Galois connections in Computational Intelligence: a short survey I.P. Cabrera, P. Cordero, M. Ojeda-Aciego Universidad de Málaga. Andalucía Tech. Dept. Matemática Aplicada. Spain {ipcabrera,pcordero,aciego}@uma.es Abstract—The construction of Galois connections between unstructured and distributed data [3]. Different computational different structures provides a number of advantages, both from intelligence tools which convert data into formal contexts and the theoretical and the applied standpoints. In this paper, we then analyse those contexts providing a concept lattice as a survey some works on Galois connections focused essentially on certain aspects of Computational Intelligence. form of visualised knowledge have been developed. Keywords: Galois connection, Computational Intelligence In this work, we will stress on the usefulness of Galois connection as a tool for Computational Intelligence, and we I. INTRODUCTION will survey some recently published papers in which Galois connections play an important role. The notion of Galois connection originated in pure mathe- The structure of this paper is the following: in Section II matics. Historically, the first occurrence of a Galois connection we recall the idea of Galois connection, and introduce the appeared in the works of Évariste Galois, hence its name, on different versions that are in use, both in the crisp and in the the solvability by radicals of polynomial equations. The main fuzzy cases; then, Section III focuses on recent applications idea was to link the algebraic solution to a polynomial equation of Galois connections in the field of Evolutive Computation; to the structure of the group of permutations associated with later, in Section IV, the focus is put on applications related the roots of the polynomial; in other works, he “moved" the to Neural Computation; and, then, in Section V, we consider problem of studying solutions to a polynomial to the realm applications within the area of Fuzzy Computation; finally, in of group theory. This is the essence of Galois connection, a Section VII, we state some conclusions and future work. passing between two (apparently disparate) worlds. The term Galois connection was coined by Ore in 1944 in II. DEFINITIONS the context of complete lattices, and then Kan introduced the The standard notion of Galois connection is defined be- adjunctions in the context of category theory in 1958. Apart tween two partially ordered sets. However, not all the authors from some particularities, both notions are closely related and, consider the same definition of Galois connection and it is in fact, are interchangeable. remarkable that not all of them are equivalent. In fact, there Applications of Galois connections to Computer Science are four different notions of Galois connection, the most often can be traced back to the eighties: in [1] Galois connections used being the “right Galois connection” (also known as are used to proof the correctness of a compiler, to solve a antitone Galois connection) and the “adjunction" (also known a data type coercion problem; and to Scott’s inverse limit as isotone Galois connections). construction for recursively defined domains. Much more Definition 1: Let A = (A; ≤) and B = (B; ≤) be posets, recently, we still can find further applications, for instance, f : A ! B and g : B ! A be two mappings. The pair (f; g) [2] introduces a binary relational combinator which mirrors the is called a linguistic structure in expressions such as “the smallest such • Right Galois Connection between A and B, denoted by number”, “the best approximation”, “the longest such list” and (f; g): A *( B if, for all a 2 A and b 2 B it holds that exploits its potential for calculating programs by optimization, a ≤ g(b) if only if b ≤ f(a) in particular, to specifications written in the form of Galois connections, in which one of the adjoints delivers the optimal • Left Galois Connection between A and B, denoted by solution being sought. (f; g): A +) B if, for all a 2 A and b 2 B it holds that An interesting example of Galois connection arises in the g(b) ≤ a if only if f(a) ≤ b field of Formal Concept Analysis (FCA), which can be seen both as a mathematical theory aiming at restructuring lattice • Adjunction between A and B, denoted by (f; g): A B theory (as stated by Rudolph Wille) and as a technology if, for all a 2 A and b 2 B it holds that of data processing which complements collective intelligence a ≤ g(b) if only if f(a) ≤ b and helps visualising the hidden information in apparently • Co-Adjunction between A and B, denoted