Arithmetizing Classes Around NC 1 and L

Abstract

Abstract. The parallel complexity class NC 1 has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al.[CMTV98] considered arithmetizations of two of these classes, #NC 1 and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in FLogDCFL, while counting proof-trees in logarithmic width formulae has the same power as #NC 1 . We also consider polynomial-degree restrictions of SC i , denoted sSC i , and show that the Boolean class sSC 1 is sandwiched between NC 1 and L, whereas sSC 0 equals NC 1 . On the other hand, the arithmetic class #sSC 0 contains #BWBP and is contained in FL, and #sSC 1 contains #NC 1 and is in SC 2 . We also investigate some closure properties of the newly defined arithmetic classes.

Small-Space Analogues of Valiant’s Classes

“…For i ≥ 0, let VsSC i denote the sub-class of families of polynomials in VSC i whose witness circuits also have syntactic degree bounded by poly(n). Analogous classes sSC i in the Boolean and counting worlds have been studied in [7]. Examining the proof of Lemma 6, we see that the main barrier in extending it to these larger classes is that when we slice C into D and C m , C m is no longer linear in the "slice variables" Z.…”

“…In [7], poly size circuits of log width and poly degree were introduced. The above definition generalises this definition to arbitrary width.…”

“…The above definition generalises this definition to arbitrary width. A notable difference is that in [7] and [8], the degree bound was on the syntactic degree of the width-bounded circuits rather than on the degree of output polynomial. This was necessary to bound the degree of the output polynomial as well as the size of its coefficients.…”

“…Our model is based on width of arithmetic circuits, and captures both succinctness of coefficients and ease of evaluating the polynomials. Special cases of this model have been studied in the past: in [7], such circuits over Z with Boolean inputs and an additional syntactic degree restriction are studied, while in [8,9] such circuits over arbitrary rings but restricted to be syntactic multilinear are studied. We show that our notion of space VSPACE(s) coincides with that of [5,6] at polynomial space with uniformity (Theorem 1), and so far avoids the pitfalls of being too powerful or too weak at logarithmic space.…”

Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits

“…GapBWBP = GapNC 1 . In Limaye et al (2010), this study was extended to bounded width circuits of small (polynomial) degree and size, i.e. sSC 0 , showing that GapNC 1 ⊆ GapsSC 0 , but it is not known whether this containment is strict or not.…”