Languages recognized by nondeterministic quantum finite automata

Languages recognized b y nondeterministic quan tum finite automata ⋆ Abuzer Y ak aryılmaz and A.C. Cem Sa y Bo˘ gazi¸ ci Universit y , Department o f Computer Engineering, Bebek 34342 ˙ Istanbul, T urkey abuzer,say @boun.edu. tr F ebruary 12, 2010 Abstract. The nondeterministic quantum finite automaton (NQF A) is the only known case where a one-wa y q uantum fin ite automaton (QF A) mo del has b een show n to b e strictly sup erior in terms of language recognition p ow er to its probabilistic counterpart. W e give a c h aracterizatio n of the class of languages recognized by NQF A’s, demonstrating th at it is equal to the class of exclusive stochastic languages. W e also charac terize the class of languages that are recognized n ecessaril y by tw o-sided error by QF A’s. It is shown that these classes remain t h e same when the QF A’s used in their definitions are replaced by sev eral different mo del v ariants that have app eared in the literature. W e pro ve several closure p roperties of the related classes. The ramifications of these results about classical and quantum sublogarithmic space comp lex ity classes are examined. 1 In t r o duction An in teresting f eature of b oth probab ilistic and quantum compu tational m o dels is that in some cases, the set of p roblems that can b e solved gets larger wh en the automaton in question is allo w ed to mak e more err or in its decisions, wh ereas in s ome other cases, s uc h a relaxation do es not increase the computational p o we r at all. When one-wa y p robabilistic finite automata (PF A’s) are required to m ake n o error in their d ecisions, they recognize exactly the class of regular languages. When they are allo wed to mak e b ounde d err or , th at is, to give the correct resp onse for ea c h input with probabilit y at least 1 2 + δ , f or a fixed δ > 0, the class of languages that are recognized remains the same. The computational p o wer of P F A’s is seen to increase on ly wh en w e allo w t w o-sided unb ounde d err or , where the only requirement is that all m em b ers of the recognized language are accepted with probabilit y greater th an the acceptance pr obabilit y of any nonmemb er . There are sev eral alternativ e mod els of quantum finite automata (QF A’s), and differences (e.g. in features regarding the form an d n u m b er of measuremen ts that can b e p erformed on the mac h in e, whether an output tap e is included or not, etc.) in their definitions, w h ic h would not affect the p o wer of classical automata, yield QF A v arian ts of differing computational p ow er. In the zero-error and b ound ed-error cases, some QF A v arian ts [28,20,30] are strictly inferior to the corresp on d ing PF A’s from the p oint of view of language recognition p ow er, whereas the most general mo d els [31,11,7,17] are equiv alent to their probabilistic counte rparts in th ose settings. In the unb ounded error case, the languages recognized with cutp oin t by the w eak est QF A mo d el [28] form a prop er sub class [6] of th e corresp onding classical class (the s tochastic languages), w hereas it was disco vered recen tly [42,43] that more generalized QF A v arian ts, includ ing th e p opular K ondacs-W atrous mo del, are equiv alen t to PF A’s in this case as w ell. ⋆ This work was partially supp orted by the Scientific and T echnological R esearch Council of T urk ey (T ¨ UB ˙ IT AK) with grant 108142 and th e Bo˜ gazi¸ ci Universit y Research F und with grant 08A102. With regard to state complexit y , su fficien tly general QF A mo d els can sim ulate all zero-error and b ound ed-error PF A’s with small ov erhead, and some r egular languages ha v e b ou n ded-error QF A’s that are exp onent ially smaller than the corresp onding PF A [3]. In the t w o-sided unboun ded error setting, quant um and pr obabilistic m ac hines can sim ulate eac h other with only a p olynomial o ve rhead in the num b er of states [43]. W e stu d y the computational p o wer of QF A’s in the one-sided un b ounded error setting, wh ere one of the t w o r esp onses that the machine can outpu t ab out the mem b ership of th e inp ut string in the recog nized language is correct with certa in t y , and the other resp onse has a nonzero proba- bilit y of b eing correct. Since the er r or b ou n d can b e impro ve d b y rep eating the computatio n, an examination of languages recognizable in this setting is significan t for und erstanding the p ow er of generalizat ions of th e und erlying mo d el to include, say , a tw o-w ay tap e head. Just lik e their classical coun terp arts, Q F A’s that recognize their languages with cutp oin t 0, (that is, w ith p ositiv e one-sided error,) are also known as nondeterministic m ac hines. It is we ll kno wn that classical nond etermin istic finite automata recognize p recisely the regular languages. In notable p revious w ork on nondeter- ministic quan tum fin ite automata (NQF A’s), Bertoni and Carp en tieri ha ve sh o wn [6] that the class of languages recognized by NQF A’s of the Mo ore-Crutchfield t yp e do es not conta in any nonempty finite languages, bu t do es con tain the nonregular language L neq = { w ∈ { a, b } ∗ | | w | a 6 = | w | b } , where | w | σ denotes the num b er of o ccurrences of the sym b ol σ in the str ing w . Nak anish i et al. [29] considered the somewhat more p ow erful Kondacs-W atrous mo d el of QF A’s, and pro v ed that NQF A’s of this t yp e can recognize all regular languages, establishing their strict sup eriority ov er their classica l counterparts. In this pap er, we giv e a full characte rization of the class of languages recognized by all NQF A v arian ts that are at least as ge neral as the Kondacs-W atrous type, demonstrating that it is equal to the class of exclusive stochastic languages. This lies prop erly b et we en the classes of languages recognized with zero error and t w o-sided un b ounded error b y QF A’s 1 . Every regular language has a NQF A with at most linearly more states than the corresp ond ing classica l n ondeterministic finite automaton (NF A), and there exist infinite families of regular languages w hic h can b e recognized b y just tunin g the transition amplitudes of a NQF A with a constant num b er of states, whereas the sizes of the corresp ondin g NF A’s grow without b ound. W e also pro ve seve ral new closur e prop erties of the related classes, and examine what these r esu lts imply ab out the comparative p o w er of probabilistic vs. quan tum T uring mac h ines with small space b ound s. The r est of this pap er is str u ctured as follo ws : Section 2 con tains the relev ant d efinitions and previously kno w n facts. In S ection 3, we give a characte rization of the class of languages recognized b y Kondacs-W atrous NQF A’s, and discuss the sup eriorit y of sev eral NQF A v arian ts o ver their classical count erparts in terms of language recognition and succinctness. An examination of the relationships among languages which can b e recog nized by QF A’s with one-sided error and those that r equire t w o-sided err or is p resen ted in S ection 4. Section 5 con tains sev eral pro ofs of closure prop erties for the classes of languages recognized with one-sided error. Section 6 is a conclusion, where we examine the consequences of the NQF A results for classical and quantum sub logarithmic space complexity classes. 