Nondeterministic Biautomata and Their Descriptional Complexity
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1 Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik Justus-Lieig-Universität Arndtstr. 2, Gießen, Germny 23. Theorietg Automten und Formle Sprchen Ilmenu, Septemer 2013
2 1 Biutomt Nondeterministic Biutomt Structurl Properties of Biutomt 2 Descriptionl Complexity Complexity of the -Property Conversion Prolems for -F-Biutomt Complexity of the F-Property 3 Conclusion M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
3 1 Biutomt Nondeterministic Biutomt Structurl Properties of Biutomt 2 Descriptionl Complexity Complexity of the -Property Conversion Prolems for -F-Biutomt Complexity of the F-Property 3 Conclusion M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
4 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 0 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
5 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 1 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
6 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 3 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
7 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 3 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
8 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 2 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
9 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 3 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
10 Biutomt...s introduced in [Kĺım, Polák, 2011] A iutomton A = (Q,Σ,,,q 0,F) is deterministic finite utomton with two reding heds (left-to-right nd right-to-left)... q 2 q 4 q 0 q 1 q 3 q 4... tht stisfies the following two restrictions:,, 1 heds move independently: (q ) = (q ) ( -property) 2 forwrd/ckwrd cceptnce: q F q F (F-property) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
11 Results on Biutomt [Kĺım, Polák, 2011/12] Similrities to ordinry deterministic finite utomt: iutomt ccept regulr lnguges hve unique miniml utomt Simple chrcteriztions of regulr lnguge clsses vi iutomt: prefix-suffix testle lnguges piecewise testle nd k-piecewise testle lnguges M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
12 Results on Biutomt [Kĺım, Polák, 2011/12] Similrities to ordinry deterministic finite utomt: iutomt ccept regulr lnguges hve unique miniml utomt Simple chrcteriztions of regulr lnguge clsses vi iutomt: prefix-suffix testle lnguges piecewise testle nd k-piecewise testle lnguges [Jirásková, Kĺım, 2012] Descriptionl Complexity: NFA/DFA/syntctic monoid iutomton M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
13 Results on Biutomt [Kĺım, Polák, 2011/12] Similrities to ordinry deterministic finite utomt: iutomt ccept regulr lnguges hve unique miniml utomt Simple chrcteriztions of regulr lnguge clsses vi iutomt: prefix-suffix testle lnguges piecewise testle nd k-piecewise testle lnguges [Jirásková, Kĺım, 2012] Descriptionl Complexity: NFA/DFA/syntctic monoid iutomton [H., J., 2013] Brzozowski-like minimiztion of iutomt Chrcteriztions of cyclic nd commuttive regulr lnguges M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
14 Biutomt...more generl nondeterministic iutomton (NBiA) A = (Q,Σ,,,I,F): Q finite set of sttes Σ input lphet : Q Σ 2 Q forwrd trnsition function : Q Σ 2 Q ckwrd trnsition function I Q set of initil sttes F Q set of finl or ccepting sttes A ccepts w Σ if w = u 1 u 2...u k v k...v 2 v 1 (u i,v i Σ ), nd [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F deterministic iutomton (DBiA): I = 1 q = 1 nd q = 1 for ll q Q, Σ M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
15 Biutomt...more generl nondeterministic iutomton (NBiA) A = (Q,Σ,,,I,F): Q finite set of sttes Σ input lphet : Q Σ 2 Q forwrd trnsition function : Q Σ 2 Q ckwrd trnsition function I Q set of initil sttes F Q set of finl or ccepting sttes A ccepts w Σ if w = u 1 u 2...u k v k...v 2 v 1 (u i,v i Σ ), nd [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F deterministic iutomton (DBiA): I = 1 q = 1 nd q = 1 for ll q Q, Σ M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
