Descriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata

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1 Desriptionl Complexity of Non-Unry Self-Verifying Symmetri Differene Automt Lurette Mris 1,2 nd Lynette vn Zijl 1 1 Deprtment of Computer Siene, Stellenosh University, South Afri 2 Merk Institute, CSIR, South Afri Astrt. We show tht, for every n 2, there is regulr lnguge L n epted y non-unry self-verifying symmetri differene utomton with n sttes, suh tht its equivlent miniml deterministi finite utomton hs 2 n 1 sttes. 1 Introdution In [1], we extended the notion of self-verifition (SV) to symmetri differene nondeterministi finite utomt (XNFA). We estlished n upper ound of 2 n 1 on the stte omplexity of SV-XNFA in the se of unry lphet, nd showed tht the ound is not tight. A lower ound of 2 n 1 1 ws lso estlished for the unry se. We showed this to e tight ound in [2], where we lso introdued more generl notion of eptne for SV-XNFA. In this pper, we onsider non-unry lphets, nd we give n upper ound of 2 n 1 nd lower ound of 2 n 1. 2 Preliminries An NFA N is five-tuple N = (Q, Σ, δ, Q 0, F ), where Q is finite set of sttes, Σ is finite lphet, δ : Q Σ 2 Q is trnsition funtion (where 2 Q indites the power set of Q), Q 0 Q is set of initil sttes, nd F Q is the set of finl, or eptne, sttes. The trnsition funtion δ n e extended to strings in the Kleene losure Σ of the lphet: δ (q, w 0 w 1... w k ) = δ(δ(... δ(q, w 0 ), w 1 ),..., w k ). For onveniene, we write δ(q, w) to men δ (q, w). An NFA N is sid to ept string w Σ if q 0 Q 0 nd δ(q 0, w) F, nd the set of ll strings (lso lled words) epted y N is the lnguge L(N) epted y N. Any NFA hs n equivlent DFA whih epts the sme lnguge. The DFA equivlent to given NFA is found y performing the suset onstrution [3]. In essene, the suset onstrution keeps trk of ll the sttes tht the NFA my e in t the sme time, nd forms the sttes of the equivlent DFA y grouping of the sttes of the DFA. In short, δ(a, σ) = δ(q, σ) q A

2 2 for ny A Q nd σ Σ. 2.1 Symmetri differene utomt (XNFA) A symmetri differene NFA (XNFA) is defined extly s n NFA, exept tht the DFA equivlent to the XNFA is found y tking the symmetri differene (in the set theoreti sense) in the suset onstrution. Tht is, for ny two sets A nd B, the symmetri differene is given y (A, B) = (A B) \ (A B). The suset onstrution is then pplied s δ(a, σ) = q A δ(q, σ) for ny A Q nd σ Σ. For lrity, the DFA equivlent to n XNFA N is termed n XDFA nd denoted with N D (with orresponding Q D, δ D, Q 0D nd F D ). It is ustomry to require tht n XDFA finl stte onsist of n odd numer of finl XNFA sttes, s n nlogy to the symmetri differene set opertion [4] this is known s prity eptne. Given prity eptne, XNFA hve een shown to e equivlent to weighted utomt over the finite field of two elements, or GF(2) [4,5]. For n XNFA N = (Q, Σ, δ, Q 0, F ), the trnsitions for eh lphet symol σ n e represented s mtrix over GF(2). Eh row represents mpping from stte q Q to set of sttes P 2 Q. P is written s vetor with one in position i if q i P, nd zero in position i if q i P. Hene, the trnsition tle is represented s mtrix M σ of zeroes nd ones (see Exmple 1). This is known s the hrteristi or trnsition mtrix for σ of the XNFA. Initil nd finl sttes re similrly represented y vetors, nd pproprite vetor nd mtrix multiplitions over GF(2) represent the ehviour of the XNFA 3. For instne, in the unry se we would hve single mtrix M tht desries the trnsitions on for some XNFA with n sttes. We enode the initil sttes Q 0 s vetor of length n over GF(2), nmely v(q 0 ) = [q 00 q q 0n 1 ], where q 0i = 1 if q i Q 0 nd 0 otherwise. Similrly, we enode the finl sttes s length n vetor v(f ) = [q F0 q F1... q Fn 1 ]. Then v(q 0 )M is vetor tht enodes the sttes rehed fter reding the symol extly one, nd v(q 0 )M k enodes the sttes rehed fter reding the symol k times. The weight of word w k of length k is given y (w k ) = v(q 0 )M k v(f ) T. We n sy tht M represents the word, nd M k = M k represents the word k. In the inry se, we would hve two mtries, M for trnsitions on nd M for trnsitions on. Reding n orresponds to multiplying y M, while reding orresponds to multiplying y M. Let M w e the result of the 3 In GF(2), = 0.

