Universal Formulae for Percolation Thresholds

A power law is postulated for both site and bond percolation thresholds. The formula writes $p_c=p_0[(d-1)(q-1)]^{-a}d^{\ b}$, where $d$ is the space dimension and $q$ the coordination number. All thresholds up to $d\rightarrow \infty$ are found to belong to only three universality classes. For first two classes $b=0$ for site dilution while $b=a$ for bond dilution. The last one associated to high dimensions is characterized by $b=2a-1$ for both sites and bonds. Classes are defined by a set of value for $\{p_0; \ a\}$. Deviations from available numerical estimates at $d \leq 7$ are within $\pm 0.008$ and $\pm 0.0004$ for high dimensional hypercubic expansions at $d \geq 8$. The formula is found to be also valid for Ising critical temperatures.Comment: 11 pages, latex, 3 figures not include

Newtonian Program Analysis via Tensor Product

Recently, Esparza et al. generalized Newton's method -- a numerical-analysis algorithm for finding roots of real-valued functions -- to a method for finding fixed-points of systems of equations over semirings. Their method provides a new way to solve interprocedural dataflow-analysis problems. As in its real-valued counterpart, each iteration of their method solves a simpler ``linearized'' problem. One of the reasons this advance is exciting is that some numerical analysts have claimed that ```all' effective and fast iterative [numerical] methods are forms (perhaps very disguised) of Newton's method.'' However, there is an important difference between the dataflow-analysis and numerical-analysis contexts: when Newton's method is used on numerical-analysis problems, multiplicative commutativity is relied on to rearrange expressions of the form ``c*X + X*d'' into ``(c+d) * X.'' Such equations correspond to path problems described by regular languages. In contrast, when Newton's method is used for interprocedural dataflow analysis, the ``multiplication'' operation involves function composition, and hence is non-commutative: ``c*X + X*d'' cannot be rearranged into ``(c+d) * X.'' Such equations correspond to path problems described by linear context-free languages (LCFLs). In this paper, we present an improved technique for solving the LCFL sub-problems produced during successive rounds of Newton's method. Our method applies to predicate abstraction, on which most of today's software model checkers rely