Connectivity of Boolean satisfiability

In this thesis we are concerned with the solution-space structure of Boolean satisfiability problems, from the view of theoretical computer science, especially complexity theory. We consider the solution graph of Boolean formulas; this is the graph where the vertices are the solutions of the formula, and two solutions are connected iff they differ in the assignment of exactly one variable. For this implicitly defined graph, we then study the st-connectivity and connectivity problems. The first systematic study of the solution graphs of Boolean constraint satisfaction problems was done in 2006 by Gopalan et al., motivated mainly by research for satisfiability algorithms. In particular, they considered CNFC(S)-formulas, which are conjunctions of constraints that arise from inserting variables and constants in relations of some finite set S. Gopalan et al. proved a computational dichotomy for the st-connectivity problem, asserting that it is either solvable in polynomial time or PSPACE-complete, and an aligned structural dichotomy, asserting that the maximal diameter of connected components is either linear in the number of variables, or can be exponential. Further, they conjectured a trichotomy for the connectivity problem: That it is either in P, coNPcomplete, or PSPACE-complete. Together with Makino et al., they already proved parts of this trichotomy. Building on this work, we here complete the proof of the trichotomy, and also correct a minor mistake of Gopalan et al., which leads to slight shifts of the boundaries. We then investigate two important variants: CNF(S)-formulas without constants, and partially quantified formulas. In both cases, we prove dichotomies for st-connectivity and the diameter analogous to the ones for CNFC(S)-formulas. For for the connectivity problem, we show a trichotomy in the case of quantified formulas, while in the case of formulas without constants, we identify fragments where the problem is in P, where it is coNP-complete, and where it is PSPACE-complete. Finally, we consider the connectivity issues for B-formulas, which are arbitrarily nested formulas built from some fixed set B of connectives, and for B-circuits, which are Boolean circuits where the gates are from some finite set B. Here, we make use of Emil Post’s classification of all closed classes of Boolean functions. We prove a common dichotomy for both connectivity problems and the diameter: on one side, both problems are in P and the diameter is linear, while on the other, the problems are PSPACE-complete and the diameter can be exponential. For partially quantified B-formulas, we show an analogous dichotomy.