Ker-I Ko

Computational Complexity of Real Functions

In this chapter we introduce the formal definitions of computational complexity of real functions. We first review the definitions of computable real numbers and computable real functions and their basic properties. Then, the oracle Turing machine is introduced as the formal model for computing real functions. This model allows us to define the complexity measures for computing real functions in a natural way. The class of polynomial time computable real functions is then defined, and several characterizations of this class will be given. Other complexity classes of real numbers and real functions, such as NP real functions and log-space computable real functions, will be defined in later chapters.

On self-reducibility and weak P-selectivity

Abstract Self-reducible sets and some low sets, including p -selective sets, and weakly p -selective sets are studied. Several different formulations of self-reducible sets are given and compared with each other. A new characterization of p -selective sets is found, and weakly p -selective sets are introduced as a generalization of p -selective sets based on this characterization. It is proved that self-reducible sets are not polynomial-time Turing reducible to these sets. As a consequence, ⩽ p m -completeness and ⩽ p T -completeness in NP are not likely to be distinguished by weakly p -selective sets.

On some natural complete operators

Abstract An operator is a mapping from integer functions to integer functions. It is known, from a result by Baker, Gill and Solovay (1975), that there exists an operator computable in polynomial time by a nondeterministic oracle Turing machine (called an NP operator) but not computable in polynomial time by any deterministic oracle Turing machine. We investigate several natural operators which share similar properties. We use the concept of completeness to give a precise classification of the complexity of these operators. For example, the question of finding maximum values of polynomial-time computable functions can be formulated as an operator complete for the class of NP operators.

On helping by robust oracle machines

Abstract The concept of helping by robust oracle Turing machines, introduced recently by Schoning, is extended to the notion of ‘one-sided helping’ and its relations to the structural properties of NP sets are investigated. Various results on who can help whom have been obtained on sets in BPP, in R, in UP and sparse sets. Sets which do not help any set are studied and two types of sets defined by structural properties are demonstrated to help no sets except those in P. Sest which help themselves are also studied and are shown to be related to self-reducible sets. Several previously known structural results on NP sets are reproved from the point of view of helping.

On the notion of infinite pseudorandom sequences

Abstract Three definitions of infinite pseudorandom sequences, with respect to polynomial time and space complexity, are introduced and compared with each other. It is shown that the first two definitions, based on Martin-Lofs notion of sequential tests and Levin and Schnorrs notion of monotonic operator complexity, are equivalent with respect to polynomial space complexity, while both are strictly stronger than the third definition, which is derived from Von Misess notion of collectives.

On the computational complexity of ordinary differential equations

The computational complexity of the solution y of the differential equation y ′( x ) = f ( x, y ( x )), with the initial value y (0) = 0, relative to the computational complexity of the function f is investigated. The Lipschitz condition on the function f is shown to play an important role in this problem. On the one hand, examples are given in which f is polynomial time computable but none of the solutions y is computable. On the other hand, if f is polynomial time computable and if f satisfies a weak form of the Lipschitz condition then the (unique) solution y is polynomial space computable. Furthermore, there exists a polynomial time computable function f which satisfies this weak Lipschitz condition such that the (unique) solution y is not polynomial time computable unless P = PSPACE.

Computational Complexity of Two-Dimensional Regions

The computational complexity of bounded sets of the two-dimensional plane is studied in the discrete computational model. We introduce four notions of polynomial-time computable sets in R

The maximum value problem and NP real numbers

^2

A polynomial-time computable curve whose interior has a nonrecursive measure

and study their relationship. The computational complexity of the winding number problem, membership problem, distance problem, and area problem is characterized by the relations between discrete complexity classes of the NP theory.

Distinguishing conjunctive and disjunctive reducibilities by sparse sets

Abstract The concept of nondeterministic computation has been playing an important role in discrete complexity theory. In this paper the concept of nondeterminism is applied to a numerical problem—finding the maximum value of a polynomial time computable real function. The class of all these maximum values is characterized as the class of nondeterministic polynomial time ( NP ) computable real numbers. The completeness of real numbers is then investigated. The result that NP real numbers cannot be polynomial time many-one complete in NP (unless P = NP ) shows a basic difference between the maximum value problem and many natural NP combinatorial problems. It is also shown that real numbers are not complete in r.e. sets or PSPACE (unless P = PSPACE) and this seems to be a general phenomenon. Finally, the relationship between NP real numbers and NP sets over a single-letter alphabet and the existence of hardest NP real numbers are also discussed.