Polynomial-time Computable Functions Research Articles

The gap between monotone and non-monotone circuit complexity is exponential

A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastnc losn. We observe that this lower bound can be improved to exp(cn1/6−o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grotschel—Lovasz—Schrijver on the Lovasz — capacity, ϑ of a graph.

Elementary properties of the class of nondeterministic polynomial time computable functions

Elementary properties of the class of nondeterministic polynomial time computable functions

Diagonalizations over polynomial time computable sets

A formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP.

How to construct random functions

A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented. This generator is a deterministic polynomial-time algorithm that transforms pairs ( g , r ), where g is any one-way function and r is a random k -bit string, to polynomial-time computable functions ƒ r : {1, … , 2 k } → {1, … , 2 k }. These ƒ r 's cannot be distinguished from random functions by any probabilistic polynomial-time algorithm that asks and receives the value of a function at arguments of its choice. The result has applications in cryptography, random constructions, and complexity theory.

On the computational complexity of best Chebyshev approximations

The best Chebyshev approximation of degree n to a continuous function f on [0, 1] is the unique polynomial ϕ of degree less than or equal to n such that the maximum difference of f and ϕ on [0, 1] is minimized. On the basis of a formal model of computation, it is shown that the question of whether the best Chebyshev approximations of polynomial-time computable functions on [0, 1] are always polynomial-time computable depends on the relationship among well-known discrete complexity classes. In particular, P = NP implies that these best approximations are polynomial-time computable, and EXP ≠ NEXP implies that these best approximations are not polynomial-time computable. It is also pointed out that the fact that the popular Remes algorithm converges fast does not conflict with the above result, since the Remes algorithm requires, in each iteration, the finding of maximal points of continuous functions on an interval [ a, b], which is, in general, provably intractable.

Some observations on the connection between counting and recursion

Based on Valiant's class # P of all functions counting the number of accepting computations of nondeterministic polynomial-time Turing machines, the polynomial-time hierarchy of counting functions is introduced. The class PHCF of all functions of this hierarchy and some of its subclasses are characterized by recursion-theoretic means. It turns out that, from the recursion-theoretic point of view, PHCF is an analogue to Kalmár's class E of elementary functions, to the class Pspace of polynomial-space computable functions as well as to the class P of polynomial-time computable functions.

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 real functions

Recursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects. We investigate the computational complexity of real functions using the methods of recursive function theory. Partial recursive real functions are defined and their domains are characterized as the recursively open sets. We define the time complexity of recursive real continuous functions and show that the time complexity and the modulus of uniform continuity of a function are closely related. We study the complexity of the roots and the differentiability of polynomial time computable real functions. In particular, a polynomial time computable real function may have a root of arbitrarily high complexity and may be nowhere differentiable. The concepts of the space complexity and nondeterministic computation are used to study the complexity of the integrals and the maximum values of real functions. These problems are shown to be related to the “P=?NP” and the “P=?PSPACE” questions.

Some negative results on the computational complexity of total variation and differentiation

The computational complexity of the following classical theorems in real analysis is examined. (1) A function f is of bounded variation if and only if it is a difference of two increasing functions. (2) An absolutely continuous function on a compact interval is of bounded variation. (3) A function of bounded variation has a derivative almost everywhere. All the results obtained here are negative in the sense that if we restrict our domain to polynomial time computable functions, the new versions of the above theorems are no longer true. For instance, the polynomial time computability of a function and its total variation function does not guarantee the polynomial time computability of its derivative.

The maximum value problem and NP real numbers

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.

Simple Gödel Numberings, Isomorphisms, and Programming Properties

Restricted classes of programming systems (Gödel numberings) are studied, where a programming system is in a given class if every programming system can be translated into it by functions in a given restricted class. For pairs of systems in various “natural” classes, results are given on the existence of isomorphisms (one-to-one and onto translations) between them from the corresponding classes of functions. The results with the most computational significance concern polynomial time programming systems. It is shown that if $\mathcal{P} = \mathcal{N}\mathcal{P}$ then every two polynomial time programming systems are isomorphic via a polynomial time computable function. If $\mathcal{P} \ne \mathcal{N}\mathcal{P}$ this result points the way to the possible existence of “natural” but intractable computational problems concerning programming systems. Results are also given concerning the relationship between the complexity of certain important and commonly used properties of programming systems (such as effective composition of programs) and the complexity of translations into the systems.