Reducibility in polynomialtime computable analysis Akitoshi Kawamura (U Tokyo) Wadern, Germany, September 24, 2015
Computable Analysis Applying computability / complexity theory to problems involving real numbers 1 2 Classify problems by computational hardness Rigorous treatment of discrete algorithms Reducibility and completeness Infinite objects that are described only by approximation Type-two computation 3 Some examples
Computable Analysis Existence / Continuity ε, δ, from a δ-approximation of the input, we can determine Computability Poly-space Poly-time NC compute easily compute foundation of validated numerical algorithms? an ε-approximation of the output
1. Some complexity classes
FP and FPSPACE P FP Relation v A x (v is a valid output for input x) decision problems A: Σ 0,1 The class of problems A: Σ Σ such that there is a polynomial-time TM M solving A. PSPACE FPSPACE M x A x (for all x dom A) Similarly, with polynomial space. We do not have a proof of P PSPACE. But we can still classify problems by
Reducibility and completeness Definition For A, B: Σ Σ, we write A B if there are r, s FP such that s B r A A u = r x, B s x. Definition A problem B: Σ Σ in class C is C-complete if A B for all B C. A PSPACE-complete problem (QBF) Tell whether a given quantified propositional formula X 1 X 2 X 3 X 4 φ ԦX is true.
NP and #P NP The class of decision problems A: Σ 0,1 such that there are a polynomial p and a polynomial-time TM M with A x = 1 there is y Σ p x with M x, y = 1. An NP-complete problem (SAT) Tell whether a given propositional formula φ ԦX is satisfiable. #P The class of problems A: Σ Σ such that there are a polynomial p and a polynomial-time TM M with A x = the number of y Σ p x with M x, y = 1. A #P-complete problem (#SAT) Tell how many assignments satisfy the given propositional formula φ ԦX. P NP PSPACE FP #P FPSPACE
2. Computing real functions
Computing real functions Definition φ: Σ Σ is a name of t R if φ(0 n ) encodes a rational within distance 2 n of t. An oracle TM M computes f: [0, 1] R if M φ is a name of f(t) for every name φ of t R. The function computed by M with oracle φ Computing f: [0, 1] R 0 n oracle machine 0 m 2 n -approx. of t 2 m -approx. of f(t) Polynomial time means polynomial in m. (Hence n = poly(m).) In general, the running time also depends on the oracle.
More generally: Computing with representations X γ Σ functions Σ Σ problem A machine M Y δ Σ representation Type-two computation Example: Representation of R: t R string 0 n (a string encoding) a 2 n -approximation (rational) of t
More generally: Computing with representations X γ Σ functions Σ Σ problem A machine M Y δ Σ representation Type-two computation Name of f C[0, 1]: 0 n 0 μ(n) (u, 0 m ) μ: modulus of continuity of f x y < 2 μ(n) f x f y < 2 n 2 m -approx. of f(u)
Second-order polynomials Definition (based on [Mehlhorn 1976, Kapron-Cook 1996]) Let Σ denote the set of φ: Σ Σ that are length-monotone, i.e., x y φ x φ y. For each φ Σ, define its length φ : N N by φ x = φ x. An oracle TM M is poly-time if the running time of M φ (x) is bounded by a second-order polynomial P( φ )( x ). an expression built from +,, and application of φ, e.g., φ 4 φ 2 x 3 2 + 5 + φ x 4. Type-two class FP (also called the Basic Feasible Functionals) Similarly FPSPACE
Type-two complexity classes NP The class of problems A: Σ Σ 0,1 such that there are a polynomial p and a polynomial-time TM M with A φ x = 1 there is y Σ p φ x with M φ x, y = 1. #P The class of problems A: Σ Σ such that there are a polynomial p and a polynomial-time TM M with A φ x = the number of y Σ p φ x with M φ x, y = 1. Unlike type-one classes, these are easy to separate. P NP PSPACE FP #P FPSPACE
Reducibility query answer poly-time query answer A B B input output poly-time A input output A B Polynomial-time Weihrauch reducibility
Formally: Definition For A, B: Σ Σ, A B if there are r, s FP such that A φ = r φ, B s φ r s B A Cf. Reduction between type-one problems. B s r A
3. Examples
Examples p p SAT formula φ with a predicate symbol p φ p satisfiable? QBF formula φ with a predicate symbol p φ p true? An NP-complete problem A PSPACE-complete problem
Examples h 0 = 0, h t = g t, h t. y g(t, y) h(t) 0 1 t If g: 0, 1 [ 1, 1] R is Lipschitz-continuous, this equation has a unique solution h = ODE g : [0,1] R.
Examples Poly-space Cauchy-Lipschitz Theorem The operator ODE: g h is in FPSPACE, and is FPSPACE-complete. Corollary Non-uniform version [Ko 1983, K 2010] If a Lipschitz g: 0, 1 1, 1 R is in FPSPACE, so is the solution h = ODE g : 0,1 R. But the same is not true for FP, unless P = PSPACE.
Examples MAX: g h h x = max y g x, y Theorem (uniform) The operator MAX is FP NP -complete. Corollary Theorem (non-uniform) [Ko, Friedman 1984] The following statement is true iff P = NP: If g: 0, 1 2 R is in FP, so is MAX g : [0,1] R.
Examples INT: g h h x = y 0,1 g x, y dy Theorem (uniform) The operator INT is FP #P -complete. Corollary Theorem (non-uniform) [Friedman 1984] The following statement is true iff FP = #P: If g: 0, 1 2 R is in FP, so is INT g : [0,1] R.
Complexity of operators Open problem: More refined analysis! (summary by Ker-I Ko) http://www.cs.sunysb.edu/~keriko/cca10.pdf