nd ψ 2 re then both in CcoSeq nd in CSeq. ψ 1 is neither sequentil nor co-sequentil nd ψ 2 is sequentil but not co-sequentil. Symmetricly to ψ 2, function ψ 3 cn be built, tht is in both CcoSeq nd CSeq nd tht is co-sequentil but not sequentil. Figure 3 shows the geogrphy of the set of rtionl functions nd of the subsets we hve defined in it. 7

8 ϕ Seq CSeq ψ 1 ψ 2 CcoSeq ψ 0 ψ 3 coseq RtF Figure 3: The geogrphy of rtionl functions 3 Rdix enumertion of rtionl lnguges As the rdix order, s well s the identity on every rtionl set, re synchronized rtionl reltions, stndrd properties of these synchronized rtionl reltions llow to prove the following result (cf. [2], [11]) : Proposition 3. The successor function of rtionl lnguge is relized by letterto-letter finite trnsducer. The im of our min result Theorem 1 is to give more insight on the successor function. For its proof let us fix some nottion nd recll some further definitions. 3.1 The synchronized product of ry utomt Ry utomt Definition 5. A lnguge L is ry lnguge if it is of the form L = uv w. Property 1. Let L be lnguge. If L hs t most one word for ech length then L is finite union of ry lnguges. Definition 6. An utomton is ry utomton if its structure shows tht it recognizes ry lnguge : it is trim, it hs unique initil stte, unique finl stte nd consists in unique circuit together with unique pth coming into the circuit nd unique pth going out of the circuit. Exmple 4. The Figure 4 shows two exmples of ry utomt, A 1 nd B 1. 8

9 b b b () A 1 (b) B 1 Figure 4: A 1 nd B Synchronized product Definition 7. Let A = Q, A, E, I, T nd B = R, A, F, J, U be two finite utomt. The synchronized product of A nd B is the trnsducer : A B = Q R, A, A, G, I J, T U where the set of trnsitions G is defined by : G = {((p, r), (, b), (q, s)) (p,, q) E, (r, b, s) F } Let A nd B be two finite utomt tht recognize respectively the rtionl lnguges L nd K. The synchronized product 1 A B relizes the reltion θ: A A such tht θ = {(u, v) u L, v K, u = v }. The trnsducer A B relizes function if nd only if K hs t most one word for every length but even if A nd B re deterministic, it is not likely to be sequentil s soon s there is stte in B which is the source of more thn one trnsition. Exmple 5. Figure 5 shows the synchronized product A 1 B 1 where A 1 nd B 1 re both deterministic, A 1 recognizes L 1 = {(b) k k N} nd B 1 recognizes K 1 = {b(b) k k N}. As the stte q of B 1 is the source of two trnsitions, the stte (p, q) in A 1 B 1 is the source of two trnsitions with sme input nd A 1 B 1 is not sequentil. The synchronized product is dditive, tht is: [ ] [ ] A x B y = x X y Y (x,y) X Y (A x B y ) Mking the synchronized product sequentil For the purpose of the construction underlying our proof, we slightly enlrge the fmily of utomt we re considering : the trnsitions of utomt re lbeled either by letter of A or by the empty word 1 A (trnsitions lbeled by 1 A re spontneous trnsitions), 1 This synchronized product is slightly different from the one considered in [11, Exerc. IV.6.17] where it recognizes the reltion whose grph is L K. 9