3 3 pproprite multiplitions of M nd M representing some w {, }, then the weight of w is given y (w) = v(q 0 )M w v(f ) T. We now show tht, in the unry se, so-lled hnge of sis is possile, where for some n n trnsition mtrix M of n XNFA nd ny non-singulr n n mtrix A, M = A 1 M A is the trnsition mtrix of n equivlent XNFA with v(q 0) = v(q 0 )A nd v(f ) T = A 1 v(f ) T. For ny word w k of length k, we hve the following: (w k ) = v(q 0)M k v(f ) T = v(q 0 )A(A 1 M A) k A 1 v(f ) T = v(q 0 )M k v(f ) T = (w k ). This lso pplies to the inry se. For some XNFA N, let M w = k i=1 M σ i represent word w = σ 1 σ 2...σ k, where M σi = M if σ i =, nd similrly for. Now, let N e n XNFA whose trnsition mtries re M = A 1 M A nd M = A 1 M A for some non-singulr A. Then w is represented y M w = k i=1 M σ i = M σ 1 M σ 2...M σ k = (A 1 M σ1 A)(A 1 M σ2 A)...(A 1 M σk A) = A 1 M σ1 M σ2...m σk A = A 1 M w A. And so the weight of ny word w k on N is (w) = v(q 0)M wv(f ) T = v(q 0 )A(A 1 M w A)A 1 v(f ) T = v(q 0 )M w v(f ) T = (w). Note tht the ove disussion does not rely on the ft tht there re only two lphet symols, nd so pplies in generl to the n-ry se s well. 2.2 Self-verifying utomt (SV-NFA) Self-verifying NFA (SV-NFA) [6 8] re utomt with two kinds of finl sttes, nmely ept sttes nd rejet sttes, s well s neutrl non-finl sttes. It is required tht for ny word, one or more of the pths for tht word reh single kind of finl stte, i.e. either ept sttes or rejet sttes re rehed, ut not oth. Consequently, self-verifying utomt rejet words expliitly if they reh rejet stte, in ontrst to NFA, where rejetion is the result of filure to reh n ept stte.

4 4 Definition 1. A self-verifying nondeterministi finite utomton (SV-NFA) is 6-tuple N = (Q, Σ, δ, Q 0, F, F r ), where Q, Σ, δ nd Q 0 re defined s for stndrd NFA. F Q nd F r Q re the sets of ept nd rejet sttes, respetively. The remining sttes, tht is, the sttes elonging to Q \ (F F r ), re lled neutrl sttes. For eh input string w in Σ, it is required tht there exists t lest one pth ending in either n ept or rejet stte; tht is, δ(q 0, w) (F F r ) for ny q 0 Q 0, nd there re no strings w suh tht oth δ(q 0, w) F nd δ(q 1, w) F r re nonempty, for ny q 0, q 1 Q 0. Sine ny SV-NFA either epts or rejets ny string w Σ expliitly, its equivlent DFA must do so too. The pth for eh w in DFA is unique, so eh stte in the DFA is n ept or rejet stte. Hene, for ny DFA stte d, there is some SV-NFA stte q i d suh tht q i F (nd hene d FD ) or q i F r (nd hene d FD r ). Sine eh stte in the DFA is suset of sttes of the SV-NFA, ept nd rejet sttes nnot our together in DFA stte. Tht is, if d is DFA stte, then for ny p, q d, if p F then q / F r nd vie vers. 2.3 Self-verifying symmetri differene utomt (SV-XNFA) In [1], self-verifying symmetri differene utomt (SV-XNFA) were defined s omintion of the notions of symmetri differene utomt nd self-verifying utomt, ut only the unry se ws exmined. We now restte the definition of SV-XNFA in order to present results on lrger lphets in Setion 4. Note, however, tht the definition is slightly mended: in [1], the impliit ssumption ws mde tht no SV-XNFA stte ould e oth n ept stte nd rejet stte. This ssumption is explored in detil for the unry se in [2], ut for our urrent purposes it suffies to sy tht suh requirement removes the equivlene etween XNFA nd weighted utomt over GF(2), whih is essentil for ertin opertions on XNFA, suh s minimistion [5]. This implies tht prity eptne pplies to SV-XNFA, where the ondition for self-verifition (SVondition) is tht for ny word, n odd numer of pths end in either ept sttes or rejet sttes, ut not oth. In terms of the equivlent XDFA, this is equivlent to requiring tht ny XDFA stte ontin either n odd numer of ept sttes or n odd numer of rejets sttes, ut not oth. If n XNFA stte is oth n ept stte nd rejet stte, it ontriutes to oth ounts. Definition 2. A self-verifying symmetri differene finite utomton (SV-XNFA) is 6-tuple N = (Q, Σ, δ, Q 0, F, F r ), where Q, Σ, δ nd Q 0 re defined s for XNFA, nd F nd F r re defined s for SV-NFA, exept tht F F r need not e empty. Tht is, eh stte in the SV-XDFA equivlent to N must ontin n odd numer of sttes from either F or F r, ut not oth, nd some SV-XNFA sttes my elong to oth F nd F r. The SV-ondition for XNFA implies tht if stte in the SV-XDFA of n SV- XNFA N ontins n odd numer of sttes from F, it my lso ontin n

