Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik Justus-Lieig-Universität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle Sprchen Ilmenu, 25.-27. Septemer 2013
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,,, 0 1... 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
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,,, 0 1... 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
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
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
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
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
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
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
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
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