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Beyond NP: The Work and Beyond NP: The Work and Legacy of Larry Stockmeyer Legacy of Larry Stockmeyer

Lance Fortnow Lance Fortnow University of Chicago University of Chicago

Larry Joseph Stockmeyer Larry Joseph Stockmeyer

• 1948

1948 – – Born in Indiana Born in Indiana

• 1974

1974 – – MIT Ph.D. MIT Ph.D.

• IBM Research at

IBM Research at Yorktown and Yorktown and Almaden for most of Almaden for most of his career his career

• 82 Papers (11 JACM)

82 Papers (11 JACM)

– – 49 Distinct Co 49 Distinct Co-

• Authors

Authors

• 1996

1996 – – ACM Fellow ACM Fellow

• Died July 31, 2004

Died July 31, 2004

The Universe The Universe

Computer of Protons Computer of Protons

The Universe The Universe

11,000,000,000 Light Years

Computer of Protons Computer of Protons

Radius 10-15 Meters

Computing with the Universe Computing with the Universe

• Universe can only have 10

Universe can only have 10123

123 proton gates.

proton gates.

• Consider the true sentences of weak

Consider the true sentences of weak monadic second monadic second-

• order theory of the natural
• rder theory of the natural

numbers with successor (EWS1S). numbers with successor (EWS1S).

– – ∃ ∃A A ∀ ∀B B ∃ ∃x (x x (x ∈ ∈ A A → → x+1 x+1 ∈ ∈ B) B)

• Cannot solve EWS1S on inputs of size 616

Cannot solve EWS1S on inputs of size 616 in universe with proton in universe with proton-

• sized gates.

sized gates.

– – Stockmeyer Ph.D. Thesis 1974 Stockmeyer Ph.D. Thesis 1974 – – Stockmeyer Stockmeyer-

• Meyer JACM 2002

Meyer JACM 2002

The Universe The Universe

11,000,000,000 Light Years

The Universe The Universe

78,000,000,000 Light Years

Computing with the Universe Computing with the Universe

• Universe can have 10

Universe can have 10123

123 proton gates.

proton gates.

Computing with the Universe Computing with the Universe

• Universe can have

Universe can have 3.5* 3.5*10 1012

125 5 proton gates.

proton gates.

Computing with the Universe Computing with the Universe

• Universe can have

Universe can have 3.5* 3.5*10 1012

125 5 proton gates.

proton gates.

• Cannot solve EWS1S on inputs of size 616

Cannot solve EWS1S on inputs of size 616 in universe with proton in universe with proton-

• sized gates.

sized gates.

Computing with the Universe Computing with the Universe

• Universe can have

Universe can have 3.5* 3.5*10 1012

125 5 proton gates.

proton gates.

• Cannot solve EWS1S on inputs of size 61

Cannot solve EWS1S on inputs of size 619 9 in universe with proton in universe with proton-

• sized gates.

sized gates.

Science Fiction? Science Fiction?

• The complexity of algorithms

The complexity of algorithms tax even the resources of sixty tax even the resources of sixty billion gigabits billion gigabits---

• --or of a
• r of a

universe full of bits; Meyer and universe full of bits; Meyer and Stockmeyer had proved, long Stockmeyer had proved, long ago, that, regardless of ago, that, regardless of computer power, problems computer power, problems existed which could not be existed which could not be solved in the life of the solved in the life of the universe. universe.

Evolution of Complexity Evolution of Complexity

Evolution of Complexity Evolution of Complexity

Turing-Church-Kleene-Post 1936

Computably Enumerable Computable

Evolution of Complexity Evolution of Complexity

Computably Enumerable

Evolution of Complexity Evolution of Complexity

Kleene 1956

Computably Enumerable

Regular Languages Finite Automata

Evolution of Complexity Evolution of Complexity

Chomsky Hierarchy1956

Computably Enumerable

Regular Languages Finite Automata

Evolution of Complexity Evolution of Complexity

Chomsky Hierarchy 1956

Computably Enumerable

Unrestricted Grammars Regular Languages Finite Automata Regular Grammars Context-Free Grammars Push-Down Automata Context-Sensitive Grammars Linear-Bounded Automata

Real Computers Real Computers

Faster Computers Faster Computers

Evolution of Complexity Evolution of Complexity

Computably Enumerable Computable

Evolution of Complexity Evolution of Complexity

Evolution of Complexity Evolution of Complexity

Computable

Evolution of Complexity Evolution of Complexity

Hartmanis-Stearns 1965

Computable

Evolution of Complexity Evolution of Complexity

Hartmanis-Stearns 1965

Computable

TIME(n2)

Evolution of Complexity Evolution of Complexity

Hartmanis-Stearns 1965

Computable

TIME(n2) TIME(2n) TIME(n5)

Evolution of Complexity Evolution of Complexity

Hartmanis-Stearns 1965

TIME(n)

Computable

Limitations of Limitations of DTIME(t(n DTIME(t(n)) ))

• Not Machine Independent.

