The Average Complexity of Moore s State Minimization Algorithm is O(n log log n)

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1 The Averge Complexity of Moore s Stte Miimiztio Algorithm is O( log log ) Julie Dvid Istitut Gsprd Moge, Uiversité Pris Est Mre-l-Vllée Cedex 2, Frce. Astrct. We prove tht the verge complexity, for the uiform distriutio o complete determiistic utomt, of Moore s stte miimiztio lgorithm is O(log log ), where is the umer of sttes i the iput utomt. 1 Itroductio Due to their efficiecy to represet regulr lguges d perform most of usul computtios they ivolve, fiite stte utomt re used i vrious fields such s liguistics, ioiformtics, progrm verifictio d dt compressio. A miiml utomt is the smllest complete determiistic utomt tht c e ssocited to regulr lguge. Becuse this utomto is uique, up to isomorphism o the lels of the sttes, it is coicl represettio of regulr lguge d permits to test the equlity etwee regulr lguges d equivlece etwee utomt. Most stte miimiztio lgorithms compute the miiml utomto of regulr lguge tkig determiistic utomto s iput, y idetifyig the idistiguishle sttes. Moore proposed the first miimiztio lgorithm[8], which is sed o the clculus of the Myhill-Nerode equivlece, y refiemets of prtitios of the set of sttes. There re t most 2 such refiemets, ech of them requirig lier ruig time: i the worst cse, the complexity is qudrtic. Though, i [1], it is proved tht the verge complexity of the lgorithm is ouded y O( log ). Sice this result does ot rely o the uderlyig grph of the utomto, it holds for y proilistic distriutio o this grph. Also, the oud is tight for ury utomt. Hopcroft s stte miimiztio lgorithm [6] is the est kow lgorithm with O( log ) worst-cse complexity. It lso uses prtitio refiemets to compute the miiml utomto, ut its descriptio is ot determiistic, mkig its lysis complicted. Differet proofs of its correctess were give [5, 7] d severl uthors [3, 4] proved the tightess of the upper oud of the complexity for differet fmilies of utomt. I this pper, we prove tht for the uiform distriutio o complete determiistic utomt, the verge complexity of the lgorithm due to Moore is This work ws completed with the support of the ANR project GAMMA umer

2 O( log log ). The rticle is orgized s follows: fter recllig the sics of utomt miimiztio (Sectio 2), we itroduce the tools we use for the verge lysis (Susectios 2.3 to 2.5). Sectio 3 is dedicted to the verge time complexity lysis of Moore s lgorithm. Due to lck of spce, the proof of Lemm 6 is ot fully detiled, ut ide of the proof is give. The pper closes with discussio o Hopcroft s lgorithm executios, which re fster th Moore s lgorithm oes, for y iput utomto d cojecture o the verge complexity of oth lgorithms, for vrious distriutios o utomt. 2 Prelimiries 2.1 Defiitios d ottios A fiite determiistic utomto A = (A, Q,, q 0, F) is quituple where Q is fiite set of sttes, A is fiite set of letters clled lphet, the trsitio fuctio is mppig from Q A to Q, q 0 Q is the iitil stte d F Q is the set of fil sttes. A utomto is complete whe its trsitio fuctio is totl. The trsitio fuctio c e exteded y morphism to ll words of A : p ε = p for y p Q d for y u, v A, p (uv) = (p u) v. A word u A is recogized y utomto whe p u F. The set of ll words recogized y A is deoted y L(A). We ote A i the words of legth i d A i the word of legth less or equl to i. A utomto is ccessile whe for y stte p Q, there exists word u A such tht q 0 u = p. A trsitio structure is utomto where the set of fil sttes is ot specified. Give such trsitio structure T = (A, Q,, q 0 ) d suset F of Q, we deote y (T, F) the utomto (A, Q,, q 0, F). For give