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- 1. Unambiguous Functions in Logarithmic Space Grzegorz Herman Michael Soltys Computability in Europe July 21, 2009
- 2. The Context
- 3. The Context nondeterminism for bounded space well understood. . .
- 4. The Context nondeterminism for bounded space well understood. . . except L vs. NL (since 2004, SL vs. NL)
- 5. The Context nondeterminism for bounded space well understood. . . except L vs. NL (since 2004, SL vs. NL) unambiguity seems to be the most useful intermediate step
- 6. The Context nondeterminism for bounded space well understood. . . except L vs. NL (since 2004, SL vs. NL) unambiguity seems to be the most useful intermediate step a breakthrough due to Reinhardt and Allender (1997): UL/poly = NL/poly
- 7. The Context nondeterminism for bounded space well understood. . . except L vs. NL (since 2004, SL vs. NL) unambiguity seems to be the most useful intermediate step a breakthrough due to Reinhardt and Allender (1997): UL/poly = NL/poly no major results since
- 8. Rationale
- 9. Rationale want to measure relative (un)ambiguity of problems
- 10. Rationale want to measure relative (un)ambiguity of problems need a meaningful notion of unambiguous nondeterministic reductions
- 11. Rationale want to measure relative (un)ambiguity of problems need a meaningful notion of unambiguous nondeterministic reductions need a well-behaved model for computing functions
- 12. Nondeterministic Function Classes: Existing Models
- 13. Nondeterministic Function Classes: Existing Models multi-valued functions, or
- 14. Nondeterministic Function Classes: Existing Models multi-valued functions, or functions expressing properties of computation graphs (e.g., #L, GapL), or
- 15. Nondeterministic Function Classes: Existing Models multi-valued functions, or functions expressing properties of computation graphs (e.g., #L, GapL), or deterministic computation with oracle queries (e.g., FNL = FLNL).
- 16. Nondeterministic Function Classes: Our Model
- 17. Nondeterministic Function Classes: Our Model nondeterministic machines with deterministic answers
- 18. Nondeterministic Function Classes: Our Model nondeterministic machines with deterministic answers oracle-based input and output
- 19. Nondeterministic Function Classes: Our Model nondeterministic machines with deterministic answers oracle-based input and output explicit failures (uncatchable exceptions)
- 20. Nondeterministic Function Classes: Our Model nondeterministic machines with deterministic answers oracle-based input and output explicit failures (uncatchable exceptions) (un)ambiguity captured by the shape of computation graphs
- 21. Reductions: The Deﬁnition
- 22. Reductions: The Deﬁnition A function φ : A → B reduces to ψ : C → D if there exist:
- 23. Reductions: The Deﬁnition A function φ : A → B reduces to ψ : C → D if there exist: uniformly unambiguous, parametrized family of input transformations: θi : A → C, and
- 24. Reductions: The Deﬁnition A function φ : A → B reduces to ψ : C → D if there exist: uniformly unambiguous, parametrized family of input transformations: θi : A → C, and a function gathering the results on the transformed inputs: ξ : D∗ → B,
- 25. Reductions: The Deﬁnition A function φ : A → B reduces to ψ : C → D if there exist: uniformly unambiguous, parametrized family of input transformations: θi : A → C, and a function gathering the results on the transformed inputs: ξ : D∗ → B, such that ξ(ψ(θ0(x)), . . . , ψ(θp(|x|)(x))) = φ(x)
- 26. Reductions: An Example
- 27. Reductions: An Example target problem: counting up to k simple paths between s and t
- 28. Reductions: An Example target problem: counting up to k simple paths between s and t reduced problem: counting up to k + 1 simple paths between s and t
- 29. Reductions: An Example target problem: counting up to k simple paths between s and t reduced problem: counting up to k + 1 simple paths between s and t restriction: class of graphs closed under edge removal
- 30. Other Beneﬁts
- 31. Other Beneﬁts relating the ambiguity of computation to that of the input graph
- 32. Other Beneﬁts relating the ambiguity of computation to that of the input graph (minor) improvements of known results
- 33. Other Beneﬁts relating the ambiguity of computation to that of the input graph (minor) improvements of known results (e.g., combining [Allender, Reinhardt] with [Buntrock et al.])
- 34. Thank You!