A Galois Framework for the Study of Analogical Classifiers

Abstract

In this paper, we survey some recent advances in the study of analogical classifiers, i.e., classifiers that are compatible with the principle of analogical inference. We will present a Galois framework induced by relation between formal models of analogy and the corresponding classes of analogy preserving functions. The usefulness these general results will be illustrated over Boolean domains, which explicitly present the Galois closed sets of analogical classifiers for diferent pairs of formal models of Boolean analogies.

Keywords

• eol>Analogical proportion
• analogical reasoning
• analogical classifier
• Galois theory

1. Motivation and Background

Analogical reasoning (AR) is a remarkable capability of human thought that exploits parallels between situations of diferent nature to infer plausible conclusions, by relying simultaneously on similarities and dissimilarities. Machine learning (ML) and artificial intelligence (AI) have tried to develop AR, mostly based on cognitive considerations, and to integrate it in a variety of ML tasks, such as natural language processing (NLP), preference learning and recommendation [ 1, 2, 3, 4, 5 ]. Also, analogical extrapolation (inference) can solve dificult reasoning tasks such as scholastic aptitude tests and visual question answering [ 6, 7, 8, 9 ]. Inference based on AR can also support dataset augmentation (analogical extension and extrapolation) for model learning, especially in environments with few labeled examples [ 10 ]. Furthermore, AR can also be performed at a meta level for transfer learning [ 11, 12 ] where the idea is to take advantage of what has been learned on a source domain in order to improve the learning process in a target domain related to the source domain. Moreover, analogy making can provide useful explanations that rely on the parallel example-counterexample [ 13 ] and guide counterfactual generation [ 14 ].

However, early works lacked theoretical and formalizational support. The situation started to change about a decade ago when researchers adopted the view of analogical proportions as statements of the form “ relates to as relates to ”, usually denoted : :: : . Such proportions are at the root of the analogical inference mechanism, and several formalisms to study this mechanism have been proposed, which follow diferent axiomatic and logical approaches [ 15, 16 ]. For instance, [ 17 ] introduces the following 4 postulates in the linguistic context as a guideline for formal models of analogical proportions: symmetry (if : :: : , then : :: : ), central permutation (if : :: : , then : :: : ), strong inner reflexivity (if : :: : , then = ), and strong reflexivity (if : :: : , then = ). Such postulates appear reasonable in the word domain, but they can be criticized in other application domains. For instance, in a setting where two distinct conceptual spaces are involved, as in wine : French :: beer : Belgian where two diferent spaces “drinks” and “nationality” are considered, the central permutation is not tolerable.

Recently, [ 18 ] proposed an algebraic framework of analogies that is naturally embedded into first-order logic via model-theoretic types. It provides a unifying setting where the different axiomatic approaches in the literature and respective domains of interpretation can be considered.

Example 1. Among the classical models of analogy on the two-element set {0, 1}, it is noteworthy to mention [ 19 ] and [ 20 ] definitions of Boolean analogy that correspond respectively to the relations 1 and 2 below. In this paper we represent a relation as a matrix whose columns are precisely the tuples belonging to the relation. Note that 2 contains only patterns of the form : :: : and : :: : , and it is often referred to as the minimal model of analogy.

Other approaches to formalizing the notion of analogy, include the factorial view of [ 21 ] and the functional view of [ 22, 23 ], except that it is not bound by the central permutation postulate. Such a framework is close to Gentner’s symbolic model of analogical reasoning [ 24 ] based on structure mapping theory and first implemented in [ 25 ]. Note that diferent axiomatic approaches entail diferent dataset augmentation procedures, and may impact diferently on several tasks related to AR.

A key task associated with AR is analogy solving, i.e. finding or extrapolating, for a given triple , , a value such that : :: : is a valid analogy. Such a task has been addressed in the framework of case-based reasoning (CBR), where solutions are generated by retrieval and adaptation [ 26, 27 ]. Following the same tracks, AR was also adapted to analogy based classification [ 28 ] where objects are viewed as attribute tuples (instances) x = ( 1, . . . , ). Indeed, if a, b, c are in analogical proportion for most of their attributes, and class labels are known for a, b, c but unknown for d, then one may infer the label for d as a solution of an analogical proportion equation. All these applications rely on the same idea: if four instances a, b, c, d are in analogical proportion for most of the attributes describing them, then it may still be the case for the other attributes (a), (b), (c), (d) (for some function ). This principle is called analogical inference principle (AIP).

