Operations on Unambiguous Finite Automata

Transcription

1 Opertions on Unmbiguous Finite Automt Jozef Jirásek, Jr., Glin Jirásková, 2,, nd Jurj Šebej Mthemticl Institute, Slovk Acdemy of Sciences Grešákov 6, 4 Košice, Slovki jirsekjozef@gmil.com, jirskov@sske.sk 2 Institute of Computer Science, Fculty of Science, P. J. Šfárik University, Jesenná 5, 4 Košice, Slovki jurj.sebej@gmil.com Abstrct. A nondeterministic finite utomton is unmbiguous if it hs t most one ccepting computtion on every input string. We investigte the complexity of bsic regulr opertions on lnguges represented by unmbiguous finite utomt. We get tight upper bounds for intersection (mn), left nd right quotients (2 n ), positive closure (3/4 2 n ), str (3/4 2 n ), shuffle (2 mn ), nd conctention (3/4 2 m+n ). To prove tightness, we use binry lphbet for intersection nd left nd right quotients, ternry lphbet for str nd positive closure, fiveletter lphbet for shuffle, nd seven-letter lphbet for conctention. We lso get some prtil results for union nd complementtion. Introduction A nondeterministic mchine is unmbiguous if it hs t most one ccepting computtion on every input string. Ambiguity ws studied intensively minly in connection with context-free lnguges nd it is well known tht the clsses of mbiguous, unmbiguous, nd deterministic context-free lnguges re ll different. Ambiguity in finite utomt ws first considered by Schmidt [22] in his unpublished thesis, where he obtined lower bound 2 Ω( (n)) on the conversion of unmbiguous finite utomt into deterministic finite utomt, s well s for the conversion of nondeterministic finite utomt into unmbiguous finite utomt. He lso developed n interesting lower bound method for the size of unmbiguous utomt bsed on the rnk of certin mtrices. Sterns nd Hunt [24] provided polynomil lgorithms for the equivlence nd continment problems for unmbiguous finite utomt (UFAs), nd they extended them to mbiguity bounded by fixed integer k. Chn nd Ibrr [5] provided polynomil spce lgorithm to decide, given nondeterministic finite utomton (NFA), whether it is finitely mbiguous. They lso showed tht it is PSPACE-complete to decide, given n NFA M nd n integer k, whether M is k-mbiguous. Reserch supported by VEGA grnt 2/84/5. Reserch supported by VEGA grnt /42/5.

2 Ibrr nd Rvikumr [3] defined the mbiguity function M (n): N N of n NFA M such tht M (n) is the mximum number of distinct ccepting computtions of M on ny string of length n, nd they proved tht the exponentil mbiguity problem is decidble for NFAs. Weber nd Seidl [25] showed tht if n n-stte NFA is finitely mbiguous, then it is t most 5 n/2 n n -mbiguous. Alluzen et l. [] considered ε-nfas, nd they proved tht, given trim ε-cycle free NFA A, it is decidble in time tht is cubic in the number of trnsitions of A, whether A is finitely, polynomilly, or exponentilly mbiguous. Rvikumr nd Ibrr [2] considered the reltionship between different types of mbiguity of NFAs to the succinctness of their representtions, nd they provided complete picture for unry nd bounded lnguges. Exponentilly nd polynomilly mbiguous NFAs were seprted by Leung [6] by providing, for every n, n exponentilly mbiguous n-stte NFA such tht every equivlent polynomilly mbiguous NFA requires 2 n sttes. The UFA-to-DFA trdeoff ws improved to the optiml bound 2 n by Leung [7]. He described, for every n, binry n-stte UFA with unique initil stte whose equivlent DFA requires 2 n sttes. Similr binry exmple with multiple initil sttes ws given by Leiss [5], nd ternry one ws presented lredy by Lupnov [8]; note tht the reverse of Lupnov s ternry witness for NFA-to-DFA conversion is deterministic. Leung [7] elborted the Schmidt s lower bound method for the number of sttes in UFA. He considered, for lnguge L, mtrix whose rows re indexed by strings x i nd columns by strings y i, nd the entry in row x i nd column y j is if x i y j L nd it is otherwise. He showed tht the rnk of such mtrix provides lower bound on the number of sttes in ny UFA for L. Using this method, he ws ble to describe for every n n n-stte finitely mbiguous NFA, whose equivlent UFA requires 2 n sttes. A lower bound method ws further elborted by Hromkovič et l. [2]. They used communiction complexity to show tht so clled exct cover of ll s with monochromtic sub-mtrices in communiction mtrix of lnguge provides lower bound on the size of ny UFA for the lnguge. This llowed them to simplify proofs presented in [22,24]. Using communiction complexity methods, Hromkovič nd Schnitger [] showed seprtion of finitely nd polynomilly mbiguous NFAs, nd even proved hierrchy for polynomil mbiguity. A survey pper on unmbiguity in utomt theory ws presented by Colcombet [6], where he considered word utomt, tropicl utomt, infinite tree utomt, nd register utomt. He showed tht the notion of unmbiguity is not well understood so fr, nd tht some chllenging problems, including complementtion of UFAs, remin open. Unry unmbiguous utomt were exmined by Okhotin [2], who proved tht the tight upper bound for UFA-to-DFA conversion in the unry cse is given by function in e Θ( 3 n(ln n) 2), while the trde-off for NFA-to-UFA conversion is e n ln n(+o()). He lso considered the opertions of str, conctention, nd complementtion on unry UFA lnguges, nd obtined the tight upper bound (n ) 2 + for str, n upper bound mn for conctention which is tight if m, n re reltively prime, nd lower bound n 2 ϵ for complementtion. 2

