Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete

Abstract

ABSTRACT. In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally restricted feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.Let gW OE0; 1 R ! R be a continuous function and consider the initial value problemThe Picard-Lindelöf (or Cauchy-Lipschitz) Theorem [15] states 1 that this equation has a unique solution hW OE0; 1 ! R if g is Lipschitz continuous (along its second argument), i. e., (2) jg.t; y 0 / g.t; y 1 /j Ä Z jy 0 y 1 j; t 2 OE0; 1; y 0 ; y 1 2 R for some constant Z independent of y 0 , y 1 and t . We are interested in the computational complexity of the solution h under this condition. Our model of computation of real functions is reviewed in Section 1. It is adopted from computable analysis and is thus consistent with the conventional notion of computability (see Section 5.3 for other perspectives on the "complexity" of similar problems). We formulate our main result in Section 2: the solution of the above equation can be polynomialspace complete, even if g is polynomial-time computable. This was open since 1983 [18]. The proof will be given in Section 3. The main idea is to regard the differential equation with the Lipschitz condition as a polynomial-space computation tableau with certain restrictions. In Section 4, we use the same techniques solve several other problems, including the ones about Volterra integral equations [20]. Section 5 discusses related results and problems.Key words and phrases. computable analysis; computational complexity; exponential space; initial value problem; Lipschitz condition; ordinary differential equations; Picard-Lindelöf Theorem; polynomial space; Volterra integral equations.This work was supported in part by the Nakajima Foundation and by the Natural Sciences and Engineering Research Council of Canada.1 There are several variants of the theorem; here is a proof sketch for ours. Let C be the set of all continuous real functions on OE0; 1. A solution of (1) is a fixed point of the operator T W C ! C defined byThis solution exists and is unique by the Banach fixed point theorem, because T is a contraction with respect to the distance d given by d.h 0 ; h 1 / D max t2OE0;1 exp. 2Zt /jh 0 .t/ h 1 .t/j for h 0 , h 1 2 C . To compute a real function f , the machine should output an approximation of f .t / to given precision 2 m by consulting the oracle for approximations of t to any precision 2 n it desires (left). An alternative picture (right) is that the machine converts any stream of improving approximations of t to a stream of improving approximations of f .t/.

Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

“…We claim that φ ∈ ALP and that for any w, w ′ ∈ Γ * , φ(ψ k (w),ψ k (w ′ )) = ψ k (w#w ′ ). The fact that φ ∈ ALP is immediate using Theorem 4.4 and the fact that n → k −n−1 is analog-polytimecomputable 28 . The second fact is follows from a calculation:…”

“…We also mention that Friedman and Ko (see [29]) proved that polynomial time computable functions are closed under maximization and integration if and only if some open problems of computational complexity (like P = NP for the maximization case) hold. The complexity of solving Lipschitz continuous ordinary differential equations has been proved to be polynomial-space complete by Kawamura [28].…”

Parametrised second-order complexity theory with applications to the study of interval computation

“…They defined the computational complexity of real numbers and of real functions on compact intervals and proved some famous hardness results for problems such as integration, maximisation, or solving ordinary differential equations (see [Fri84,Ko83], cf. also [Kaw10]). This line of research is summarised nicely in Ko's book [Ko91].…”

Function Spaces for Second-Order Polynomial Time

“…computable translations. Various results on individual operators have been obtained in this new framework [13,16,17,28], leaving the field at a very similar state as the early investigation of computability in analysis: While some indicators are available what good choices of representations are, an overall theory of representations for computational complexity is missing. Our goal here is to provide the first steps towards such a theory by investigating the role of admissibility and the presence of function spaces for polynomial-time computability.…”