Not Machine Independent.

• Separations are by diagonalization and not

Separations are by diagonalization and not by natural problems. by natural problems.

• No clear notion of efficient computation.

No clear notion of efficient computation.

Evolution of Complexity Evolution of Complexity

Cobham 1964 Edmonds 1965

Computable

Evolution of Complexity Evolution of Complexity

Cobham 1964 Edmonds 1965

Computable P=∪DTIME(nk)

Evolution of Complexity Evolution of Complexity

Cobham 1964 Edmonds 1965

Computable P=∪DTIME(nk)

Matching

Evolution of Complexity Evolution of Complexity

Computable P

Evolution of Complexity Evolution of Complexity

Computable P

Cook 1971 Levin 1973 Karp 1972

NP

SAT Clique Partition Max Cut

State of Complexity 1972 State of Complexity 1972

Computable P NP

Enter Larry Stockmeyer Enter Larry Stockmeyer

• January 1972

January 1972 – – Bachelors/Masters at MIT Bachelors/Masters at MIT

– – Bounds on Polynomial Evaluation Algorithms Bounds on Polynomial Evaluation Algorithms

• Can we find natural hard problems?

Can we find natural hard problems?

– – Diagonalization methods do not lead to natural Diagonalization methods do not lead to natural problems. problems. – – There are natural NP There are natural NP-

• complete problems but

complete problems but cannot prove them not in P. cannot prove them not in P. – – With Advisor Albert Meyer With Advisor Albert Meyer

Regular Expressions with Squaring Regular Expressions with Squaring

• (0+1)*00(0+1)*00(0+1)*

(0+1)*00(0+1)*00(0+1)*

– – All strings with two sets of consecutive zeros. All strings with two sets of consecutive zeros.

• Allow Squaring operator: r

Allow Squaring operator: r2

2=

=rr rr

• (0+1)*(0

(0+1)*(02

2(0+1)*)

(0+1)*)2

2

• No more expressive power but can be much

No more expressive power but can be much shorter when used recursively. shorter when used recursively.

– – ((((((0 ((((((02

2)

)2

2)

)2

2)

)2

2)

)2

2)

)2

2)=

)= 0000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

Computable P NP

REGSQ = { R | L(R) ≠ Σ*}

EXPSPACE

REGSQ

PSPACE

Regular Expressions with Squaring Regular Expressions with Squaring

• Meyer and Stockmeyer,

Meyer and Stockmeyer, “ “The Equivalence The Equivalence Problem for Regular Expressions with Problem for Regular Expressions with Squaring Requires Exponential Space Squaring Requires Exponential Space” ” – – SWAT 1972 SWAT 1972

• MINIMAL

MINIMAL

– – Set of Boolean formulas with no smaller Set of Boolean formulas with no smaller equivalent formula. equivalent formula.

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

Computable P NP

Complexity of MINIMAL

MINIMAL

MINIMAL MINIMAL

• MINIMAL

MINIMAL

– – Set of Boolean formulas with no smaller Set of Boolean formulas with no smaller equivalent formula. equivalent formula.

• MINIMAL in NP?

MINIMAL in NP?

– – Can Can’ ’t check all smaller formulas. t check all smaller formulas.

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

Computable P NP

Complexity of MINIMAL

MINIMAL MINIMAL

MINIMAL MINIMAL

• MINIMAL

MINIMAL

– – Set of Boolean formulas with no smaller Set of Boolean formulas with no smaller equivalent formula. equivalent formula.

• MINIMAL in NP?

MINIMAL in NP?

– – Can Can’ ’t check all smaller formulas. t check all smaller formulas.

• MINIMAL in NP?

MINIMAL in NP?

– – Can Can’ ’t check equivalence. t check equivalence.