3 In this pper, we continue this reserch nd study the complexity of bsic regulr opertions on lnguges represented by unmbiguous finite utomt. First, we restte the lower bound method from [7,22]. Using the notions of rechble nd so clled co-rechble sttes in n NFA N, we ssign mtrix M N to the NFA N in such wy tht the rnk of M N provides lower bound on the number of sttes in ny UFA for the lnguge L(N). We use this to get ll our lower bounds. To get upper bounds, we first construct n NFA for the lnguge resulting from n opertion, nd then we pply the (incomplete) subset construction to this NFA to get n incomplete DFA, so lso UFA, for the resulting lnguge. As result, we get tight upper bounds for intersection (mn), left nd right quotients (2 n ), positive closure (3/4 2 n ), str (3/4 2 n ), shuffle (2 mn ), nd conctention (3/4 2 m+n ). To prove tightness, we lwys use smll fixed lphbets. Since the reverse of n mbiguous finite utomt is unmbiguous, the complexity of lnguge nd its reverse is the sme. Finlly, we get some prtil results for complementtion nd union. 2 Preliminries We ssume tht the reder is fmilir with bsic notions in forml lnguges nd utomt theory. For detils nd ll the unexplined notions, the reder my refer to [,23,26]. Let Σ be finite lphbet of symbols. Then Σ denotes the set of strings over the lphbet Σ including the empty string ε. The length of string w is denoted by w, nd the number of occurrences of symbol in string w by # (w). A lnguge is ny subset of Σ. For finite set X, the crdinlity of X is denoted by X, nd its power-set by 2 X. A nondeterministic finite utomton (NFA) is 5-tuple N = (Q, Σ,, I, F ), where Q is finite nonempty set of sttes, Σ is finite nonempty input lphbet, Q Σ Q is the trnsition reltion, I Q is the set of initil sttes, nd F Q is the set of finl sttes. Ech element (p,, q) of is clled trnsition of N. A computtion of N on n input string n is seguence of trnsitions (q,, q )(q, 2, q 2 ) (q n, n, q n ). The computtion is ccepting if q I nd q n F ; in such cse we sy tht the string n is ccepted by N. The lnguge ccepted by the NFA N is the set of strings L(N) = {w Σ w is ccepted by N}. An NFA N = (Q, Σ,, I, F ) is unmbiguous (UFA) if it hs t most one ccepting computtion on every input string, nd it is deterministic (DFA) if I = nd for ech stte p in Q nd ech symbol in Σ, there is t most one stte q in Q such tht (p,, q) is trnsition of N. Let us emphsize tht we llow NFAs to hve multiple initil sttes, nd DFAs to be incomplete. The trnsition reltion my be viewed s function from Q Σ to 2 Q, which cn be extended to the domin 2 Q Σ in the nturl wy. We denote this function by. Using this nottion we get L(N) = {w Σ I w F }. 3

4 Every NFA N = (Q, Σ,, I, F ) cn be converted to n equivlent incomplete DFA N = (2 Q \{ }, Σ,, I, F ), where F = {R 2 Q \{ } R F }, nd for ech R in 2 Q \ { } nd ech in Σ, the prtil trnsition function is defined s follows: R = R if R nd R is undefined otherwise. We cll the DFA N the incomplete subset utomton of NFA N. Since every incomplete DFA is UFA, we get the following observtion. Proposition. If lnguge L is ccepted by n n-stte NFA, then L is ccepted by UFA of t most 2 n sttes. The reverse w R of string w is defined by ε R = ε nd (v) R = v R where Σ nd v Σ. The reverse of lnguge L is the lnguge L R defined by L R = {w R w L}. The reverse of n utomton N = (Q, Σ,, I, F ) is the NFA N R obtined from N by swpping the role of initil nd finl sttes nd by reversing ll the trnsitions. Formlly, we hve N R = (Q, Σ, R, F, I), where q R = {p Q q p } for ech stte q in Q nd ech symbol in Σ. The NFA N R ccepts the reverse of the lnguge L(N). Let N = (Q, Σ,, I, F ) be n NFA. We sy tht set S is rechble in N if there is string w in Σ such tht S = I w. Next, we sy tht set T is co-rechble in N if T is rechble in N R. In wht follows we re interesting in non-empty rechble nd co-rechble sets, nd we use the following nottion: R = {S Q S is rechble in N nd S }, () C = {T Q S is co-rechble in N nd T }. (2) The next observtion uses the notions of rechble nd co-rechble sets in n NFA to get chrcteriztion of unmbiguous utomt. Proposition 2. Let R nd C be the fmilies of non-empty rechble nd corechble sets in n NFA N. Then N is unmbiguous if nd only if S T for ech S in R nd ech T in C. Proof. (If) Assume tht S T for ech S in R nd ech T in C. Suppose for contrdiction, tht there is string w, on which N hs two distinct ccepting computtions. Let be the first symbol in w such tht fter reding, the two corresponding sttes in these computtions re distinct; denote the two sttes by p nd q, respectively. Then w = uv for some strings u nd v. Let S = I u nd T = F R v R. Then S R, T C, nd S T {p, q}, contrdiction. (Only if) For contrdiction, suppose tht N is unmbiguous nd there re two distinct sttes p, q in S T. Let S be rechble by u nd T be co-rechble by v. Then there re two distinct ccepting computtions on uv R u v : s p R f u v nd s 2 q R f 2 for some sttes s, s 2 in I nd f, f 2 in F, contrdiction. If N R is deterministic, then ech co-rechble set in N is of size one, nd we get the following result. Corollry 3. Let N be n NFA. If N R is deterministic, then N is unmbiguous. 4