1 T o our know ledge, this is the only case where these three classes have b een sh own to b e distinct for any automaton mod el, b e it qu antum or classical. 2 Preliminaries 2.1 Automata In the follo w in g, Σ denotes the inp ut alphab et, not con taining the end-markers ¢ and $, and Γ is the tap e alph ab et, s uc h that Γ = Σ ∪ { ¢ , $ } . Definition 1. A (1-way) pr ob abilistic finite automaton (PF A) with n ∈ Z + states is a 4-tuple P = ( S, Σ , { A σ ∈ Γ } , F ) , wher e 1. S = { s 1 , · · · , s n } i s th e set of states, and s 1 is the start state, 2. A σ is the n × n r e al-value d sto chast ic tr ansition matrix for symb ol σ , that is, A σ ( i, j ) is the value of the tr ansition pr ob ability fr om state s i to state s j when r e ading symb ol σ , 3. F ⊆ S is the set of ac c epting states. The probab ility distribution of P ’s states at an y p oint during the pro cessing of the in put string can b e traced using an n -elemen t r o w vect or. F or an in put string w ∈ Σ ∗ , w = ¢ w $, v 0 = (1 , 0 , · · · , 0) 1 × n denotes the initial state ve ctor. The effect of reading the i th tap e sym b ol can b e calculated by multiplying the ve ctor v i − 1 b y the matrix A w i , y ieldin g v i . v | w | = v 0 A w 1 · · · A w | w | denotes the final state v ector. Th e acceptance pr ob ab ility of w by P is f P ( w ) = X s i ∈ F v | w | ( i ) , (1) where v | w | ( i ) denotes the i th en try of v | w | . Definition 2. A gener alize d pr ob abilistic finite automaton (GPF A ) with n ∈ Z + states is a 5-tuple G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) , wher e 1. S = { s 1 , · · · , s n } i s th e set of states, 2. A σ is the n × n r e al-value d tr ansition matrix for symb ol σ , that is, A σ ( i, j ) is the (p oss ibly ne gative) “ weight” of the tr ansition fr om state s i to state s j when r e ading symb ol σ , 3. v 0 is the r e al-value d initial 1 × n v e ctor, and, 4. f i s the r e al-value d final n × 1 ve ctor. A GPF A G is asso ciated by a fun ction f G : Σ ∗ → R , in the follo win g wa y: F or an inp u t string w ∈ Σ ∗ , f G ( w ) = v 0 A w 1 · · · A w | w | f . (2) Definition 3. A (1-way) Kondacs-Watr ous quantum finite automaton (KWQF A) [20] with n ∈ Z + states is a 5-tuple M = ( Q, Σ , { U σ ∈ Γ } , Q acc , Q r ej ) , wher e 1. Q = { q 1 , · · · q n } i s the set of states, and q 1 is the initial state, 2. U σ is the n × n c omplex-value d unitary tr ansition matrix for symb ol σ , that is, U σ ( j, i ) is the amplitude of the tr ansition fr om q i to q j when r e ading the symb ol σ , 3. Q acc and Q r ej , disjoint subsets of Q , ar e the sets of ac c epting and r eje cting states, and Q non = Q \ ( Q acc ∪ Q r ej ) is the set of non-halting states. The amplitude distribution of the states of a quant um automaton is represent ed by an n - elemen t c olumn vec tor. | u 0 i , the initial state v ector, equals (1 , 0 , · · · , 0) T 1 × n . Note the difference with probabilistic automata. F or a giv en inp ut string w ∈ Σ ∗ , M sca ns the tap e, con taining w = ¢ w $, from the left to the righ t. During the pro cessing of eac h symb ol, the machine un dergo es t wo op erations: First, its state v ector ev olv es according to the unitary trans f ormation associated with th e scanned symb ol, that is, | u i i = U w i | u i − 1 i . Then, the machine is observ ed to see wh ether it has accepted, rejected, or not halted y et. A t this p oin t, eac h accepting state with amplitude α adds | α | 2 to the o v erall acceptance pr obabilit y f M ( w ) of the input 2 . In the particular K W QF A’s that will b e describ ed in this pap er, in termediate observ ations can yield the result “reject”, but the accepting states can b e en tered only at the end of the co mputation, afte r scanning the r igh t end-mark er $. Halting states “drop out” of th e state v ector | u i i , their amplitudes b eing replaced with zeros, and the h ead mo ves on to the next symbol. As w e hav e defin ed them, PF A’s pro cess all of the inpu t string b efore deciding on acce ptance or rejection, wh ereas KW QF A’s can halt b efore reac hing the en d of the input. (The QF A v arian t that precisely corresp onds to Defin ition 1 is th e M o or e- Crutchfield QF A (MCQF A) [28 ].) This difference should n ot distract the reader, s in ce it is easy to show that the classes of languages recognized by PF A’s, b oth with general cutp oint , and with cutp oint 0, (to b e d efi ned in the next subsection,) do not c h ange wh en the mod el is mo d ified to give it this additional capabilit y 3 . Th is is tr ue for all PF A v arian ts that ma y b e obtained by app ropriately r econfiguring Definition 1 to corresp ond to the v arious Q F A mo dels that are cited in this pap er. The only cru cial d istinction b et w een Definitions 1 and 3 is the one b et wee n classical and quant um. 2.2 Languages Definition 4. A n automato n A define d over alph ab et Σ divides Σ ∗ into thr e e disjoint subsets with cutp oint λ ∈ R : 1. L ( A , < λ ) = { w ∈ Σ ∗ | f A ( w ) < λ } , 2. L ( A , = λ ) = { w ∈ Σ ∗ | f A ( w ) = λ } , 3. L ( A , > λ ) = { w ∈ Σ ∗ | f A ( w ) > λ } . A dditional ly, we define L ( A , 6 = λ ) = L ( A , < λ ) ∪ L ( A , > λ ) . Definition 5. The p air ( A , λ ) is e qu i valent under cutp oint sep ar ation to the p air ( A ′ , λ ′ ) , denote d as ( A , λ ) ≡ ( A ′ , λ ′ ) , if L ( A , < λ ) = L ( A ′ , < λ ′ ) L ( A , = λ ) = L ( A ′ , = λ ′ ) L ( A , > λ ) = L ( A ′ , > λ ′ ) , 2 This is the b eha vior allo w ed b y the general KWQF A definition. 