In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the class of polynomials that can be approximated arbitrarily closely by polynomials in VP_e. We describe VP_e-bar with a strikingly simple complete polynomial (in characteristic different from 2) whose recursive definition is similar to the Fibonacci numbers. Further understanding this polynomial seems to be a promising route to new formula lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992 Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP_e-bar. As a natural continuation of this work we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.
@cite_24: We establish a lower bound for the formula size of quolynomials over arbitrary fields. Our basic formula operations are addition, subtraction, multiplication and division. The proof is based on Neciporuk’s [Soviet Math. Doklady, 7 (1966), pp. 999–1000] lower bound for Boolean functions and uses formal power series. This result immediately yields a lower bound for the formula size of rational functions over infinite fields. We also show how to adapt Neciporuk’s method to rational functions over finite fields. These results are then used to show that, over any field, the @math determinant function has formula size at least @math . We thus have an algebraic analogue to the @math lower bound for the Boolean determinant due to Kloss [Soviet Math. Doklady, 7 (1966), pp. 1537–1540].
@cite_37: We study arithmetic proof systems @math and @math operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field @math . We establish a series of structural theorems about these proof systems, the main one stating that @math proofs can be balanced: if a polynomial identity of syntactic degree @math and depth @math has a @math proof of size @math , then it also has a @math proof of size @math in which every circuit has depth @math . As a corollary, we obtain a quasi-polynomial simulation of @math by @math . Using these results we obtain the following: consider the identities @math where @math , and @math are @math square matrices and @math is a triangular matrix with $z_...
In the algebraic setting the smallest formula for the determinant has size @math , which can be deduced from e.g. @cite_37 . The best known lower bound on the formula size of @math is @math by @cite_24 . That paper also gives a quadratic lower bound for an explicit polynomial (note that the lower bound for the determinant is not quadratic in the number of variables).