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Beyond NP: the Work and Legacy of Larry Stockmeyer BeyondBeyond NP:NP: TheThe WorkWork andand LegacyLegacy ofof LarryLarry StockmeyerStockmeyer LanceLance FortnowFortnow UniversityUniversity ofof ChicagoChicago LarryLarry JosephJoseph StockmeyerStockmeyer 19481948 –– BornBorn inin IndianaIndiana 19741974 –– MITMIT Ph.D.Ph.D. IBMIBM ResearchResearch atat YorktownYorktown andand AlmadenAlmaden forfor mostmost ofof hishis careercareer 8282 PapersPapers (11(11 JACM)JACM) – 49 Distinct Co-Authors 19961996 –– ACMACM FellowFellow DiedDied JulyJuly 31,31, 20042004 TheThe UniverseUniverse ComputerComputer ofof ProtonsProtons TheThe UniverseUniverse 11,000,000,000 Light Years ComputerComputer ofof ProtonsProtons Radius 10-15 Meters ComputingComputing withwith thethe UniverseUniverse UniverseUniverse cancan onlyonly havehave 1010123 protonproton gates.gates. ConsiderConsider thethe truetrue sentencessentences ofof weakweak monadicmonadic secondsecond--orderorder theorytheory ofof thethe naturalnatural numbersnumbers withwith successorsuccessor (EWS1S).(EWS1S). –– ∃∃AA ∀∀BB ∃∃xx (x(x ∈∈ AA →→ x+1x+1 ∈∈ B)B) CannotCannot solvesolve EWS1SEWS1S onon inputsinputs ofof sizesize 616616 inin universeuniverse withwith protonproton--sizedsized gates.gates. –– StockmeyerStockmeyer Ph.D.Ph.D. ThesisThesis 19741974 –– StockmeyerStockmeyer--MeyerMeyer JACMJACM 20022002 TheThe UniverseUniverse 11,000,000,000 Light Years TheThe UniverseUniverse 78,000,000,000 Light Years ComputingComputing withwith thethe UniverseUniverse UniverseUniverse cancan havehave 1010123 protonproton gates.gates. ComputingComputing withwith thethe UniverseUniverse UniverseUniverse cancan havehave 3.5*3.5*1010125 protonproton gates.gates. ComputingComputing withwith thethe UniverseUniverse UniverseUniverse cancan havehave 3.5*3.5*1010125 protonproton gates.gates. CannotCannot solvesolve EWS1SEWS1S onon inputsinputs ofof sizesize 616616 inin universeuniverse withwith protonproton--sizedsized gates.gates. ComputingComputing withwith thethe UniverseUniverse UniverseUniverse cancan havehave 3.5*3.5*1010125 protonproton gates.gates. CannotCannot solvesolve EWS1SEWS1S onon inputsinputs ofof sizesize 616199 inin universeuniverse withwith protonproton--sizedsized gates.gates. ScienceScience Fiction?Fiction? TheThe complexitycomplexity ofof algorithmsalgorithms taxtax eveneven thethe resourcesresources ofof sixtysixty billionbillion gigabitsgigabits------oror ofof aa universeuniverse fullfull ofof bits;bits; MeyerMeyer andand StockmeyerStockmeyer hadhad proved,proved, longlong ago,ago, that,that, regardlessregardless ofof computercomputer power,power, problemsproblems existedexisted whichwhich couldcould notnot bebe solvedsolved inin thethe lifelife ofof thethe universe.universe. EvolutionEvolution ofof ComplexityComplexity EvolutionEvolution ofof ComplexityComplexity Turing-Church-Kleene-Post 1936 Computably Enumerable Computable EvolutionEvolution ofof ComplexityComplexity Computably Enumerable EvolutionEvolution ofof ComplexityComplexity Kleene 1956 Computably Enumerable Regular Languages Finite Automata EvolutionEvolution ofof ComplexityComplexity Chomsky Hierarchy 1956 Computably