10 b b q b b b b b p (p, q) Figure 5: Synchronized product of A 1 nd B 1 the subsets I nd T of Q for initil nd finl sttes re replced by (prtil) functions from Q into A, still denoted by I nd T, nd their vlue I(p) nd T(q) re rther denoted I p nd T q. f The lbel c of computtion c : i t is then c = I i.f.t t. We denote by l(c) A the number of trnsitions of the computtion c nd we cll it the length of c. When it is needed we cll clssicl utomton n utomton without spontneous trnsitions nd with finl nd initil sttes. It is well known tht the fmily of lnguges recognized by such utomt is not lrger thn RtA nd tht ny clssicl utomton cn be seen s utomt choosing T t = 1 A when t is finl, T t = otherwise, I i = 1 A if i is initil nd I i = otherwise. The definition of this lrger clss of utomt is tken in view of the following construction. Proposition 4. Every ry utomton B is trnsformed into n equivlent ry utomton B with the following two properties: (i) Every stte of B is the source of t most one trnsition. (ii) If the computtions c in B nd c in B hve sme lbel then l(c) = l(c ). Proof. Property (i) is obtined by replcing the unique pth going out of the circuit by finl function on the stte which is source of this pth; Property (ii) by dding dequte number of spontneous trnsitions t the beginning of the pth coming in the circuit (s it cn be seen on Figure 6 for B 1 ). b b 1 1 b b () B 1 (b) B 1 Figure 6: B 1 nd B 1 10

11 The definition of synchronized product goes over this lrger clss of utomt nd it now holds: A B = Q R, A {1 A }, A {1 A }, G, I J, T U where: I J (p,q) = (I p, J q ), T U (p,q) = (T p, U q ) nd G = {((p, r), (x, y), (q, s)) (p, x, q) E, (r, y, s) F }. If A is clssicl utomton, the synchronized product A B relizes reltion which ssocite with u in L ll words in K tht re lbel of computtions in B whose length is equl to u. Figure 7 shows the synchronized product of A 1 nd B b b b 1 b 1 b b b 1 Figure 7: Synchronized product of A 1 nd From Proposition 4 follow the two properties : Property 2. Let A nd B be two ry utomt. Then A B nd A B re equivlent. Property 3. Let A nd B be two deterministic ry utomt. Then A B is sequentil trnsducer with single initil nd single finl stte. Proposition 5. Let L nd K be two rtionl lnguges with t most one word for ech length. The function ϕ tht mps every word of L onto the word of K with the sme length is relized by finite union of sequentil trnsducers with single finl stte (in prticulr, ϕ CSeq). Proof. A rtionl lnguge with t most one word for ech length is union of disjoint ry lnguges nd then cn be relized by union of deterministic ry utomt. Let A 1,..., A l be the deterministic ry utomt whose union recognizes L nd B 1,..., B k the deterministic ry utomt whose union recognizes K. Then K is lso recognized by the union of B 1,..., B k. For ll i nd j, A i B j is sequentil trnsducer nd from Property 2 relizes the sme function tht A i B j. The function ϕ is relized by the union of A i B j, thus by the union of A i B j, which proves the sttement. The previous propositions nd their proofs re obviously symmetricl nd the following result holds : 11 B 1

12 Proposition 6. Let L nd K be two rtionl lnguges with t most one word for ech length. The function ϕ tht mps every word of L onto the word of K with the sme length is relized by finite union of co-sequentil trnsducers with single initil stte (in prticulr, ϕ CcoSeq). 3.2 The uniform length successor function Let L be lnguge of A. Let us denote the sets of miniml words nd of mximl words of L (for ech length) by Min(L) nd Mx(L) respectively : Min(L) = {u L v L, u = v u v} Mx(L) = {u L v L, u = v v u} If L is rtionl, so re Min(L) nd Mx(L) (cf. [10, 12] for instnce). The min building block of our construction is the synchronized product, which yields trnsducers tht relizes length-preserving functions. As Succ L is not lengthpreserving function (it holds u Mx(L) u Succ L (u) ). We slightly chnge both the function nd the lnguge tht we study. If L is lnguge of A, let ULSucc L be the uniform length successor function, tht is the restriction of Succ L to (A A) : u L, ULSucc L (u) = Succ L (u) u = Succ L (u) u Mx(L), nd if u is in Mx(L), then ULSucc L (u) is undefined. For n ordered lphbet A, let A $ = A {$} where $ is letter tht does not belongs to A nd which is by ssumption smller thn every letter in A. We ssocite with every lnguge L of A the lnguge K = $ L of A $ nd it holds: Succ L (u) = v!k ULSucc K ($ k u) = v. (2) Proposition 7. If ULSucc K is relized by finite union of co-sequentil trnsducers, then so is Succ L. Proof. Let π : A $ A be the projection tht erses the $ symbol. With slight buse of nottion we write : π [ULSucc K ($ A A )] = Succ L. Let τ 1,...,τ k be the co-sequentil trnsducers whose union relizes ULSucc K nd for every τ i let : τ i = τ i ($ A A ) The trnsducer τ i is co-sequentil s ($ A A ) is recognizble reltion (cf. [11]). By (2), it holds tht for ny word u in A, if $ k u is in Dom(τ i ) then for no l k, $ l u is in Dom(τ i ). For every stte q of τ i from which cn be red word u in A to the finl stte, there exists t most one (finite) pth with word of $ s input nd 12

