From: AAAI Technical Report WS-99-14. Compilation copyright © 1999, AAAI (www.aaai.org). All rights reserved. First-Order Context and Formal Concept Analysis Laurent Chaudron and Nicolas Maille ONERACERT 2 avenue Edouard Belin BP 4025 - 31055 Toulouse Cedex 04 - FRANCE Phone: +33 5 62 25 26 55 Fax: +33 5 62 25 25 64 {chaudron,maille}©cert, fr Key words: Formal Concept Analysis, Context Based use numerical features to capture symbolic notions in KnowledgeExploration, First-Order Lattices, Concepts order to keep the semantic on each piece of the model. Discovery, Context Based Rules, Logic Programming This methodological constraint leads to cautiously define the elements of the model dedicated to the Abstract representation of the basic elements of the context and their exploitations. This is the purpose of our work. Formal Concept Analysis -also called "Galois The relations between FCAand first-order logic have Lattices"- is an algebraic modelbased on proposi- tional calculus that is used for symbolicknowledge been studied, especially in (Zickwolf 1991); our work exploration from a formal context. The aim of this is more focused on the formal prerequisites and the paper is to design the theoretical modelsrequired programmingconditions of the definition of a consist- for the extension of Formal Concept Analysis to ent model of first-order Logic FCA (say: 1LFCA). first-order logic so as to improveboth the expres- This leads to search for first-order corresponding sion power as a knowledgemining tool upon first operators to the fundamental ones in FCA: set union, order contexts, and the relevanceof its results. set intersection, and the Galois connections (one must Our contribution consists in: i) a synthesis of the notice that nothing else than these three operators basic notions of FCA,ii) the design of the Cube is needed to generate the FCAtheory). This is the Modeldedicated to the conjunctions of literals, purpose of the Cube lattice model thanks to which iii) the designof a completefirst-order logic formal concept analysis of first-order contexts. The ap- relevant definitions of 1LFCAcan be formulated and proach is described from the theoretical point of implemented. view, implementations in logic programmingand As it has been the case for FCA, the study of 1LFCA applications are also briefly presented. has created some results which maybe of a theoretical interest for themselves, even if one may also expect 1LFCAto offer more powerful exploration tools and Introduction in particular more suitable "numerical ~symbolic" As far as knowledge information is concerned, the translations for the contexts of the considered applic- induction of concepts from a context described by data ations. Thus, the motivations of this work are both and information is a pivotal topic; generally, if numer- pragmatic and theoretical. ical valuations (belief measures, preferences...) can defined on the considered data, numerical or mixed Foreword: In the next section, the foundations of clas- methods can be directly used: rough sets (Pawlak 1991; sical FCAare recalled in a revised new form which is Skowron & Polkowski 1998), Cartesian space model adapted to the extension to first-order logic. In the se- (Ichino ~ Yaguchi 1998; Ichino & Ono 1998)... quel, a classical term is underlined, while a term defined some cases, requirements or accessibility constraints by the authors is quoted within a box. In the context imply to rest only on symbolic attributes. Thus, of the paper the proofs are omitted (they can be found more fundamental models and techniques are required; in (Chaudron & Maille 1999)). Propositions, lemmas Formal Concept Analysis (Ganter & Wille 1996) is and theorems are labeled within the same sequence. suitable candidate for such a purpose. Unfortunately, FCAtheory relies only on propositional calculus and Formal Concept Analysis an extension is required so as to characterize the context with more expressive attributes. Indeed, The basic notions of FCA our applications (cooperative system design, activity Formal Concept Analysis (say: FCA) is a set- modeling, symbolic fusion...) require means to rep- theoretical model for concepts that reflects the resent both symbolic information as predicates and philosophical understanding of a concept as a context- numerical parameters. But in our approach, we do not based unit of thought consisting in two parts: the 46 extend, which contains all the entities (the objects, A’ is the set of attributes commonto all the objects the examples...) belonging to the concept, and the in A, and B’ is the set of the objects possessing their intend, which is the collection of all the attributes attributes in B. (the characteristics, the properties...) shared by the entities (Arnaud & Nicole 1662). For example, the Example 