Monad Transformers for Backtracking Search

This paper extends Escardo and Oliva's selection monad to the selection monad transformer, a general monadic framework for expressing backtracking search algorithms in Haskell. The use of the closely related continuation monad transformer for similar purposes is also discussed, including an implementation of a DPLL-like SAT solver with no explicit recursion. Continuing a line of work exploring connections between selection functions and game theory, we use the selection monad transformer with the nondeterminism monad to obtain an intuitive notion of backward induction for a certain class of nondeterministic games.Comment: In Proceedings MSFP 2014, arXiv:1406.153

Analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms

The computational complexity of solving random 3-Satisfiability (3-SAT) problems is investigated. 3-SAT is a representative example of hard computational tasks; it consists in knowing whether a set of alpha N randomly drawn logical constraints involving N Boolean variables can be satisfied altogether or not. Widely used solving procedures, as the Davis-Putnam-Loveland-Logeman (DPLL) algorithm, perform a systematic search for a solution, through a sequence of trials and errors represented by a search tree. In the present study, we identify, using theory and numerical experiments, easy (size of the search tree scaling polynomially with N) and hard (exponential scaling) regimes as a function of the ratio alpha of constraints per variable. The typical complexity is explicitly calculated in the different regimes, in very good agreement with numerical simulations. Our theoretical approach is based on the analysis of the growth of the branches in the search tree under the operation of DPLL. On each branch, the initial 3-SAT problem is dynamically turned into a more generic 2+p-SAT problem, where p and 1-p are the fractions of constraints involving three and two variables respectively. The growth of each branch is monitored by the dynamical evolution of alpha and p and is represented by a trajectory in the static phase diagram of the random 2+p-SAT problem. Depending on whether or not the trajectories cross the boundary between phases, single branches or full trees are generated by DPLL, resulting in easy or hard resolutions.Comment: 37 RevTeX pages, 15 figures; submitted to Phys.Rev.

Hybrid solvers for the Boolean Satisfiability problem: an exploration

The Boolean Satisfiability problem (SAT) is one of the most extensively researched NP-complete problems in Computer Science. This thesis focuses on the design of feasible solvers for this problem. A SAT problem instance is a formula in propositional logic. A SAT solver attempts to find a solution for the formula. Our research focuses on a newer solver paradigm, hybrid solvers, where two solvers are combined in order to gain the benefits from both solvers in the search for a solution. Our hybrid solver, AmbSAT, combines two well-known solvers: the systematic Davis-Putnam-Logemann-Loveland solver (DPLL) and the stochastic WalkSAT solver. AmbSAT\u27s design is original and differs from the hybrid solver designs in the research literature. AmbSAT utilizes a DPLL algorithm to lead the search and WalkSAT at appropriate points to aid in the search process. Central to AmbSAT\u27s design is the notion of ambivalence. Essentially, ambivalence attempts to formally identify the points in time when the DPLL solver might be well served by further guidance from WalkSAT. In this thesis, we present three different ambivalence notions and analyze their performance against a pure DPLL solver. Our results are promising, and indicate that AmbSAT performs better than a pure DPLL solver on a diverse collection of SAT problem instances