13 tht ends in q. Since Dom(τ i ) $ A, this pth begins with n initil stte. Let us note v the output of this pth. For every τ i, let τ i be the trnsducer built from τ i by replcing the finite pths lbeled by $ k v by n initil function with output v on the stte they rrive: τ i is equivlent to π(τ i ) nd is lso co-sequentil. Since Dom(τ i ) Dom(τ i) nd since for ny u in A there is t most unique k such tht $ k u is in Dom(τ i), it holds tht the domins of the τ i re pirwise disjoint. An utomton is sid to be stndrd (resp. co-stndrd) if it hs single initil (resp. finl) stte without ny incoming (resp. outgoing) trnsition. Trnsducers re (specil) utomt nd the sme definitions pply. The conctention of two reltions ϕ nd ϕ is denoted 2 by ϕ ϕ nd is the reltion whose grph is the product (in A A ) of the grphs of ϕ nd ϕ : if (u, v) ϕ nd (u, v ) ϕ then (uu, vv ) ϕ ϕ. The clssicl opertion of conctention of two utomt A nd A consists in dding spontneous trnsition from every finl stte of A to every initil stte of A. The sme construction pplies for the conctention of two trnsducers τ nd τ with the supplementry ingredient tht the outputs of the dded spontneous trnsitions re the products of the vlues of finl function of τ by the vlues of the initil function of τ. The conctention of τ nd τ relizes the conctention of the reltions relized by τ nd τ. Moreover, it is esy to verify the following property. Property 4. If τ is co-sequentil nd co-stndrd trnsducer nd if τ is cosequentil trnsducer with single initil stte then τ τ is co-sequentil trnsducer. 3.3 The lst step Proposition 8. If L is rtionl lnguge of A, then ULSucc L is union of cosequentil functions. Proof. Let A = Q, A, δ, i, T be deterministic utomton tht recognizes L nd tht is fixed for the remining of the proof. We lso note q = p.w if q = δ(p, w), L p = {w A p.w T } nd L p = {w A i.w = p}. The L p nd L p re rtionl nd the L p re pirwise disjoint s A is deterministic. Anlysis: Let u in L nd v = ULSucc L (u). Let w = u v the longest prefix common to u nd v : u = wu nd v = wbv with b (this holds since u = v ). Both u nd bv belong to L p nd w to L p. Let K p, be the set of mximl words of L p tht begin with n : K p, = Mx(( 1 L p )) 2 The conctention of words is usully denoted by simple juxtposition but in order to void confusion with the composition of functions, tht is lso usully denoted tht wy, we prefer to hve different nd explicit, nottion. 13

