Weakly Bounded Probabilistic Polytime is Contained in POLYSIZE

It is known that bounded error probabilistic polynomial time (BPP) languages are accepted by polynomial size circuit families (POLYSIZE). We sharpen and extend this result to WBPP for which the BPP error bound ffl ? 0 is weakened to ffl(n) =\Omega\Gamma3 =n O(1) ) for length n inputs. The WBPP result is obtained by using Turing randomness to avoid involved counting arguments. 1 Introduction Complexity theory is the part of computer science that identifies computing resources and establishes quantitative relationships among them. In this way, one resource can be measured in terms of others. We will be concerned with measuring randomness in terms of Boolean circuit size. Additional details about the notions used here may be obtained from [2]. It will be convenient to express computation in terms of language acceptance. Languages will be subsets of f0; 1g + . The output of a Turing machine M on input x will be designated by M(x). If a Turing machine M with inputs over f0; 1g + ha..