determiistic trsitio structure with sttes there re exctly 2 distict determiistic utomt tht c e uilt from this trsitio structure. Ech of them correspods to choice of set of fil sttes. I the followig we oly cosider complete determiistic utomt d complete determiistic trsitio structures, the ccessiility is ot gurteed. Cosequetly these ojects will most of the time just e clled respectively utomt or trsitio structures. The set Q of -stte trsitio structure will e deoted y {1,..., }. The set of utomt d the set of trsitio structures with sttes will respectively e deoted A d T. Also, sice there re k trsitios d sice for ech trsitio, there re possile rrivl sttes, we hve T = k d A = 2 k (whe E is the crdil of the set E). The term 2 comes from the choice of the set of fil sttes. The militry order o words, oted < mil, is defied s follows: u, v A, u < mil v if u < v or u = v d u is smller th v i the lexicogrphicl order. Let Cod e Boole coditio, the Iverso rcket [Cod] is equl to 1 if the coditio Cod is stisfied d 0 otherwise. For y o-egtive iteger i, two sttes p, q Q re i-equivlet, deoted y p i q, whe for ll words u A i, [p u F ] = [q u F ]. Two sttes p d q re equivlet (oted p q) whe for ll u A, [p u F ] = [q u F ]. This

3 equivlece reltio o Q is clled Myhill-Nerode equivlece [9]. This reltio is sid to e right ivrit, meig tht for ll u A d ll p, q Q, p q p u q u. Propositio 1. Let A = (A, Q,, q 0, F) e determiistic utomto with sttes. The followig properties hold: (1) For ll i N, i+1 is prtitio refiemet of i, tht is, for ll p, q Q, if p i+1 q the p i q. (2) For ll i N d for ll p, q Q, p i+1 q if d oly if p i q d for ll A, p i q. (3) If for some i N (i + 1)-equivlece is equl to i-equivlece the for every j i, j-equivlece is equl to Myhill-Nerode equivlece. For y iteger 1 d y m N, we deote y A m the set of utomt with sttes for which m is the smllest iteger such tht the m-equivlece m is equl to Myhill-Nerode equivlece. It is well kow tht m Moore s Stte Miimiztio Algorithm I this sectio we descrie Moore s lgorithm [8] to compute the miiml utomto of regulr lguge represeted y determiistic utomto. It uilds the prtitio of the set of sttes correspodig to Myhill-Nerode equivlece d mily relies o properties (2) d (3) of Propositio 1: The prtitio π is iitilized ccordig to the 0-equivlece 0, the t ech itertio the prtitio correspodig to the (i + 1)-equivlece i+1 is computed from the oe correspodig to the i-equivlece i usig property (2). The lgorithm hlts whe o ew prtitio refiemet is otied, d the result is Myhill- Nerode equivlece ccordig to property (3). The miiml utomto c the e computed from the resultig prtitio sice it is the quotiet utomto y Myhill-Nerode equivlece. Accordig to Propositio 1, if utomto is miimized i more th l prtitio refiemets, the there exists t lest pir of sttes p, q d word u of legth l + 1, such tht p l q d p u 0 q u, tht is to sy t lest two sttes re seprted durig the l + 1-th prtitio refiemet. I the remider of this sectio we itroduce the depedecy tree d modifictio of the depedecy grph itroduced i [1]. Those tools will llow us to give upper oud o the umer of utomt miimized i more th l prtitio refiemets, which is useful for the verge complexity lysis. 2.3 The Depedecy Tree I the followig, we itroduce the depedecy tree to model set of trsitio structures. To egi with, we expli how depedecy tree R(p) c e otied from fixed trsitio structure τ d fixed stte p d the how this oject will help to estimte the crdil of set of trsitio structures. For