MINIMAL MINIMAL

• MINIMAL

MINIMAL

– – Set of Boolean formulas with no smaller Set of Boolean formulas with no smaller equivalent formula. equivalent formula.

• MINIMAL in NP?

MINIMAL in NP?

– – Can Can’ ’t check all smaller formulas. t check all smaller formulas.

• MINIMAL in NP?

MINIMAL in NP?

– – Can Can’ ’t check equivalence. t check equivalence.

• MINIMAL is in NP with an

MINIMAL is in NP with an “ “oracle

• racle”

” for for equivalence. equivalence.

MINIMAL in NP with Equivalence Oracle MINIMAL in NP with Equivalence Oracle (x ∨ y) ∧ (x ∨ y) ∧ z Equivalence

(x ∧ z , (x ∨ y) ∧ (x ∨ y) ∧ z)

Guess: x ∧ z EQUIVALENT

MINIMAL MINIMAL

• MINIMAL is in NP with an

MINIMAL is in NP with an “ “oracle

• racle”

” for for equivalence or non equivalence or non-

• equivalence.

equivalence.

MINIMAL MINIMAL

• MINIMAL is in NP with an

MINIMAL is in NP with an “ “oracle

• racle”

” for for equivalence or non equivalence or non-

• equivalence.

equivalence.

• Since non

Since non-

• equivalence is in NP we can

equivalence is in NP we can solve MINIMAL in NP with NP oracle. solve MINIMAL in NP with NP oracle.

MINIMAL MINIMAL

• MINIMAL is in NP with an

MINIMAL is in NP with an “ “oracle

• racle”

” for for equivalence or non equivalence or non-

• equivalence.

equivalence.

• Since non

Since non-

• equivalence is in NP we can

equivalence is in NP we can solve MINIMAL in NP with NP oracle. solve MINIMAL in NP with NP oracle.

• Suggests a

Suggests a “ “hierarchy hierarchy” ” above NP. above NP.

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

P NP

The Polynomial Time Hierarchy

NPNP

MINIMAL

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

P NP=Σ1

p

The Polynomial Time Hierarchy

NPNP

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

P NP=Σ1

p

The Polynomial Time Hierarchy

NPNP =Σ2

p

NPΣ2p =Σ3

p

NPΣ3p =Σ4

p

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

P Σ1

p=NP

The Polynomial Time Hierarchy

Σ2

p

Σ3

p

Σ4

p

co-NP=Π1

p

co-NPNP=Π2

p

MINIMAL

co-NPΣ2p =Π3

p

co-NPΣ3p =Π4

p

Meyer Meyer-

• Stockmeyer 1972

Stockmeyer 1972

PNP=∆2

p

Σ1

p=NP

The Polynomial Time Hierarchy

Σ2

p

Σ3

p

Σ4

p

co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

PΣ2p=∆3

p

PΣ3p=∆4

p

PH

Properties of the Hierarchy Properties of the Hierarchy

• Meyer

Meyer-

• Stockmeyer,

Stockmeyer, “ “The Equivalence The Equivalence Problem for Regular Expressions with Problem for Regular Expressions with Squaring Requires Exponential Space Squaring Requires Exponential Space” ”, , SWAT 1972 SWAT 1972

• Stockmeyer,

Stockmeyer, “ “The Polynomial The Polynomial-

• Time

Time Hierarchy Hierarchy” ”, , TCS, 1977. TCS, 1977.

• Wrathall

Wrathall, , “ “Complete Sets and the Complete Sets and the Polynomial Polynomial-

• Time Hierarchy

Time Hierarchy” ”, TCS, 1977. , TCS, 1977.

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

Σ3

p=∆3 p

∆4

p

PH PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

Σ3

p=∆3 p

∆4

p

PH PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Co-NP=Π1

p

Π2

p

P=∆1

p

PH=Σ3

p=∆3 p=Π3 p

PSPACE

Properties of the Hierarchy Properties of the Hierarchy

P=NP=PH PSPACE

If P = NP

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH=PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH=PSPACE

Properties of the Hierarchy Properties of the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Co-NP=Π1

p

Π2

p

P=∆1

p

PSPACE=PH=Σ3

p=∆3 p=Π3 p

Quantifier Characterization Quantifier Characterization

A language L is in A language L is in Σ Σ3

3 P P if for all x in

if for all x in Σ Σ* *

x is in L x is in L ⇔ ⇔ ∃ ∃u u ∀ ∀v v ∃ ∃w w P(x,u,v,w P(x,u,v,w) )