5 Recll tht the stte complexity of regulr lnguge L, sc(l), is the smllest number of sttes in ny complete DFA ccepting the lnguge L. The stte complexity of regulr opertion is the mximl stte complexity of lnguges resulting from the opertion, considered s the function of stte complexities of the rguments. The nondeterministic stte complexity of lnguges nd opertions is defined nlogously using NFA representtion of lnguges. We define the unmbiguous stte complexity of regulr lnguge L, usc(l), s the smllest number of sttes in ny UFA for L. To prove tht DFA is miniml, we only need to show tht ll its sttes re rechble from the initil stte, nd tht no two distinct sttes re equivlent. To prove minimlity of NFAs, fooling set lower bound method my be used [2,8]. To prove lower bound for the size of UFA, method bsed on rnks of certin mtrices ws developed by Schmidt [22, Theorem 3.9], Leung [7, Theorem 2] nd Hromkovič et l. [2]. We use it in the following sttement. Proposition 4 ([2,7,22]). Let L be ccepted by n NFA N. Let R nd C be the fmilies of non-empty rechble nd co-rechble sets in N, respectively. Let M N be the mtrix in which the rows re indexed by sets in R, the columns re indexed by sets in C, nd in the entry (S, T ), we hve / if S nd T re/ re not disjoint. Then usc(l) rnk(m N ). Proof. Let A be miniml UFA ccepting L. Consider mtrix M A, in which rows re indexed by the sttes of A nd columns re indexed by strings generting the co-rechble sets in C. The entry (q, w) is if w R is ccepted by A from q, nd it is otherwise. Then every row of M N is sum of the rows of M A corresponding to the sttes in S: Notice tht since A is UFA, for every column there is t most one such row tht contins. Thus every row of M N is liner combintion of rows in M A, nd therefore rnk(m N ) rnk(m A ) usc(l). Throughout our pper, we use the following observtion from [6] nd its corollry stted in the proposition below. Lemm 5 ([6, Lemm 3]). Let Q = n nd M n be 2 n 2 n mtrix over the field with chrcteristic 2 with rows nd columns indexed by non-empty subsets of Q such tht M n (S, T ) = if S T nd M n (S, T ) = otherwise. Then the rnk of M n is 2 n. Proposition 6. Let L be ccepted by n NFA N. Let R be the fmily of ll non-empty rechble sets in N. If ech non-empty set is co-rechble in NFA N, then usc(l) R. Proof. Consider the mtrix M N given by Proposition 4. Notice tht M N contins R rows of the mtrix M n given in Lemm 5. By Lemm 5, the rnk of M n is 2 n, so the rows of M n re linerly independent. Therefore ll the rows of M N must be linerly independent, nd we hve rnk(m N ) = R. Hence usc(l) rnk(m N ) = R by Proposition 4. 5

6 3 Opertions on Unmbiguous Finite Automt We strt with the reversl nd intersection opertions. Then we continue with left nd right quotients. Notice tht if n NFA N is unmbiguous then lso N R is unmbiguous. Hence we get the following result. Theorem 7 (Reversl). Let L be regulr lnguge. Then usc(l R ) = usc(l). Proof. If L is ccepted by n unmbiguous NFA N, then L R is ccepted by N R which is unmbiguous s well, nd hs the sme number of sttes s N. Hence usc(l R ) usc(l). On the other hnd, the lnguge L R cnnot be ccepted by ny smller UFA becuse otherwise the lnguge L = (L R ) R would be ccepted by smller UFA s well. Theorem 8 (Intersection). Let K nd L be lnguges over Σ with usc(k)=m nd usc(l) = n. Then usc(k L) mn, nd the bound is tight if Σ 2. Proof. To get the upper bound, let K nd L be ccepted m-stte nd n-stte UFAs A = (Q A, Σ, A, I A, F A ) nd B = (Q B, Σ, B, I B, F B ), respectively. Construct the product utomton N = (Q A Q B, Σ,, I A I B, F A F B ), where (p, q) = p A q B. Then N is n mn-stte UFA for K L. To prove tightness, consider lnguges K = {w {, b} # (w) = m }, nd L = {w {, b} # b (w) = n }, which re ccepted by UFAs A = ({,,, m }, {, b}, A, {}, {m }), where i A = i + if i m 2 nd i A b = i for ech stte i, nd B = ({,,, n }, {, b}, B, {}, {n }), where j B = j for ech j nd j B b = j + if j n 2, respectively. Construct the product utomton N s described bove, nd notice tht ech singleton set {(i, j)} is rechble nd co-rechble in N. Hence M N is n mn mn identity mtrix, nd the theorem follows by Proposition 4. The left quotient of lnguge L by string w is w\l = {x w x L}, nd the left quotient of lnguge L by lnguge K is the lnguge K\L = w K w\l. The stte complexity of the left quotient opertion is 2n [27], nd its nondeterministic stte complexity is n+ [4]. In both cses, the witness lnguges re defined over binry lphbet. Our next result shows tht the tight upper bound for UFAs is 2 n. To prove tightness we use binry lphbet. Theorem 9 (Left Quotient). Let K, L Σ, usc(k) = m, nd usc(l) = n. Then usc(k \ L) 2 n, nd the bound is tight if Σ 2. Proof. To get n upper bound, let A be n n-stte UFA for L. Construct n n- stte NFA N for K \ L from A by mking initil ll sttes of A tht re rechble from the initil set by some string in K. By Proposition, usc(k \ L) 2 n. For tightness, let K = { k k m } nd L be the lnguge ccepted by the n-stte DFA A = ({,,, n }, {, b}, {}, {,,, n }) shown in Fig.. Notice tht ech stte of A is rechble by some string in K. Construct n n-stte NFA N for K \ L from A by mking ll the sttes initil. Hence the initil set of N is {,,, n }. Next, we cn shift every rechble subset right 6

