Unambiguous functions in logarithmic space - CiE 2009

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Unambiguous functions in logarithmic space - CiE 2009

  1. 1. Unambiguous Functions in Logarithmic SpaceGrzegorz Herman Michael SoltysComputability in EuropeJuly 21, 2009
  2. 2. The Context
  3. 3. The Contextnondeterminism for bounded space well understood. . .
  4. 4. The Contextnondeterminism for bounded space well understood. . .except L vs. NL (since 2004, SL vs. NL)
  5. 5. The Contextnondeterminism for bounded space well understood. . .except L vs. NL (since 2004, SL vs. NL)unambiguity seems to be the most useful intermediate step
  6. 6. The Contextnondeterminism for bounded space well understood. . .except L vs. NL (since 2004, SL vs. NL)unambiguity seems to be the most useful intermediate stepa breakthrough due to Reinhardt and Allender (1997):UL/poly = NL/poly
  7. 7. The Contextnondeterminism for bounded space well understood. . .except L vs. NL (since 2004, SL vs. NL)unambiguity seems to be the most useful intermediate stepa breakthrough due to Reinhardt and Allender (1997):UL/poly = NL/polyno major results since
  8. 8. Rationale
  9. 9. Rationalewant to measure relative (un)ambiguity of problems
  10. 10. Rationalewant to measure relative (un)ambiguity of problemsneed a meaningful notion of unambiguous nondeterministicreductions
  11. 11. Rationalewant to measure relative (un)ambiguity of problemsneed a meaningful notion of unambiguous nondeterministicreductionsneed a well-behaved model for computing functions
  12. 12. Nondeterministic Function Classes: Existing Models
  13. 13. Nondeterministic Function Classes: Existing Modelsmulti-valued functions, or
  14. 14. Nondeterministic Function Classes: Existing Modelsmulti-valued functions, orfunctions expressing properties of computation graphs(e.g., #L, GapL), or
  15. 15. Nondeterministic Function Classes: Existing Modelsmulti-valued functions, orfunctions expressing properties of computation graphs(e.g., #L, GapL), ordeterministic computation with oracle queries(e.g., FNL = FLNL).
  16. 16. Nondeterministic Function Classes: Our Model
  17. 17. Nondeterministic Function Classes: Our Modelnondeterministic machines with deterministic answers
  18. 18. Nondeterministic Function Classes: Our Modelnondeterministic machines with deterministic answersoracle-based input and output
  19. 19. Nondeterministic Function Classes: Our Modelnondeterministic machines with deterministic answersoracle-based input and outputexplicit failures (uncatchable exceptions)
  20. 20. Nondeterministic Function Classes: Our Modelnondeterministic machines with deterministic answersoracle-based input and outputexplicit failures (uncatchable exceptions)(un)ambiguity captured by the shape of computation graphs
  21. 21. Reductions: The Definition
  22. 22. Reductions: The DefinitionA function φ : A → B reduces to ψ : C → D if there exist:
  23. 23. Reductions: The DefinitionA function φ : A → B reduces to ψ : C → D if there exist:uniformly unambiguous, parametrized family of inputtransformations:θi : A → C, and
  24. 24. Reductions: The DefinitionA function φ : A → B reduces to ψ : C → D if there exist:uniformly unambiguous, parametrized family of inputtransformations:θi : A → C, anda function gathering the results on the transformed inputs:ξ : D∗ → B,
  25. 25. Reductions: The DefinitionA function φ : A → B reduces to ψ : C → D if there exist:uniformly unambiguous, parametrized family of inputtransformations:θi : A → C, anda function gathering the results on the transformed inputs:ξ : D∗ → B, such thatξ(ψ(θ0(x)), . . . , ψ(θp(|x|)(x))) = φ(x)
  26. 26. Reductions: An Example
  27. 27. Reductions: An Exampletarget problem: counting up to k simple pathsbetween s and t
  28. 28. Reductions: An Exampletarget problem: counting up to k simple pathsbetween s and treduced problem: counting up to k + 1 simple pathsbetween s and t
  29. 29. Reductions: An Exampletarget problem: counting up to k simple pathsbetween s and treduced problem: counting up to k + 1 simple pathsbetween s and trestriction: class of graphs closed under edge removal
  30. 30. Other Benefits
  31. 31. Other Benefitsrelating the ambiguity of computationto that of the input graph
  32. 32. Other Benefitsrelating the ambiguity of computationto that of the input graph(minor) improvements of known results
  33. 33. Other Benefitsrelating the ambiguity of computationto that of the input graph(minor) improvements of known results(e.g., combining [Allender, Reinhardt] with [Buntrock et al.])
  34. 34. Thank You!

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