Beyond NP: The Work and Beyond NP: The Work and Legacy of Larry - PowerPoint PPT Presentation

MINIMAL in NP with Equivalence Oracle MINIMAL in NP with Equivalence Oracle (x ∨ y) ∧ (x ∨ y) ∧ z Equivalence Guess: x ∧ z (x ∧ z , (x ∨ y) ∧ (x ∨ y) ∧ z) EQUIVALENT

MINIMAL MINIMAL � MINIMAL is in NP with an � MINIMAL is in NP with an “ “oracle oracle” ” for for equivalence or non- -equivalence. equivalence. equivalence or non

MINIMAL MINIMAL � MINIMAL is in NP with an � MINIMAL is in NP with an “ “oracle oracle” ” for for equivalence or non- -equivalence. equivalence. equivalence or non � 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 oracle” ” for for equivalence or non- -equivalence. equivalence. equivalence or non � 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- -Stockmeyer 1972 Stockmeyer 1972 Meyer The Polynomial Time Hierarchy NP NP MINIMAL NP P

Meyer- -Stockmeyer 1972 Stockmeyer 1972 Meyer The Polynomial Time Hierarchy NP NP NP= Σ 1 p P

Meyer- -Stockmeyer 1972 Stockmeyer 1972 Meyer The Polynomial Time Hierarchy NP Σ 3p = Σ 4 p NP Σ 2p = Σ 3 p NP NP = Σ 2 p NP= Σ 1 p P

Meyer- -Stockmeyer 1972 Stockmeyer 1972 Meyer The Polynomial Time Hierarchy Σ 4 co-NP Σ 3p = Π 4 p p Σ 3 co-NP Σ 2p = Π 3 p p Σ 2 p co-NP NP = Π 2 p MINIMAL Σ 1 p =NP co-NP= Π 1 p P

Meyer- -Stockmeyer 1972 Stockmeyer 1972 Meyer The Polynomial Time Hierarchy PH Σ 4 p Π 4 p P Σ 3p = ∆ 4 p Σ 3 p Π 3 p P Σ 2p = ∆ 3 p Σ 2 p Π 2 p P NP = ∆ 2 p Σ 1 p =NP co-NP= Π 1 p P= ∆ 1 p

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” ” , TCS, 1977. Hierarchy , TCS, 1977. � Wrathall � Wrathall, , “ “Complete Sets and the Complete Sets and the Polynomial- -Time Hierarchy Time Hierarchy” ”, TCS, 1977. , TCS, 1977. Polynomial

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH Σ 4 p Π 4 p ∆ 4 p Π 3 p Σ 3 p = ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH Σ 4 p Π 4 p ∆ 4 p Π 3 p Σ 3 p = ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH= Σ 3 p = ∆ 3 p = Π 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy If P = NP PSPACE P=NP=PH

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PSPACE PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PH=PSPACE Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PH=PSPACE Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Properties of the Hierarchy Properties of the Hierarchy PSPACE=PH= Σ 3 p = ∆ 3 p = Π 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Quantifier Characterization Quantifier Characterization Σ 3 Σ * A language L is in Σ if for all x in Σ P if for all x in P * A language L is in 3 ⇔ ∃ ∃ u ∀ v ∃ w x is in L ⇔ u ∀ v ∃ w P(x,u,v,w P(x,u,v,w) ) x is in L Π 3 Σ * A language L is in Π if for all x in Σ P if for all x in P * A language L is in 3 ⇔ ∀ ∀ u ∃ v ∀ w x is in L ⇔ u ∃ v ∀ w P(x,u,v,w P(x,u,v,w) ) x is in L

Complete Sets Complete Sets � We define B � We define B 3 by the set of true quantified 3 by the set of true quantified formula of the form formula of the form ∃ x ∃ x ∃ x ∀ y ∀ y ∃ z ∃ z ∃ 1 ∃ … ∃ n ∀ … ∀ n ∃ … ∃ … … … x 1 x 2 x n y 1 y n z 1 z n 2 1 1 n ϕ (x … ,x … ,y … ,z ϕ , … , … , … (x 1 ,x n ,y 1 ,y n ,z 1 ,z n ) 1 , n ,y 1 , n ,z 1 , n )

Complete Sets in the Hierarchy Complete Sets in the Hierarchy PSPACE PH Σ 4 p B 4 Π 4 B 4 p ∆ 4 p Σ 3 p B 3 Π 3 B 3 p ∆ 3 p Σ 2 p B 2 Π 2 B 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 B 1 =SAT B 1 p P= ∆ 1 p