7 b b b n 2 n Fig.. The UFA of lnguge L with usc(k \ L) = 2 n, where K = m. by one (modulo n) by reding, nd we cn remove the stte n from ny subset contining stte n by reding b. Therefore ech non-empty set is rechble in N. To construct N R, we only need to reverse the trnsitions on in N. The initil subset of N R is {,,, n }, nd we cn gin shift ny subset nd remove one stte s before. It follows tht every non-empty set is rechble in N R, tht is, co-rechble in NFA N. By Proposition 6, we hve usc(k \ L) 2 n. The right quotient of lnguge L by string w is L/w = {x x w L}, nd the right quotient of lnguge L by lnguge K is L/K = w K L/w. If lnguge L is ccepted by n n-stte DFA or NFA A, then the lnguge L/K is ccepted by n utomton tht is exctly the sme s A, except for the set of finl sttes tht consists of ll sttes of A, from which some string in K is ccepted by A [27]. Thus sc(l/k) n nd nsc(l/k) n. The tightness of the first upper bound hs been shown using binry lnguges in [27]. The second upper bound is met by unry lnguges m nd n. Our next im is to show tht the tight upper bound for unmbiguous finite utomt is 2 n, with witnesses defined over binry lphbet. Theorem (Right Quotient). Let K, L Σ, usc(k) = m, nd usc(l) = n. Then usc(l / K) 2 n, nd the bound is tight if Σ 2. Proof. To get n upper bound, let A be n n-stte UFA for L. Construct n n- stte NFA for L / K s described bove. By Proposition, usc(l / K) 2 n. To prove tightness, let K = { k k m } nd L be the lnguge ccepted by the n-stte NFA A = ({,,, n }, {, b}, {,,, n }, {n }) shown in Fig. 2. Since the utomton A R is deterministic, the NFA A is unmbiguous by Corollry 3. Since string in K is ccepted by A from ech stte of A, we construct n NFA N for L / K from A by mking ll the sttes of A finl. Notice tht we obtin the sme NFA s in the proof of the previous lemm, thus by the sme rguments usc(l / K) 2 n. b b b n 2 n Fig. 2. The UFA of lnguge L with usc(l / K) = 2 n, where K = m. 7

8 cd df df df d d, f d, f, c, c, c m c c, f c, f cf cf c c d d d b b b b, d b, d b, d n b cf c b Fig. 3. Witness UFAs for shuffle meeting the upper bound 2 mn (left) nd sketch of the resulting NFA N. Now let us continue with the shuffle nd conctention opertions. The shuffle of two strings u nd v over n lphbet Σ is defined s the set of strings u v = {u v u k v k u = u u k, v = v v k, u,, u k, v,, v k Σ }. The shuffle of lnguges K nd L over Σ is defined by KL = u K,v L u v. The stte complexity of the shuffle opertion on lnguges represented by incomplete deterministic utomt ws studied by Câmpenu et l. [3]. They proved tht 2 mn is tight upper bound for tht cse. Here we show tht the sme upper bound is tight lso for UFAs, nd to prove tightness, we use lmost the sme lnguges s in [3, Theorem ]. Up to our best knowledge, the problem is still open for complete deterministic utomt. Theorem (Shuffle). Let K, L Σ, usc(k) = m, nd usc(l) = n. Then usc(l K) 2 mn, nd the bound is tight if Σ 5. Proof. Let A = (Q A, Σ, A, I A, F A ) nd B = (Q B, Σ, B, I B, F B ) be m- nd n-stte UFAs for K nd L respectively. Then K L is ccepted by n mnstte NFA N = (Q A Q B, Σ,, I A I B, F A F B ), where for ech stte (p, q) in Q A Q B nd ech symbol in Σ, we hve (p, q) = (p A {q}) ({p} q B ). Hence usc(k L) 2 mn by Proposition. To prove tightness, let Σ = {, b, c, d, f}. Let K nd L be the regulr lnguges ccepted by DFAs A = ({,,, m }, Σ, A, {}, {m }) nd B = ({,,, n }, Σ, B, {}, {n }) shown in Fig. 3 (left); notice tht these DFAs re the sme s in [3, Theorem ] up to the position of finl sttes. Construct n NFA N for K L s described bove. Fig. 3 (right) shows sketch of the resulting NFA. It is shown in [3] tht ech non-empty set is rechble in N: The initil set {(, )} goes to the full set {,,, m } {,,, n } by c m d n, nd for ech subset S with (i, j) S, we hve S m i b n j f i b j = S \ {(i, j)}. Next, in N R we hve {(m, n )} R c m d n = {,,, m } {,,, n }, nd S R i b j f m i b n j = S \ {(i, j)} for ech subset S with (i, j) S. It follows tht ech non-empty set is co-rechble in N, so usc(l) 2 mn. 8

