Happy Birthday

Michael !!

Probabilistic &NondeterministicFinite Automata

Avi WigdersonInstitute for Advanced Study

Very old (1996) joint work withAnne CondonLisa HelersteinSam Pottle

Pick a computational model.Study the relative power of its variants:Deterministic,Non-deterministic,ProbabilisticPolynomial 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=LFinite automata! (= constant memory)[GMR,B] Arthur-Merlin, [F,D,BV] Quantum

(part of) Rabin’s legacy

€

⊆

Deterministic,Non-deterministic,ProbabilisticArthur-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)

Deterministic,Non-deterministic,ProbabilisticArthur-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

L languageML infinite binary matrix

x,y lexical order y 1101…[Myhill-Nerode] L regular 0110… iff ML has x … L(xy)

finite number of rows iff 1’s of ML have

finite partition/cover by 1-tiles

Communication Complexity [Yao]

111…1…111…1…… … …111…1…… … …

1-tile

x y

Q: states|Q|=s

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…… … …111…1…… … …

x y

Q: states|Q|=s

111…1…111…1…

111…1…

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…… … …111…1…… … …

x y

Q: states|Q|=s

1…1…1111…1…111

… 111…1…… 111…1…

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…… … …111…1…… … …

x y

Q: states|Q|=s

111…1…111…1…

111…1…

L accepted by 2DFA [RS] 2DFA = Regular Tile per crossingSequence c∈Q2s

{x: c consistent with x} X{y: c cons with y & c Acc} s2s tiles (partition)

Proofs

111…1…111…1…… … …111…1…… … …

x y

Q: states|Q|=s

111…1…111…1…

111…1…

L accepted by 2NFA [S] 2NFA = Regular Tile per crossingSequence c∈Q2s

{x can Start c } X{y can c Accept } s tiles (cover)

Proofs

111…1…111…1…… … …111…1…… … …

x y

Q: states|Q|=s

1…1…1111…1…111

… 111…1…… 111…1…

L accepted by 2PFA* y[DS,KF] 2PFA* = Regular

Tile per O(s)-state Markovchain 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”

Proofs

111…1…111…1…… … …111…1…… … …

x y

Q: states|Q|=s

111…1…111…1…

111…1…

L, Lc accepted by 2AMFA*[CHPW] L is Regular Tile per O(s)-state Markovchain 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 ML(n)

(log n)O(s) 0-tiles (cover) of ML(n)

[AUY,MS] Rank(ML(n)) = no(1)

Proofs

111…1…111…1…… … …111…1…… … …

x y

Q: states|Q|=s

1…1…1111…1…111

… 111…1…… 111…1…

111…0…00...0…… … …00…0…… … …

0…0…000…0…00

[CHPW] L not Regular Rank(ML(n)) = n infinitely often

[Frobenius ‘1894][Iohvidov ‘1969]Special case when L is unaryML Hankel matrix

Main Thm

1 1 1 0 1 0 01 1 0 1 0 01 0 1 0 00 1 0 01 0 00 00

What is the power of interactive proofs when the verifier has constant memory?

2AMFA* = Regular ??

Open question

Happy Birthday

Michael !!