THE COMPUTATIONAL COMPLEXITY OF SOME LOGICAL THEORIES

Upper and lower bounds on the inherent computational complexity of the decision problem for a number of logical theories are established. A general form of Ehrenfeucht game technique for deciding theories is developed which involves analyzing the expressive power of formulas with a given quantifier depth. The method allows one to decide the truth of sentences by limiting quantifiers to range over finite sets. In particular for cn 2 the theory of integer addition an upperbound of space 2 is obtained; c''n 2 this is close to the known lower bound of nondeterministic time 2 . A general development of decision procedures for theories of product structures is presented, which allows one to conclude in most cases that if the theory of a structure is elementary recursive, then the theory of its weak direct power (as well as other kinds of direct products) is elementary recursive. In particular, for the theory of the weak direct power of , cn 2 2 and hence for integer multiplication, an upper bound of space 2 is c''n 2 2 obtained. The known lower bound is nondeterministic time 2 . Finally, the complexity of the theories of pairing functions is discussed and it is shown that no collection of pairing functions has an elementary recursive theory.