Enumerable Regular Languages Finite Automata EvolutionEvolution ofof ComplexityComplexity Chomsky Hierarchy 1956 Computably Enumerable Unrestricted Grammars Context-Sensitive Grammars Linear-Bounded Automata Context-Free Grammars Push-Down Automata Regular Languages Finite Automata Regular Grammars RealReal ComputersComputers FasterFaster ComputersComputers EvolutionEvolution ofof ComplexityComplexity Computably Enumerable Computable EvolutionEvolution ofof ComplexityComplexity EvolutionEvolution ofof ComplexityComplexity Computable EvolutionEvolution ofof ComplexityComplexity Hartmanis-Stearns 1965 Computable EvolutionEvolution ofof ComplexityComplexity Hartmanis-Stearns 1965 Computable TIME(n2) EvolutionEvolution ofof ComplexityComplexity Hartmanis-Stearns 1965 Computable TIME(2n) TIME(n5) TIME(n2) EvolutionEvolution ofof ComplexityComplexity Hartmanis-Stearns 1965 Computable TIME(n) LimitationsLimitations ofof DTIME(t(nDTIME(t(n)))) NotNot MachineMachine Independent.Independent. SeparationsSeparations areare byby diagonalizationdiagonalization andand notnot byby naturalnatural problems.problems. NoNo clearclear notionnotion ofof efficientefficient computation.computation. EvolutionEvolution ofof ComplexityComplexity Cobham 1964 Edmonds 1965 Computable EvolutionEvolution ofof ComplexityComplexity Cobham 1964 Edmonds 1965 Computable P=∪DTIME(nk) EvolutionEvolution ofof ComplexityComplexity Cobham 1964 Edmonds 1965 Computable Matching P=∪DTIME(nk) EvolutionEvolution ofof ComplexityComplexity Computable P EvolutionEvolution ofof ComplexityComplexity Cook 1971 Levin 1973 Karp 1972 Computable SAT Clique NP Partition Max Cut P StateState ofof ComplexityComplexity 19721972 Computable NP P EnterEnter LarryLarry StockmeyerStockmeyer JanuaryJanuary 19721972 –– Bachelors/MastersBachelors/Masters atat MITMIT –– BoundsBounds onon PolynomialPolynomial EvaluationEvaluation AlgorithmsAlgorithms CanCan wewe findfind naturalnatural hardhard problems?problems? –– DiagonalizationDiagonalization methodsmethods dodo notnot leadlead toto naturalnatural problems.problems. –– ThereThere areare naturalnatural NPNP--completecomplete problemsproblems butbut cannotcannot proveprove themthem notnot inin P.P. –– WithWith AdvisorAdvisor AlbertAlbert MeyerMeyer RegularRegular ExpressionsExpressions withwith SquaringSquaring (0+1)*00(0+1)*00(0+1)*(0+1)*00(0+1)*00(0+1)* –– AllAll stringsstrings withwith twotwo setssets ofof consecutiveconsecutive zeros.zeros. AllowAllow SquaringSquaring operator:operator: rr2==rrrr (0+1)*(0(0+1)*(02(0+1)*)(0+1)*)2 NoNo moremore expressiveexpressive powerpower butbut cancan bebe muchmuch shortershorter whenwhen usedused recursively.recursively. –– ((((((0((((((02))2))2))2))2))2)=)= 0000000000000000000000000000000000000000000000000000000000000000 MeyerMeyer--StockmeyerStockmeyer 19721972 REGSQ = { R | L(R) ≠Σ*} Computable REGSQ EXPSPACE PSPACE NP P RegularRegular ExpressionsExpressions withwith SquaringSquaring MeyerMeyer andand Stockmeyer,Stockmeyer, ““TheThe EquivalenceEquivalence ProblemProblem forfor RegularRegular ExpressionsExpressions withwith SquaringSquaring RequiresRequires ExponentialExponential SpaceSpace”” –– SWATSWAT 19721972 MINIMALMINIMAL –– SetSet ofof BooleanBoolean