5 5 even numer of sttes from F r, nd hene elong to FD, nd vie vers. An SV-XDFA stte my ontin ny numer of neutrl sttes from N. The hoie of F nd F r for given SV-XNFA N is lled n SV-ssignment of N. An SV-ssignment where either F or F r is empty, is lled trivil SVssignment. Otherwise, if oth F nd F r re nonempty, the SV-ssignment is non-trivil. 3 XNFA nd liner feedk shift registers In [9] it is shown tht unry XNFA re equivlent to liner feedk shift registers (LFSRs). Speifilly, mtrix M with hrteristi polynomil (X) is ssoited with ertin yle struture of sets of XNFA sttes (or of XDFA sttes), nd the hoie of Q 0 determines whih yle represents the ehviour of speifi unry XNFA. The yle struture is indued y (X), so ny mtrix tht hs (X) s its hrteristi polynomil hs the sme yle struture, lthough the sttes ourring in the yles differ ording to eh speifi mtrix. For the n-ry se, the trnsition mtrix for eh symol is ssoited with its own yle struture, nd the hoie of Q 0 determines whih yle is relised in the n-ry XNFA for eh symol. There re 2 n 1 possile hoies for Q 0 (we exlude the empty set). Evidently, the yles ssoited with eh symol might overlp, nd so the struture of the n-ry XNFA would not e yli itself, lthough the trnsitions for eh symol would exhiit yli ehviour. Speifilly, for n n-ry XNFA N nd some symol σ Σ, we refer to the yle struture of N on σ s the yle struture resulting from onsidering only trnsitions on σ. Our min results will e derived from exmining the yle struture indued y eh symol of the lphet of the utomton, s well s the wys in whih the yles overlp. For ny (X) = X n + n 1 X n X + 0 there is ompnion mtrix M of the form given elow, suh tht (X) = det(xm I), where I is the identity mtrix. We sy tht M is in nonil form. M = n 2 n 1 In the next lemm, it will e onvenient to represent XDFA sttes d s Q s s = s n 1, s n 2,..., s 1, s 0, where s i = 1 if q i d s nd 0 otherwise. The lemm is dpted from [10] on the sis of the equivlene etween unry XNFA nd LFSRs. Lemm 1. Let M σ e trnsition mtrix representing trnsitions on σ for some XNFA N, with hrteristi polynomil σ (X), nd let M σ e in nonil form. Let f e ijetion of the sttes of the equivlent XDFA N D onto polynomils of degree n 1, suh tht f mps the stte s = s n 1, s n 2,..., s 1, s 0 into the