16 Biutomt...more generl nondeterministic iutomton (NBiA) A = (Q,Σ,,,I,F): Q finite set of sttes Σ input lphet : Q Σ 2 Q forwrd trnsition function : Q Σ 2 Q ckwrd trnsition function I Q set of initil sttes F Q set of finl or ccepting sttes A ccepts w Σ if w = u 1 u 2...u k v k...v 2 v 1 (u i,v i Σ ), nd [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F deterministic iutomton (DBiA): I = 1 q = 1 nd q = 1 for ll q Q, Σ M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
17 Structurl Properties of Biutomt -property: for ll q Q,, Σ: (q ) = (q ) F-property: for ll q Q, Σ: (q ) F (q ) F -F-DBiA re iutomt s introduced in [Kĺım, Polák, 2011] M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
18 Structurl Properties of Biutomt -property: for ll q Q,, Σ: (q ) = (q ) F-property: for ll q Q, Σ: (q ) F (q ) F -F-DBiA re iutomt s introduced in [Kĺım, Polák, 2011] Simpler cceptnce conditions generl: A ccepts w if w = u 1 u 2...u k v k...v 2 v 1 (u i,v i Σ ) nd [((...((((I u 1 ) v 1 ) u 2 ) v 2 )...) u k ) v k ] F with : A ccepts w if w = uv (u,v Σ ) nd [(I u) v] F (equivlent: [(I v) u] F ) with, F: A ccepts w if [I w] F (equivlent: [I w] F ) M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
19 1 Biutomt Nondeterministic Biutomt Structurl Properties of Biutomt 2 Descriptionl Complexity Complexity of the -Property Conversion Prolems for -F-Biutomt Complexity of the F-Property 3 Conclusion M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
20 Complexity of the -Property Lemm For ny given n-stte -NBiA A one cn construct n equivlent NFA tht hs O(n 2 ) sttes. In prticulr L(A) is regulr. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
21 Complexity of the -Property Lemm For ny given n-stte -NBiA A one cn construct n equivlent NFA tht hs O(n 2 ) sttes. In prticulr L(A) is regulr. Lemm 1 Let G = (N,T,P,S) e liner context-free grmmr in normlform. Then one cn construct n equivlent ( N +1)-stte NiBA A. Automton A my stisfy the F-property. 2 Let A e n n-stte NBiA. Then one cn construct n equivlent liner context-free grmmr G in normlform with (n + 1) nonterminls. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
22 Complexity of the -Property Lemm For ny given n-stte -NBiA A one cn construct n equivlent NFA tht hs O(n 2 ) sttes. In prticulr L(A) is regulr. Lemm 1 Let G = (N,T,P,S) e liner context-free grmmr in normlform. Then one cn construct n equivlent ( N +1)-stte NiBA A. Automton A my stisfy the F-property. 2 Let A e n n-stte NBiA. Then one cn construct n equivlent liner context-free grmmr G in normlform with (n + 1) nonterminls. Theorem The trde-off etween NBiAs nd -NBiAs is non-recursive. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
23 n2 n 2(n 1) Conversions to -F-DBiA From [Jirásková, Kĺım 2012]: 1 NFA -F-DBiA 2 DFA -F-DBiA 3 SynMon -F-DBiA SynMon n 2 -F-DBiA 2 2n 2(2 n 1) NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
24 n2 n 2(n 1) Conversions to -F-DBiA Theorem For ny given n-stte NBiA one cn construct n equivlent DBiA tht hs 2 n sttes. Further, this construction preserves the - nd F-properties. Proof Ide: Power-set construction. -F-NBiA SynMon n 2 -F-DBiA 2 n 2 2n 2(2 n 1) NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
25 n2 n 2(n 1) Conversions to -F-DBiA Lemm For ll m 1 there is lnguge L m ccepted y -F-NBiA with 3m+2 sttes nd for which every -F-DBiA needs t lest 2 2m +1 sttes. Proof Ide: L m = Σ Σ m 1 Σ -F-NBiA SynMon 2 Θ(n) n 2 2 2n 2(2 n 1) -F-DBiA NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
26 n2 n 2(n 1) Conversions to -F-NBiA SynMon n 2 -F-NBiA 2 Θ(n) -F-DBiA 2 2n 2(2 n 1) NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
27 n2 n 2(n 1) Conversions to -F-NBiA Theorem For ny given n-stte NFA one cn construct n equivlent -F-NBiA tht hs n 2 sttes. Proof Ide: Construct product of forwrd nd ckwrd utomt. -F-NBiA n 2 SynMon 2 Θ(n) n 2 n 2 2 2n 2(2 n 1) -F-DBiA NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