4 Algorithm 1: Moore s lgorithm 1 if F = the 2 retur (A, {1},,1, ) 3 if F = {1,..., } the 4 retur (A, {1},,1, {1}) 5 forll p {1,..., } do 6 π [p] = [p F ] 7 π = udefied 8 while π π do 9 π = π 10 compute π from π 11 retur the quotiet of A y π The computtio of the ew prtitio is doe usig the followig property o ssocited equivlece reltios: ( p i q p i+1 q p i q A To ech stte p is ssocited sigture s[p] such tht p i+1 q if d oly if s[p] = s[q]. The sttes re the sorted ccordig to their sigture, i order to compute the ew prtitio. The use of lexicogrphic sort provides complexity of Θ(k) for this prt of the lgorithm. I this descriptio of Moore s lgorithm, deotes the fuctio such tht 1 = 1 for ll A. Lies 1-4 correspod to the specil cses where F = or F = Q. I the process, π is the ew prtitio d π the former oe. Lies 5-6 is the iitiliztio of π to the prtitio of 0, π is iitilly udefied. Lies 8-10 re the mi loop of the lgorithm where the ew prtitio is computed, usig the secod lgorithm elow. The umer of itertios of Moore s lgorithm is the umer of times those lies re executed. Algorithm 2: Computig π from π 1 forll p {1,..., } do 2 s[p] = (π[p], π[p 1],..., π[p k ]) 3 compute the permuttio σ tht sorts the sttes ccordig to s[] 4 i = 0 5 π [σ(1)] = i 6 forll p {2,..., } do 7 if s[p] s[p 1] the i = i π [σ(p)] = i 9 retur π Fig.1. Descriptio of Moore s lgorithm fixed trsitio structure with sttes over k-letter lphet d fixed stte p, we defie the fuctio isode mppig A to {0, 1} s follows: { 0 if v A such tht p w = p v d v < mil w, isode(w) = 1 otherwise. R(p) is tree i which odes d leves of depth h re lelled y words of legth h. It is uilt recursively, usig redth-first trversl of the odes of the tree strtig from the ode p. For ech ode of depth h lelled y w, d ech letter i the lphet, we dd ode lelled y w t depth h + 1 if isode(w) is equl to 1, d lef otherwise. Figure 2 gives exmple of depedecy tree. Note tht this costructio resemles the method used i [2] to rdomly geerte ccessile utomt, except the uthors use depth-first trversl. It is esy to see tht some depedecy trees c e otied from severl fixed trsitio structures d sttes. I the remider of the pper, we

5 ,, ε 2 4 6, 8 10,, , () () Fig.2. Let () e the fixed trsitio structure d 2 e the fixed stte, () is the ssocited depedecy tree R(2). We hve S 2(2) = {ε,,,,}, L 2(2) = {, } d s 2(2) = {2, 3, 4,5, 6}. chrcterize sets of trsitio structures correspodig to prticulr depedecy trees. We itroduce some ottios ssocited to depedecy tree R(p): S h (p) deotes the set of ll odes of depth less or equl to h, L h (p) deotes the set of ll the odes t give depth h. Sice every ode i the tree is lelled y word, we ote w S h (p) or w L h (p) if w is word lellig ode i those sets. We lso defie the set s h (p) of ll the sttes tht re reched from stte p y followig pth lelled y word of less or equl to h. For ll the trsitio structures ssocited to depedecy tree R(p), we hve s h (p) = S h (p). Lemm 1. For y fixed stte p, if depedecy tree R(p) cotis f leves t depth less or equl to h, the the umer of ssocited trsitio structures ) f. is ouded ove y T ( Sh (p) Proof. We recll tht T is equl to the product of the crdils of the sets of possile rrivl sttes, for ech trsitio. Let w e the lel of lef t depth less th h. For every trsitio structure ssocited to the tree R(p), the trsitio lelled y outgoig from the stte p w eds i stte p v, with v S h (p). Therefore, the umer of possile rrivl sttes for this trsitio is ouded ove y S h (p) isted of. This is rough upper oud ut sufficiet for the eeds of the proof. 2.4 The T -Depedecy grph We itroduce other model for sets of trsitio structures. For two fixed sttes p d q, two fixed x-tuples (x is fixed iteger) of o-empty words u = (u 1,..., u x ) d v = (v 1,..., v x ), two fixed sets ϕ p d ϕ q of pirs of words