3 Note that, when comparing tw o Q F A v ariants with each other, this kind of difference is v ery imp ortant, since it usually affects t he computational p ow er of the mo dels. wher e A , A ′ ar e automata and λ, λ ′ ∈ R ar e cutp oints. Definition 6. The language r e c o gnize d by automaton A with cutp oint λ ∈ R is define d as L ( A , λ ) = L ( A , > λ ) . L ( A , λ ) is said to b e r e c o gnize d by automaton A with one-si de d cutp oint λ ∈ R if L ( A , < λ ) = ∅ . Definition 7. [32] 1. The languages r e c o gnize d b y PF A’s with cu tp oint λ ∈ [0 , 1) c onstitute the class of sto chastic languages (S > ). The c ol le ction of languages whose c omplements ar e sto chastic is the class c o- S > . 2. L anguages of the form L ( P , = λ ) , for any PF A P , and any λ ∈ [0 , 1] , c onstitute the class S = . 3. L anguages of the form L ( P , 6 = λ ) , for any PF A P , and any λ ∈ [0 , 1] , c onstitute the class exclusive sto chast ic languages (S 6 = ). R emark 1. In the study of complexit y classes defined in terms of T ur ing mac hines, “recognition with cutp oin t” is used synonymously with “unboun ded-error recognition”. This u sage do es n ot provi de an appr opriate co ve rage of the intuitiv e concept of unb ounded-error computation that we d escrib ed in Section 1 in th e case of P F A’s: Giv en a PF A P w hic h recognize s a language L with cutp oint, one can b uild a new PF A P ′ for the complemen t of L b y just switc hing the stat uses of the accepting and n on-accepting states of P . Since P ′ accepts any member of L with greater probabilit y than any nonmem b er, w e sa y th at it recognize s L with unb ounded error. How ev er, since S > is not kno wn to b e closed under complemen tation, w e do not know in general w hether L is sto c hastic or not. F or this reason, we take S > ∪ co-S > to b e the class of languages recognized w ith unboun ded error by PF A’s 4 . Definition 8. 1. The languages r e c o gnize d by KWQF A’s (MCQF A’s) with cutp oint λ ∈ [0 , 1) c onstitute the class QL (M CL). 2. The languages r e c o gnize d by KWQF A’s (MCQF A ’ s) with cutp oint 0 , i.e., those of the form L ( M , 0) , for any KWQF A (MCQF A) M , c onstitute the class NQL (NMCL). As mentio ned b efore, n on d eterministic computation corresp onds to recognition with cutp oint 0, and so NMCL and NQL denote the classes of languages recognized by n ondeterministic MCQF A’s and K W QF A’s, resp ective ly . 4 Note that S > r at , the class of languages recognized with cutp oint 1 2 by PF A ’s whose transition matrices contain only rational numbers, is kn o wn [37 ] to b e closed un der complemen tation. I t is how ever customary to defin e PF A’s and QF A ’s with general transition probabilities/amplitudes, as we did in D efinitions 1 and 3, in the fi nite automata literature, and we follo w th is conv ention. S ee Section 6 for more on this p oint. F a ct 21. [36] Let G 1 b e a GPF A and λ 1 ∈ R b e a cutp oint. F or any cutp oint λ 2 ∈ R , there exists a GPF A G 2 suc h th at ( G 1 , λ 1 ) ≡ ( G 2 , λ 2 ). F a ct 22. [32] Let P 1 b e a PF A and λ 1 ∈ [0 , 1) b e a cutp oin t. F or an y cutp oint λ 2 ∈ (0 , 1), th ere exists a P F A P 2 suc h th at ( P 1 , λ 1 ) ≡ ( P 2 , λ 2 ). F a ct 23. [43] F or any PF A P , ther e exists a KWQF A M suc h that ( P , 1 2 ) ≡ ( M , 1 2 ). F a ct 24. [23,43] F or an y KWQF A M and cutp oint λ ∈ [0 , 1), th ere exists a GPF A G su c h that ( M , λ ) ≡ ( G , λ ). F a ct 25. [36] F or an y GPF A G and cu tp oin t λ 1 ∈ R , there exist a PF A P and a cutp oint λ 2 ∈ (0 , 1) suc h th at ( G , λ 1 ) ≡ ( P , λ 2 ). F a ct 26. [6] MCL ( S > . F a ct 27. [ 8] Any MCQF A with n states can b e simulated b y a KWQF A with 2 n states, so MCL ⊆ QL and NMCL ⊆ NQL. F a ct 28. [32] The class of regular languages is a prop er subset of b oth S 6 = and S = . F a ct 29. [32] S 6 = ( S > and S > \ S = 6 = ∅ . F a ct 210. [9,29] The class of regular languages is a prop er subset of NQ L. By F acts 23-25, QL = S > , PF A’s and KWQF A’s ha v e the same language r ecognitio n p o w er with general cutp oin t and with u n b ounded error (Remark 1). It has in f act b een shown [42] th at all one-w ay QF A m o dels [31,11,7,18] that generaliz e the KW QF A are also equiv alen t to the PF A in this regard. (S ee Subsection 3.2 for more on this.) W e are intereste d in the case of one-sided unboun ded error, where one of the tw o resp onses that the mac hine can outpu t ab out the members h ip of the inp u t string in the recognized language is correct with certain t y , and the other resp onse has a nonzero probabilit y of b eing correct. W e sa y that s u c h an automa ton has p ositive one-side d err or if it rejects n on-mem b ers of its language with certaint y . This corresp onds to recognition with cutp oin t 0. Th e opp osite case is called ne gative one-side d err or , where the language in question is of the form L ( A , 6 = 1), recalling that, when A is a P F A or a QF A, f A has range [0 , 1]. PF A’s can r ecognize all and only the regular languages with cutp oin t 0 [26]. KWQF A’s can do more than th at, as will b e charac terized in the n ext section. 3 Languages Recognized with One-sided Error W e start the pr esentati on of our main result b y stating a fact wh ic h will b e u seful in sev eral pro ofs in the p ap er. Lemma 1. F or any language L , L ∈ S 6 = if and only if ther e exists a GPF A that r e c o gnizes L with one-side d cutp oint 0 . Pr o of. The forw ard direction is pro v en on page 171 of [32]. In th e rev erse direction, if a GPF A recognizes L with one-sided cu tp oin t 0, then L ∈ S 6 = b y F act 25. 3.1 A c haracterization of NQL Lemma 2. S 6 = ⊆ NQL. Pr o of. If L ∈ S 6 = , then there exists an n -state PF A P = ( S, Σ , { A σ ∈ Γ } , F ) such that L = L ( P , 6 = 1 2 ). W e define S ′ , v ′ 0 , and { A ′ σ ∈ Γ } as f ollo ws: 1. S ′ = S ∪ { s n +1 , s n +2 , s n +3 } ; 2. v ′ 0 = (1 , 0 , · · · , 0) is a 1 × ( n + 3)-dimensional r o w v ector; 3. Eac h A ′ σ is a ( n + 3) × ( n + 3)-dimensional matrix: A ′ ¢ =      1 2 A ¢ [ r 1 ] 0 0 1 2 1 0 · · · 0 0 0 0 . . . . . . 