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_47: We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (SPS) arithmetic circuits over a field F is polynomial time equivalent to identity testing of depth 2 (PS) arithmetic circuits over U2(F), the algebra of upper-triangular 2 2 matrices with entries from F. Such a connection is a bit surprising since we also show that, as computational models, PS circuits over U2(F) are strictly ‘weaker’ than SPS circuits over F. The equivalence further implies that PIT of SPS circuits reduces to PIT of width-2 commutative Algebraic Branching Programs(ABP). Further, we give a deterministic polynomial time identity testing algorithm for a PS circuit of size s over commutative algebras of dimension O(log s log log s) over F. Over commutative algebras of dimension poly(s), we show that identity testing of PS circuits is at least as hard as that of SPS circuits over F.
@cite_39: It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by polynomial-length algebraic straight-line programs that use only three registers. This was previously known for Boolean formulas [D. A. Barrington, J. Comput. System Sci., 38 (1989), pp. 150–164], which are equivalent to algebraic formulas over the ring @math . For formulas over arbitrary rings, the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial-length straight-line programs. Moreover, the straight-line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective. A consequence of this is that the problem of determining the iterated product of @math matrices is complete (under P-projections) for algebraic @math . Also, when the ring is @math , the programs that arise in the c...
Ben-Or and Cleve @cite_39 proved that a family of polynomials has polynomially bounded formula size if and only if it is computable by width-3 ABPs of polynomial size. An excellent exposition on the history of small-width computation can be found in @cite_47 , along with an explicit polynomial that cannot be computed by width-2 ABPs: @math . Saha, Saptharishi and Saxena [Cor. 14] SSS:09 showed that @math cannot be computed by width-2 ABPs that correspond to the iterated matrix multiplication of upper triangular matrices.