Theoretically, it is quite challenging to find and characterize situations where AIP can be soundly applied. A first step toward explaining the analogical mechanism consists in characterizing the set of functions for which AIP is sound (i.e., no error occurs) no matter which triplets of examples are used. In case of Boolean attributes and for the minimal model 2, it was shown in [ 10 ] that these so-called “analogy-preserving” (AP) functions coincide exactly with the set of afine Boolean functions. Moreover, it was also shown that, when the function is not afine, the prediction accuracy remains high if the function is close to being afine [

These results were later extended to nominal (finite) underlying sets when taking the minimal model of analogy in both the domain and codomain of classifiers [ 30 ].

Intuitively, this class will change when adopting diferent models of analogy. This motivated a deeper study of the relation between formal models of analogy and the corresponding class of AP functions, and which culminated in a Galois theory of analogical classifiers [ 31 ]. In this paper, we briefly survey this Galois framework for analogical classifiers, and we illustrate these results for Boolean classifiers by revisiting diferent formal Boolean models of analogy, and describe the corresponding Galois closed sets of analogical classifiers.

This paper is organized as follows. We first briefly survey the universal algebraic framework pertaining to relational preservation in Section 2. We then adapt the latter to the framework of analogical preservation in Section 3, and recall the Galois theory for analogical classifiers presented in [ 31 ]. We illustrate these results by considering two classical models of Boolean analogies and describing sets of analogical classifiers accordingly. As a by-product, it follows that they actually correspond to the set of afine Boolean functions. 2. Galois Theories for Functions Let and be nonempty sets. A function of several arguments from to is a mapping the rule and 1, . . . , ∈ ℱ :

→ for some natural number called the arity of . Denote by ℱ -ary functions of several arguments from to , and let ℱ := ⋃︀ ∈N ℱ ( ).

In the case when = we speak of operations on , and we use the notation and := ℱ . For any set

⊆ ℱ , the -ary part of is ( ) := ( ), then the composition ( 1, . . . , ) belongs to ℱ ( ) the set of all ∩ ℱ ( )

( ). If ∈ ℱ and is defined by ( ) := ℱ ( ) ( ) ( 1, . . . , )(a) := ( 1(a), . . . , (a)) for all a ∈ . . We denote by the set of all projections on .

The -th -ary projection pr( )

( ) is defined by pr( )( 1, . . . , ) := for all 1, . . . , ∈

The notion of functional composition can be extended to sets of functions as follows. Let ⊆ ℱ and

⊆ ℱ . The composition of with is the set := { ( 1, . . . , ) ∈ ℱ | ∈ ( ), 1, . . . , ∈ ( )}. in symbols, i.e., the smallest clone on containing .

A clone on is a set ⊆ and ⊆ . For

⊆ , we denote by ⟨ ⟩ the clone generated by , ⊆ that is closed under composition and contains all projections, We say that ∈ ℱ ( ) is a minor of ∈ ℱ ( ), ≤ , if ∈ { } . The minor relation ≤ minor-closed classes or minions. Equivalently, a set is a quasi-order (a reflexive and transitive relation) on ℱ . Downsets of (ℱ , ≤) are called is a minion if ⊆ . A set ⊆ ℱ

for every finite subset is -locally closed if for all ∈ ℱ ⊆ of size at most , there exists a ∈ such that | = | . A (say is -ary), it holds that ∈ whenever

Subsets of are called -ary relations on . Denote by ℛ on , and let ℛ := ⋃︀ ∈N ℛ ( ). Let set is said to be locally closed if it is -locally closed for every positive integer . ( ) the set of all -ary relations ( ). We say that the function preserves the relation (or is a polymorphism of , or is an invariant of ), and we write componentwise application of to the tuples a = ( 1, . . . , ), i.e.: ◁ , if for all a1, . . . , a ∈ , we have (a1, . . . , a ) ∈ , where (a1, . . . , a ) denotes the (a1, . . . , a ) := ( ( 11, . . . , 1), . . . , ( 1 , . . . , )).

Inv : ( ) → (ℛ ) given by the following rules: for all ℛ ⊆ ℛ and ℱ ⊆ , The preservation relation ◁ induces a Galois connection between the sets and ℛ of operations and relations on . Its polarities are the maps Pol : (ℛ ) → ( ) and Pol ℛ := { ∈ | ∀

∈ ℛ : ◁ Inv ℱ := { ∈ ℛ | ∀ ∈ ℱ : ◁ } , } .

Under this Galois connection, the closed sets of operations are precisely the locally closed clones. The closed sets of relations, known as relational clones, are precisely the locally closed sets of relations that contain the empty relation and the diagonal relations and are closed under formation of primitively positively definable relations. This was first shown for finite base sets in [ 32, 33, 34 ] and later extended for arbitrary sets in [ 35, 36 ].