28 A Lower Bound Technique for -F-NBiA Lemm A set S = {(x i,y i,z i ) x i,y i,z i Σ,1 i n} is i-fooling set for L Σ if 1 for 1 i n it is x i y i z i L, nd 2 for 1 i,j n, with i j, it is x i y j z i / L or x j y i z j / L. Then ny -F-NBiA for L hs t lest S sttes. Proof Ide: Let A = (Q,Σ,,,I,F) e -F-NBiA for L. If x i y i z i L then there is q i [(I x i ) z i ] such tht (q i y i ) F. If Q < S then there re i j with q i = q j. Then x i y j z i L nd x j y i z j L, contrdiction. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
29 Lower Bound for DFA -F-NBiA Theorem For every n 1 there is lnguge L n ccepted y n n-stte DFA, such tht ny -F-NBiA needs n 2 sttes to ccept L n. Sketch of Proof: Consider A n,,, n 1 with the i-fooling set S = {(x i,j,y i,j,z i,j ) 0 i,j n 1}, where x i,j = i, y i,j = n i n 2 j+2, z i,j = n j. Note tht x i,j y i,j z i,j = n n 2 n+2 L(A). Consider (i,j) (i,j ): i = i : Word x i,j y i,j z i,j = n n 2 n+2+j j L(A). i i : Assume (j j) mod n n 1, then x i,j y i,j z i,j = n i +i n 2 n+2+j j L(A). Cse (j j) mod n = n 1 similr. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
30 Lower Bound for DFA -F-NBiA Theorem For every n 1 there is lnguge L n ccepted y n n-stte DFA, such tht ny -F-NBiA needs n 2 sttes to ccept L n. Sketch of Proof: Consider A n,,, n 1 with the i-fooling set S = {(x i,j,y i,j,z i,j ) 0 i,j n 1}, where x i,j = i, y i,j = n i n 2 j+2, z i,j = n j. Note tht x i,j y i,j z i,j = n n 2 n+2 L(A). Consider (i,j) (i,j ): i = i : Word x i,j y i,j z i,j = n n 2 n+2+j j L(A). i i : Assume (j j) mod n n 1, then x i,j y i,j z i,j = n i +i n 2 n+2+j j L(A). Cse (j j) mod n = n 1 similr. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
31 Conversions to -F-NBiA SynMon -F-NBiA -F-DBiA n 2 n2 n 2(n 1) n 2 2 Θ(n) n 2 2 2n 2(2 n 1) NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
32 Conversions to -F-NBiA Theorem Let L e regulr lnguge given y syntctic monoid of size n. Then one cn construct -F-NBiA for L tht hs t most n sttes. This ound cn e reched for every n 1. n2 n 2(n 1) -F-NBiA n 2 n SynMon 2 Θ(n) n 2 n 2 2 2n 2(2 n 1) -F-DBiA NFA 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
33 Conversions to -F-NBiA Theorem For ny given regulr expression of lphetic width n one cn construct n equivlent -F-NBiA tht hs (n+1) 2 sttes. n2 n 2(n 1) RegExp Θ(n 2 ) n -F-NBiA SynMon 2 Θ(n) n 2 -F-DBiA Lemm For ll n 1 there is regulr lnguge L n with lphetic width n, such tht ny -F-NBiA needs n 2 sttes to ccept L n. n 2 NFA n 2 2 2n 2(2 n 1) 2 n DFA M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
34 Complexity of the F-Property The F-property is esy for -NBiAs, Lemm For ny given n-stte -NBiA one cn construct n equivlent -F-NBiA tht hs O(n 4 ) sttes. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
35 Complexity of the F-Property The F-property is esy for -NBiAs, Lemm For ny given n-stte -NBiA one cn construct n equivlent -F-NBiA tht hs O(n 4 ) sttes....nd hrd for -DBiAs. Lemm For every n 1 there is regulr lnguge L n ccepted y -DBiA with O(n 2 ) sttes nd for which every -F-DBiA needs Ω(2 n ) sttes. M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
36 1 Biutomt Nondeterministic Biutomt Structurl Properties of Biutomt 2 Descriptionl Complexity Complexity of the -Property Conversion Prolems for -F-Biutomt Complexity of the F-Property 3 Conclusion M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
37 Conclusion Summry nondeterminism for iutomt structurl properties:, F descriptionl complexity of conversion prolems Further Reserch tight ound for determiniztion of -F-NBiAs tight ounds for -BiA -F-BiA nd -BiA NFA/DFA understnding the F-property (lso concerning minimiztion prolems) further chrcteriztions of suregulr lnguge clsses residul iutomt... M. Holzer, S. Jkoi (JLU Gießen) Nondeterministic Biutomt 23. Theorietg, Ilmenu, Septemer 2013
38 Thnk you for your ttention!
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