1 0 · · · 0 0 0 0      , A ′ σ ∈ Σ =         A σ 0 n × 3 1 0 0 0 3 × n 0 1 0 0 0 1         , A ′ $ =         A $ 0 n × 3 1 0 0 0 3 × n 0 1 0 0 0 1                  t 1 , 1 t 1 , 2 0 0 n × n . . . t n, 1 t n, 2 0 1 0 0 0 3 × n 0 1 0 − 1 2 1 2 0          , where A ¢ [ r 1 ] is the fir st row of A ¢ ; t i, 1 = 1 and t i, 2 = 0 when s i ∈ F , and t i, 1 = 0 and t i, 2 = 1 when s i / ∈ F for 1 ≤ i ≤ n . F or a giv en input w ∈ Σ ∗ , w = ¢ w $, let v ′ | w | = v ′ 0 A ′ ¢ A ′ w 1 · · · A ′ w | w | A ′ $ . It is ea sily v erified that this computation “imitates” the pro cessing of w by P ; the first n en tries of the m anipulated ve ctor v ′ con tain exactly the state v ector of P (m ultiplied by 1 2 ) in the corresp onding steps of its execution. The last matrix m ultiplication results in v ′ | w | =  0 1 × n 2 f P ( w ) − 1 4 , 3 − 2 f P ( w ) 4 , 0  . The ( n + 1) st en try of v ′ | w | equals 0 if and only if w / ∈ L . Using a mo d ifi ed version of the PF A sim ulation metho d d escrib ed in [43], w e can construct a KW QF A M = ( Q, Σ , { U σ ∈ Γ } , Q acc , Q r ej ) recognizing L with cutp oint 0: F or eac h σ ∈ Γ , U σ is built according to the template U T σ =         c σ A ′ σ c σ B σ c σ C σ D σ         , where B σ = [ b i,j ] is a lo wer triangular matrix, and C σ = [ c i,j ] is a diagonal matrix. The ent ries of U σ are computed iterativ ely using the follo wing pro cedur e: 1. The en tries of B σ and C σ are set to 0. 2. The en tries of B σ are u p dated to make the ro ws of  A ′ σ | B σ  pairwise orthogonal. Sp ecifically , for i = 1 · · · n + 2 set b i,i = 1 for j = i + 1 · · · n + 3 set b j,i to some v alue so that the i th and j th rows b ecome orthogonal set l max to the maxim um of the lengths (norms ) of the rows of  A ′ σ | B σ  3. The d iagonal entries of C σ are u p dated to m ak e the length of eac h r o w of  A ′ σ | B σ | C σ  equal to l max . Sp ecifically , for i = 1 · · · n + 3 set l i to the cu r ren t length of the i th ro w of  A ′ σ | B σ  set c i,i to q l 2 max − l 2 i 4. Set c σ to 1 l max . 5. The entries of D σ are selected to mak e U T σ a un itary matrix. The transp ose accounts for the difference b et w een th e p robabilistic and quan tu m vect or notations. The s tate s et Q = Q non ∪ Q acc ∪ Q r ej is sp ecified as: 1. q n +1 ∈ Q acc corresp onds to state s n +1 ; 2. q n +2 ∈ Q r ej corresp onds to state s n +2 ; 3. { q 1 , · · · , q n , q n +3 } ∈ Q non corresp ond to the r emaining states of S ′ , where q 1 is the start state; 4. All the new states that are d efined durin g the construction of { U σ ∈ Γ } are rejecting ones. M simulate s the computation of P for a given inpu t string w ∈ Σ ∗ b y repr esen ting the p robabilit y of eac h stat e s j b y the amplitud e of the corresp onding state q j ; sp ecifically , th is amplitude equals c ¢  Q k i =1 c w i  × v ′ ( j ) imm ed iately after the ( k + 1) st step of the compu tation [43], w here k ≤ | w | . The trans itions from the 2 n + 6 states added during the construction of U σ ∈ Γ for ensuring u nitarit y do not in terfere with this sim ulation, s ince the computation halts immediately on th e “branc hes” where th ese states are en tered. T herefore, th e top n + 3 en tries of the state v ector of M equal c ¢   | w | Y i =1 c w i   c $  0 1 × n 2 f P ( w ) − 1 4 , 3 − 2 f P ( w ) 4 , 0  T just b efore th e last measuremen t on the right end-marker. Since the amplitude of the only accepting state is nonzero if and only if w ∈ L , L is recognized by M with cutp oin t 0. Lemma 3. NQ L ⊆ S 6 = . Pr o of. By F act 24, there exists a GPF A w ith one-sided cutp oin t 0 for an y mem b er L of NQL . By Lemma 1, L is an exclusive stochastic language . Theorem 1. S 6 = = N QL. Corollary 1. S = is pr e cisely the class of languages tha t c an b e r e c o gnize d with ne gative one-si de d err or by KWQF A’s. The sup eriorit y of KW QF A’s o v er PF A’s in the one-sided error sett ing no w follo ws from F act 28. By F act 29, there exist languages that KW QF A’s can recognize with t w o-sided, but not one-sided error. Th e class of these languages is precisely (S > ∪ co-S > ) \ (S 6 = ∪ S = ) (Remark 2.1). Note that the ab o ve results also establish that the class of languages r ecognized by NQF A’s is n ot closed under complemen tation (F act 52). 3.2 More general QF A models Sev eral one-wa y Q F A mo dels (lik e [30,31,11,7,18], and the one-w ay v ersion of the mac hin es of [4],) that generalize the KW QF A ha ve app eared in th e literature. In the b ounded-err or case, some of these generalized mac hin es recognize more languages than the KW Q F A. W e claim that the classes of languages recognize d by the n ondeterministic v ers ions of all th ese automata are iden tical to eac h other, and they coincide w ith NQL. W e demonstrate this f act for one of the most general mo d els, n amely , the quan tum finite au- tomaton with ancilla qubits (QF A-A) [31], whic h can sim u late all kn own one-w a y QF A mo dels. Let us giv e the name QF A-A 0 to the class of languages recognized with cutp oin t 0 b y QF A-A’s. F or an y QF A-A M , there exists a GPF A that compu tes exactly the s ame acceptance probabilit y function as M [45], so QF A-A 0 ⊆ S 6 = b y Lemma 1. Since any KW Q F A can b e simulate d b y a QF A-A, NQL ⊆ QF A-A 0 . Therefore, QF A-A 0 = NQL= S 6 = . 3.3 Space e fficiency of QF A’s with cutp oint 0 It is well kno wn [3,27] that some infinite families of languages can b e recognized with one-sided b ound ed error by just tun ing th e transition amplitudes of a QF A with a constant n u m b er of states, whereas the s izes of the corresp ond ing PF A’s grow without b ound. After a simple examp le, we w ill argue that th is adv an tage is also v alid in the unboun d ed error case. Definition 9. F or m ∈ Z + , L m ⊆ { a } ∗ is define d as L m = { a i | i mod ( m ) 6 = 0 } . Theorem 2. F or m > 1 , L m c an b e r e c o gnize d by a 2-state MCQF A 5 with cutp oint 0 . Pr o of. M b egins the computation at state q 0 , and eac h transition with the sym b ol a corresp onds to a rotation 6 b y angle π m in the | q 0 i - | q 1 i plane, where q 1 is the accepting state. F or an y p ositiv e n , it is kn o wn [26] that ev ery n -state PF A w ith cutp oin t 0 has an equiv alent nondeterministic finite automat on with the same n u m b er of states. Th er efore, only finitely man y distinct languages can b e recognized with one-sided unb ounded error b y PF A’s with at most n states. Com b ining this with th e f act that any n -state PF A with c utp oint 0 can b e sim ulated by a KW QF A w ith 2 n + 4 states us in g a simple adaptation of th e tec hnique of [43 ], th e sup er iority of QF A’s o ver PF A’s in this regard is established 7 . 