6 6 polynomil f(s) = s n 1 X n 1 + s n 2 X n s 1 X + s 0. Then f mps the stte M σ s into the polynomil Xf(s) mod σ (X). For ny (X) over GF(2) with degree n, the polynomil lger modulo (X) is representtion of GF(2 n ), the finite field of 2 n elements. Lemm 1 provides mpping etween these elements nd the sttes of XDFA. Furthermore, the XDFA stte rrived t fter trnsition from stte s on σ orresponds to the polynomil whih results from multipying f(s) y X within this representtion of GF(2 n ). The non-zero elements of GF(2 n ) form yli multiplitive group [10]. Exmple 1. Let N e inry XNFA (shown in Fig. 1), where M is the ompnion mtrix of (X) = X 4 + X 2 + X + 1 nd M is the ompnion mtrix of (X) = X 4 + X 3 + X + 1. M nd M re given in Fig.?? nd Fig.??, respetively. The resulting XDFA is shown in Fig. 2, while some exmples ompring stte trnsitions nd polynomil multiplition re shown in Tle 1. Note tht, for now, the fous is on the yli ehviour of the equivlent XDFA, nd so we do not refer to ny finl sttes M = M = , q 1, strt q 0, strt q 0, q 0,, q 1,, q 0, q 1,, Fig. 1. Exmple 1: N q 1,,,, q 0, q 1, Fig. 2. Exmple 1: N D

7 7 Tle 1. Trnsitions on δ orrespond to multiplition y X δ(σ, s) Xf(s) mod σ(x) δ({q 0}, ) = {q 1} X(1) = X δ({}, ) = {q 0, q 1, } X(X 3 ) = X 4 mod (X) = X 3 + X δ({q 0,, }, ) = {q 0,, } X(X 3 + X 2 + 1) = X 4 + X 3 + X mod (X) = X 3 + X δ({q 1}, ) = {} X(X) = X 2 δ({q 0, q 1, }, ) = {q 0,, } X(X 3 + X + 1) = X 4 + X 3 + X mod (X) = X 3 + X 2 + X δ({q 1,, }, ) = {q 0, q 1, } X(X 3 + X 2 + X) = X 4 + X 3 + X mod (X) = X 2 + X Non-unry SV-XNFA The upper ound on stte omplexity is simply 2 n 1, sine this is the numer of non-empty susets for ny set of n XNFA sttes. We now work towrds estlishing lower ound on stte omplexity. First, we restte the following lemm from [1] for the unry se. Lemm 2. Let (X) = (X + 1)φ(X) e polynomil of degree n with nonsingulr ompnion mtrix M, nd let N e n XNFA with trnsition mtrix M nd Q 0 = {q 0 }. Then the equivlent XDFA N D hs the following properties: 1. Q D > n 2. d is odd for d Q D 3. [q 0 ], [q 1 ],..., [q n 1 ] Q D d is the numer of XNFA sttes in the XDFA stte d Q, or the numer of one s in the representtion of d s s n 1, s n 2,..., s 1, s 0 where s i = 1 if q i d nd 0 otherwise. Theorem 1. Let M σ1, M σ2,..., M σr e the ompnion mtries of r polynomils σ1 (X) = (X + 1)φ σ1 (X), σ2 (X) = (X + 1)φ σ2 (X),..., σr (X) = (X + 1)φ σr (X), respetively, nd let M σ1, M σ2,..., M σr e the trnsition mtries of some n-ry XNFA N with Σ = {σ 1, σ 2,..., σ r } nd Q 0 = {q 0 }. Then the numer of sttes in the equivlent XDFA N D does not exeed 2 n 1. Furthermore, N hs n SV-ssignment. Proof. See Appendix A. The following lemm provides further informtion on the yle struture indued y polynomils with X + 1 s ftor. Lemm 3. Let σ (X) = (X + 1)φ(X). Then, in the ompnion mtrix M σ of σ (X), whih is the trnsition mtrix on some symol σ for n XNFA, the stte mpped to φ(x) s desried in Lemm 1, i.e. d φ, is the stte ontined in the yle of length one, when onsidering only trnsitions on σ.