6 (w,w ) such tht w < mil w, we defie the set T (p, q, ϕ p, ϕ q, u, v ) s follows: T (p, q, ϕ p, ϕ q, u, v ) = {τ T (w p, w p ) ϕ p, p w p = p w p, (w q, w q ) ϕ q, q w q = q w q, i x, p u i = q v i } We defie the x-tuples of words u = (u 1,..., u x) d v = (v 1,..., v x) d the x-tuples of letters α = (α 1,..., α x) d β = (β 1,..., β x), such tht for ll i x we hve u i = u i α i d v i = v i β i. From T (p, q, ϕ p, ϕ q, u, v ), oe c defie the udirected grph G (p, q, ϕ p, ϕ q, u, v ), clled the T -depedecy grph, s follows: its vertices re pirs (r, ), with r Q d A, tht model trsitios. There is edge ((r, ), (t, )) i G (p, q, ϕ p, ϕ q, u, v ) if d oly if for ll τ T (p, q, ϕ p, ϕ q, u, v ), r = t. The T -depedcy grph G (p, q, ϕ p, ϕ q, u, v ) stisfies the two followig properties: For ll i x, there exists edge ((p u i, α i), (q v i, β i)). For ll (w 1, w 2 ) ϕ p (resp. (w 3, w 4 ) ϕ q ), we hve w 1 = w 1 1 d w 2 = w 2 2 with 1, 2 A d such tht there exists edge ((p w 1, 1 ), (p w 2, 2)) (resp. ((q w 3, 3), (q w 3, 3))). Lemm 2. If G (p, q, ϕ p, ϕ q, u, v ) cotis cyclic sugrph iduced y suset of odes with j edges, the: T (p, q, ϕ p, ϕ q, u, v ) T j Proof. Two trsitios i the sme coected compoets of G (p, q, ϕ p, ϕ q,- u, v ) shre the sme rrivl stte. Hece if x is the umer of coected compoets i the grph, the T (p, q, ϕ p, ϕ q, u, v ) x. If G (p, q, ϕ p, ϕ q, u, v ) cotis cyclic sugrph with exctly j edges, the there is t most k j coected compoets. 2.5 The F-Depedecy Grph I this susectio, we slightly modify the otio of depedecy grph itroduced i [1]. Let τ e fixed trsitio structure with sttes d l e iteger such tht 1 l <. Let p, q e two sttes of τ such tht p q d u word of legth l. We defie F τ (p, q, u) s the set of sets of fil sttes F for which i the utomto (τ, F) the sttes p d q re seprted y the word u. Tht is to sy : F τ (p, q, u) = {F {1,..., } for ll (τ, F), p u 1 q, [p u F ] [q u F ]} From the set F τ (p, q, u) oe c defie the udirected grph G τ (p, q, u), clled the F-depedecy grph, s follows:

7 1 2 8, ε () () Fig.3. () is fixed trsitio structure d () the F-depedecy grph for p = 3, q = 9 d u =. Thks to (), we kow tht ll sttes i sme coected compoet will e either ll fil or ll o-fil. Hece, there re t most 2 4 possile sets of fil sttes, isted of 2 9. its set of vertices is {1,...,}, the set of sttes of τ; there is edge (s, t) etwee two vertices s d t if d oly if there exists word w of legth less th l such tht s = p w d t = q w d for ll F F τ (p, q, u), [s F ] = [t F ]. The F-depedecy grph cotis some iformtio tht is sic igrediet of the proof: it is coveiet represettio of ecessry coditios for set of fil sttes to e i F τ (p, q, u), tht is, for Moore s lgorithm to require more th u itertios ecuse of p, q d u. Figure 3 shows exmple of F- depedecy grph. Lemm 3. [1] If G τ (p, q, u) cotis cyclic sugrph with t lest i edges, the F τ (p, q, u) 2 i. The otios of depedecy grphs d depedecy tree will e used i susectios 3.2 d 3.3 to oti upper ouds o the crdil of sets of utomt with give properties d prove tht their cotriutio to the verge complexity is egligile. 3 Moore s Algorithm: verge cse lysis I [1], it is proved tht the verge complexity of Moore s stte miimiztio lgorithm is O( log ). Sice the result is otied y studyig oly properties o the sets of fil sttes of utomt miimized i give complexity, it holds for y distriutio o the set of trsitio structures. I this pper, i order to improve the upper oud o the verge complexity, we lso hve to study some properties of trsitio structures. Sice the eumertio of ccessile utomt with give properties is ope prolem, we focus our study o the uiform distriutio over the set of complete determiistic utomt.