4 S 6 = , S = , and Languages Recognized wit h Tw o-sided Error T o gain a b etter understanding of the classes of languages recognizable by p ositiv e one-sided, neg- ativ e one-sided, and necessarily tw o-sided error by QF A’s, w e examine some examples fr om eac h of those families. Bertoni and C arp ent ieri [6] sh o wed that L neq is in NMCL, and that its complement, sa y , L eq = { w ∈ { a, b } ∗ | | w | a = | w | b } , is not in MCL. No w that w e h a ve T heorem 1, w e can use the w ell-known results [32, ? ] from the P F A literature that state that L eq ∈ S = , L neq ∈ S 6 = , but not vice v ersa, to conclude that stronger QF A v ariant s also can not recognize L eq with p ositiv e on e-sided error, and neither can they r ecognize L neq with negativ e one-sided err or. S imilarly , L¯ ace et al. [21] pro v ed recen tly th at the complement of th e palindrome language L pal = { w ∈ { a, b } ∗ | w = w r } is in NQL. W e can sh o w the corresp on d ing resu lt f or L pal using the follo wing fact: F a ct 41. [14] Let L ∈ S = . Then there exists a natural num b er n ≥ 1, suc h that f or an y s trings u, v , y ∈ Σ ∗ , if uv , uy v , · · · , uy n − 1 v ∈ L, then uy ∗ v ⊆ L. Theorem 3. L pal / ∈ S 6 = . 5 There is an equiv alent 4-state KWQF A. 6 F or det ails of a similar construction for a nonregular language, see [6]. 7 Note that a QF A -A can realize this simulation with just n states. Pr o of. Supp ose that L pal ∈ S 6 = . Then L pal ∈ S = . Let u = a n b , y = a , and v = ε . a n b, a n ba, · · · , a n ba n − 1 ∈ L pal imply that a n ba n ∈ L pal b y F act 41. Since this string is actually a member of L pal , we hav e a con tradiction. W e will n o w exhib it some languages which can only b e recognize d by t wo-sided error by a QF A. Theorem 4. L = { aw 1 ∪ bw 2 | w 1 ∈ L eq , w 2 ∈ L neq } ∈ S > \ ( S = ∪ S 6 = ) . Pr o of. Supp ose that L ∈ S 6 = , then th ere exists a GPF A G = ( S, Σ , { A σ ∈{ a,b } } , v 0 , f ) recognizing L with one-sided cu tp oin t 0. Th e GPF A G ′ = ( S, Σ , { A σ ∈{ a,b } } , v 0 A a , f ) recognizes L eq with one-sided cutp oin t 0, meaning that L eq ∈ S 6 = . This con tradicts the well-kno wn fact men tioned in the first paragraph of this section. Supp ose now that L ∈ S = , then L = { ε ∪ aw 2 ∪ bw 1 | w 1 ∈ L eq , w 2 ∈ L neq } is in S 6 = , whic h also results in a contradict ion for the same reason. Since b oth L eq and its complemen t are s tochastic, it is not difficult to s ho w that L is sto c h astic. Lemma 4. L lt = { w ∈ { a, b } ∗ | | w | a < | w | b } / ∈ ( S = ∪ S 6 = ) . Pr o of. Supp ose that L lt ∈ S = . Let u = ε , y = a , and v = b n . b n , ab n , · · · , a n − 1 b n ∈ L lt imply th at a n b n ∈ L lt b y F act 41. Sin ce this string is actually a mem b er of L lt , we ha v e a con tr a- diction. Similarly , supp ose that L lt ∈ S 6 = , or L lt ∈ S = . Let u = a n , y = b , and v = b . a n b, a n b 2 , · · · , a n b n ∈ L lt imply th at a n b n +1 ∈ L lt b y F act 41. Sin ce this string is actuall y a memb er of L lt , w e ha v e a con tradiction. Corollary 2. L lt ∈ S > \ ( S = ∪ S 6 = ) . Pr o of. This follo ws from Lemma 4 and the fact that L lt ∈ S > [33,19]. Theorem 5. L eq · b = L eq · b + ∈ S > \ ( S = ∪ S 6 = ) . Pr o of. The p ro of of L eq · b / ∈ (S = ∪ S 6 = ) uses the setup presen ted in Lemma 4 , i.e., 1. select u = ε , y = a , and v = b n to con tradict w ith L eq · b ∈ S = , 2. select u = a n , y = b , and v = b to contradict with L eq · b ∈ S 6 = . An y strin g w is a mem b er of L eq · b if and only if it has th e f ollo wing thr ee pr op erties: – w ends with b . – w ∈ L lt . – Let u b e the longest p refix of w ending with a ( u = ε if w ∈ { b ∗ } ). Then, u ∈ L lt . Since these prop erties can b e chec k ed easily by a t w o-wa y PF A with b ounded error, L eq · b ∈ S > [33,19]. W e conclude this section b y sho wing th e sto chastic it y of an imp ortan t family of languages. Definition 10. [24] The wor d pr oblem f or a gr oup is the pr oblem o f de ciding whether or not a pr o duct of gr oup elements is e qual to the identity element. Definition 11. L et G k = ( G, ◦ ) b e a finitely g e ner ate d fr e e gr oup with a b asis Σ = { σ 1 , . . . , σ k , σ − 1 1 , . . . , σ − 1 k } , wher e k ∈ Z + is the r ank of G k . L w p ( G k ) ⊆ Σ ∗ is the language define d as L w p ( G k ) = { w = w 1 · · · w | w | | w i ∈ Σ , 1 ≤ i ≤ | w | , w 1 ◦ · · · ◦ w | w | = ı } , wher e ı ∈ G is the identity element of G k . F a ct 42. (P age 1 of [25]) Let G k 1 1 and G k 2 2 b e finitely generated free groups. Then G k 1 1 and G k 2 2 are isomorphic if and only if k 1 = k 2 . Corollary 3. L w p ( G k 1 1 ) and L w p ( G k 2 2 ) ar e isomorph ic if and only if k 1 = k 2 , wher e G k 1 1 and G k 2 2 ar e finitely gener ate d fr e e gr oups. As a generic n ame, L w p ( k ) can b e used instead of L w p ( G k ) due to Corollary 3, where k ∈ Z + . F a ct 43. [39] L w p (1) ∈ S > . F a ct 44. [9] L w p ( k ) ∈ co-NMCL, the class of languages whose complements are in NMCL, for any k ∈ Z + . Corollary 4. L w p ( k ) ∈ S = for any k ∈ Z + . W e will no w pr o vide a pro of of the follo wing theorem. Theorem 6. L w p ( k ) ∈ S > for any k ≥ 2 . In fact, Th eorem 6 wa s stated as a corollary on p age 1463 of [9], but the pu rp orted p ro of there w as b ased on the claim that co-NMCL ⊆ MCL ⊆ S > . It is how ev er known [6], as we m en tioned ab o ve, that a member of co-NMCL ( L eq ) lies outside MCL. F urtherm ore, th e s ame demonstration can b e easily extended to L w p ( k ), where k ∈ Z + . Corollary 5. L w p ( k ) / ∈ M CL for any k ∈ Z + . Since it is still an op en pr oblem wh ether S = ⊆ S > or not, w e cannot use Corollary 4 directly to pro v e Th eorem 6. In stead, we w ill fo cus on a su b class of S = that is kno wn to b e a sub s et of S > . Definition 12. [37] S = r at is the class of the languages of the form L ( G , = λ ) , wher e G is a r ational GPF A, (i.e. one whose tr ansition matric es and initial and final ve ctors c ontain only r ational num- b ers,) and λ is a r ational numb er. A dditional ly, S 6 = r at is the class of languages whose c omplements ar e in S = r at . F a ct 45. [37] S = r at ( S > . Definition 13. SO 3 ( Q ) is the gr oup of r otations on R 3 that ar e 3 × 3 dimensiona l ortho gonal matric es having only r ational entries with determinant +1 . F a ct 46. [34,13] F or any k ≥ 2, SO 3 ( Q ) con tains a free subgroup with rank k , namely S k . Pr o of of The or em 6. F or L w p ( k ), we d efine a rational GPF A G k = ( { s 1 , s 2 , s 3 } , Σ , { A σ ∈ Σ } , v 0 , f ), where 1. Σ = { R 1 , . . . , R k , R − 1 1 , . . . , R − 1 k } is a basis of S k ; 2. A σ = σ for eac h σ ∈ Σ ; 3. v 0 = (1 , 0 , 0); 4. f = (1 , 0 , 0) T . It is ob vious that w ∈ L w p ( k ) if and only if A 1 . . . A w = I 3 × 3 if and only if f G k ( w ) = v 0 A 1 . . . A w f = 1, where w ∈ Σ ∗ . Th u s, L w p ( k ) ∈ S = r at b y selecti ng the cutp oin t as 1. W e can conclude with F act 45. 