8 8 Proof. Consider the following: (X + 1)φ(X) = σ (X) Xφ(X) + φ(x) = σ (X) Xφ(X) = φ(x) + σ (X) Therefore, Xφ(X) = φ(x) in the representtion of GF (2 n ) s polynomils over GF(2) modulo σ (X). By Lemm 1, this orresponds to δ(d φ, σ) = d φ. We now present witness lnguge for ny n to show tht 2 n 1 is lower ound on the stte omplexity of SV-XNFA with non-unry lphets. Lemm 4. Let φ(x) = X n 1 + φ n 2 X n φ 1 X + φ 0 e ny primitive polynomil of degree n 1. Let N e inry XNFA, nd let the trnsition mtrix on e the ompnion mtrix of (X) = (X +1)φ(X) nd the trnsition mtrix on e the ompnion mtrix of (X) = X n +φ(x). Then the equivlent XDFA of the XNFA with Q 0 = {q 0 } ontins extly 2 n 1 odd-sized sttes. Proof. We write (X) nd (X) in the following wy: (X) = X n + n 1 X n X + 0 (X) = X n + φ n 1 X n 1 + φ n 2 X n φ 1 X + φ 0 Sine φ(x) is primitive, it hs no roots in GF(2), inluding 1, so it must hve n odd numer of non-zero terms. Therefore, y Lemm 1, d φ is odd. Furthermore, (X) hs n even numer of non-zero terms, nd so hs 1 s root. Consequently, (X) hs X + 1 s ftor. The trnsition mtries M nd M re given elow. Let Q 0 = {q 0 }. Then y Theorem 6 of [1], the yle struture on is equivlent to n XDFA yle with 2 n 1 1 sttes, ll of whih, y Lemm 2, hve odd size. Also, y Lemm 3, d φ is not ontined in this yle. This mens tht on, every odd-sized stte in the XDFA is rehed exept for d φ. Now, from M it follows diretly tht M = n 2 n M = φ 0 φ 1... φ n 2 φ n 1 δ({q n 1 }, ) = d φ. Furthermore, sine X + 1 is ftor of, every trnsition from n odd-sized stte on is to n odd-sized stte. Consequently, the inry XNFA N is equivlent to n XDFA tht rehes ll 2 n 1 odd-sized sttes nd none other.

9 9 Theorem 2. For ny n 2, there is lnguge L n so tht some n-stte inry SV-XNFA epts L n nd the miniml SV-XDFA tht epts L n hs 2 n 1 sttes. Proof. Let (X) = (X +1)φ(X) nd = X n +φ(x), where φ(x) is primitive polynomil nd let (X) nd (X) hve degree n. We onstrut n SV-XNFA N with n sttes whose equivlent N D hs 2 n 1 sttes s in Lemm 4, nd let F = {q 0 } nd F r = Q \ F. Let L 1 n = (2n 1 1)i+j for i 0 nd j J, where J is some set of integers, represents suset of the lnguge epted y N tht onsists only of strings ontining. Now, from the trnsition mtrix of N it follows tht 0, n J, while 1, 2,..., n 1 / J, sine q 0 δ(q 0, n ) nd q 0 / δ(q 0, m ) for m < n. If there is n N D with fewer thn 2n 1 1 sttes tht epts L 1 n, then there must e some d j {q 0 } Q D suh tht q 0 d j, q 0 δ(d j, n ) nd there is no m < n so tht q 0 δ(d j, m ). Let d k e ny stte in N D suh tht d k {q 0 }. Let mx(d k ) e the lrgest susript of ny SV-XNFA stte in d k. Then mx(d k ) > 0. Let m = n mx(d k ), so m < n. Then, from the trnsition mtrix of N, it follows tht q 0 δ(d k, m ). Tht is, for ny d k there is n m < n so tht q 0 δ(d k, m ). Therefore, there is no N D with fewer thn 2n 1 1 sttes tht epts L 1 n. Now, let L 2 n = n, whih is lso suset of the lnguge epted y N. In order to ept this lnguge, fter reding n, stte must hve een rehed wherefter every trnsition on must result in n ept stte, i.e. n XDFA stte ontining q 0. But there is only one suh stte, nd tht is d φ, sine δ(d φ, ) = d φ, whih is exluded from the yle needed to ept L 1 n. Therefore, ll 2 n 1 odd-sized sttes re neessry to ept L 1 L 2. Let L n e the lnguge epted y N, then sine L 1 n L 2 n L n, t lest 2 n 1 sttes re neessry to ept L n. Theorem 3. For ny n 2, there is lnguge L n so tht some n-stte n- ry SV-XNFA epts L n nd the miniml SV-XDFA tht epts L n hs 2 n 1 sttes. Proof. See Appendix A. The following theorem shows tht ny given SV-XNFA n e used to otin nother one vi so-lled hnge of sis. Theorem 4. Given ny SV-XNFA N = (Q, Σ, δ, Q 0, F, F r ) with n sttes nd trnsition mtries M σ1, M σ2,..., M σr, nd ny non-singulr n n mtrix A, we enode Q 0 s vetor v(q 0 ) of length n over GF(2) nd F nd F r s vetors v(f ) nd v(f r ) respetively. Then there is n SV-XNFA N = (Q, Σ, δ, Q 0, F, F r ) where M σ i = A 1 M σi A for 0 i r, v(q 0) = v(q 0 )A, v(f ) T = A 1 v(f ) T nd v(f r ) T = A 1 v(f r ) T, nd N epts the sme lnguge s N. Proof. In the disussion in Setion 2.1 we showed tht for XNFA, the hnge of sis desried on n XNFA N tht results in N, (w) = (w). We extend

