1. Beyond NP: The Work andLegacy of Larry Stockmeyer Lance Fortnow University of Chicago

2. Larry Joseph Stockmeyer • 1948 – Born in Indiana • 1974 – MIT Ph.D. • IBM Research at Yorktown and Almaden for most of his career • 82 Papers (11 JACM) • 49 Distinct Co-Authors • 1996 – ACM Fellow • Died July 31, 2004

3. The Universe

4. Computer of Protons

5. The Universe 11,000,000,000 Light Years

6. Computer of Protons Radius 10-15 Meters

7. Computing with the Universe • Universe can only have 10123 proton gates. • Consider the true sentences of weak monadic second-order theory of the natural numbers with successor (EWS1S). • A B x (x  A  x+1  B) • Cannot solve EWS1S on inputs of size 616 in universe with proton-sized gates. • Stockmeyer Ph.D. Thesis 1974 • Stockmeyer-Meyer JACM 2002

8. The Universe 11,000,000,000 Light Years

9. The Universe 78,000,000,000 Light Years

10. Computing with the Universe • Universe can have 10123 proton gates.

11. Computing with the Universe • Universe can have 3.5*10125proton gates.

12. Computing with the Universe • Universe can have 3.5*10125proton gates. • Cannot solve EWS1S on inputs of size 616 in universe with proton-sized gates.

13. Computing with the Universe • Universe can have 3.5*10125proton gates. • Cannot solve EWS1S on inputs of size 619 in universe with proton-sized gates.

14. Science Fiction? • The complexity of algorithms tax even the resources of sixty billion gigabits---or of a universe full of bits; Meyer and Stockmeyer had proved, long ago, that, regardless of computer power, problems existed which could not be solved in the life of the universe.

15. Evolution of Complexity

16. Evolution of Complexity Turing-Church-Kleene-Post 1936 Computably Enumerable Computable

17. Evolution of Complexity Computably Enumerable

18. Evolution of Complexity Kleene 1956 Computably Enumerable Regular Languages Finite Automata

19. Evolution of Complexity Chomsky Hierarchy1956 Computably Enumerable Regular Languages Finite Automata

20. Evolution of Complexity 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

21. Real Computers

22. Faster Computers

23. Evolution of Complexity Computably Enumerable Computable

24. Evolution of Complexity

25. Evolution of Complexity Computable

26. Evolution of Complexity Hartmanis-Stearns 1965 Computable

27. Evolution of Complexity Hartmanis-Stearns 1965 Computable TIME(n2)

28. Evolution of Complexity Hartmanis-Stearns 1965 Computable TIME(2n) TIME(n5) TIME(n2)

29. Evolution of Complexity Hartmanis-Stearns 1965 Computable TIME(n)

30. Limitations of DTIME(t(n)) • Not Machine Independent. • Separations are by diagonalization and not by natural problems. • No clear notion of efficient computation.

31. Evolution of Complexity Cobham 1964 Edmonds 1965 Computable

32. Evolution of Complexity Cobham 1964 Edmonds 1965 Computable P=DTIME(nk)

33. Evolution of Complexity Cobham 1964 Edmonds 1965 Computable Matching P=DTIME(nk)

34. Evolution of Complexity Computable P

35. Evolution of Complexity Cook 1971 Levin 1973 Karp 1972 Computable SAT Clique NP Partition Max Cut P

36. State of Complexity 1972 Computable NP P

37. Enter Larry Stockmeyer • January 1972 – Bachelors/Masters at MIT • Bounds on Polynomial Evaluation Algorithms • Can we find natural hard problems? • Diagonalization methods do not lead to natural problems. • There are natural NP-complete problems but cannot prove them not in P. • With Advisor Albert Meyer

38. Regular Expressions with Squaring • (0+1)*00(0+1)*00(0+1)* • All strings with two sets of consecutive zeros. • Allow Squaring operator: r2=rr • (0+1)*(02(0+1)*)2 • No more expressive power but can be much shorter when used recursively. • ((((((02)2)2)2)2)2)= 0000000000000000000000000000000000000000000000000000000000000000

39. Meyer-Stockmeyer 1972 REGSQ = { R | L(R)  *} Computable REGSQ EXPSPACE PSPACE NP P

40. Regular Expressions with Squaring • Meyer and Stockmeyer, “The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space” – SWAT 1972 • MINIMAL • Set of Boolean formulas with no smaller equivalent formula.

41. Meyer-Stockmeyer 1972 Complexity of MINIMAL Computable MINIMAL NP P

42. MINIMAL • MINIMAL • Set of Boolean formulas with no smaller equivalent formula. • MINIMAL in NP? • Can’t check all smaller formulas.

43. Meyer-Stockmeyer 1972 Complexity of MINIMAL Computable MINIMAL MINIMAL NP P

44. MINIMAL • MINIMAL • Set of Boolean formulas with no smaller equivalent formula. • MINIMAL in NP? • Can’t check all smaller formulas. • MINIMAL in NP? • Can’t check equivalence.

45. MINIMAL • MINIMAL • Set of Boolean formulas with no smaller equivalent formula. • MINIMAL in NP? • Can’t check all smaller formulas. • MINIMAL in NP? • Can’t check equivalence. • MINIMAL is in NP with an “oracle” for equivalence.

46. MINIMAL in NP with Equivalence Oracle (x  y)  (x  y)  z Equivalence Guess: x  z (x  z , (x  y)  (x  y)  z) EQUIVALENT

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

48. MINIMAL • MINIMAL is in NP with an “oracle” for equivalence or non-equivalence. • Since non-equivalence is in NP we can solve MINIMAL in NP with NP oracle.

49. MINIMAL • MINIMAL is in NP with an “oracle” for equivalence or non-equivalence. • Since non-equivalence is in NP we can solve MINIMAL in NP with NP oracle. • Suggests a “hierarchy” above NP.

50. Meyer-Stockmeyer 1972 The Polynomial Time Hierarchy NPNP MINIMAL NP P