5 Closure Pr op erties The p reviously disco vered closure prop erties of S > , S 6 = and S = are listed b elo w. F a ct 51. 1. S > is not closed under union and intersectio n [16 ,15,22]. 2. S > is closed under union and in tersection w ith a regular language [10,35]. 3. S > is closed under rev ersal [36]. 4. S > is not closed under concate nation, K leene closure, and homomorphism [12,38]. 5. S > is closed under complemen tation ov er u n ary alphab ets [16]. F a ct 52. 1. Both S 6 = and S = are closed under union and in tersection [32 ]. 2. Neither S 6 = nor S = is closed under complemen tation [14]. 3. S > is closed under in tersection with a member of S 6 = [32]. W e will pro v e several new non trivial closure pr op erties of the “one-sided” classes S 6 = and S = . 5.1 Dissimilar closure prop erties of S 6 = and S = The p ro ofs of the n ext few th eorems use the capabilit y of GPF A’s to implement n ondeterministic branc hing by ju st adding the tr an s ition matrices of the branches, and the nice prop erties of com- putation with one-sided cutp oin t 0. Theorem 7. S 6 = is close d under c onc atenation. Pr o of. If L 1 , L 2 ∈ S 6 = , then there exist t wo GPF A’s G 1 = ( S ′ , Σ , { A ′ σ ∈ Σ } , v ′ 0 , f ′ ) and G 2 = ( S ′′ , Σ , { A ′′ σ ∈ Σ } , v ′′ 0 , f ′′ ) suc h that L 1 and L 2 are r ecognize d with one-sided cutp oin t 0 b y G 1 and G 2 , resp ectiv ely . Let n 1 and n 2 b e th e sizes of the s tate sets S 1 and S 2 , resp ectiv ely . W e construct a new GPF A G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) recognizing L = L 1 L 2 (concatenat ion of L 1 and L 2 ) with one-sided cutp oint 0. The details of G are as follo w s: 1. The size of S is n = n 1 + n 2 ; 2. v 0 is a 1 × n row v ector, (a) v 0 = ( v ′ 0 | v ′′ 0 ) if ε (empty string) b elongs to L 1 , and (b) v 0 = ( v ′ 0 | 0 1 × n 2 ) if ε / ∈ L 1 ; 3. f is a n × 1 column vec tor, (a) f = (( f ′ ) T | ( f ′′ ) T ) T if ε ∈ L 2 , and (b) f = (0 1 × n 1 | ( f ′′ ) T ) T if ε / ∈ L 2 ; 4. { A σ ∈ Σ } is the set of n × n matrices, A σ =  A ′ σ X σ 0 n 2 × n 1 A ′′ σ  , (3) where X σ is an n 1 × n 2 matrix, defined as X σ =     v ′′ 0 (1) A ′ σ f ′ v ′′ 0 (2) A ′ σ f ′ · · · v ′′ 0 ( n 2 ) A ′ σ f ′ | {z } column 1 | {z } column 2 | {z } column n 2     . (4) The idea b ehind the construction is that for a giv en input string w ∈ Σ ∗ , eac h prefix of w , sa y , u ∈ Σ ∗ ( w = uv ), is chec k ed for b elonging to L 1 , and if so, the rest, v , is c heck ed for b elonging to L 2 . G sim u lates G 1 in the firs t n 1 p ositions of its s tate v ector. If G 1 accepts an in p ut prefix u ending with σ , the r esu lt of the m u ltiplicatio n b et wee n th at 1 × n 1 ro w ve ctor d escribing the d istribution after pro cessing the fir st | u | − 1 inpu t symb ols and the column v ector A ′ σ f ′ , that is, f G 1 ( u ), will b e p ositiv e. Otherwise, f G 1 ( u ) = 0. By Equations 3 and 4, the v ector f G 1 ( u ) . v ′′ 0 will b e added to the last n 2 p ositions of G ’s state v ector, m eaning th at G 2 will run on the remainder v of the inpu t. The con tribu tion of this b r anc h of the computation to f G ( w ) is jus t the p ro duct of the v alue f G 2 ( v ) and the co efficien t f G 1 ( u ), and will b e p ositiv e if b oth s ubstrings are accepted b y the resp ectiv e mac hines, and zero otherwise. Since G 2 starts run ning in this manner in ea c h step, its part of the o ve rall state v ector con tains in general th e sum of many 1 × n 2 v ectors, multiplied by th eir resp ectiv e co efficien ts, at any intermediate step of the compu tation. The cases where the empt y string app ears in L 1 or L 2 are handled appropriately . In other w ords, f G ( w ) = X uv = w f G 1 ( u ) f G 2 ( v ) , and G recognizes the concatenation of L 1 with L 2 . Theorem 8. S = is not c lose d u nder c onc atenation. Pr o of. L eq and { b } + are in S = , but L eq · b = L eq . { b } + is not, due to Theorem 5. Theorem 9. S 6 = is close d under Kle ene closur e. Pr o of. If L ∈ S 6 = , then there exists a GPF A G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) suc h that L ∪ { ε } is recognized b y G with one-sided cutp oin t 0. Let n b e the size of the s tate set S . W e construct a new GPF A G ′ = ( S, Σ , { A ′ σ ∈ Σ } , v 0 , f ) r ecognizing L ∗ (Kleene closure of L ) with one-sided cutp oin t 0. Each element of { A ′ σ ∈ Σ } is defined as A ′ σ = A σ + X σ , (5) where X σ is an n × n matrix, defined as X σ =     v 0 (1) A σ f v 0 (2) A σ f · · · v 0 ( n ) A σ f | {z } column 1 | {z } column 2 | {z } column n     . (6) F or a giv en in p ut strin g w ∈ Σ ∗ , | w | = l , f G ′ ( w ) = v 0 [ A w 1 + X w 1 ][ A w 2 + X w 2 ] · · · [ A w l + X w l ] f , and s o f G ′ ( w ) = X u 1 u 2 ··· u k = w k Y i =1 f G ( u i ) ! , where 1 ≤ k ≤ l and ea c h u i ∈ Σ ∗ . Therefore, if w can b e divided, i.e., w = u 1 · · · u k , suc h that eac h u i ∈ L ( f G ( u i ) > 0), then f G ′ ( w ) > 0. On the ot her hand, if there is no suc h division, then f G ′ ( w ) = 0. Lemma 5. L et L eq ′ = { w ∈ { a, b } ∗ | | w | a + 1 = | w | b } . Then, L + eq ′ = L lt . Pr o of. It is ob vious that if w ∈ L + eq ′ , then w ∈ L lt . If w ∈ L lt , then ∃ k ∈ Z + suc h th at | w | b = | w | a + k . Then, there m ust exist k + 1 indices, i 0 = 0 < i 1 < i 2 < · · · < i k = | w | , s uc h that eac h prefi x of w of length i j has j m ore b ’s than a ’s, where 1 ≤ j ≤ k . In other wo r ds, w can b e partitioned into k consecutiv e s u bstrings, u 1 , u 2 , · · · , u k , satisfying 1. w = u 1 u 2 · · · u k , and, 2. | u j | = i j − i j − 1 , that is, u j b egins with the ( i j − 1 + 1)th s ym b ol of w and ends with the i j th sym b ol of w , wh ere 1 ≤ j ≤ k . Since eac h u j is a mem b er of L eq ′ , w e can conclud e that w ∈ L + eq ′ . Theorem 10. S = is not c lose d u nder Kle ene closur e. Pr o of. It can b e sho wn easily that L eq ′ is in S = . Ho wev er, L ∗ eq ′ , wh ich is L lt ∪ { ε } by Lemma 5, is not in S = due to Corollary 2. Lemma 6. L et h : Σ → Σ \ { κ } b e a homomorphism such that h ( σ ) =  σ , σ 6 = κ ε , σ = κ , wher e κ is a sp e cific symb ol in Σ . If L ⊆ Σ ∗ is in S 6 = , then so is h ( L ) . Pr o of. Let G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) b e the GPF A recognizing L with one-sided cutp oin t 0, and let Σ ′ = Σ \ { κ } . F or any w ∈ h ( L ), there exists a u ∈ L , suc h that h ( u ) = w , i.e., u = κ c 0 w 1 κ c 1 · · · κ c | w |− 1 w | w | κ c | w | for some nonnegativ e integ er c i ’s, where 0 ≤ i ≤ | w | . In fact, we can b ound all c i ’s by a natural n um b er, sa y n L , due to F act 41: Supp ose that none of the strings in { κ c 0 w 1 κ c 1 · · · κ c | w |− 1 w | w | κ c | w | | 0 ≤ i ≤ | w | , 0 ≤ c i ≤ n L } are m em b ers of L (so th ey are all mem b ers of L ∈ S = ). Then, b y u sing F act 41, κ ∗ w 1 κ c 1 · · · κ c | w |− 1 w | w | κ c | w | ⊆ L, (1 ≤ i ≤ | w | , 0 ≤ c i ≤ n L ) κ ∗ w 1 κ ∗ · · · κ c | w |− 1 w | w | κ c | w | ⊆ L, (2 ≤ i ≤ | w | , 0 ≤ c i ≤ n L ) . . . κ ∗ w 1 κ ∗ · · · κ ∗ w | w | κ ∗ ⊆ L. W e conclude that w / ∈ h ( L ), wh ic h is a con tradiction. Therefore, for an y input s tr ing w ∈ ( Σ ′ ) ∗ , we can sim ulate the computation of G on some u ’s, where eac h c i is guessed nond eterministically fr om the set { 0 , 1 , · · · , n L − 1 } . The follo wing matrix can b e defined to imp lemen t the non d eterministic b ranc hin g of th e computation: X κ = I + n L − 1 X j = 1 A j κ . By em b edd ing X κ in a conv enien t wa y in the definition of G , w e can get the GPF A G ′ = ( S, Σ ′ , { A ′ σ ∈ Σ ′ = A σ X κ } , v 0 X κ , f ) , whic h r ecognize s h ( L ) with one-sided cutp oin t 0. Hence, f G ′ ( w ) can b e calculated as f G ′ ( w ) = X u ∈{ κ c 0 w 1 κ c 1 ··· κ c | w |− 1 w | w | κ c | w | } f G ( u ) for the inp ut string w ∈ ( Σ ′ ) ∗ , where 0 ≤ c i ≤ n L − 1, and 0 ≤ i ≤ | w | . Since the computation p aths resulting in u / ∈ L pro duce f G ( u ) = 0, f G ′ ( w ) > 0 is satisfied only when there is a computation path resulting in u ∈ L . Lemma 7. L et h : Σ → Υ ∗ b e a homo morphism such tha t | h ( σ ) | > 0 for al l σ ∈ Σ . If L ⊆ Σ ∗ is in S 6 = , then so is h ( L ) . Pr o of. Let G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) b e the GPF A recognizing L with one-sided cutp oin t 0. W e will sho w that there exists a GPF A G ′ = ( S ′ , Υ , { A ′ γ ∈ Υ } , v ′ 0 , f ′ ) recognizing h ( L ) with one-sided cutp oin t 0. G ′ runs G on a nondeterministically c h osen input u = u 1 u 2 · · · u | u | ∈ Σ ∗ , w hile chec king whether its o w n inpu t s tr ing w matc hes h ( u 1 ) h ( u 2 ) · · · h ( u | u | ) or not. F or eac h suc h nondeterministic com- putation path, w e hav e the follo win g cases: 1. at least one of the m atc hes f ail, h ( u ) 6 = w , then all ent ries rep r esen ting the state vecto r of G in this branc h are set to zero, 2. all substitutions succeed, h ( u ) = h ( u 1 ) h ( u 2 ) · · · h ( u | u | ) = w , then (a) f G ( u ) = 0 for u / ∈ L , and (b) f G ( u ) > 0 for u ∈ L . f G ′ ( w ) will aga in b e defined as the summation o v er all computation paths, i.e., f G ′ ( w ) = P { u | h ( u )= w } f G ( u ). Hence, w ∈ h ( L ) only if there is at least one su ccessful s ubstitution, f ( u ) = w , and u ∈ L . The tec hnical d etails of G ′ are as follo ws: 1. F or eac h σ ∈ Σ , w e will u s e a separate 1 × | h ( σ ) || S | dimensional regio n in the s tate vec tor to trace the substitutions.    ( · · · · · · ) | {z } σ 1 ( · · · · · · ) | {z } σ 2 · · · ( · · · · · · ) | {z } σ | Σ |    (7) It is easily form ulated that | S ′ | = | S |  P σ ∈ Σ | h ( σ ) |  . 2. Eac h A ′ γ ∈ Υ is defined with resp ect to the separation ab o ve . A ′ γ =       A ′ γ ,σ 1 T · · · T T A ′ γ ,σ 2 · · · T . . . . . . . . . . . . T T · · · A ′ γ ,σ | Σ |       (8) Eac h T is almost a zero matrix, except a case whic h will b e describ ed b elo w. 3. Let σ ∈ Σ . Su p p ose that h ( σ ) = γ 1 γ 2 · · · γ l ∈ Υ ∗ and l > 0. Then, the r egion corresp onding to σ in G ′ ’s state v ector can b e partitioned to l 1 × | S | blo cks: ( · · · |{z} 1 | · · · |{z} 2 | · · · | · · · |{z} l ) . (9) F or γ ∈ Υ , A ′ γ ,σ can b e partitioned in to b lo c ks of dimension | S | × | S | : If l = 1, A ′ γ ,σ = ( T l ) , (10) and if l > 1, A ′ γ ,σ =        0 T 1 0 · · · 0 0 0 T 2 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · T l − 1 T l 0 0 · · · 0        , (11) where f or γ i = γ , T i = I , and T i = 0 otherwise, (0 < i < l ); for γ l = γ , T l = A σ , an d T l = 0 otherwise. Additionally , for γ l = γ , the b ottom-leftmost blo c ks of all T ’s that are on the same ro w with A ′ γ ,σ in (8) are equal to A σ ; for γ l 6 = γ , all those blocks con tain all 0’s. 4. v ′ 0 =    ( v 0 | 0 , · · · , 0) | {z } σ 1 ( v 0 | 0 , · · · , 0) | {z } σ 2 · · · ( v 0 | 0 , · · · , 0) | {z } σ | Σ |    , (12) f ′ =    ( f | 0 , · · · , 0) | {z } σ 1 ( f | 0 , · · · , 0) | {z } σ 2 · · · ( f | 0 , · · · , 0) | {z } σ | Σ |    T . (13) When σ ∈ Σ is b eing guessed , where h ( σ ) = γ 1 γ 2 · · · γ l ∈ Υ ∗ and l > 0, the simulat ed state vect or of G is wr itten in the first slot of (9). Whenev er the matc hes s u cceed for γ 1 , · · · , γ l − 1 , the sim ulated state v ector of G is transferred to the next slot in (9); in any other case, it is set to 0. When γ l is successfully sub stituted, this sim u lated state v ector is up dated as if G has read th e symbol σ , and the result is transferred anew in the fir st slots of all the regions corresp onding to symb ols in Σ ; otherwise, it is set to 0. Theorem 11. S 6 = is close d under homomo rphism. Pr o of. Let h : Σ → Υ ∗ b e a h omomorphism, L ⊆ Σ ∗ , and L ∈ S 6 = . If h is a homomorph ism of the form in Lemma 7, then the pro of is complete. Otherwise, sup p ose that there are k ≥ 1 symb ols in Σ , i.e., σ 1 , σ 2 , · · · , σ k , suc h that h ( σ i ) = ε , where 1 ≤ i ≤ k . So, we can defin e k h omomorphisms of the f orm in Lemma 6: h 1 : Σ → Σ 1 , where Σ 1 = Σ \ { σ 1 } h 2 : Σ 1 → Σ 2 , where Σ 2 = Σ 1 \ { σ 2 } . . . h k : Σ k − 1 → Σ k , where Σ k = Σ k − 1 \ { σ k } . Additionally , we defi n e h k +1 : Σ k → Υ ∗ , where h k +1 ( σ ) = h ( σ ) for σ ∈ Σ k . Since h is the comp osi- tion of the h i ’s (1 ≤ i ≤ k + 1), h ( L ) is also in S 6 = (Lemma 6 and 7). Theorem 12. S = is not c lose d u nder homomorphism . Pr o of. Consider the languages L 1 = { w 1 cw 2 | w 1 ∈ L eq , w 2 ∈ b + } and L 2 = { w 1 cw 2 | w 1 ∈ L eq , w 2 ∈ b ∗ } . It is not hard to sho w that b oth languages are in S = . Let h 1 and h 2 b e t w o homomorphisms defined as – h 1 ( a ) = a , h 1 ( b ) = b , h 1 ( c ) = ε , and – h 2 ( a ) = a , h 2 ( b ) = b , h 2 ( c ) = b . L eq · b = h 1 ( L 1 ) = h 2 ( L 2 ), and so S = is not close d un der ( ε -free) homomorphism due to Theorem 5. 