9 The conctention of lnguges K nd L is KL = {uv u K nd v L}. The stte complexity of conctention is m2 n 2 n, nd its nondeterministic stte complexity is m + n. In both cses, the witnesses re defined over binry lphbet [9,4,9,27]. In the next theorem we get tight upper bound for conctention on UFAs. To prove tightness, we use seven-letter lphbet. Theorem 2 (Conctention). Let K, L Σ, usc(k)=m, nd usc(l)= n, where m, n 2. Then usc(kl) 3/4 2 m+n, nd the bound is tight if Σ 6. Proof. Let A = (Q A, Σ, A, I A, F A ) nd B = (Q B, Σ, B, I B, F B ) be UFAs for lnguges K nd L, respectively. Let Q A = m, F A = k, Q B = n, I B = l. Construct n NFA N = (Q A Q B, Σ,, I, F B ) for KL, where for ech q in Q A Q B nd ech in Σ, q A, if q Q A nd q A F A = ; q = q A I B, if q Q A nd q A F A ; q B, if q Q B, nd { I A, if I A F A = ; I = I A I B, otherwise. Notice tht if set S is rechble in the NFA N nd S F A, then I B S. It follows tht the number of rechble sets is 2 m k 2 n +(2 m 2 m k )2 n l, which is mximl if l =. In such cse, this number equls (2 m + 2 m k )2 n, which is mximl if k =. After excluding the empty set, we get the upper bound. For tightness, let K nd L be lnguges over {, b, c, d, α, β, γ} ccepted by utomt A nd B shown in Fig. 4, where Q A = {q, q,, q m } nd Q B = {,,, n }. Notice tht A R nd B re deterministic, so A nd B re unmbiguous. Construct n NFA N for KL from utomt A nd B s described bove, tht is, by dding trnsitions on, b, c from stte q m 2 to stte, nd by dding trnsitions on α, β, γ from stte q m to stte. The initil stte of N is q nd the unique finl stte is n. α, β, γ, b, c, d α, β, γ, d α, β, γ, d α, β, γ q, b q, b, c, b, c q m 2, b, c q m, b, c, d, β, b, c, d, γ, b, c, d, γ, b, c, d, γ α α, β α, β n 2 α, β n α, β Fig. 4. Witness UFAs for conctention meeting the upper bound 3/4 2 m+n. 9

10 First, let us show tht the fmily of ll the non-empty sets tht re rechble in N contins ech subset S T, where S Q A, T Q B nd if q m S then T. Consider severl cses: () Let S = {q m } T, where T Q B nd T. The set {q m, } is reched from the initil set {q } by m. Next, ech set {q m,, j 2,, j k } of size k + is reched from the set {q m,, j 3 j 2 +,, j k j 2 + } of size k by the string αβ j 2. This proves cse () by induction. (2) Let S Q A, T Q B, nd T. The set {q } T is reched by from the set {q m } T which is considered in cse (). Next, in the NFA A, ech subset S of Q A with q / S is reched from set S of the sme size nd with S by string in, nd ech set such set S is reched from smller set by string in cb. This proves cse (2) by induction since there is loop on, b, c in ech stte of T, nd we hve T. (3) Let S Q A nd q m / S. Let T Q B nd / T. Then the set S T is reched by γ from the set S T {} which is considered in cse (2). It follows tht N hs 3/4 2 m+n non-empty rechble sets. Now let us show tht ech non-empty set is co-rechble in N. First, notice tht in B R, we cn shift ech set cycliclly by using string in α, nd we cn eliminte stte from ech set contining using γ. This mens tht ech subset of Q B is reched from Q B in B R. Similrly, ech string in performs cyclic shift of ny set in A R. Next, we cn eliminte stte q i from ny set S contining the stte q i nd stte q j with j > i: We shift S cycliclly so tht stte q i is moved to q, then we use string in c + to merge originl sttes q i nd q j. After tht reding string in results in eliminting the stte q i from the originl set S. Therefore, ech non-empty subset of Q A is rechble from Q A in A R. The empty set is reched from {q } by c in A R. It follows tht for ech S Q A with q m S nd ech T Q B, there exist string u S over {, c} nd string v T over {α, γ} such tht in N R, we hve {n } αn m α Q A {n } u S S {n } αn β n u S Q T B S T ; recll tht there is loop on, b, c in ech stte of Q B, nd there is loop on α, β, γ in ech stte of Q A. Next if q m / S, then for ech T Q B the set S T is reched from S {q m } T by d since we hve loop on d in ech stte of A nd B, except for stte q m. Hence ech non-empty set is co-rechble in NFA N, nd our proof is complete. Now we consider the Kleene closure (str) nd positive closure opertions. For lnguge L, the str of L is the lnguge L = i Li, where L = {ε} nd L i+ = L i L. The positive closure of L is L + = i Li. The stte complexity of the str opertion is 3/4 2 n with binry witness lnguges [9,27]. In the unry cse, the tight upper bound is (n ) 2 + [4,27]. The nondeterministic stte complexity of str is n +, with witnesses defined over unry lphbet [9]. We first consider the positive closure of UFA lnguges, nd we get tight upper bound 3/4 2 n on its complexity. Our worst-cse exmple is defined over ternry lphbet.