14 nd let H p, be the set of miniml words of L p tht begin with letter c greter thn : H p, = Min( c>c(c 1 L p )). The sets K p, nd H p, re both rtionl sets. Clim 1 : If w L p then u K p, nd bv H p,. Proof of Clim 1. If w L p nd wu is recognized by A then u ( 1 L p ). If u K p, then there exists u ( 1 L p ) with u = u nd such tht u u. Thus u = wu wu wbv = v, contrdiction with v = Succ L (u). We prove tht bv H p, symmetriclly. The lnguges K p, nd H p, hve both t most one word for every length. By Proposition 5 nd Proposition 6 the synchronized product of utomt tht recognizes them is finite union of sequentil or co-sequentil trnsducers τ 1,..., τ k with pirwise disjoint domins. Let us choose the co-sequentil option. There exists exctly one i such tht u belongs to Dom(τ i ) nd then bv = (u )τ i. Let K p be co-deterministic nd co-stndrd utomton recognizing L p nd let κ p be the co-sequentil nd co-stndrd trnsducer tht relizes the identity on L p. The conctention θ p,i = κ p τ i is co-sequentil trnsducer nd it holds v = uθ p,i. Synthesis : Let θ p,i = κ p τ i be co-sequentil trnsducer built s bove nd let r nd s be two words of A such tht: r = sθ p,i. First, s belongs to Dom(θ p,i ) = Dom(κ p )Dom(τ i ) nd necessrily s = ww with w L p nd w K p, L p. s thus belongs to L. It follows then tht r = ww with w in H p, nd w = w. Clim 2 : If w K p,, w H p,, nd w = w, then, for ll w in L p : ww = ULSucc L (ww ). Proof of Clim 2. From the definition of K p, nd H p,, it holds tht ww nd ww re in L, ww = ww nd ww ww nd hence w is prefix of the longest common prefix of ww nd its successor. Since w is in K p, (the set of mximl words of L p beginning with ) the longest common prefix of ww nd its successor is prefix of w. Hence the longest common prefix of ww nd its successor is w. From Clim 1 nd since K p, nd H p, hve t most one word for every length, ww = ULSucc L (ww ). As ULSucc L is function, the domins of the θ p,i re pirwise disjoint, nd this completes the proof of Proposition 8 nd thus of Theorem 1. Remrk 4. Every construction involved in the proof of Proposition 8 is symmetricl in the sense tht the utomt cn be chosen to be deterministic or co-deterministic 14

15 nd the trnsducers sequentil or co-sequentil (for instnce, the τ i ) but for one point: the utomt K p cn be chosen to be co-deterministic nd co-stndrd but it is not possible to ssume, in generl, tht they cn be chosen deterministic nd co-stndrd (which would be necessry to complete the symmetricl construction). And indeed (cf. Proposition 2) the successor function for A is not in CSeq. References [1] Berstel, J. Trnsductions nd Context-Free Lnguges. Teubner, [2] Berthé, V., Frougny, Ch., Rigo, M., nd Skrovitch, J. On the cost nd complexity of the successor function. In Proc. WORDS 2007 (P. Arnoux, N. Bédride, nd J. Cssigne, eds.), Tech. Rep., Institut de mthémtiques de Luminy (Mrseille), (2007) [3] Berthé, V., Frougny, Ch., Rigo, M., nd Skrovitch, J. On the concrete complexity of the successor function. in preprtion. [4] Choffrut, Ch. Une crctéristion des fonctions séquentielles et des fonctions sous-séquentielles en tnt que reltions rtionnelles, Theoret. Computer Sci., vol. 5 (1977), [5] Eilenberg, S. Automt, Lnguges nd Mchines, vol. A, Acdemic Press, [6] Frougny, Ch. Fiboncci representtions nd finite utomt. IEEE Trns. on Infor. Theory. 37 (1991) [7] Frougny, Ch. On the sequentility of the successor function. Inform. nd Computtion 139 (1997) [8] Lecomte, P. nd Rigo, M. Numertion systems on regulr lnguge, Theory Comput. Syst. 34 (2001) [9] Perrin, D. Finite utomt, in: Hndbook of Theoreticl Computer Science (J. vn Leeuwen, ed.) Elsevier (1990) Vol B, [10] Skrovitch, J. Deux remrques sur un théorème de S. Eilenberg. RAIRO Theor. Informtics nd Appl. 17 (1983), [11] Skrovitch, J. Eléments de théorie des utomtes. Vuibert, English trnsltion: Elements of Automt Theory, Cmbridge University Press, to pper. [12] Shllit, J. Numertion systems,liner recurrences, nd regulr sets. Inform. nd Comput. 113 (1994),

