In the last decade, Answer Set Programming (ASP) and Satisfiability (SAT) have been used to solve combinatorial search problems and practical applications in which they arise. In each of these formalisms, a tool called a solver is used to solve problems. A solver takes as input a specification of the problem – a logic program in the case of ASP, and a CNF theory for SAT – and produces as output a solution to the problem. Designing fast solvers is important for the success of this general-purpose approach to solving search problems. Classes of instances that pose challenges to solvers can help in this task. In this dissertation we create challenging yet simple benchmarks for existing solvers in ASP and SAT.We do so by providing models of simple logic programs as well as models of simple CNF theories. We then randomly generate logic programs as well as CNF theories from these models. Our experimental results show that computing answer sets of random logic programs as well as models of random CNF theories with carefully chosen parameters is hard for existing solvers. We generate random logic programs with 2-literals, and our experiments show that it is hard for ASP solvers to obtain answer sets of purely negative and constraint-free programs, indicating the importance of these programs in the development of ASP solvers. An easy-hard-easy pattern emerges as we compute the average number of choice points generated by ASP solvers on randomly generated 2-literal programs with an increasing number of rules. We provide an explanation for the emergence of this pattern in these programs. We also theoretically study the probability of existence of an answer set for sparse and dense 2-literal programs. We consider simple classes of mixed Horn formulas with purely positive 2- literal clauses and purely negated Horn clauses. First we consider a class of mixed Horn formulas wherein each formula has m 2-literal clauses and k-literal negated Horn clauses. We show that formulas that are generated from the phase transition region of this class are hard for complete SAT solvers. The second class of Mixed Horn Formulas we consider are obtained from completion of a certain class of random logic programs. We show the appearance of an easy-hard-easy pattern as we generate formulas from this class with increasing numbers of clauses, and that the formulas generated in the hard region can be used as benchmarks for testing incomplete SAT solvers.

Despite much work over the previous decade, the Satisfiability Threshold Conjecture remains open. Random k-SAT, for constant k >= 3, is just one family of a large number of constraint satisfaction problems that are conjectured to have exact satisfiability thresholds, but for which the existence and location of these thresholds has yet to be proven. Of those problems for which we are able to prove an exact satisfiability threshold, each seems to be fundamentally different than random 3-SAT. This thesis defines a new family of constraint satisfaction problems with constant size constraints and domains and which contains problems that are NP-complete and a.s.\ have exponential resolution complexity. All four of these properties hold for k-SAT, k >= 3, and the exact satisfiability threshold is not known for any constraint satisfaction problem that has all of these properties. For each problem in the family defined in this thesis, we determine a value c such that c is an exact satisfiability threshold if a certain multi-variable function has a unique maximum at a given point in a bounded domain. We also give numerical evidence that this latter condition holds. In addition to studying the satisfiability threshold, this thesis finds exact thresholds for the efficient behavior of DPLL using the unit clause heuristic and a variation of the generalized unit clause heuristic, and this thesis proves an analog of a conjecture on the satisfiability of (2+p)-SAT. Besides having similar properties as k-SAT, this new family of constraint satisfaction problems is interesting to study in its own right because it generalizes the XOR-SAT problem and it has close ties to quasigroups.

Despite much work over the previous decade, the Satisfiability Threshold Conjecture remains open. Random k-SAT, for constant k >= 3, is just one family of a large number of constraint satisfaction problems that are conjectured to have exact satisfiability thresholds, but for which the existence and location of these thresholds has yet to be proven. Of those problems for which we are able to prove an exact satisfiability threshold, each seems to be fundamentally different than random 3-SAT. This thesis defines a new family of constraint satisfaction problems with constant size constraints and domains and which contains problems that are NP-complete and a.s.\ have exponential resolution complexity. All four of these properties hold for k-SAT, k >= 3, and the exact satisfiability threshold is not known for any constraint satisfaction problem that has all of these properties. For each problem in the family defined in this thesis, we determine a value c such that c is an exact satisfiability threshold if a certain multi-variable function has a unique maximum at a given point in a bounded domain. We also give numerical evidence that this latter condition holds. In addition to studying the satisfiability threshold, this thesis finds exact thresholds for the efficient behavior of DPLL using the unit clause heuristic and a variation of the generalized unit clause heuristic, and this thesis proves an analog of a conjecture on the satisfiability of (2+p)-SAT. Besides having similar properties as k-SAT, this new family of constraint satisfaction problems is interesting to study in its own right because it generalizes the XOR-SAT problem and it has close ties to quasigroups.