11 Theorem 3 (Positive Closure). Let L be lnguge over Σ with usc(l) = n, where n 2. Then usc(l + ) 3/4 2 n, nd the bound is tight if Σ 3. Proof. To get n upper bound, let A = (Q, Σ,, I, F ) be n n-stte UFA for L. Construct n NFA N = (Q, Σ, +, I, F ) for L + where the trnsition function + is defined s{ q I, if q F ; q + = q, otherwise for ech stte q in Q nd ech symbol in Σ. Notice tht if set S is rechble in N nd S F, then I S. Hence set contining some finl stte of A nd not contining some initil stte of A cnnot be rechble in N. Denote this fmily of unrechble sets by Unrech. Let us count the number of such sets. To this im, denote I \ F = i, I F = j, nd F \ I = k. Since A is unmbiguous, we must hve j. Consider three cses: () Let j =. Then we must hve i, k since A hs t lest one initil nd t lest one finl stte. A set in Unrech contins non-empty subset of F, non-universl subset of I, nd ny subset of the remining sttes. Thus we hve Unrech = (2 i )(2 k )2 n i k = ( 2 i )( 2 k )2n /4 2 n. (2) Let j = nd i = k =. Then A hs one initil stte, which is lso the only finl stte of A. Then L = L +, nd so usc(l + ) = usc(l). (3) Let j = nd let t lest one of i, k be positive. A set in Unrech which contins the unique initil nd finl stte must contin non-universl subset of I \ F. A set which does not contin this stte must contin non-empty subset of F \ I. Thus we hve Unrech = ( (2 i )2 k + 2 i (2 k ) ) 2 n i k = ( 2 k+ 2 i+ ) 2 n /4 2 n. In cse (2), we hve n 3/4 2 n since n 2. Next, in cses () nd (3), t lest /4 2 n subsets re unrechble. The empty subset is not in Unrech. It follows tht there re t most 3/4 2 n rechble non-empty subsets in N. This proves the upper bound. To prove tightness, let L be the lnguge ccepted by the ternry DFA A shown in Fig. 5 (top). Construct the NFA N for L + s described bove. Notice tht the DFA A restricted to the lphbet {, b} is the sme s the witness DFA for the str opertion from [27, Theorem 3.3, Fig. 4] In prticulr, this mens tht N hs 3/4 2 n non-empty rechble subsets. Let us show tht ech non-empty set is co-rechble in N. To this im, we use only the trnsitions on nd c. Fig. 5 (bottom) shows these trnsitions in N R. Notice, tht we cn shift ech subset S of {,,, n } cycliclly by one to the set {(s ) mod n s S}: we use if / S or if both nd n re in S, nd we use c otherwise. Next, we cn eliminte stte i from ny set contining sttes i nd i +. It follows tht ech non-empty set is rechble from {,,, n } in N R. To conclude the proof, notice tht the initil set {n } of N R goes to {,,, n } by the string n2.

12 bc c c c, b, b, b, b n 3 n 2 n b c c c c n 3 n 2 n c Fig. 5. The witness UFA for positive closure meeting the upper bound 3/4 2 n (top), nd the trnsitions on, c in the NFA N R (bottom). Finlly, let us show tht the tight upper bound for str on UFAs is 3/4 2 n, nd tht it is met by the lnguge ccepted by the UFA shown in Fig. 5 (top). Theorem 4 (Str). Let L be regulr lnguge over Σ with usc(l) = n, where n 2. Then usc(l ) 3/4 2 n, nd the bound is tight if Σ 3. Proof. The upper bound follows from the previous theorem since if ε L, then L + = L, nd otherwise we only need to dd one more initil nd finl stte to the UFA for L + to ccept the empty string. The resulting utomton is unmbiguous since the new stte ccepts only the empty string which is not ccepted by UFA for L +. For tightness, consider the lnguge L ccepted by the UFA A shown in Fig. 5 (top). Construct n NFA N for L from UFA A by dding new initil nd finl stte q, nd by dding the trnsitions on, b from n 2 to, nd the trnsition by c from n to. As shown in [27, Theorem 3.3] the NFA N hs 3/4 2 n rechble sets: the initil set {q, }, ll the subsets of {,,, n } contining stte, nd ll the non-empty subsets of {, 2,, n 2}. Next, consider the NFA N R. The initil set of N R is {q, n }. Next, s we hve shown in the proof of Theorem 3, ech non-empty set is rechble in N R. Now consider the 3/4 2 n 2 n mtrix M N. {q, n } T {n } T = {n } {q, } S n S S n / S Tble. Mtrix M N for NFA N for the str of the lnguge from Fig. 5 (top). 2