16 [13] Schützenberger, M. P. Sur une vrinte des fonctions séquentielles, Theoret. Computer Sci. vol. 4 (1977),

17 A Exmple For L rtionl lnguge, the proof of the theorem gives us n lgorithm to build union of co-sequentil trnsducers relizing Succ L : 1. Choose A deterministic utomton recognizing L. 2. Add loop lbeled by $ on the initil stte of A to hve B recognizing lnguge is K = $ L. 3. For every stte p of B nd every letter, build utomt for K p, nd H p, nd split them into union of ry utomt. 4. Co-determinize utomt of K p, nd trnsform ny utomton H of H p, into H. 5. H p, K p, is the union τ p,,1,...,τ p,,k of the synchronized products. 6. Compute K p (nd κ p) for ll stte p of B nd conctente κ p with τ p,,i for ech letter. 7. Delete the trnsition with $ s input nd mke stte p initil with u s output whenever p could be reched from initil stte by pth lbeled $ k u. Let us now continue nd use this lgorithm on exmple 2. Let A = {0, 1, 2} nd L 1 = A \ {A (21 2)A 0A }. We build in this nnexe union of right sequentil trnsducers tht relizes Succ L1. Let K 1 = $ L 1, the Figure 8 shows utomt to recognize L nd K (steps (1) nd (2)). We chose the utomton for L 1 to be stndrd in order to simplify the modifiction to obtin utomton for K 1. 0, 1 p 1 2 q 0 r 2 1 $ 0, 1 p 1 2 q 0 r 2 1 () Deterministic utomton for L 1 (b) Deterministic utomton for K 1 Figure 8: Automt for L 1 nd K 1. The Figure 9 gives the result for step (3), tht is, for ech stte s nd letter, the utomt to recognize the mximl lnguge recognizble from s beginning with nd the miniml lnguge recognizble from s beginning by letter greter tht. These utomt recognizes lnguges with t most one word per length nd cn be splitted into ry-utomt (s seen on Figure 9). 17

18 It cn be seen tht ll components of K s, for s stte nd letter of {0, 1, 2, $} re lredy co-deterministic. In this prticulr exmple (nd in order to simplify the finl result), even if some components of H s, hve input degrees greter thn 1, we skip step (4). This cn be done in this specil cse becuse the synchronized products re nonetheless co-sequentil (s it cn be seen on Figure 10). Figure 11 shows the κ s for every sttes s, tht is co-sequentil nd co-stndrd trnsducer tht relizes the identity on the lnguge K s. The Figure 12 gives the finl result fter steps (6) nd (7). It cn be seen tht trnsducers of Figure 12 re not relly co-sequentil since there re severl finl sttes. We indeed merged the conctentions of κ s nd K q, H q,, for every letters, into the sme trnsducer to simplify the result (note tht if we restrin to single finl stte then the trnsducers re co-sequentil). 18

19 $ $ 0 $ () K p,$ (b) H p,$ (c) K p,1 (d) H p, (e) K q,0 (f) H q,0 (g) K q,1 (h) H q, (i) K r,0 (j) H r,0 Figure 9: 19

20 $ 1 $ () K p,$ H p,$ = τ 0 τ 1 (b) K p,1 H p,1 = τ 2 τ 3 (c) K q,0 H q,0 = τ 4 τ (d) K q,1 H q,1 = τ 6 τ 7 (e) K r,0 H r,0 = τ 8 τ 9 Figure 10: Synchronized products τ i $ $ $ $ $ $ $ $ $ $ $ $ $ $ () κ p (b) κ q (c) κ r Figure 11: 20

21 , () θ p,0 θ p,1 θ p,2 θ p,3 (b) θ q,4 θ q,5 θ q,6 θ q, (c) θ r,8 θ r,9 Figure 12: Trnsducers whose union relizes Succ L1 21