8 3.1 Mi Result d Decompositio of the Proof The mi result of this pper is the followig. Theorem 1 For y fixed iteger k 2 d for the uiform distriutio over the determiistic d complete utomt with sttes over k-letter lphet, the verge complexity of Moore s stte miimiztio lgorithm is O( log log ). Recll tht the umer of prtitio refiemets mde y Moore s stte miimiztio lgorithm is smller or equl to 2 d tht A i is the set of utomt of A for which i is the smllest iteger such tht i is equl to Myhill-Nerode equivlece. The verge umer of prtitio refiemets is give y: ( N = 1 2 ) (i + 1) A i A i=0 We defie λ = log k log , which will e used i the sequel. We gther the sets A i, i order to oti the followig upper oud: 5log 2 N λ + 1 A i A + (5 log 2 + 1) A i A A i A i λ i=λ+1 i=5 log }{{}}{{} 2 +1 }{{} S1 S2 S3 S1 is less th λ d equl to O(log log ). I [1], it is proved tht 2 i=5 log 2 +1 Ai A. Therefore we kow tht S3 is equl to O(1). Hece, i the followig, we prove tht: S2 = (5 log 2 + 1) A 5log 2 i=λ+1 A i = O(log log ) (1) For y l > 0, we defie the set A (p, q, l) s the set of utomt with sttes, where the sttes p d q re seprted durig the l-th prtitio refiemet: A (p, q, l) = {(τ, F) A τ T, F {1,..., }, p l 1 q, p l q} Remrk 1. Note tht if i the utomto (τ, F), for ll letter A, p = q, the either p 0 q or p q. Therefore (τ, F) / A (p, q, l) with l > 0. Cosequetly, i the remider of the proof, i ll sets of trsitio structures where p d q re fixed, there exists letter such tht p q. The followig sttemet comes from the defiitio of the sets it ivolves: A i = A (p, q, λ + 1) i>λ p,q {1,...,} Let τ e trsitio structure, d p, q e two distict sttes. Recllig tht s h (p) is defied i Sectio 2.3, if A is k-letter lphet, d µ positive iteger, we defie two properties ssocited to trsitio structures:

9 (1) lrgetree(τ, p, µ) is true if d oly if s µ (p) k µ 1. Note tht this implies tht s µ 2 (p) k µ 2 1. (2) oitersectio(τ, p, q) is true if d oly if s λ 2 (p) s λ 2 (q) =. For fixed sttes p d q, utomto is i A (p, q, λ + 1) if its ssocited trsitio structure is i oe of the three distict sets we re out to defie: X is the set of ll trsitio structures τ such tht: there exists stte r Q such tht lrgetree(τ, r, λ) is flse. Note tht this set does ot rely o the vlues of p d q. Y (p, q) is the set of trsitio structures τ such tht: for ll stte r Q, the property lrgetree(τ, r, λ) is true, for ll words w A 2, the property oitersectio(τ, p w, q w) is flse. α (p, q) is the set of trsitio structures τ such tht: for ll stte r Q, the property lrgetree(τ, r, λ) is true, there exists w A 2 such tht oitersectio(τ, p w, q w) is true. 3.2 Trsitio Structures with Huge F-Depedecy Grph Lemm 4. For y fixed trsitio structures τ α (p, q), d fixed word u of legth λ+1, the followig property holds: every F-depedecy grph G τ (p, q, u) cotis cyclic sugrph with t lest k λ 2 1 edges. Proof. Let G e the sugrph G τ (p, q, u) defied s follows: there exists edge (p wv, q wv) i G, if d oly if v lels ode i S λ 2 (p w). G cotis exctly s λ 2 (p w) edges, sice for ll v S λ 2 (p w), the sttes p wv re ll pirwise distict. Sice lrgetree(τ, r, λ) is true for ll stte r Q, we hve s λ 2 (p w) k λ 2 1. G is cyclic: ideed, the property oitersectio(τ, p w, q w) idictes tht the set of odes coected to t lest oe edge forms iprtite grph (the odes of s λ 2 (p w) o oe side d the odes of s λ 2 (q w) o the other), where ll the odes of s λ 2 (p w) re oly coected to oe edge. Corollry 1. For λ = log k log , the umer ( of utomt i A (p, q, λ+ 1) whose trsitio structures re i α (p, q) is O T 2 log ). 