Cédric Piette

Local-search Extraction of MUSes

SAT is probably one of the most-studied constraint satisfaction problems. In this paper, a new hybrid technique based on local search is introduced in order to approximate and extract minimally unsatisfiable subformulas (in short, MUSes) of unsatisfiable SAT instances. It is based on an original counting heuristic grafted to a local search algorithm, which explores the neighborhood of the current interpretation in an original manner, making use of a critical clause concept. Intuitively, a critical clause is a falsified clause that becomes true thanks to a local search flip only when some other clauses become false at the same time. In the paper, the critical clause concept is investigated. It is shown to be the cornerstone of the efficiency of our approach, which outperforms competing ones to compute MUSes, inconsistent covers and sets of MUSes, most of the time.

On Approaches to Explaining Infeasibility of Sets of Boolean Clauses

These last years, the issue of locating and explaining contradictions inside sets of propositional clauses has received a renewed attention due to the emergence of very efficient SAT solvers. In case of inconsistency, many such solvers merely conclude that no solution exists or provide an upper approximation of the subset of clauses that are contradictory. However, in most application domains, only knowing that a problem does not admit any solution is not enough informative, and it is important to know which clauses are actually conflicting. In this paper, the focus is on the concept of minimally unsatisfiable subformulas (MUSes), which explain logical inconsistency in terms of minimal sets of contradictory clauses. Specifically, various recent results and computational approaches about MUSes and related concepts are discussed.

Using local search to find MSSes and MUSes

In this paper, a new complete technique to compute Maximal Satisfiable Subsets (MSSes) and Minimally Unsatisfiable Subformulas (MUSes) of sets of Boolean clauses is introduced. The approach improves the currently most efficient complete technique in several ways. It makes use of the powerful concept of critical clause and of a computationally inexpensive local search oracle to boost an exhaustive algorithm proposed by Liffiton and Sakallah. These features can allow exponential efficiency gains to be obtained. Accordingly, experimental studies show that this new approach outperforms the best current existing exhaustive ones.

Revisiting clause exchange in parallel SAT solving

Managing learnt clause database is known to be a tricky task in SAT solvers. In the portfolio framework, the collaboration between threads through learnt clause exchange makes this problem even more difficult to tackle. Several techniques have been proposed in the last few years, but practical results are still in favor of very limited collaboration, or even no collaboration at all. This is mainly due to the difficulty that each thread has to manage a large amount of learnt clauses generated by the other workers. In this paper, we propose new efficient techniques for clause exchanges within a parallel SAT solver. In contrast to most of the current clause exchange methods, our approach relies on both export and import policies, and makes use of recent techniques that proves very effective in the sequential case. Extensive experimentations show the practical interest of the proposed ideas.

MUS-based generation of arguments and counter-arguments

Most of the approaches of computational argumentation define an argument as a pair consisting of premises and a conclusion, where the latter is entailed by the former. However, the matter of computing arguments and counter-arguments remains largely unsettled. We propose here a method to compute arguments and counter-arguments in the context of propositional logic, by using the concept of a MUS (Minimally Unsatisfiable Subset). The idea relies on the fact that reduction ad absurdum is valid in propositional logic: 〈Φ,α〉 is an argument induced from a knowledge base Δ iff Φ ⋃ {¬α} is inconsistent. Therefore, if Φ ⋃ {¬α} is a MUS of Δ ⋃ {¬α} that contains ¬α then 〈Φ,α〉 is an argument from Δ. Not only do we present an algorithm that generates arguments, we also present an algorithm generating the complete argumentation tree induced by a given argument. We include a report on computational experimentations with both algorithms.

Tracking MUSes and Strict Inconsistent Covers

In this paper, a new heuristic-based approach is introduced to extract minimally unsatisfiable subformulas (in short, MUSes) of SAT instances. It is shown that it often outperforms competing methods. Then, the focus is on inconsistent covers, which represent sets of MUSes that cover enough independent sources of infeasibility for the instance to regain satisfiability if they were repaired. As the number of MUSes can be exponential with respect to the size of the instance, it is shown that such a concept is often a viable trade-off since it does not require us to compute all MUSes but provides us with enough mutually independent infeasibility causes that need to be addressed in order to restore satisfiability

ON FINDING MINIMALLY UNSATISFIABLE CORES OF CSPs

When a Constraint Satisfaction Problem (CSP) admits no solution, it can be useful to pinpoint which constraints are actually contradicting one another and make the problem infeasible. In this paper, a recent heuristic-based approach to compute infeasible minimal subparts of discrete CSPs, also called Minimally Unsatisfiable Cores (MUCs), is improved. The approach is based on the heuristic exploitation of the number of times each constraint has been falsified during previous failed search steps. It appears to enhance the performance of the initial technique, which was the most efficient one until now.

Efficient combination of decision procedures for MUS computation

In recent years, the problem of extracting a MUS (Minimal Unsatisfiable Subformula) from an unsatisfiable CNF has received much attention. Indeed, when a Boolean formula is proved unsatisfiable, it does not necessarily mean that all its clauses take part to the inconsistency; a small subset of them can be conflicting and make the whole set without any solution. Localizing a MUS can thus be extremely valuable, since it enables to circumscribe a minimal set of constraints that represents a cause for the infeasibility of the CNF. In this paper, we introduce a novel, original framework for computing a MUS. Whereas most of the existing approaches are based on complete algorithms, we propose an approach that makes use of both local and complete searches. Our combination is empirically evaluated against the current best techniques on a large set of benchmarks.

MUST: provide a finer-grained explanation of unsatisfiability

In this paper, a new form of explanation and recovery technique for the unsatisfiability of discrete CSPs is introduced. Whereas most approaches amount to providing users with a minimal number of constraints that should be dropped in order to recover satisfiability, a finer-grained alternative technique is introduced. It allows the user to reason both at the constraints and tuples levels by exhibiting both problematic constraints and tuples of values that would allow satisfiability to be recovered if they were not forbidden. To this end, the Minimal Set of Unsatisfiable Tuples (MUST) concept is introduced. Its formal relationships with Minimal Unsatisfiable Cores (MUCs) are investigated. Interestingly, a concept of shared forbidden tuples is derived. Allowing any such tuple makes the corresponding MUC become satisfiable. From a practical point of view, a two-step approach to the explanation and recovery of unsatisfiable CSPs is proposed. First, a recent approach proposed by Hemery et al.s is used to locate a MUC. Second, a specific SAT encoding of a MUC allows MUSTs to be computed by taking advantage of the best current technique to locate Minimally Unsatisfiable Sub-formulas (MUSes) of Boolean formulas. Interestingly enough, shared tuples coincide with protected clauses, which are one of the keys to the efficiency of this SAT-related technique. Finally, the feasibility of the approach is illustrated through extensive experimental results.

Let the Solver Deal with Redundancy

Handling redundancy in propositional reasoning and search is an active path of theoretical research. For instance, the complexity of some redundancy-related problems for CNF formulae and for their 2-SAT and Horn SAT fragments have been recently studied. However, this issue is not actually addressed in practice in modern SAT solvers, and is most of the time just ignored. Dealing with redundancy in CNF formulae while preserving the performance of SAT solvers is clearly an important challenge. In this paper, a self-adaptative process is proposed to manage redundant clauses, enabling redundant information to be discriminated and to keep only the one that proves useful during the search.