Computability in Europe 2006 Clocking Type-2 Computation in the Unit Cost Model

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1 Computability in Europe 2006 Clocking Type-2 Computation in the Unit Cost Model Chung-Chih Li Illinois State University Normal, Illinois, USA July 4,

2 Types Bertrand Russell ( ) Type-0: N (natural numbers). 0, 1, 2, 3... Type-1: functions, N N. f(x) = x 2, g(x) = 1 3 x3 Type-2: functionals, (N N) (N N). Mapping from type-1 objects to type-1 objects. ( x 2 ) = 1 3 x3 Type-0 Type-1 Type-2. 2

3 Some more Examples 1. Γ(f) = f f. Γ : (N N) (N N) 2. F (f, x) = f(x). F : (N N) N N 3. G(f, x) = x i=0 f(i). G : (N N) N N 4. H(f, x) = µi [f(i) = x]. H : (N N) N N How difficult are they to compute? 3

4 Formalisms for computing at type-1 A function (N N) is computable iff it is λ-definable (A. Church), or recursively definable (K. Gödel/S. Kleene), or Turing machine computable (A. Turing), or you name it!! Church-Turing Thesis: All algorithms are computable!! While any reasonable formalisms yields the same computable class, a machine model can provide an intuitive understanding of computational complexity. 4

5 The Foundations of Classical Type-1 Complexity Theory ϕ i i N : an acceptable indexing, Φ i i N : a complexity measure. 1. Complexity Class [HS65]: { C(t) = f e [ ϕ e = f x t : N N, a computable resource bound. ( )]} Φ e (x) t( x ). x, the size of the representation of x, e.g., x log 2 (x). 2. Axiomization (Abstract Complexity Measure [Blu67]) Axiom 1: e, x N[ϕ e (x) Φ e (x) ]. { } Axiom 2: (e, x, m) Φ e (x) m is recursive. 5

6 Type-1 Complexity Theory t g Gap t Union Speedup Compression (Honesty) Polynomial Hierarchy P feasible NP? 6

7 By Union Theorem (McCreight and Meyer [MM69]): For any computable f : N N, there is a computable t : N N such that C(t) = O(f), where O(f) = C(af + b). a,b N There is a computable t : N N such that, C(t) = O(1) O(n) O(n 2 ) O(n 3 ) = P. There is a computable t : N N such that, C(t) = EXP. 7

8 How about Type-2 Computation? Good News: Oracle Turing Machine (OTM) a widely accepted machine model for type-2 computation. 8

9 Oracle Turing Machine [Tur39] Input, x N Oracle f query TM M answer M f (x) = y y N, output Cook Reduction [Coo71] M SAT (x) = 1 if x TSP ; 0 otherwise, 9

10 Computable F : (N N) N N Oracle Turing Machines (OTMs) Type-1 input, f : N N Type-0 input, x N Oracle f query TM M answer y N, output F (f, x) = M f (x) = y 10

11 Bad News: There is no corresponding Church-Turing thesis at type-2. if f = g then return 1; else return 0; How to handle finitely many type-1 inputs? How to patch our program on finitely many bad type-1 inputs? What should be a workable notion of asymptotic computations? How to clock a machine? to shut down? What is the sort of clocks (bounds) for type-2 computations? Why POTM (Polynomial-time OTM) [CK89, Coo91] isn t so nice? 11

12 Type-2 Complexity Theory C(T ) =? T =?????? 40 years after, still stuck!!? L Feasible? BFF 2 feasible 12

13 Our solutions: (Time/Space) constructible bounds (clocks) Type-2 Time Bounds (T 2 TB) T 2 TB: F N N... convergent... Finite compact in... Let β T 2 TB. C(β) = where E e,β = {(f, x) { } ϕ e E e,β is compact, } ϕ e,β (f, x). 13

14 A Clocked Oracle Turing Machine Type-1 input, f : N N Type-0 input, x N Clock β Oracle f query answer TM, M e $Budget$ if budget 0 bϕ e,β (f, x) y N, output bϕ e,β (f, x) = bϕ e (f, x) = y bϕ e,β : (N N) N N 14

15 β T 2 TB is Locking Detectable if β has a locking detector l such that, l : F N {0, 1} is recursive and for every total function f, finite function σ, and x N, we have 1. l(σ, x) = 1 = β(σ, x). 2. lim σ f l(σ, x) = 1. Theorem: There is an effective operator Θ : T 2 TB T 2 TB such that, for every β T 2 TB, we have C(β) = C(Θ(β)) 15

16 Theorem: Basic Hierarchy [ ] β 1, β 2 T 2 TB β 1 β 2 = C(β 1 ) C(β 2 ). Question: Is there any effective operator Θ such that, for any β T 2 TB, we have C(β) C(Θ(β)) C(Θ 2 (β)) C(Θ 3 (β))? The classical Operator Gap Theorem [Con72, You73] says NO to the type-1 complexity classes. 16