10 10 this to SV-XNFA y defining two new funtions. Rell tht M w represents the sequene of mtrix multiplitions for some w of length k, nd tht M w = A 1 M w A. Then, let ept(w) = v(q 0 )M w v(f ) T rejet(w) = v(q 0 )M w v(f r ) T. The SV-ondition is tht ept(w) rejet(w) for ny w Σ. Similr to (w), we hve nd ept (w) = v(q 0)M wv(f ) T = v(q 0 )A(A 1 M w A)A 1 v(f ) = v(q 0 )M w v(f ) = ept(w) rejet (w) = v(q 0)M wv(f r ) T = v(q 0 )A(A 1 M w A)A 1 v(f r ) = v(q 0 )M w v(f r ) = rejet(w) Clerly, the SV-ondition is met y ept nd rejet, nd so N is n SV- XNFA tht epts the sme lnguge s N. The numer of non-singulr n n mtries over GF(2) (inluding the identity mtrix) is GL(n, Z 2 ) = n 1 k=0 (2n 2 k ), nd so for ny SV-XNFA nother GL(n, Z 2 ) 1 equivlent SV-XNFA n e found. Exmple 2. Let N e n SV-XNFA with lphet Σ = {,, }, nd the following trnsition mtries: M is the ompnion mtrix of (X) = X 4 +X 3 +X 2 +1, M is the ompnion mtrix of X 4 +X 3 +X +1, nd M is the ompnion mtrix of (X) = X 4 + X 2 + X + 1. Let Q 0 = {q 0 }, F = {q 0, } nd F r = {q 1, }. Fig. 3 shows N nd the equivlent XDFA N D is given in Fig. 5, where doule edge indites n ept stte nd thik edge indites rejet stte. Consider the following mtrix A: A = We use A to mke hnge of sis from N to N. Let N e n XNFA with Σ = {,, }, where M = A 1 M A, M = A 1 M A nd M = A 1 M A. Furthermore, let v(q 0) = v(q 0 )A, i.e. Q 0 = {q 1,, }. Finlly, let v(f ) T = A 1 v(f ) T nd v(f r ) T = A 1 v(f r ) T, i.e. F = {q 0, } nd F r = {, }. Fig. 4, shows N, with doule edge inditing n ept stte, thik edge

11 11 inditing rejet stte nd thik doule edge inditing stte tht is oth n ept stte nd rejet stte. Fig. 6 gives the equivlent XDFA N D. It is worth noting tht, lthough N hs different struture thn N, N D hs the sme struture s N D, nd epts the sme lnguge. Also, note tht in N D, the stte {q 0, q 1, } is rejet stte, euse it ontins n even numer of ept sttes, nmely q 0 nd, ut n odd numer of rejet sttes, nmely.,, q 1,,, q 1 strt q 0,,,,, strt q 0,,,,,,,,, Fig. 3. Exmple 2: N,, Fig. 4. Exmple 2: N 5 Conlusion We hve given n upper ound of 2 n 1 on the stte omplexity of SV-XNFA with lphets lrger thn one, nd lower ound of 2 n 1. We hve lso shown tht, given ny SV-XNFA with n sttes, it is possile to find nother GL(n, Z 2 ) 1 equivlent SV-XNFA vi hnge of sis. Referenes 1. Mris, L., vn Zijl, L.: Unry Self-verifying Symmetri Differene Automt. In: Proeedings of Desriptionl Complexity of Forml Systems: 18th IFIP WG 1.2 Interntionl Conferene, DCFS 2016, Buhrest, Romni, July 5-8, Springer Interntionl Pulishing (2016) Mris, L., Vn Zijl, L.: Stte omplexity of Unry SV-XNFA with Different Aeptne Conditions. Sumitted for pulition 3. Hoproft, J.E., Ullmn, J.D.: Introdution to Automt Theory, Lnguges, nd Computtion. 1st edn. Addison-Wesley Longmn Pulishing Co., In., Boston, MA, USA (1990)