by (f; g): A Partially supported by Spanish Ministry of Science projects TIN2014- B if, for all a 2 A and b 2 B it holds that 59471-P and TIN2015-70266-C2-1-P, co-funded by the European Regional Development Fund (ERDF) g(b) ≤ a if only if b ≤ f(a) It is noteworthy that this definition is also compatible with • Symmetric if ρU (a; b) = ρU (b; a) for all a; b 2 U. the case of A = (A; ≤) and B = (B; ≤) being preordered • Antisymmetric if ρU (a; b) = ρU (b; a) = > implies a = b, sets. for all a; b 2 U. @ Taking into account the dual order, by which A = (A; ≥), We can now introduce the notions of fuzzy poset and fuzzy it is not difficult to check that the following conditions are preposet as follows: equivalent: Definition 3: An L-fuzzy poset is a pair U = (U; ρU ) in which ρU is a reflexive, antisymmetric and transitive -fuzzy @ L 1) (f; g): A *( B. 3) (f; g): A B . relation on U. 2) (f; g): @ +) @ . 4) (f; g): @ . A B A B An L-fuzzy preposet is a pair U = (U; ρU ) in which ρU is a reflexive and transitive L-fuzzy relation on U. Focusing on one particular notion, say Right Galois Con- From now on, when no confusion arises, we will omit the nection, there are many other equivalent definitions in terms prefix “L-”. of particular properties of the components of the connection We can now recall the extension to the fuzzy case provided f and g. by Yao and Lu, also used in [6],which can be stated as follows: The different characterizations for each of the notions of Definition 4 ( [7]): Let A = hA; ρAi and B = hB; ρBi (right- or left-) Galois connection and (co-)adjunction are be fuzzy preposets. A pair of mappings f : A ! B and summarized in Table I (taken from [4]), where we assume that g : B ! A forms a Galois connection between A and B, the standard notions of (crisp) order theory are known by the denoted (f; g): A B if, for all a 2 A and b 2 B, the max(C) reader. The only non-standard notation is that of p- equality ρA(a; g(b)) = ρB(f(a); b) holds. to refer to the set of maximum elements of subset C of a A further step towards generalization to the fuzzy realm preordered set (note that the absence of antisymmetry leads to is possible when considering fuzzy equivalence relations in the possible existence of several different elements fulfilling each of the involved sets instead of the mere equality relation. the properties of being a maximum for C). This leads to a notion of fuzzy Galois connection in which Galois connections in the fuzzy case the mappings f and g can be seen, in some sense, as fuzzy mappings instead of being crisp ones. A more interesting framework to work with Galois connec- The additional consideration of an underlying fuzzy equiv- tions for Computational Intelligence is to consider potential alence relation suggests considering the following notions: extensions of the notion to the fuzzy case. As usual, we will Definition 5: consider a complete residuated lattice = (L; ≤; >; ?; ⊗; !) L fuzzy structure A = hA; ≈ i A as underlying structure for considering the generalization to a (i)A A is a set endowed with ≈ fuzzy framework. As usual, supremum and infimum will be a fuzzy equivalence relation A. morphism A B denoted by _ and ^, respectively (ii)A between two fuzzy structures and is f : A ! B a ; a 2 A An -fuzzy set is a mapping from the universe set, say X, to a mapping such that for all 1 2 the L (a ≈ a ) ≤ (f(a ) ≈ the lattice L, i.e. X : U ! L, where X(u) means the degree following inequality holds: 1 A 2 1 B f(a )) f : A!B in which u belongs to X. We will denote LA to refer to the 2 . In this case, we write , and we say f compatible ≈ ≈ set of all mappings from A to L. that is with A and B. Given X and Y two L-fuzzy sets, X is said to be included We can now introduce the notion of fuzzy preordered in Y , denoted as X ⊆ Y , if X(u) ≤ Y (u) for all u 2 U. The structure as follows: subsethood degree S(X; Y ), by which X is a subset of Y , is Definition 6: Given a fuzzy structure A = hA; ≈Ai, the V pair = hA; ρAi will be called a ⊗-≈A- fuzzy preordered defined by S(X; Y ) = u2U X(u) ! Y (u) . A The first notion of fuzzy Galois connection was given by structure or simply fuzzy preordered structure (when there is Belohlávek,ˇ and it can be rewritten as follows: no risk of confusion), if ρA is a fuzzy relation that is ≈A- Definition 2 ( [5]): An (L-)fuzzy Galois connection between reflexive, ⊗-≈A-antisymmetric and ⊗-transitive, where A B B A and B is a pair of mappings f : L ! L and g : L ! (i) ≈A-reflexive means (a1 ≈A a2) ≤ ρA(a1; a2) for all A A B L such that, for all X 2 L and Y 2 L it holds that a1; a2 2 A.