3 Proof. This follows directly from Lemms 3 d 4 : for y distict sttes p d q, y word u of legth λ + 1, d y fixed trsitio structure τ α (p, q), we hve F τ (p, q, u) = O (2 log 3) 2 For fixed trsitio structure τ α (p, q), sice the umer of words i A λ+1 is O(log ), the umer of choices of sets of fil sttes such tht the utomt re i A (p, q, λ + 1) is ouded ove y: ( 2 ) log F τ (p, q, u) = O 3 u A λ+1

10 3.3 Negligile Sets of Trsitio Structures ( Lemm 5. The umer of trsitio structure i X is O T log5 Proof. For fixed stte r d fixed iteger µ {1,...,λ}, we defie the sets X (r, µ) of ll trsitio structures τ for which µ is the smllest iteger such tht the property lrgetree(τ, r, µ) is flse. We hve : X = X (r, µ) r {1,...,} µ {1,...,λ} For ll trsitio structures i X (r, µ), the depedecy tree R(r) cotis t lest two leves of depth less or equl to µ. Ideed, if R(r) cotis t most oe lef of depth less th µ, the there exist k 1 letters A such tht R(r ) does ot coti y lef of depth less th µ. Therefore we hve S µ (p) k µ 1. We decompose the possile depedecy trees R(r) ito two differet kids: 1. All leves re t level µ. Let k e the size of the lphet d f the umer of leves, the umer of trees of this kid is equl to k µ ( k µ ) f=2 f. 2. There exists exctly oe lef of depth h (with h < µ), d t lest oe of depth µ. The umer of trees of this kid is t most µ 1 h=1 (k h k ( µ k µ ) ) f=1. Usig the upper oud of Lemm 1 o the umer of trsitio structures couted y ech tree, we oti: [ k µ (k ) ( ) ] [ µ f µ 1 Sµ (r) k µ ( ) ( ) ] k X (r, µ) T + k h µ f+1 Sµ (r) T f f f=2 h=1 f=1 Sice µ λ, we hve S µ (r) < k λ+1 d: [ k λ (k ) ( ) λ X (r, µ) < T k λ+1 f ] + λk2λ+1 f f f=2 Sice we hve ( ) ( ) f ( ) k f: λ k λ+1 k 2λ+1 X (r, µ) < T k4λ+2 2 k λ f=1 ( ) k 2λ+1 f + λk4λ+2 f=0 2 ). [ (k ) ( ) λ k λ+1 f ] f ( ) k 2λ+1 f f=0 ) X (r, µ) = O ( T λk4λ = O ( T log4 3 log log 3 ) 2 Sice this upper oud holds for y µ {1,..., λ} d y r Q, we oti: X X (r, µ) = O ( T log4 3 log 2 log 3 ) r {1,...,} µ {1,...,λ} 2 f

11 Lemm 6. For ( y distict sttes p d q, the umer of trsitio structures i Y (p, q) is O T log6 ). 3 Proof. For y trsitio structure i Y (p, q), for ll words w A 2, there exist two words u, v A λ 2 such tht p wu = q wv. We prtitio the set Y (p, q) ccordig to the leves the depedecy trees coti. Both trees do ot coti lef of depth less or equl to λ : let E e the set of letters, such tht for ll E, p = q. We defie e = E. Accordig to Remrk 1, we hve e < k. For ll A \ E d ll c A, oitersectio(τ, p c, q c) is flse. For u d v of size x = e + k(k e), such tht for ll 1 j e, we hve u j = v j = j with j E d such tht for ll e < j x, w j is prefix of u j d v j, where w j is word of the form c. This suset is icluded i: T (p, q,,, u, v ) u, v E A E, p =q There re 2 k 1 possile susets E. There re less th k 2λ(x e) possile choices for u i, v i A λ, for e < i x. For ll j, l x, j l, sice u j d u l (resp. v j d v l ) lel odes i R(p) (resp. R(q)), settig u j = u j α j d u l = u l α l, we hve (p u j, α j) (p u l, α l) d there is o pth etwee (p u j, α j) d (p u l, α l) sice if would imply tht p u j = p u l d tht either u j or u l lels lef. Therefore, G (p, q,,, u, v ) cotis cyclic grph with x edges ((p u j, α j), (q v j, β j)). Sice x k + 1 (for e = k 1), usig Lemm 2, we oti the upper oud stted ove. At lest oe tree cotis lef of depth less or equl to λ : due to lck of spce, we will ot descrie this set. The ide is tht, just like i the previous cse, we re le to gurtee tht T -depedecy grph lwys cotis cyclic sugrph with k + 1 edges d tht there is O(log 2(k+1) ) possile grphs. 