The preservation relation can be adapted for functions of several arguments from to ; we now need to consider pairs of relations. Let ℛ ( ) := ℛ ( ) × ℛ ( ) and ℛ := ⋃︁ ℛ

( ) ∈N the sets ℱ for all ⊆ ℛ maps Pol : (ℛ and ℛ

) → (ℱ and ℱ ⊆ ℱ

, be the set of all ( -ary) relational constraints from to . of functions and relational constraints from to . Its polarities are the ) and Inv : (ℱ ) → (ℛ

) given by the following rules: Pol := { ∈ ℱ Inv ℱ := {(, ) ∈ ℛ | ∀(,

) ∈ : ◁ (, | ∀ ∈ ℱ : ◁ (, ) , } ) . } some ℱ ⊆ ℱ The sets Pol

and Inv ℱ of the form Pol for some ⊆ ℛ are said to be definable by relational constraints and functions, respectively. are said to be defined by

and ℱ , respectively. Sets of functions and sets of relational constraints of the form Inv ℱ for

The closed sets of functions under this Galois connection were described for finite base sets in [ 37 ] and later for arbitrary sets [ 38 ]. This result was refined in [ 39 ] for sets of functions definable by relations of restricted arity.

Theorem 2 ([ 38, 39 ]). Let and be arbitrary nonempty sets, and let ⊆ ℱ . 1. is definable by constraints if and only if is a locally closed minion. 2. is definable by constraints of arity if and only if is an -locally closed minion.

The closed sets of relational constraints were described in terms of closure conditions that parallel those for relational clones. The description of the dual objects of constraints on possibly infinite sets and

was also provided in [ 38 ] and inspired by those given in [ 34, 35, 36, 37 ] and given in terms of positive primitive first-order relational definitions applied simultaneously on antecedents and consequents. Sets of constraints that are closed under such formation schemes are said to be closed under conjunctive minors. Moreover, every function satisfies the empty (∅, ∅) and the equality (= , = ) constraints, and if a function satisfies a constraint (, ), then also satisfies its relaxations ( ′, ′) such that ′ ⊆ and ′ ⊇ .

As for functions, in the infinite case, we also need to consider a “local closure” condition to describe the dual closed sets of relational constraints on and . A set and is -locally closed if it contains every relaxation of its members whose antecedent has of constraints on size at most , and it is locally closed if it is -locally closed for every positive integer . relational constraints on and .

Theorem 3 ([ 38, 39 ]). For arbitrary nonempty sets and , and let ⊆ ℛ be a set of 1. is definable by some set ⊆ ℱ

2. is definable by some set ⊆ ℱ

if and only if it is locally closed, contains the binary ( ) of -ary functions if and only it is -locally closed, equality and the empty constraints, and it is closed under relaxations and conjunctive minors. contains the binary equality and the empty constraints, and it is closed under relaxations and conjunctive minors.

Let

and let 1 and 2 be clones on and . We say that is stable under right composition with 1 if 2 ⊆ . We say that is ( 1, 2)-stable or a ( 1, 2)-clonoid, if 1 ⊆ , and we say that is stable under left composition with 2 if 1 ⊆ and 2 ⊆ .

Motivated by earlier results on linear definability of equational classes of Boolean functions [ 40 ] which were described in terms of stability under compositions with the clone of constant preserving afine functions, [ 41 ] introduced a Galois framework for describing sets of functions ℱ ⊆ ℱ

stable under right and left compositions with clones 1 on and 2 on , respectively. For that they restricted the defining dual objects to relational constraints (, ) where and invariant under 1 and 2, respectively, i.e., referred to as ( 1, 2)-constraints. We denote by ℛ ∈ Inv 1 and ∈ Inv 2. These were ( 1, 2) the set of all ( 1, 2)-constraints.

Theorem 4 ([ 41 ]). Let and be arbitrary nonempty sets, and let 1 and 2 clones on and , respectively. A set ⊆ ℱ

is definable by some set of ( 1, 2)-constraints if and only if is locally closed and stable under right and left composition with 1 and 2, respectively, i.e., it is a locally closed ( 1, 2)-clonoid. Dually, a set of ( 1, 2)-constraints is definable by a set ⊆ ℱ if = Inv ∩ ℛ To describe the dual closed sets of ( 1, 2)-constraints, [ 41 ] observed that conjunctive minors of ( 1, 2)-constraints are themselves ( 1, 2)-constraints. However, this is not the case for relaxations. They thus proposed the following variants of local closure and of constraint

A set 0 of ( 1, 2)-constraints is said to be ( 1, 2)-locally closed if the set relaxations of the various constraints in 0 is locally closed. A relaxation ( 0, 0) of a relational of all constraint (,

) is said to be a ( 1, 2)-relaxation if ( 0, 0) is a ( 1, 2)-constraint.