13 Assume tht the first row is indexed by {q, }, then there re rows indexed by rechble sets contining n, nd then by rechble sets not contining n. Next, ssume tht the first column is indexed by the initil co-rechble set {q, n }, the lst column is indexed by {n }, nd ll the remining columns re indexed by the remining co-rechble sets. This is illustrted in Tble. Then the rnk of the sub-mtrix obtined from M N by removing the first row nd the first column is 3/4 2 n. Next, notice tht the first nd the lst columns differ only in the entry in the first row. Hence the first row cnnot be expressed s liner combintion of ny others rows. Therefore rnk(m N ) = 3/4 2 n, nd the theorem follows by Proposition 4. 4 Prtil Results for Complementtion nd Union In this section we present prtil results for the complementtion nd union opertions on UFA lnguges. The complement of lnguge L over Σ is the lnguge L c = Σ \ L. A lnguge nd its complement hve the sme stte complexity since to get DFA for the complement of L, we only need to interchnge the sets of finl nd non-finl sttes in DFA for L. For NFAs, the tight upper bound for complementtion is 2 n with witnesses defined over binry lphbet [9,4]. For unry UFAs, the problem ws studied by Okhotin who provided lower bound n 2 o() for complementtion of unry UFAs [2, Theorem 6]. In the next theorem we del with n upper bound. Then we consider union. Theorem 5 (Complementtion: Upper Bound). Let L be regulr lnguge with usc(l) = n, where n 7. Then usc(l c ) 2.79n+log n. Proof. Let A be n n-stte UFA for L nd R nd C be the sets of non-empty rechble nd co-rechble sets of A. First, we show tht usc(l c ) min{ R, C }. We hve usc(l c ) R since we cn get DFA for L c by pplying the subset construction to A nd by interchnging the sets of finl nd non-finl sttes in the resulting DFA tht hs R rechble sttes. Next, we hve usc(l c ) C since the NFA A R is unmbiguous, so usc((l R ) c ) C which mens tht usc(l c ) C since complement nd reversl commutes nd the reverse of UFA is UFA. Next, let k = mx{ X X R}, nd pick set S in R of size k. Then ech set in R hs size t most k, nd ech set in C my hve t most one element in S by Proposition 2. Thus R ( ) n + ( ) n ( ) n nd C (k + )2 n k. k If k n/2, then C (n/2 + ) 2 n/2 2.5n+log n, nd the theorem follows. Now ssume tht k < n/2. Then R k ( ) n k n( en k )k nd C n2 n k. Let r(k) = n( en k )k nd c(k) = n2 n k. Then r(k) increses, while c(k) decreses with k. It follows tht if we pick k such tht k < n/2, then usc(l c ) r(k ) if k k, nd usc(l c ) c(k ) otherwise. By setting k = nx nd by solving ( en nx )nx = 2 n nx, we get x =.244, k =.244n, r(k ) n+log n, nd c(k ) n+log n. This completes our proof. 3

14 b b b p p p 2 p m d d d q c q c q 2 c c q n c Fig. 6. The UFAs for union meeting the bound mn + m + n; the loops on c, d resp. on, b in ech stte of A nd B, respectively, re not displyed. Proposition 6 (Union). Let K nd L be lnguges over Σ with usc(k) = m nd usc(l) = n, where m n. Then () usc(k L) m + n usc(k c ) m + n2.79n+log n ; (b) the bound mn + m + n is met if Σ 7. Proof. () The clim follows from the equlity K L = K (L K c ), where denotes disjoint union, since we hve usc(l K c ) n usc(l c ) by Theorem 8, nd, moreover, the NFA for disjoint union of UFAs is unmbiguous. The second inequlity is given by Theorem 5. (b) Let K nd L be the lnguges over {, b, c, d} ccepted by DFAs A nd B, such tht we hve loop on c, d in ech stte of A nd loop on, b in ech stte of B; the remining trnsitions re shown in Fig. 6. Construct n NFA N for K L from A nd B so tht the set of initil sttes is {p, q } nd the set of finl sttes is {p,, p m, q,, q n }. Then the sets {p i, q j }, {p i }, {q j } with i m nd j n re rechble in N. To conclude the proof notice tht ech set is co-rechble in N since we cn shift ny set in A cycliclly by reding nd eliminte one stte by reding b, while ech stte of B remins in itself. The sme cn by done symmetriclly with the sets in B. 5 Conclusions We investigted the complexity of bsic regulr opertions on lnguges represented by unmbiguous finite utomt. Since the reverse of n unmbiguous utomton is unmbiguous, lnguge nd its reversl hve the sme complexity for UFAs. Next, we obtined tight upper bounds for intersection (mn), left nd right quotients (2 n ), positive closure (3/4 2 n ), str (3/4 2 n ), shuffle (2 mn ), nd conctention (3/4 2 m+n ). To get upper bounds, we constructed n NFA for the lnguge resulting from n opertion, nd pplied the (incomplete) subset construction to it. For lower bounds, we defined witness lnguges in such wy tht we were ble to ssign mtrix to resulting lnguge. The rnk of this mtrix provided lower bound on the unmbiguous stte complexity of the resulting lnguge. 4

15 To prove tightness, we used binry lphbet for intersection nd left nd right quotients, ternry lphbet for str nd positive closure, five-letter lphbet for shuffle, nd seven-letter lphbet for conctention. For complementtion nd union, we provided upper bounds 2.79 n+log n nd m + n2.79 n+log n, respectively. Finlly, we got lower bound mn + m + n for union in the quternry cse. The exct complexity of complementtion nd union on UFAs remins open. All our result re summrized nd compred to the known results on the stte complexity nd nondeterministic stte complexity of the considered opertions in Tble 2. In the cse of complementtion, we tried to use fooling set lower bound method, but we were ble to describe fooling set for the complement of n n-stte UFA lnguge only of size n + log n. Moreover, it seems tht every such fooling set is of size which is liner in n [7]. Thus the fooling set technique cnnot be used to get lrger lower bound. Neither the method bsed on the rnk of mtrices cn be used here since the mtrices of lnguge nd its complement hve the sme rnk, up to one. Therefore, to get lrger lower bound for complementtion, some other techniques should be developed. sc Σ nsc Σ usc Σ reversl 2 n 2 n + 2 n intersection mn 2 mn 2 mn 2 left quotient 2 n 2 n n 2 right quotient n n 2 n 2 positive closure 3/4 2 n 2 n 3/4 2 n 3 str 3/4 2 n 2 n + 3/4 2 n 3 shuffle 2 (m )(n ) 5 mn 2 2 mn 5 conctention (m /2) 2 n 2 m + n 2 3/4 2 m+n 7 complement n 2 n n+log n n 2 ϵ [2] union mn 2 m + n + 2 m + n2.79n+log n mn + m + n 4 Tble 2. The complexity of regulr opertions: stte complexity, nondeterministic stte complexity, unmbiguous stte complexity, nd the sizes of lphbets for worst-cse exmples. References. Alluzen, C., Mohri, M., Rstogi, A.: Generl lgorithms for testing the mbiguity of finite utomt nd the double-tpe mbiguity of finite-stte trnsducers. Int. J. Found. Comput. Sci. 22(4), (2), S