17 Answer-Length cost model vs. Unit cost model Type-1 input, f : N N Type-0 input, x N Clock β Oracle f query TM, M e $Budget$ if budget 0 bϕ e,β (f, x) y N, output bϕ e,β (f, x) = bϕ e (f, x) = y C A (β) vs. C U (β) 17

18 Theorem: (Weak Type-2 Gap Theorem) Given any strictly increasing recursive function g : N N, there exists β T 2 TB such that, C A (β) = C A (g β). Theorem: (Anti-Gap Theorem I) For every β T 2 TB, there exists a recursive function g : N N such that C U (β) C U (g β). Theorem: (Weak Anti-Gap Theorem II) Suppose g : N N is recursive and, for every x N, g(x) 3x. Then, for every strong β T 2 TB, C U (β) C U (g β). 18

19 Theorem: (Inflation Theorem) There is a recursive operator Θ such that, for each β T 2 TB, we have Θ(β) T 2 TB and C(β) C(Θ(β)). Corollary: (Compression Theorem) C(β) C(Θ(β)) C(Θ 2 (β)) C(Θ 3 (β)) Hold in both models 19

20 Just like at type-1, the union of arbitrary complexity classes fail to be a complexity class in general. Theorem: There exist β 1, β 2 T 2 TB such that, α T 2 TB, C(α) C(β 1 ) C(β 2 ). Theorem: BFF 2 is not type-2 complexity class, i.e., β T 2 TB[C(β) BFF 2 ]. 20

21 The Union Theorem (McCreight and Meyer [MM69]) Theorem: Given any sequence of recursive functions f 0, f 1, f 2,... such that, (i) λi, x.f i (x) is recursive, and (ii) f 0 f 1 f 2 there is a recursive function g such that C(g) = i N The conditions are rather weak. C(f i ). P, P SP ACE, EXP, Big-O(t), ect., are all complexity classes. 21

22 Definitions: (Oh! They are too ugly to be seen clearly!!) Let β i denote a sequence of type-2 time bounds β 0, β 1, β 2, We say that β i is uniform if and only if λi, σ, x.β i (σ, x) is recursive. 2. We say that β i is ascending if and only if, for all i N, β i β i We say that β i is useful if and only if, for all i N, β i is useful. 4. We say that β i is convergent if and only if, for every (f, x) T N, there is a σ f such that, β i (σ, x) for every i N. 5. We say that β i is uniformly convergent if and only if, for every n N and (σ, x) F N, if β n (σ, x), then for all i N, β i (σ, x). 6. We say that β i is strongly convergent if and only if β i is uniformly convergent and there is a recursive locking detector for β 0. 22

23 Under answer-length cost model Theorem: (Type-2 Union Theorem) Suppose that β i is (i) uniform, (2) ascending, (3) useful, (4) strongly convergent. Then, there is an α T 2 TB such that, C A (α) = C A ( β i ). 23

24 Type-2 Big-O Given β T 2 TB, define { } O(β) = ϕ e ϕ e C(cβ + d) for some c, d N. Under unit cost model Theorem: For every β T 2 TB, O(β) is not a complexity class under unit-cost model. 24

25 Conclusion and current and future study The two different models give two very different type-2 complexity theories. What is the notion of honesty at type-2? Speedup theorems? 25

26 References [Blu67] Manuel Blum. A machine-independent theory of the complexity of recursive functions. Journal of the ACM, 14(2): , [CK89] Stephen A. Cook and Bruce M. Kapron. Characterization of the basic feasible functions of finite type. Proceedings of the 30th Annual IEEE Symposium on the Foundations of Computer Science, pages , [Con72] Robert L. Constable. The operator gap. Journal of the ACM, 19: , [Coo71] [Coo91] [HS65] [MM69] [Tur39] [You73] Stephen Cook. The complexity of theorem proving procedures. Proceedings of the 3rd Annual ACM Symposium on the Theory of Computing, pages , Stephen A. Cook. Computability and complexity of higher type functions. In Y. N. Mpschovakis, editor, Logic from Computer Science, pages Springer-Verlag, J. Hartmanis and R. E. Stearns. On the computational complexity of algorithms. Transitions of the American Mathematics Society, pages , May E. McCreight and A. R. Meyer. Classes of computable functions defined by bounds on computation. Proceedings of the First ACM Symposium on the Theory of Computing, pages 79 88, Alan M. Turing. System of logic based on ordinals. Proceedings of the London Mathematical Society, 45: , Paul Young. Easy construction in complexity theory: Gap and speed-up theorems. Proceedings of the American Mathematical Society, 37(2): , February