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!