formulasformulas withwith nono smallersmaller equivalentequivalent formula.formula. MeyerMeyer--StockmeyerStockmeyer 19721972 Complexity of MINIMAL Computable MINIMAL NP P MINIMALMINIMAL MINIMALMINIMAL –– SetSet ofof BooleanBoolean formulasformulas withwith nono smallersmaller equivalentequivalent formula.formula. MINIMALMINIMAL inin NP?NP? –– CanCan’’tt checkcheck allall smallersmaller formulas.formulas. MeyerMeyer--StockmeyerStockmeyer 19721972 Complexity of MINIMAL Computable MINIMAL MINIMAL NP P MINIMALMINIMAL MINIMALMINIMAL –– SetSet ofof BooleanBoolean formulasformulas withwith nono smallersmaller equivalentequivalent formula.formula. MINIMALMINIMAL inin NP?NP? –– CanCan’’tt checkcheck allall smallersmaller formulas.formulas. MINIMALMINIMAL inin NP?NP? –– CanCan’’tt checkcheck equivalence.equivalence. MINIMALMINIMAL MINIMALMINIMAL –– SetSet ofof BooleanBoolean formulasformulas withwith nono smallersmaller equivalentequivalent formula.formula. MINIMALMINIMAL inin NP?NP? –– CanCan’’tt checkcheck allall smallersmaller formulas.formulas. MINIMALMINIMAL inin NP?NP? –– CanCan’’tt checkcheck equivalence.equivalence. MINIMALMINIMAL isis inin NPNP withwith anan ““oracleoracle”” forfor equivalence.equivalence. MINIMALMINIMAL inin NPNP withwith EquivalenceEquivalence OracleOracle (x ∨ y) ∧ (x ∨ y) ∧ z Equivalence Guess: x ∧ z (x ∧ z , (x ∨ y) ∧ (x ∨ y) ∧ z) EQUIVALENT MINIMALMINIMAL MINIMALMINIMAL isis inin NPNP withwith anan ““oracleoracle”” forfor equivalenceequivalence oror nonnon--equivalence.equivalence. MINIMALMINIMAL MINIMALMINIMAL isis inin NPNP withwith anan ““oracleoracle”” forfor equivalenceequivalence oror nonnon--equivalence.equivalence. SinceSince nonnon--equivalenceequivalence isis inin NPNP wewe cancan solvesolve MINIMALMINIMAL inin NPNP withwith NPNP oracle.oracle. MINIMALMINIMAL MINIMALMINIMAL isis inin NPNP withwith anan ““oracleoracle”” forfor equivalenceequivalence oror nonnon--equivalence.equivalence. SinceSince nonnon--equivalenceequivalence isis inin NPNP wewe cancan solvesolve MINIMALMINIMAL inin NPNP withwith NPNP oracle.oracle. SuggestsSuggests aa ““hierarchyhierarchy”” aboveabove NP.NP. MeyerMeyer--StockmeyerStockmeyer 19721972 The Polynomial Time Hierarchy NPNP MINIMAL NP P MeyerMeyer--StockmeyerStockmeyer 19721972 The Polynomial Time Hierarchy NPNP p NP=Σ1 P MeyerMeyer--StockmeyerStockmeyer 19721972 The Polynomial Time Hierarchy Σ p p NP 3 =Σ4 Σ p p NP 2 =Σ3 NP p NP =Σ2 p NP=Σ1 P MeyerMeyer--StockmeyerStockmeyer 19721972 The Polynomial Time Hierarchy p Σ Σ p p 4 co-NP 3 =Π4 p Σ Σ p p 3 co-NP 2 =Π3 p Σ NP p 2 MINIMAL co-NP =Π2 Σ p=NP p 1 co-NP=Π1 P MeyerMeyer--StockmeyerStockmeyer 19721972 The Polynomial Time Hierarchy PH Σ p p 4 Π4 p Σ3 p Σ p P =∆4 p 3 Π3 Σ p p P 2 =∆3 Σ p p 2 Π2 NP p P =∆2 Σ p=NP p 1 co-NP=Π1 p P=∆1 PropertiesProperties ofof thethe HierarchyHierarchy MeyerMeyer--Stockmeyer,Stockmeyer, ““TheThe EquivalenceEquivalence ProblemProblem forfor RegularRegular ExpressionsExpressions withwith SquaringSquaring RequiresRequires ExponentialExponential SpaceSpace””,, SWATSWAT 19721972 Stockmeyer,Stockmeyer, ““TheThe PolynomialPolynomial--TimeTime HierarchyHierarchy””,, TCS,TCS, 1977.1977. 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