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. ∪ 20 ∪ 15)) ∪ ((2 ∪ 5)+(7 ∪ 9)) (50+(40 ∪ 20 ∪ 15)) ∪ ((2 ∪ 5)+(7 ∪ 9)) (50+(40 � Meyer � Meyer- -Stockmeyer 1973 Stockmeyer 1977 Stockmeyer 1973 Stockmeyer 1977 Σ 2 INEQ is Σ 2p p - – N N- -INEQ is -complete complete – � Umans 1999 � Umans 1999 Σ 2 Succinct Set Cover is Σ 2p p - – Succinct Set Cover is -complete complete – � Schafer 1999 � Schafer 1999 Σ 3 Succinct VC Dimension is Σ p - – Succinct VC Dimension is 3p -complete complete –

ω - The ω -jump of the Hierarchy jump of the Hierarchy The � Meyer � Meyer- -Stockmeyer 1973, Stockmeyer 1977 Stockmeyer 1973, Stockmeyer 1977 ∪ B = ∪ B ω B k B ω = 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.

ω - Complexity of ω -jump jump Complexity of B ω (TQBF) PSPACE PH Σ 4 p B 4 Π 4 B 4 p ∆ 4 p Σ 3 p B 3 Π 3 B 3 p ∆ 3 p Σ 2 p B 2 Π 2 B 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 B 1 =SAT B 1 p P= ∆ 1 p

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 Acc Rej Rej Rej Rej Acc Acc

Alternation Alternation ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀

Alternation Alternation ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ Acc Acc Acc Acc Acc Acc Acc AccAccAcc 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 ⊆ DSPACE(t(n )) ⊆ � ATIME(t(n � ATIME(t(n)) DSPACE(t(n)) )) ⊆ ATIME(s )) ⊆ � NSPACE(s(n � 2 (n)) NSPACE(s(n)) ATIME(s 2 (n)) ∪ DTIME(c )) = ∪ � ASPACE(s(n � s(n) ) ) ASPACE(s(n)) = DTIME(c s(n ) L ⊆ P ⊆ PSPACE ⊆ EXP ⊆ EXPSPACE ⊆ … = = = = ⊆ ⊆ ⊆ ⊆ … AL AP APSPACE AEXP

Σ 2 Alternate Characterization of Σ p 2p Alternate Characterization of ∃ ∃ ∃ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ ∀ Rej Rej Rej Rej Rej Rej Rej Rej Rej Acc Rej Rej Acc Acc Acc Acc

Other Alternating Models Other Alternating Models Chandra-Kozen-Stockmeyer 1981 � 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 O(n) O(n) ) – Accepts exactly E=DTIME(2 ) – – Strictly stronger than Strictly stronger than PDAs PDAs – – Inclusion due to Inclusion due to Ladner Ladner- -Lipton Lipton- -Stockmeyer Stockmeyer ’ ’78 78 –

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 Couting Couting Approximate � #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 – Σ 2 with an oracle for Σ p . 2p . with an oracle for � Toda 1991 � Toda 1991 – Every language in PH reducible to #P Every language in PH reducible to #P –

Complexity of #P Complexity of #P PSPACE P #P Perm PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 Approx-#P p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 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, Smolensky Smolensky and many others. and many others. Razborov,

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- -time hierarchy infinite. time hierarchy infinite. alternating log � 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 � BH 1 = NP 1 = NP � BH � BH k+1 = { B- -C | B in NP and C in C | B in NP and C in BH BH k } k+1 = { B k } � { ( � { (G,k G,k) | Max clique of G has size k} in BH ) | Max clique of G has size k} in BH 2 2 Σ 3 then PH= Σ � Kadin � p . Kadin: If : If BH BH k =BH k+1 p . k =BH k+1 then PH= 3

Probabilistic Computation Probabilistic Computation PSPACE P #P PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

Probabilistic Computation Probabilistic Computation Sipser-Gács-Lautemann 1983 PSPACE P #P PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p BPP ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p

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 Babai-Moran 1988 PSPACE P #P PH Σ 4 p Π 4 p ∆ 4 p Σ 3 p Π 3 p ∆ 3 p Σ 2 p Π 2 p AM MA BPP ∆ 2 p Σ 1 p =NP Co-NP= Π 1 p P= ∆ 1 p