2: in C1, if A = {Objl}, Then A’ = pair: ({B7~7, A3XX, DC9} , {wings, engines})may {Prol, Pro2, Pro5}, and (A" = A). If B {Prol, Pr o2}, naturally induce a simple and commonconcept... then B’ = {Objl}, and (B" = X). One can also notice Based upon Galois connections, FCA was first de- the case of the empty set: as asubset ofO: 0~ = P, and scribed in (Barbut & Monjardet 1970) and in the 80’s conversely, as a subset of P: 0~ = O. R. Wille designed a dedicated theory and program at the University of Darmstadt. An introduction can be Remarks: found in (Davey & Priestley 1990) and FCA theory ¯ Depending on the properties of sets O and P, the and applications are now described in the reference dual operators are called polarities in (Birkhoff 1940), book (Ganter & Wille 1996). FCAis frequently used whereas when defined by their basic properties (see as a preprocessing tool for classification (Carpineto Proposition 1) they are traditionally knownas Galois Romano 1996), but in our approach, we stay closer connections. to the original purpose of FCA. In FCA, the basic ¯ Notation ~ is the same for subsets of O and P as both notion that models the knowledge about a specific play a symmetric role in the theory. This symmetry domain is the formal context -described as a binary disappears when P deals with predicates instead of relation between two finite sets- from which concepts propositions. and conceptual double hierarchies can be formally ¯ Usually the context is suppose to verify O N P = 0 derived so as to form the mathematical structure of but all cases can be considered. a lattice 1 with respect to a subconcept-superconcept ¯ The attributes are atomic positive formulas. In FCA relation. FCAis used for self-emergent classification the negation operator is not considered explicitly. of objects, detection of hidden implications between ¯ In this section, the definitions, the propositions objects, construction of concept sequences, object and the proofs are based on the usual set operations. recognition, aggregation of data and information, More precisely the definitions and arguments rely knowledge representation and analysis. only on the set of properties induced by the fact that (7)(0), C, U, M) and (7)(P), C, U, M) are lattices. allow the extension of FCAto first-order logic to be Definitions and properties of FCA "smoothly" achieved. Definition 1 A formal context C is defined as a triple (O, P, ~) where O (objects or entities) and Proposition 1 the dual operators verify the following (properties or attributes) are finite sets and ~ is properties: for all Ai subset of O (the symmetric prop- mapping from O onto ~’(P). For o in O, ¢(o) indicates erties stand for any subset of P): the subset of attributes possessed by o (¢ is frequently (i)A1 C A2 ~ A~ C A~, synthetised in a table). (ii)A C A’, (iii)A’ = A", Example 1: Let C1 = (O1,P1), 01 = Objqe[1.a ], P1 = (iv)(A1 N ~ D A’I U A’2, Projde[1.5], (v)(A1 U A2)’ = A~ M ¢ ( Objl )-= {Prol,Pro2,Prob} Definition 3 Given A C O and B C P, the pair ¢( Obj2)={Pro2,Pro3,Pro$,Prob} ~ ¢ ( Obj3)= {Prol ,Pro3,Prob (A, B) is concept if fde/ A’= Band B =A. A is the extend o--~-the concept and B is the intend. In the sequel we assume that (O, P, ¢) denotes the working context. The set of all concepts defined on the context (O, P, () is denoted as L. Definition 2 The dual operators (denoted as _~ between O and P are defined as follows: Example 3: in C1, ({Obj2,Objl},{Pro5,Pro2}) and (VA ({Obj3,Objl},{Pro5,Prol}) are concepts. O)(VB C P), A’ =d¢f No~A~(o), Two questions arise: what is the structure of L and B’ = ef {o e OIBc ¢(o)}. how to determine it? This is the role of the next 1given two internal operators M(infimum) and kJ (su- theoretical results. premum)on a set E, (E,M,U) is a lattice, iffd~f: Mand are idempotent, associative, commutativeand they verify Definitions 4 The supremum V and infimum A op- the absorption law x M(x U y) = x and x [3 (x V1y) = erators are defined on L as follows: for all concepts lattice is alwaysan ordered set: the relation ~ defined on E (A1,B1) and (A2,B2) in as: (x < y) ~--~def (X [7 y) = is an order rel ation forwhich n and t3 represent the greatest lower bound and the least (A1, BI) V (A2, B2) =d~l ((A1 U A2)", B1 n B~) upper bound (Birkhoff 1940). (A1,B1) A (A2, B2) =def (A1 M A2, (B1 U B2)") 47 Theorem2 (L, <<, V, A) is a complete lattice. for such a process (the specific details of the application are not detailled here). The formal definitions match with the intuitive notion Given a set of information systems characterized by of concepts as a pair of a collection of examples and functionalities (e.g. medical database access, languages their characteristics: the more objects there are the less translations, transmission capability), they are sup- characteristics they share.