On functions computable in nondeterministic polynomial time: Some characterizations

Abstract

As it is well known, the class DTIMEF(Pol _Σ) of functions computable in deterministic polynomial time is the smallest class of functions that contains the projection functions, zero functions of arities zero and one, successor functions and length multiplication, and is closed under substitution and limited recursion. In this paper it is shown that by adding one more basic function or a further closure operator one obtains the class NTIMEF(Pol _Σ) of functions computable in nondeterministic polynomial time. The additional basic function one has to take is the guess function. The operators that are studied are nondeterministic branching, bounded unordered search, which is a generalization of bounded minimization, and limited inversion. Except in the case of nondeterministic branching, with respect to the guess function and each of these operators the functions in NTIMEF(Pol _Σ) possess a normal form which says that they can be generated from functions in DTIMEF(Pol _Σ) by only one application of this additional function or operator. In order to obtain a characterization of NTIMEF(Pol _Σ) that does not use limited recursion, time and space bounded versions of the iteration operator and the operator of taking the reflexive and transive closure of a function are considered. It is shown that NTIMEF(Pol _Σ) is also the smallest class of functions that contains the length multiplication and projection, zero and successor functions and is closed under substitution, nondeterministic branching, the operation of taking the limited inverse and one of these operators. If in their definition the time restriction is skipped, one obtains a characterization of the functions computable in nondeterministic polynomial space, and if, moreover, instead of length multiplication length addition is taken as basic function, then the functions computable in nondeterministic linear space are characterized. A normal form theorem is derived which implies that in any of these cases the characterized functions can be generated by only one application of these limited iteration and/or closure operators from functions computable in nondeterministic linear time.

Parts of this work have been done while the author was visiting the Department of Computer Science of the University of Pisa, supported by a grant of the Consiglio Nazionale delle Ricerche and the Siemens Corporate Laboratories for Research and Technology.