The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [arXiv:1312.4524]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: on one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement

A Casual Tour Around a Circuit Complexity Bound

I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News, September 201

Fuzzy Maximum Satisfiability

In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to {\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the problem of finding an assignment to the variables in {\Phi} that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.Comment: 10 page

Boolean Satisfiability in Electronic Design Automation

Boolean Satisfiability (SAT) is often used as the underlying model for a significant and increasing number of applications in Electronic Design Automation (EDA) as well as in many other fields of Computer Science and Engineering. In recent years, new and efficient algorithms for SAT have been developed, allowing much larger problem instances to be solved. SAT “packages” are currently expected to have an impact on EDA applications similar to that of BDD packages since their introduction more than a decade ago. This tutorial paper is aimed at introducing the EDA professional to the Boolean satisfiability problem. Specifically, we highlight the use of SAT models to formulate a number of EDA problems in such diverse areas as test pattern generation, circuit delay computation, logic optimization, combinational equivalence checking, bounded model checking and functional test vector generation, among others. In addition, we provide an overview of the algorithmic techniques commonly used for solving SAT, including those that have seen widespread use in specific EDA applications. We categorize these algorithmic techniques, indicating which have been shown to be best suited for which tasks

Generating and Searching Families of FFT Algorithms

A fundamental question of longstanding theoretical interest is to prove the lowest exact count of real additions and multiplications required to compute a power-of-two discrete Fourier transform (DFT). For 35 years the split-radix algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic operations on real numbers for a size-n DFT, and was widely believed to be the best possible. Recent work by Van Buskirk et al. demonstrated improvements to the split-radix operation count by using multiplier coefficients or "twiddle factors" that are not n-th roots of unity for a size-n DFT. This paper presents a Boolean Satisfiability-based proof of the lowest operation count for certain classes of DFT algorithms. First, we present a novel way to choose new yet valid twiddle factors for the nodes in flowgraphs generated by common power-of-two fast Fourier transform algorithms, FFTs. With this new technique, we can generate a large family of FFTs realizable by a fixed flowgraph. This solution space of FFTs is cast as a Boolean Satisfiability problem, and a modern Satisfiability Modulo Theory solver is applied to search for FFTs requiring the fewest arithmetic operations. Surprisingly, we find that there are FFTs requiring fewer operations than the split-radix even when all twiddle factors are n-th roots of unity.Comment: Preprint submitted on March 28, 2011, to the Journal on Satisfiability, Boolean Modeling and Computatio