3.4 Cocludig the proof Recll tht we wt to prove Equtio 1. We defie X, α (p, q) d Ỹ(p, q) s the sets of utomt whose trsitio structure re respectively i X, α (p, q) d Y (p, q). We hve: A i = A (p, q, λ + 1) X α (p, q) Ỹ(p, q) i>λ p,q {1,...,} Usig Lemms 4,5 d 6 we oti: i>λ A i T 2 log 5 (5 log 2 + 1) A p,q {1,...,} ( + 2 T 2 log 3 + T 2 log 6 ) 3 5log 2 i=λ+1 ( log 7 ) A i = O = O (log log ) A log6 Hece N = O(log log ), this cocludes the proof of the mi theorem.

12 4 Coclusio I this pper, we otied ew upper oud o the verge complexity of Moore s stte miimiztio lgorithm, for the uiform distriutio o complete determiistic utomt. Also, it is possile to descrie set of Hopcroft s lgorithm executios which, for y determiistic utomt, compute the equivlece i less steps th Moore s lgorithm (due to lck of spce, we re ot givig the descriptio of those executios i this pper). Hece, for the uiform distriutio o complete determiistic utomt with sttes, there exists executio of Hopcroft s lgorithm whose verge complexity is O( log log ). This pper is first step to prove the cojecture mde i the coclusio of [1]: for the uiform distriutio o complete determiistic ccessile utomt, the verge complexity of Moore lgorithm is Θ( log log ). To prove this cojecture is ot esy tsk, sice it requires etter kowledge of the verge size of the ccessile prt i complete determiistic utomto, ut lso the verge umer of miiml utomt mogst the complete determiistic d ccessile. I would like to thk Phillipe Ducho for the fruitful discussio o upper oud of the crdil of the set X, ut lso Cyril Nicud d Frederique Bssio for their dvices d commets. Refereces 1. Frederique Bssio, Julie Dvid, d Cyril Nicud. O the verge complexity of Moore s stte miimiztio lgorithm. I Suse Alers d Je-Yves Mrio, editors, 26th Itertiol Symposium o Theoreticl Aspects of Computer Sciece (STACS 2009), Freiurg, Germy., volume 3 of LIPIcs, pge Schloss Dgstuhl - Leiiz-Zetrum fuer Iformtik, Germy, Frederique Bssio d Cyril Nicud. Eumertio d rdom geertio of ccessile utomt. Theor. Comput. Sci., 381:86 104, Je Berstel d Olivier Crto. O the complexity of Hopcroft s stte miimiztio lgorithm. I M. Domrtzki, A. Okhoti, K. Slom, d S. Yu, editors, CIAA O4, volume 3317 of Lecture Notes i Computer Sciece, pges Spriger, Giusi Cstiglioe, Atoio Restivo, d Mriell Sciortio. O extreml cses of Hopcroft s lgorithm. I S. Meth, editor, CIAA 09, volume 5642 of Lecture Notes i Computer Sciece, pges Spriger, Dvid Gries. Descriig lgorithm y Hopcroft. Act If., 2:97 109, Joh E. Hopcroft. A log lgorithm for miimizig sttes i fiite utomto. Techicl report, Stford Uiversity, Stford, CA, USA, Timo Kuutil. Re-descriig lgorithm y Hopcroft. Theor. Comput. Sci., 250(1-2): , Edwrd F. Moore. Gedke experimets o sequetil mchies. I Automt Studies, pges Priceto U., Ail Nerode. Lier utomto trsformtio. I Proc. Americ Mthemticl Society, pges , 1958.