A language L is in A language L is in Π Π3

3 P P if for all x in

if for all x in Σ Σ* *

x is in L x is in L ⇔ ⇔ ∀ ∀u u ∃ ∃v v ∀ ∀w w P(x,u,v,w P(x,u,v,w) )

Complete Sets Complete Sets

• We define B

We define B3

3 by the set of true quantified

by the set of true quantified formula of the form formula of the form ∃ ∃x x1

1∃

∃x x2

2 … …∃

∃x xn

n∀

∀y y1

1 … …∀

∀y yn

n∃

∃z z1

1 … …∃

∃z zn

n

ϕ ϕ(x (x1

1,

,… …,x ,xn

n,y

,y1

1,

,… …,y ,yn

n,z

,z1

1,

,… …,z ,zn

n)

)

Complete Sets in the Hierarchy Complete Sets in the Hierarchy

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE B3 B4 B2 B1=SAT B4 B3 B2 B1

Natural Complete Sets Natural Complete Sets

• N

N-

• INEQ

INEQ – – Inequivalence of Integer Inequivalence of Integer Expressions with union and addition. Expressions with union and addition.

(50+(40 (50+(40∪ ∪20 20∪ ∪15)) 15))∪ ∪((2 ((2∪ ∪5)+(7 5)+(7∪ ∪9)) 9))

• Meyer

Meyer-

• Stockmeyer 1973 Stockmeyer 1977

Stockmeyer 1973 Stockmeyer 1977

– – N N-

• INEQ is

INEQ is Σ Σ2

2p p-

• complete

complete

• Umans 1999

Umans 1999

– – Succinct Set Cover is Succinct Set Cover is Σ Σ2

2p p-

• complete

complete

• Schafer 1999

Schafer 1999

– – Succinct VC Dimension is Succinct VC Dimension is Σ Σ3

3p p-

• complete

complete

The The ω ω-

• jump of the Hierarchy

jump of the Hierarchy

• Meyer

Meyer-

• Stockmeyer 1973, Stockmeyer 1977

Stockmeyer 1973, Stockmeyer 1977 B Bω

ω=

=∪ ∪B Bk

k

• Quantified Boolean Formula with an

Quantified Boolean Formula with an unbounded number of alterations. unbounded number of alterations.

• Now called QBF or TQBF.

Now called QBF or TQBF.

Complexity of Complexity of ω ω-

• jump

jump

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE B3 B4 B2 B1=SAT B4 B3 B2 B1 Bω (TQBF)

Alternation Alternation

• Chandra

Chandra-

• Kozen

Kozen-

• Stockmeyer JACM 1981

Stockmeyer JACM 1981

• Chandra

Chandra-

• Stockmeyer STOC 1976

Stockmeyer STOC 1976

• Kozen

Kozen FOCS 1976 FOCS 1976

Alternation Alternation

∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Acc Acc

Alternation Alternation

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀

Alternation Alternation

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀

Acc AccAccAcc Acc Acc Acc Acc Acc Acc Acc Acc Acc Acc Acc Acc

Alternation Alternation

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Rej Acc Acc Acc Acc

Alternation Theorems Alternation Theorems

• Chandra

Chandra-

• Kozen

Kozen-

• Stockmeyer

Stockmeyer

• ATIME(t(n

ATIME(t(n)) )) ⊆ ⊆ DSPACE(t(n DSPACE(t(n)) ))

• NSPACE(s(n

NSPACE(s(n)) )) ⊆ ⊆ ATIME(s ATIME(s2

2(n))

(n))

• ASPACE(s(n

ASPACE(s(n)) = )) = ∪ ∪DTIME(c DTIME(cs(n

s(n) ))

) L ⊆ P ⊆ PSPACE ⊆ EXP ⊆ EXPSPACE ⊆ …

= = = =

AL AP APSPACE ⊆ ⊆ AEXP ⊆ ⊆ …

Alternate Characterization of Alternate Characterization of Σ Σ2

2p p

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Other Alternating Models Other Alternating Models

• Log

Log-

• Space Hierarchy

Space Hierarchy

– – Collapses to NL (Immerman Collapses to NL (Immerman-

• Szelepcs

Szelepcsé ényi nyi ’ ’88) 88)

• Alternating Finite State Automaton

Alternating Finite State Automaton

– – Same power as DFA but doubly exponential Same power as DFA but doubly exponential blowup in states. blowup in states.

