The Pattern Basis Approach to Circuit Complexity

We describe and motivate a proposed new approach to the problem of finding a “complexity formula” which lower-bounds the circuit complexity (over circuits in a given class) of an arbitrary boolean function, given its truth table, and which can be used to prove nontrivial lower bounds on the complexity of specific functions of interest. This requires addressing the “natural proofs barrier” [Razborov & Rudich 1997], which (roughly) rules out any formula which can be computed in time polynomial in the truth table size. The proposed kind of formula would take as long to compute in general as a brute-force search among all possible circuits, but has a form amenable to proving lower bounds on its value for well-understood explicit functions. This form is based on a new formalization of “patterns”, as elements of a special basis of the vector space of all truth table properties. We prove that any “pattern basis” gives a nonnegative and subadditive complexity formula, which (after scaling to fit primitive circuit costs) lower-bounds circuit complexity (though often trivially). We give specific conditions on the pattern basis which would lead to formula values having a useful range (in which all small circuits have low values, but most functions have high values). We discuss some of the issues involved in satisfying those conditions, and the features a pattern basis would need to make proving explicit-function lower bounds practical. This paper is intended for all mathematicians interested in the general phenomenon of complexity, including those in other fields which might be relevant to realizing this new approach (such as linear algebra, combinatorics, and finite group representation theory). ∗email: [email protected]; blog: http://oresmus.github.io 1 ar X iv :1 60 6. 05 33 1v 1 [ cs .C C ] 1 6 Ju n 20 16