Yakoub Salhi

A SAT-based approach for discovering frequent, closed and maximal patterns in a sequence

In this paper we propose a satisfiability-based approach for enumerating all frequent, closed and maximal patterns with wildcards in a given sequence. In this context, since frequency is the most used criterion, we introduce a new polynomial inductive formulation of the cardinality constraint as a Boolean formula. A nogood-based formulation of the anti-monotonicity property is proposed and dynamically used for pruning. This declarative framework allows us to exploit the efficiency of modern SAT solvers and particularly their clause learning component. The experimental evaluation on real world data shows the feasibility of our proposed approach in practice.

The Top- k frequent closed itemset mining using Top- k SAT problem

In this paper, we introduce a new problem, called Top-k SAT, that consists in enumerating the Top-k models of a propositional formula. A Top-k model is defined as a model with less than k models preferred to it with respect to a preference relation. We show that Top-k SAT generalizes two well-known problems: the partial Max-SAT problem and the problem of computing minimal models. Moreover, we propose a general algorithm for Top-k SAT. Then, we give the first application of our declarative framework in data mining, namely, the problem of enumerating the Top-k frequent closed itemsets of length at least min (FCIMkmin). Finally, to show the nice declarative aspects of our framework, we encode several other variants of FCIMkmin into the Top-k SAT problem.

Boolean satisfiability for sequence mining

In this paper, we propose a SAT-based encoding for the problem of discovering frequent, closed and maximal patterns in a sequence of items and a sequence of itemsets. Our encoding can be seen as an improvement of the approach proposed in [8] for the sequences of items. In this case, we show experimentally on real world data that our encoding is significantly better. Then we introduce a new extension of the problem to enumerate patterns in a sequence of itemsets. Thanks to the flexibility and to the declarative aspects of our SAT-based approach, an encoding for the sequences of itemsets is obtained by a very slight modification of that for the sequences of items.

Enumerating Prime Implicants of Propositional Formulae in Conjunctive Normal Form

In this paper, a new approach for enumerating the set prime implicants (PI) of a Boolean formula in conjunctive normal form (CNF) is proposed. It is based on an encoding of the input formula as a new one whose models correspond to the set of prime implicants of the original theory. This first PI enumeration approach is then enhanced by an original use of the boolean functions or gates usually involved in many CNF instances encoding real-world problems. Experimental evaluation on several classes of CNF instances shows the feasibility of our proposed framework.

On Measuring Inconsistency Using Maximal Consistent Sets

An important problem in knowledge-based systems is inconsistency handling. This problem has recently been attracting a lot of attention in AI community. In this paper, we tackle the problem of evaluating the amount of conflicts in knowledge bases, and provide a new fine grained inconsistency measure, denoted MCSC, based on maximal consistent sets. In particular, it is suitable in systems where inconsistency results from multiple consistent sources. We show that our measure satisfies several rational postulates proposed in the literature. Moreover, we provide an encoding in integer linear programming for computing MCSC.

Decomposition Based SAT Encodings for Itemset Mining Problems

Recently, several constraint programming (CP)/propositional satisfiability (SAT) based encodings have been proposed to deal with various data mining problems including itemset and sequence mining problems. This research issue allows to model data mining problems in a declarative way, while exploiting efficient and generic solving techniques. In practice, for large datasets, they usually lead to constraints network/Boolean formulas of huge size. Space complexity is clearly identified as the main bottleneck behind the competitiveness of these new declarative and flexible models w.r.t. specialized data mining approaches. In this paper, we address this issue by considering SAT based encodings of itemset mining problems. By partitioning the transaction database, we propose a new encoding framework for SAT based itemset mining problems. Experimental results on several known datasets show significant improvements, up to several orders of magnitude.

Extending modern SAT solvers for models enumeration

In this paper, we address the problem of enumerating all models of a Boolean formula in conjunctive normal form (CNF). We propose an extension of Conflict Driven Clause Learning (CDCL) based SAT solvers to deal with this fundamental problem. Then, we provide an experimental evaluation of our proposed SAT model enumeration algorithms on both satisfiable SAT instances taken from the last SAT challenge and on instances from the SAT-based encoding of sequence mining problems.

Sequent calculi and decidability for intuitionistic hybrid logic

In this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic (IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability.

Label-free natural deduction systems for intuitionistic and classical modal logics

In this paper we study natural deduction for the intuitionistic and classical (normal) modal logics obtained from the combinations of the axioms T, B, 4 and 5. In this context we introduce a new multi-contextual structure, called T-sequent, that allows to design simple labelfree natural deduction systems for these logics. After proving that they are sound and complete we show that they satisfy the normalization property and consequently the subformula property in the intuitionistic case.

Symmetries in itemset mining

In this paper, we describe a new framework for breaking symmetries in itemset mining problems. Symmetries are permutations between items that leave invariant the transaction database. Such kind of structural knowledge induces a partition of the search space into equivalent classes of symmetrical itemsets. Our proposed framework aims to reduce the search space of possible interesting itemsets by detecting and breaking symmetries between items. Firstly, we address symmetry discovery in transaction databases. Secondly, we propose two different approaches to break symmetries in a preprocessing step by rewriting the transaction database. This approach can be seen as an original extension of the symmetry breaking framework widely used in propositional satisfiability and constraint satisfaction problems. Finally, we show that Apriori-like algorithms can be enhanced by dynamic symmetry reasoning. Our experiments clearly show that several itemset mining instances taken from the available datasets contain such symmetries. We also provide experimental evidence that breaking such symmetries reduces the size of the output on some families of instances.