• Alternating Push

Alternating Push-

• Down Automaton

Down Automaton

– – Accepts exactly E=DTIME(2 Accepts exactly E=DTIME(2O(n)

O(n))

) – – Strictly stronger than Strictly stronger than PDAs PDAs – – Inclusion due to Inclusion due to Ladner Ladner-

• Lipton

Lipton-

• Stockmeyer

Stockmeyer ’ ’78 78

Chandra-Kozen-Stockmeyer 1981

Alternation as a Game Alternation as a Game

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Alternation as a Game Alternation as a Game

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Alternation as a Game Alternation as a Game

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Alternation as a Game Alternation as a Game

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Alternation as a Game Alternation as a Game

∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∃ ∃ ∃

Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Complete Sets Via Games Complete Sets Via Games

• Stockmeyer

Stockmeyer-

• Chandra 1979

Chandra 1979

• Can use problems based on games to get

Can use problems based on games to get completeness results for PSPACE and EXP. completeness results for PSPACE and EXP.

• Create a combinatorial game that is EXP

Create a combinatorial game that is EXP-

• complete and thus not decidable in P.

complete and thus not decidable in P.

• First complete sets for PSPACE and EXP

First complete sets for PSPACE and EXP not based on machines or logic. not based on machines or logic.

Checkers Checkers

Generalized Checkers Generalized Checkers

Generalized Checkers Generalized Checkers

• PSPACE

PSPACE-

• hard

hard

– – Fraenkel Fraenkel et al. 1978 et al. 1978

• EXP

EXP-

• complete

complete

– – Robson 1984 Robson 1984

Approximate Approximate Couting Couting

• #P

#P – – Valiant 1979 Valiant 1979

– – Functions that count solutions of NP problems. Functions that count solutions of NP problems. – – Permanent is #P Permanent is #P-

• complete

complete

• Stockmeyer 1985 building on Sipser 1983

Stockmeyer 1985 building on Sipser 1983

– – Can approximate any #P function f in Can approximate any #P function f in polytime polytime with an oracle for with an oracle for Σ Σ2

2p p.

.

• Toda 1991

Toda 1991

– – Every language in PH reducible to #P Every language in PH reducible to #P

Complexity of #P Complexity of #P

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE P#P Perm Approx-#P

Legacy of Larry Stockmeyer Legacy of Larry Stockmeyer

• Circuit Complexity

Circuit Complexity

• Infinite Hierarchy Conjecture

Infinite Hierarchy Conjecture

• Probabilistic Computation

Probabilistic Computation

• Interactive Proof Systems

Interactive Proof Systems

Circuit Complexity Circuit Complexity

• Baker

Baker-

• Gill

Gill-

• Solovay

Solovay ’ ’75: 75: Relativization Relativization Paper Paper

– – Open: Is PH infinite relative to an oracle? Open: Is PH infinite relative to an oracle?

• Sipser

Sipser ’ ’83: Strong lower bounds on depth d 83: Strong lower bounds on depth d circuits simulating depth d+1 circuits. circuits simulating depth d+1 circuits.

• Yao

Yao ’ ’85: 85: “ “Separating the Polynomial Separating the Polynomial-

• Time

Time Hierarchy by Oracles Hierarchy by Oracles” ”

• Led to future circuit results by H

Led to future circuit results by Hå åstad, stad, Razborov, Razborov, Smolensky Smolensky and many others. and many others.

Infinite Hierarchy Conjecture Infinite Hierarchy Conjecture

• Is the Polynomial

Is the Polynomial-

• Time Hierarchy Infinite?

Time Hierarchy Infinite?

• Best Evidence:

Best Evidence: Yao Yao’ ’s s result shows result shows alternating log alternating log-

• time hierarchy infinite.

time hierarchy infinite.

• Many complexity results

Many complexity results

– – If PROP then the polynomial If PROP then the polynomial-

• time hierarchy

time hierarchy collapses. collapses. – – If PH is infinite then NOT PROP. If PH is infinite then NOT PROP.

• Gives evidence for NOT PROP.

Gives evidence for NOT PROP.

If Hierarchy is Infinite If Hierarchy is Infinite … …

• SAT does not have small circuits.

SAT does not have small circuits.

– – Karp Karp-

• Lipton 1980

Lipton 1980

• Graph isomorphism is not NP

Graph isomorphism is not NP-

• complete.

complete.

