On Arithmetic Branching Programs

The model of arithmetic branching programs is an algebraic model of computation generalizing the model of modular branching programs. We show that, up to a polynomial factor in size, arithmetic branching programs are equivalent to complements of dependency programs, a model introduced by Pudll ak and Sgall 20]. Using this equivalence we prove that dependency programs are closed under conjunction over every eld, answering an open problem of 20]. Furthermore, we show that span programs, an algebraic model of computation introduced by Karchmer and Wigderson 16], are at least as strong as arithmetic programs; every arithmetic program can be simulated by a span program of size not more than twice the size of the arithmetic program. Using the above results we give a new proof that NL/poly L/poly, rst proved by Wigderson 25]. Our simulation of NL/poly is more eecient, and it holds for logspace counting classes over every eld.