5.2 Common closure prop erties of S 6 = and S = Theorem 13. S 6 = and S = ar e close d under inverse homomorphism. Pr o of. Let h : Σ → Υ ∗ b e a homomorphism, L ⊆ Υ ∗ , and L ∈ S 6 = , su c h th at the GPF A G = ( S, Υ , { A γ ∈ Υ } , v 0 , f ) recognizes L with one-sided cutp oint 0. It is easily v erified that G ′ = ( S, Σ , { A ′ σ ∈ Σ } , v 0 , f ) , where A ′ σ ∈ Σ =  A u 1 · · · A u | h ( σ ) | , h ( σ ) = u 1 · · · u | h ( σ ) | 6 = ε I , h ( σ ) = ε , recognizes h − 1 ( L ) w ith one-sided cutp oin t 0. T he same setup ca n b e extended to an y language in S = . Theorem 14. S 6 = and S = ar e close d under r eversal. Pr o of. W e us e the same idea as [36]. If L ∈ S 6 = , then th ere exists a GPF A G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) suc h that L is recognize d by G with one-sided cutp oint 0. It is easily seen that G ′ = ( S, Σ , { A T σ ∈ Σ } , f T , v T 0 ) recognizes th e r ev erse of L with one-sided cutp oin t 0. The same setup can b e extended to an y lan- guage in S = . Theorem 15. S 6 = and S = ar e close d under wor d quotient. Pr o of. If L ∈ S 6 = , then there exists a GPF A G = ( S, Σ , { A σ ∈ Σ } , v 0 , f ) such that L is r ecognized by G with one-sided cu tp oin t 0. F or any giv en w ∈ Σ ∗ , 1. GPF A G 1 = ( S, Σ , { A σ ∈ Σ } , v 0 A w 1 · · · A w | w | , f ) recognizes the language { y | w y ∈ L } with one- sided cutp oint 0; 2. GPF A G 2 = ( S, Σ , { A σ ∈ Σ } , v 0 , A w 1 · · · A w | w | f ) recognizes the language { z | z w ∈ L } with one- sided cutp oint 0. The s ame setup can b e extended to any language in S = . Theorem 16. S 6 = and S = ar e not close d u nder differ enc e. Pr o of. There exists an L ∈ S 6 = suc h th at L / ∈ S 6 = . 1. Σ ∗ and L are in S 6 = , but Σ ∗ \ L = L is not; 2. Σ ∗ and L are in S = , but Σ ∗ \ L = L is not. Theorem 17. S 6 = and S = ar e close d under differ enc e with a r e gular language. Pr o of. Regular languages are closed u nder complement ation, and S 6 = and S = are closed under in- tersection. F or completeness, we list b elo w the follo wing easy facts ab out MCL and NMCL: 1. NMCL is closed under b oth union and intersectio n . 2. Neither MCL nor NMCL is closed u nder complementa tion [6 ]. 3. Both MCL and NMCL are closed un der inv erse homomorph ism [28]. 4. Both MCL and NMCL are closed un der word quotien t [9]. 6 Concluding Remarks In this pap er, we ga v e a full charac terizatio n of the class of languages recognized b y all QF A mo dels whic h are at least as p o werful as the Kondacs-W atrous QF A with cutp oint 0. This is the only known case where the language recognition p ow er of one-w a y QF A’s h as b een p ro ven to b e strictly greater than that of their pr obabilistic coun terp arts 8 . The sup eriorit y of Q F A’s o ver PF A’s w ith regard to space efficiency in this setting was demonstrated. W e also exa mined the limitations of recognition with one-sided error for these mo dels. Seve ral new closure pr op erties of th e related classes S 6 = and S = w ere p ro ven. The r elationship b etw een nondeterministic qu an tum complexit y classes and coun ting classes h as b een studied in d etail. It is kno wn that NQP = co-C = P [46]. More r elev antly f or our w ork, W atrous [40] has shown that NQSP A CE( s ) = co-C = SP A CE( s ) for s = Ω (log( n )). Note that the sub set of S 6 = defined using PF A’s that only con tain effici en tly computable transition probabilities and cutp oint 1 2 equals co-C = SP A CE(1), so we ha ve pr o v en 9 that co-C = SP A CE(1) ⊆ NQSP ACE(1) , and whether the inclusion is strict or not dep ends on whether a tw o-w a y head would increase the computational p o w er of a NQF A 10 . The sup eriority of NQF A’s o ve r classical NF A’s has ramifications ab out r elationships among classical and quan tum nondeterministic sp ace complexit y classes for all subloga rithmic b ounds. Although the f ollo wing r esult f ollo ws from a com bination of pr eviously kno wn facts, w e hav e not seen it stated anywhere: Theorem 18. NSP ACE( s ) ( NQSP ACE( s ) for s = o (log ( n )) . Pr o of. (Sk etc h.) Quantum T uring m ac hines can simulate probabilistic T uring mac hines easily for an y common space b ound [41]. There exists a NQF A with efficientl y compu table amplitudes (i.e. a constan t-space nond etermin istic quan tu m T u ring mac hine) whic h recognizes the language L neq = { w ∈ { a, b } ∗ | | w | a 6 = | w | b } [6]. It is easily seen that L neq is a nonregular deterministic con text-free language (DCFL). It is known that n o nonregular DCFL is in NSP ACE( s ) for s = o (log ( n )) [1]. F or space b ound s s ∈ Ω (log( n )), all we kno w in this regard is the trivial f act that NSP ACE( s ) ⊆ NQSP A CE( s ) [40]. The succinct QF A mo dels alluded to in S ection 3.3 form the basis of a demonstration [44] of the fact that tw o-w a y QF A’s can ha ve a s im ilar state complexity adv ant age ov er b oth their one-wa y v ersions, and tw o-w ay classical non d eterministic automata. One imp ortant Q F A v arian t th at w as not considered in this pap er is the Latvian QF A [2], wh ic h is a generalization of the MCQ F A not thought to b e as p o werful as the KWQF A. An examination of the corresp onding classes f or this mo del w ould b e interesting. Some other op en qu estions related to this w ork are listed b elo w. 8 F rom a “pedagogical” p oint of view, this seems to us to b e one of the simplest setup s in whic h a qu an tum compu- tational mo del can b e demonstrated to out perform the corresp onding probabilistic mo del. 9 All our proofs stand when the transition probabilities and amplitudes are restricted to b e efficien tly computable num b ers, as mentioned in [5]. W e can in fact pro ve that the collection of languages recognized by the most general mod el of NQF A’s [31] is precisely the clas s S 6 = r at (see D efinition 12) when all the amplitudes of the NQF A are restricted to b e rational numbers. 10 F or any tw o- wa y PF A M and cutp oin t λ 1 ∈ [0 , 1), there exist a one-wa y PF A P and a cutp oint λ 2 ∈ [0 , 1) such that ( M , λ 1 ) ≡ ( P , λ 2 ) [19 ], whereas tw o-wa y QF A’s are more p ow erful than one-w a y Q F A’s in the general unb ounded error setting [43]. Op en Problem 1. Is MCL closed under union? In tersection? Op en Problem 2. Do NMCL and MCL coincide? Op en Problem 3. Do es S 6 = ∩ S = con tain a n onregular language? Op en Problem 4. Is S 6 = coun table or uncount able? Op en Problem 5. Is S > closed und er complemen tation? (page 158 of [32]) Op en Problem 6. Is S = a subset of S > ? (page 173 of [32]) Op en Problem 7 . 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