– – Goldreich Goldreich-

• Micali

Micali-

• Wigderson

Wigderson 1991 1991 – – Goldwasser Goldwasser-

• Sipser 1989

Sipser 1989 – – Boppana Boppana-

• H

Hå åstad stad-

• Zachos 1987

Zachos 1987

• Boolean hierarchy is infinite.

Boolean hierarchy is infinite.

– – Kadin Kadin 1988 1988

Boolean Hierarchy Boolean Hierarchy

• BH

BH1

1 = NP

= NP

• BH

BHk+1

k+1 = { B

= { B-

• C | B in NP and C in

C | B in NP and C in BH BHk

k}

}

• { (

{ (G,k G,k) | Max clique of G has size k} in BH ) | Max clique of G has size k} in BH2

2

• Kadin

Kadin: If : If BH BHk

k=BH

=BHk+1

k+1 then PH=

then PH=Σ Σ3

3 p p.

.

Probabilistic Computation Probabilistic Computation

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE P#P

Probabilistic Computation Probabilistic Computation

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE P#P BPP

Sipser-Gács-Lautemann 1983

Interactive Proof Systems Interactive Proof Systems

• Papadimitriou 1985

Papadimitriou 1985 – – Alternation between Alternation between nondeterministic and probabilistic players nondeterministic and probabilistic players

• Interactive Proof Systems

Interactive Proof Systems

– – Public Coin: Babai Public Coin: Babai-

• Moran 1988

Moran 1988 – – Private Coin: Private Coin: Goldwasser Goldwasser-

• Micali

Micali-

• Rackoff

Rackoff 1989 1989 – – Equivalent: Goldwasser Equivalent: Goldwasser-

• Sipser 1989

Sipser 1989

Interactive Proof Systems Interactive Proof Systems

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE P#P BPP

Babai-Moran 1988

MA AM

Interactive Proof Systems Interactive Proof Systems

∆2

p

Σ1

p=NP

Σ2

p

Σ3

p

Σ4

p

Co-NP=Π1

p

Π2

p

Π3

p

Π4

p

P=∆1

p

∆3

p

∆4

p

PH PSPACE=IP P#P BPP

LFKN, Shamir 1992

MA AM

Interactive Proof Systems Interactive Proof Systems

• Hardness of Approximation

Hardness of Approximation

– – Feige Feige-

• Goldwasser

Goldwasser-

• Lov

Lová ász sz-

• Safra

Safra-

• Szegedy

Szegedy 1996 1996

• Probabilistically Checkable Proofs

Probabilistically Checkable Proofs

– – NP in PCPs with NP in PCPs with O(log O(log n) coins and constant n) coins and constant number of queries. number of queries. – – Arora Arora-

• Lund

Lund-

• Motwani

Motwani-

• Sudan

Sudan-

• Szegedy

Szegedy 1998 1998

• Interactive Proofs with Finite State Verifiers

Interactive Proofs with Finite State Verifiers

– – Dwork Dwork and Stockmeyer and Stockmeyer

Other Work Other Work

• Larry Stockmeyer contributed much more to

Larry Stockmeyer contributed much more to complexity and important work in other complexity and important work in other areas including automata theory and parallel areas including automata theory and parallel and distributed computing. and distributed computing.

• Most Cited Article (

Most Cited Article (CiteSeer CiteSeer): ):

– – Dwork Dwork, Lynch, and Stockmeyer, , Lynch, and Stockmeyer, “ “Consensus in Consensus in the presence of partial synchrony the presence of partial synchrony” ” JACM, 1988. JACM, 1988.

Conclusion Conclusion

• What natural problems can

What natural problems can’ ’t we compute? t we compute?

• Led to exciting work on polynomial

Led to exciting work on polynomial-

• time

time hierarchy, alternation, approximation and hierarchy, alternation, approximation and much more. much more.

• These idea affect much of computational

These idea affect much of computational complexity today and the legacy will complexity today and the legacy will continue for generations in the future. continue for generations in the future.

Remembering Remembering

• Other members of our community that we

Other members of our community that we have recently lost have recently lost… …

George George Dantzig Dantzig

Shimon Even Shimon Even

Seymour Ginsburg Seymour Ginsburg

Frank Frank Harary Harary

Leonid Leonid Khachiyan Khachiyan

Clemens Clemens Lautemann Lautemann

Carl Smith Carl Smith

Larry Stockmeyer Larry Stockmeyer