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Happy Birthday Michael !!

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Probabilistic & Nondeterministic Finite Automata Avi Wigderson Institute for Advanced Study Very old (1996) joint work with Anne Condon Lisa Helerstein Sam Pottle

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Pick a computational model. Study the relative power of its variants: Deterministic,Non-deterministic,Probabilistic Polynomial Time: NP=P? BPP=P? [BM,Y,NW,IW] BPP=P unless NP is easy Log Space: NL=L? BPL=L? [S] NL L 2 [IS] NL=coNL [N] BPL SC [SZ] BPL=L 3/2 [R] SL=L Finite automata! (= constant memory) [GMR,B] Arthur-Merlin, [F,D,BV] Quantum (part of) Rabins legacy

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Deterministic,Non-deterministic,Probabilistic Arthur-Merlin, Quantum &1-way vs. 2-way read. 10 language classes… Regular = 1DFA, 1NFA, 1PFA, 1AMFA, 1QFA 2DFA, 2NFA, 2PFA, 2AMFA, 2QFA [Rabin-Scott 59] 1NFA = 2DFA = 1DFA [Rabin 63] 1PFA = 1DFA Comment : we shall not discuss relative succinctness Finite Automata (FA)

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Deterministic,Non-deterministic,Probabilistic Arthur-Merlin, Quantum & 1-way vs. 2-way read [Rabin-Scott 59] 1NFA = 2DFA = Regular [Rabin 63] 1PFA = Regular [Shepherdson59] 2NFA = Regular [Freivalds 81] 2PFA can compute {a n b n } !! (But in exp time) FA* : automaton runs in expected poly-time [Dwork-Stockmeyer,Keneps-Frievalds 90] 2PFA*= Regular [Condon-Hellerstein-Pottle-W 96] 1AMFA = Regular [CHPW 96] 2AMFA* co2AMFA* = Regular [Watrous 97] 2QFA* compute {a n b n }, {a n b n c n }!! (linear time) OPEN: 2AMFA* = Regular ?? Results

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L language M L infinite binary matrix x,y lexical order y 1101… [Myhill-Nerode] L regular 0110… iff M L has x … L(xy) finite number of rows iff 1s of M L have finite partition/cover by 1-tiles Communication Complexity [Yao] 111…1… … … … 111…1… … … … 1-tile x y Q: states |Q|=s

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L accepted by 1DFA [Fact] 1DFA = Regular y Tile per state q Q x {x : Start q } X {y : q Accept } s tiles (partition) Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 111…1…

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L accepted by 1NFA y [RS] 1NFA = Regular x Tile per state q Q {x can Start q } X {y can q Accept } s tiles (cover) Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 1…1…111 … 111…1…

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L accepted by 1PFA [R] 1PFA = Regular Tile per probability distribution p [10s] s {x : Start ~ p} X {y : p Accept w.p.> 2/3} (10s) s tiles (partition) Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 111…1…

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L accepted by 2DFA [RS] 2DFA = Regular Tile per crossing Sequence c Q 2s {x: c consistent with x} X {y: c cons with y & c Acc} s 2s tiles (partition) Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 111…1…

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L accepted by 2NFA [S] 2NFA = Regular Tile per crossing Sequence c Q 2s {x can Start c } X {y can c Accept } s tiles (cover) Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 1…1…111 … 111…1…

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L accepted by 2PFA* y [DS,KF] 2PFA* = Regular Tile per O(s)-state Markov chain m [log n] O(s) {x: m x-consistent} X {y: m y-cons & Pr[m Acc]> 2/3 } (log n) O(s) tiles (partition) of M L (n) [Karp,DS,KF] M L (n) has large nonregularity Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 111…1…

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L, L c accepted by 2AMFA* [CHPW] L is Regular Tile per O(s)-state Markov chain m [log n] O(s) {x can be m-consistent} X {y can be m-cons& Pr[m Acc]> 2/3 } (log n) O(s) 1-tiles (cover) of M L (n) (log n) O(s) 0-tiles (cover) of M L (n) [AUY,MS] Rank(M L (n)) = n o(1) Proofs 111…1… … … … 111…1… … … … x y Q: states |Q|=s 1…1…111 … 111…1… 111…0… 00...0… … … … 00…0… … … … 0…0…00

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[CHPW] L not Regular Rank(M L (n)) = n infinitely often [Frobenius 1894] [Iohvidov 1969] Special case when L is unary M L Hankel matrix Main Thm 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0

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What is the power of interactive proofs when the verifier has constant memory ? 2AMFA* = Regular ?? Open question

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Happy Birthday Michael !!

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