A method for improving Algorithms of Formal Concepts extraction using Prime Numbers Afef Selmi Faculty of Economic Sciences and Management of Sousse Laboratory RIADI Manouba, Tunisia Email: selmi.afef@yahoo.fr Mohamed Mohsen Gammoudi Higher Institute of Multimedia Arts of Manouba Laboratory RIADI Manouba, Tunisia Email: gammoudimomo@gmail.com Farah Harrathi Higher Institute of Multimedia Arts of Manouba Laboratory RIADI Manouba, Tunisia Email: Harrathi.farah@gmail.com Abstract—The extraction of all formal concepts from a binary relation remains a crucial problem in Formal Concept Analysis (FCA), considering that it has to be done with a low cost. In this paper, we propose a Prime Number method for improving the performance of formal concept extraction algorithms. An assignment is proposed where Objects and attributes in a formal context could be represented by an ordered list of prime numbers. The use of prime numbers allows performing the run time of two operations: intersection and union between sets considering the assertion that the comparison of numbers is faster than the comparison between strings. Index Terms—Formal Concept Analysis; Formal Context; Formal concept; Prime Numbers I. I NTRODUCTION Formal Concept Analysis (FCA) is still used in many fields: Information Retrieval [4] [12], Machine Learning [19], Software Engineering [22], etc. These fields employ FCA for three main tasks: (1) to extract formal concepts [1] [2] [6] [11] [23] ; (2) to construct the Galois lattice [5] [8] [10] [17] or (3) to extract concepts graph [13] [14]. The algorithms of extracting all concepts set are highly costed, considering the fact that they are NP-complete problems [9]. The majority of algorithms proposed in the literature tend to propose approximate solutions with a polynomial complexity [1] [2] [6] [11] [16] [23]. Those algorithms are concurrent with each other minimizing their run-times. To pursue this concurrence, we propose a new method where Prime Numbers are used in the encoding of objects and attributes of a formal context. This paper begins with an overview of some FCA basic notions. In section 3, extraction concepts algorithms and Prime Numbers studies are presented. In section 4, we present our approach of extraction concepts using Prime Numbers. In section 5, we show experimental results to evaluate our contribution. II. FORMAL CONCEPT ANALYSIS Formal Concept Analysis is a set of techniques that allows to structure and analyse data. It represents a duality called Galois connection which expresses the linkage between two kinds of items: objects and attributes. In the following, we introduce some basic notions related to FCA: formal context, Galois connection, formal concept, partial order relation and Galois lattice are introduced above [3]. A formal context is a triple (O, A, R), where O is a finite set of objects, A is a finite set of attributes and R ⊆ O × A. For X i ⊂ O and Y i ⊂ A, the two Galois connection opera- tors f and g are defined as follows: f (X i )= {y ∈ A | (x, y) ∈ R, ∀x ∈ X i } and g(Y i )= {x ∈ O | (x, y) ∈ R, ∀y ∈ Y i }. In a formal context (O, A, R), a couple (A, B), with A ⊆ O and B ⊆ A, is called formal concept if and only if f(A) = B and g(B) = A. For a concept C = (A, B), the set of objects A is called extension of C, denoted A = ext(C), and the set of attributes B is called intention of C, denoted B = int(C). Let (A1, B1) and (A2, B2) be two concepts extracted from the relation R, we call  a partial order on concepts if and only if (A1, B1)  (A2, B2), A1 ⊆ A2 and B2 ⊆ B1. Let R be a binary relation, T is the set of concepts extracted from this relation. The Galois lattice associated to R is the set of formal concepts provided with the order relation , denoted (T, ). III. RELATED WORKS In this section, we present a list of algorithms allowing formal concepts extraction mentioning the complexity of each one. A. Algorithms of formal concepts extraction In [11], a non-incremental algorithm is proposed. This algorithm uses the method of Divide and Conquer. In fact, it divides the set of all concepts into several subsets in order to minimize the search space. The mentioned algorithm has a theoretical complexity in the order of O(|O 3 ||A||L|) where |O| is the number of objects, |A| is the number of attributes and |L| is the number of concepts extracted from the relation R. This algorithm tends to show a good performance with small contexts. The algorithm proposed by [6], is an incremental one. It considers the binary relation as a binary array; it looks over it line by line to extract the formal concepts. The algorithm complexity in means of time is O(|O 2 ||A||L|). The algorithm proposed in [2] is a non-incremental algo- rithm. Since, this algorithm computes closure for only some