Polynomials and combinatorial definitions of languages

Using polynomials to represent languages and Boolean functions has opened up a new vein of mathematical insight into fundamental problems of computational complexity theory. Many notable advances in the past ten years have been obtained by working directly with these polynomials. This chapter surveys important results and open problems in this area, with special attention to low-level circuit classes and to the issues of “strong” vs. “weak” representations raised by Barrington, Smolensky, and others. Other combinatorial representations for languages besides polynomials are worthy of attention, and a new example characterizing parity-of(ands-of)-threshold circuits is presented in the last section.

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