Chapter 16. Worst-Case Upper Bounds

The chapter is a survey of ideas and techniques behind satisfiability algorithms with the currently best asymptotic upper bounds on the worst-case running time. The survey also includes related structural-complexity topics such as Schaefer’s dichotomy theorem, reductions between various restricted cases of SAT, the exponential time hypothesis, etc.

Chapter 10. Random Satisfiabiliy

In the last twenty years a significant amount of effort has been devoted to the study of randomly generated satisfiability instances. While a number of generative models have been proposed, uniformly random k-CNF formulas are by now the dominant and most studied model. One reason for this is that such formulas enjoy a number of intriguing mathematical properties, including the following: for each k≥3, there is a critical value, rk, of the clauses-to-variables ratio, r, such that for r<rk a random k-CNF formula is satisfiable with probability that tends to 1 as n→∞, while for r>rk it is unsatisfiable with probability that tends to 1 as n→∞. Algorithmically, even at densities much below rk, no polynomial-time algorithm is known that can find any solution even with constant probability, while for all densities greater than rk, the length of every resolution proof of unsatisfiability is exponential (and, thus, so is the running time of every DPLL-type algorithm). By now, the study of random k-CNF formulas has also attracted attention in areas such as mathematics and statistical physics and is at the center of an area of intense research activity. At the same time, random k-SAT instances are a popular benchmark for testing and tuning satisfiability algorithms. Indeed, some of the better practical ideas in use today come from insights gained by studying the performance of algorithms on them. We review old and recent mathematical results about random k-CNF formulas, demonstrating that the connection between computational complexity and phase transitions is both deep and highly nuanced.

Chapter 28. Pseudo-Boolean and Cardinality Constraints

Pseudo-Boolean and cardinality constraints are a natural generalization of clauses. While a clause expresses that at least one literal must be true, a cardinality constraint expresses that at least n literals must be true and a pseudo-Boolean constraint states that a weighted sum of literals must be greater than a constant. These contraints have a high expressive power, have been intensively studied in 0/1 programming and are close enough to the satisfiability problem to benefit from the recents advances in this field. Besides, optimization problems are naturally expressed in the pseudo-Boolean context. This chapter presents the inference rules on pseudo-Boolean constraints and demonstrates their increased inference power in comparison with resolution. It also shows how the modern satisfiability algorithms can be extended to deal with pseudo-Boolean constraints.

Efficient Model-Based Diagnosis of Sequential Circuits

In Model-Based Diagnosis (MBD), we concern ourselves with the health and safety of physical and software systems. Although we often use different knowledge representations and algorithms, some tools like satisfiability (SAT) solvers and temporal logics, are used in both domains. In this paper we introduce Finite Trace Next Logic (FTNL) models of sequential circuits and propose an enhanced algorithm for computing minimal-cardinality diagnoses. Existing state-of-the-art satisfiability algorithms for minimal diagnosis use Sorting Networks (SNs) for constraining the cardinality of the diagnostic candidates. In our approach we exploit Multi-Operand Adders (MOAs). Based on extensive tests with ISCAS-89 circuits, we found that MOAs enable Conjunctive Normal Form (CNF) encodings that are significantly more compact. These encodings lead to 19.7 to 67.6 times fewer variables and 18.4 to 62 times fewer clauses. For converting an FTNL model to CNF, we could achieve a speed-up ranging from 6.2 to 22.2. Using SNs fosters 3.4 to 5.5 times faster on-line satisfiability checking though. This makes MOAs preferable for applications where RAM and off-line time are more limited than on-line CPU time.

Satsisfiability and Systematicity

We introduce a new notion of systematicity for satisfiability algorithms with restarts, saying that an algorithm is strongly systematic if it is systematic independent of restart policy but weakly systematic if it is systematic for some restart policies but not others. We show that existing satisfiability engines are generally only weakly systematic, and describe FLEX, a strongly systematic algorithm that uses an amount of memory polynomial in the size of the problem. On large number factoring problems, FLEX appears to outperform weakly systematic approaches.

New Methods for Deriving All Minimal Diagnostic Using Satisfiability Algorithms

In the model-based diagnosis reasoning, diagnosis in two steps,they are generating all minimal conflict sets of conflict identification and Generate all the minimal hitting sets of candidate generation.In this paper, we propose new method based on SAT solver generates all minimal diagnostic.Firstly the normal behavior,system model and obtained observations are described in conjunctive normal form.,then all related clauses of Pending diagnostic system put into SAT solvers. The decision circuit failure problem is converted to satisfiability problem. Hence combine CSSE-tree for solving minimal diagnostic.It can determine the point of failure without first solving the conflict set and then with hitting set algorithm.A method is presented to directly slove the minimum fault sets,which is quite different from the traditional model-based diagnosis.The diagnosis is only finished in one step.