Theorem 5 ([ 41 ]). Let and be arbitrary nonempty sets, and let 1 and 2 clones on and , respectively. A set

of ( 1, 2)-constraints is definable by some set ⊆ ℱ is ( 1, 2)-locally closed and contains the binary equality constraint, the empty constraint, and it if and only if it is closed under ( 1, 2)-relaxations and conjunctive minors.

In this paper, we will focus on relational constraints whose antecedent and consequent are derived from analogies, and that we will refer to as analogical constraints. We will denote the set of all analogical constraints from to

by . 3. Galois Theory for Analogical Classifiers As mentioned in the Introduction, analogical inference yields competitive results in classification and recommendation tasks. However, the justification of why and when a classifier is compatible with the analogical inference principle (AIP) remained rather obscure until the work [ 10 ]. In this paper the authors considered the minimal Boolean analogy model (see 2 in Example 1) and addressed the problem of determining those Boolean classifiers for which the AIP always holds, that is, for which there are no classification errors. Surprisingly, they showed that they correspond to “analogy preserving” and that they constitute the clone of afine functions. This result was later generalized to binary classification tasks on nominal (finite) domains in [ where the authors considered the more stringent notion of “hard analogy preservation”. By taking the same minimal analogy model on both the domain and the label set, the authors showed that in this case the sets of hard analogy preserving functions constitute Burle’s clones [ 42 ].

Definition 6.

Let and be sets, and let and be analogical proportions defined on the two sets, respectively. A function :

) if for all a, b, c, d ∈ : ︀( (a, b, c, d) and -solv( (a), (b), (c)))︀ =⇒ ( (a), (b), (c), (d)), where (a, b, c, d) is a shorthand for ( , , , ) -solv( (a), (b), (c)) means that there is an ∈ for all ∈

{1, . . . , } and with ( (a), (b), (c), ). Denote by ) the set of all analogy-preserving functions relative to (,

→ is analogy-preserving (AP for short) relative to Proposition 7. Let and be analogical proportions defined on sets and , respectively. Then ′) we need to introduce some variants of the closure conditions discussed in Section 2. Let ℛ matrix whose columns belong to a relation be set of -ary relations on . An ∈ ℛ , is called an ℛ -locality. Let ⊆ ℛ × , and let 1 := { if for all ∈ ℱ ∈ ℛ | ∃ ∈ ℛ such that (,

) ∈ } . A set ⊆ ℱ (say is -ary), it holds that ∈ whenever for every 1-locality , either is -locally closed 1. there exists a ∈ such that 2. for any relation in 1 such that = , or

⪯ and for any ∈ { ∈ ℛ | (, ) ∈ , ⊆ } we have that

⊆ .

Let ′ := { ′ | ∈ }, and let := × ′ . We refer to the elements of analogical constraints from to . The set of analogical constraints that are ( 1, 2)-constraints as will be denoted by ( 1, 2) := ∩ ℛ

A set is said to be ( 1, 2)-analogically locally closed if it is that = ( , ), and in this case we simply say that is analogically locally closed. ( 1, 2)-locally closed. Note Theorem 9. Let and be arbitrary nonempty sets, and let 1 and 2 be clones on and , respectively.

1. A set ⊆ ℱ 2. A set ⊆ ℱ analogically locally closed ( 1, 2)-clonoid. locally closed minion.

is definable by analogical ( 1, 2)-constraints if and only if it is a ( 1, 2)is definable by analogical constraints if and only if it is an analogically 4. Application: Description of Boolean Analogical Classifiers w.r.t. Example 1 In this section we illustrate the use of the Galois theory described in Section 3 to determine the classes of analogical classifiers AP( , ) = Pol( , ′ ) for , ∈ {1, 2} (see Example 8 and Equation (1)).

Recall that, up to permutation of arguments, the binary Boolean functions are the following: the constant 0 and 1 functions, denoted respectively by 0 and 1, the first projection pr1 : ( 1, 2) ↦→ 1 and its negation ¬1 = pr1, the conjunction ∧ and its negation ↑, the disjunction ∨ and its negation ↓, the implication → and its negation ↛, and the addition + modulo 2 and its negation ↔. Note that ