Happy Birthday Michael !!

1. Happy Birthday Michael !!

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

3. 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⊆L2 [IS] NL=coNL [N] BPL⊆SC [SZ] BPL=L3/2[R] SL=L Finite automata! (= constant memory) [GMR,B] Arthur-Merlin, [F,D,BV] Quantum (part of) Rabin’s legacy

4. Finite Automata (FA) 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

5. Deterministic,Non-deterministic,Probabilistic Arthur-Merlin, Quantum & 1-way vs. 2-way read [Rabin-Scott ’59] 1NFA = 2DFA = Regular [Rabin ’63] 1PFA = Regular [Shepherdson’59] 2NFA = Regular [Freivalds ’81] 2PFA can compute {anbn} !! (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 {anbn}, {anbncn}!! (linear time) OPEN: 2AMFA* = Regular ?? Results

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

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

8. y x 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) Q: states |Q|=s Proofs 1…1…111 1…1…111 111…1… 111…1… … … … 111…1… … … … … 111…1… … 111…1…

9. y x 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) Q: states |Q|=s Proofs 111…1… 111…1… 111…1… 111…1… … … … 111…1… … … … 111…1…

10. y x L accepted by 2DFA [RS] 2DFA= Regular Tile per crossing Sequence c∈Q2s {x: cconsistent with x}X {y: ccons with y & c Acc} s2s tiles (partition) Q: states |Q|=s Proofs 111…1… 111…1… 111…1… 111…1… … … … 111…1… … … … 111…1…

11. y x L accepted by 2NFA [S] 2NFA= Regular Tile per crossing Sequence c∈Q2s {x can Start  c }X {y can c  Accept } s tiles (cover) Q: states |Q|=s Proofs 1…1…111 1…1…111 111…1… 111…1… … … … 111…1… … … … … 111…1… … 111…1…

12. y x 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 ML(n) [Karp,DS,KF] ML(n) has large “nonregularity” Q: states |Q|=s Proofs 111…1… 111…1… 111…1… 111…1… … … … 111…1… … … … 111…1…

13. y x L, Lc accepted by 2AMFA* [CHPW] L is Regular Tile per O(s)-state Markov chain m∈[log n]O(s) {x canbe m-consistent}X {y canbe m-cons& Pr[m Acc]>2/3} (log n)O(s) 1-tiles (cover)of ML(n) (log n)O(s) 0-tiles (cover) of ML(n)  [AUY,MS] Rank(ML(n)) = no(1) Q: states |Q|=s Proofs 0…0…00 0…0…00 111…0… 00...0… … … … 00…0… … … … 1…1…111 1…1…111 111…1… 111…1… … … … 111…1… … … … … 111…1… … 111…1…

14. [CHPW] L not Regular  Rank(ML(n)) = n infinitely often [Frobenius ‘1894] [Iohvidov ‘1969] Special case when L is unary ML 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 0 0

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

16. Happy Birthday Michael !!