12 12 strt q 0,, strt q 1,,,, q 0, q 1, q 1 q 0 q 0,,,,, q 1,,,, q 0, q 1,,,,,,,, q 0, q 1, q 1, q 0,, q 0, q 1, Fig. 5. Exmple 2: N D Fig. 6. Exmple 2: N D 4. Vuillemin, J., Gm, N.: Compt Norml Form for Regulr Lnguges s Xor Automt. In: Proeedings of the 14th Interntionl Conferene on the Implementtion nd Applition of Automt (CIAA 2009), Sydney, Austrli, July 14-17, Springer, Berlin, Heidelerg (2009) Vn der Merwe, B., Tmm, H., Vn Zijl, L.: Miniml DFA for Symmetri Differene NFA. In: Proeedings of the 14th Interntionl Conferene on the Desriptionl Complexity of Forml Systems (DCFS 2012), Brg, Portugl, July 23-25, Springer, Berlin, Heidelerg (2012) Assent, I., Seiert, S.: An upper ound for trnsforming self-verifying utomt into deterministi ones. RAIRO-Theoretil Informtis nd Applitions- Informtique Théorique et Applitions 41(3) (2007) Hromkovič, J., Shnitger, G.: On the Power of Ls Vegs II. Two-Wy Finite Automt. In: Proeedings of Automt, Lnguges nd Progrmming: 26th Interntionl Colloquium, ICALP 99 Prgue, Czeh Repuli, July 11 15, Springer, Berlin, Heidelerg (1999) Jirásková, G., Pighizzini, G.: Optiml simultion of self-verifying utomt y deterministi utomt. Informtion nd Computtion 209(3) (2011) Speil Issue: 3rd Interntionl Conferene on Lnguge nd Automt Theory nd Applitions (LATA 2009). 9. Vn Zijl, L., Hrper, J.P., Olivier, F.: The MERLin Environment Applied to -NFAs. In: Implementtion nd Applition of Automt: 5th Interntionl Conferene, CIAA 2000 London, Ontrio, Cnd, July 24 25, 2000 Revised Ppers. Springer, Berlin, Heidelerg (2001)

13 Stone, H.S.: Disrete Mthemtil Strutures nd their Applitions. Siene Reserh Assoites Chigo (1973) A Appendix Theorem 1. Let M σ1, M σ2,..., M σr e the ompnion mtries of r polynomils σ1 (X) = (X+1)φ σ1 (X), σ2 (X) = (X+1)φ σ2 (X),..., σr (X) = (X+1)φ σr (X), respetively, nd let M σ1, M σ2,..., M σr e the trnsition mtries of some n- ry XNFA N with Σ = {σ 1, σ 2,..., σ r } nd Q 0 = {q 0 }. Then the numer of sttes in the equivlent XDFA N D does not exeed 2 n 1. Furthermore, N hs n SV-ssignment. Proof. By Lemm 2, d is odd for d Q D in the unry se. Tht is, for ny symol with trnsition mtrix whose polynomil hs X + 1 s ftor, trnsition from n odd-sized XDFA stte is to nother odd-sized XDFA stte. Sine Q 0 = {q 0 } nd {q 0 } is odd, nd σ1 (X), σ2 (X),... σr (X), hve X + 1 s ftor, only odd-sized sttes re rehle on ny trnsition. The numer of XDFA sttes d suh tht d is odd is 2 n /2 = 2 n 1, nd so N D n hve t most 2 n 1 sttes. Sine every XDFA stte ontins n odd numer of XNFA sttes, ny hoie of F nd F r suh tht F F r = Q nd F F r = is n SV-ssignment. Theorem 3. For ny n 2, there is lnguge L n so tht some n-stte n- ry SV-XNFA epts L n nd the miniml SV-XDFA tht epts L n hs 2 n 1 sttes. Proof. In the proof of Theorem 2, lnguge L n for inry SV-XNFA is presented tht rehes 2 n 1 sttes. We onvert the inry SV-XNFA N into n n-ry SV-XNFA N in the following wy: for every σ Σ \ {, }, let M σ e the ompnion mtrix of some σ (X) = (X + 1)φ σ (X) with degree n. Then only odd-sized sttes re rehed on ny input. Let L n e the lnguge epted y N. L n requires 2 n 1 sttes, nd sine L n L n, so does L n.