16 2. Birget, J.: Prtil orders on words, miniml elements of regulr lnguges nd stte complexity. Theor. Comput. Sci. 9(2), (993), 6/ (93)96-U 3. Câmpenu, C., Slom, K., Yu, S.: Tight lower bound for the stte complexity of shuffle of regulr lnguges. Journl of Automt, Lnguges nd Combintorics 7(3), 33 3 (22) 4. Čevorová, K.: Kleene str on unry regulr lnguges. In: Jürgensen, H., Reis, R. (eds.) Descriptionl Complexity of Forml Systems - 5th Interntionl Workshop, DCFS 23, London, ON, Cnd, July 22-25, 23. Proceedings. Lecture Notes in Computer Science, vol. 83, pp Springer (23), 5. Chn, T., Ibrr, O.H.: On the finite-vluedness problem for sequentil mchines. Theor. Comput. Sci. 23, 95 (983), (88) Colcombet, T.: Unmbiguity in utomt theory. In: Shllit, J., Okhotin, A. (eds.) Descriptionl Complexity of Forml Systems - 7th Interntionl Workshop, DCFS 25, Wterloo, ON, Cnd, June 25-27, 25. Proceedings. Lecture Notes in Computer Science, vol. 98, pp Springer (25), 7/ _ 7. Eliáš, P.: Fooling sets for complements of UFA lnguges. Unpublished mnuscript (26) 8. Glister, I., Shllit, J.: A lower bound technique for the size of nondeterministic finite utomt. Inf. Process. Lett. 59(2), (996), 6/2-9(96) Holzer, M., Kutrib, M.: Nondeterministic descriptionl complexity of regulr lnguges. Int. J. Found. Comput. Sci. 4(6), 87 2 (23), Hopcroft, J.E., Ullmn, J.D.: Introduction to Automt Theory, Lnguges nd Computtion. Addison-Wesley (979). Hromkovič, J., Schnitger, G.: Ambiguity nd communiction. Theory Comput. Syst. 48(3), (2), 2. Hromkovič, J., Seibert, S., Krhumäki, J., Kluck, H., Schnitger, G.: Communiction complexity method for mesuring nondeterminism in finite utomt. Inf. Comput. 72(2), (22), 3. Ibrr, O.H., Rvikumr, B.: On sprseness, mbiguity nd other decision problems for cceptors nd trnsducers. In: Monien, B., Vidl-Nquet, G. (eds.) STACS 86, 3rd Annul Symposium on Theoreticl Aspects of Computer Science, Orsy, Frnce, Jnury 6-8, 986, Proceedings. Lecture Notes in Computer Science, vol. 2, pp Springer (986), Jirásková, G.: Stte complexity of some opertions on binry regulr lnguges. Theor. Comput. Sci. 33(2), (25), Leiss, E.L.: Succint representtion of regulr lnguges by boolen utomt. Theor. Comput. Sci. 3, (98), S (8) Leung, H.: Seprting exponentilly mbiguous finite utomt from polynomilly mbiguous finite utomt. SIAM J. Comput. 27(4), (998), dx.doi.org/.37/s

17 7. Leung, H.: Descriptionl complexity of nf of different mbiguity. Int. J. Found. Comput. Sci. 6(5), (25), S Lupnov, O.B.: A comprison of two types of finite utomt. Problemy Kibernetiki 9, Kibernetiki (963), (in Russin) Germn trnsltion: Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, (966) 9. Mslov, A.N.: Estimtes of the number of sttes of finite utomt. Soviet Mth. Dokldy, (97) 2. Okhotin, A.: Unmbiguous finite utomt over unry lphbet. Inf. Comput. 22, 5 36 (22), 2. Rvikumr, B., Ibrr, O.H.: Relting the type of mbiguity of finite utomt to the succinctness of their representtion. SIAM J. Comput. 8(6), (989), Schmidt, E.M.: Succinctness of description of context-free, regulr, nd finite lnguges. Ph. D. thesis. Cornell University (978) 23. Sipser, M.: Introduction to the theory of computtion. PWS Publishing Compny (997) 24. Sterns, R.E., Hunt, H.B.: On the equivlence nd continment problems for unmbiguous regulr expressions, regulr grmmrs nd finite utomt. SIAM J. Comput. 4(3), (985), Weber, A., Seidl, H.: On the degree of mbiguity of finite utomt. Theor. Comput. Sci. 88(2), (99), B 26. Yu, S.: Hndbook of Forml Lnguges: Volume Word, Lnguge, Grmmr, chp. Regulr Lnguges, pp. 4. Springer Berlin Heidelberg, Berlin, Heidelberg (997), Yu, S., Zhung, Q., Slom, K.: The stte complexities of some bsic opertions on regulr lnguges. Theor. Comput. Sci. 25(2), (994), doi.org/.6/ (92)-f 7