Expressibility and Parallel Complexity

It is shown that the time needed by a concurrent read concurrent write parallel random access machine CRAM to check if an input has a certain property is the same as the minimal depth of a rst order inductive de nition of the property This in turn is equal to the number of iterations of a rst order sentence needed to express the property The second contribution of this paper is the introduction of a purely syntactic uniformity notion for circuits It is shown that an equiva lent de nition for the uniform circuit classes AC i is given by rst order sentences iterated log n times Similarly uniform AC is de ned to be the rst order expressible properties which in turn is equal to constant time on a CRAM by our main theorem A corollary of our main result is a new characterization of the Polynomial Time Hierarchy PH PH is equal to the set of languages accepted by a CRAM using exponentially many processors and con stant time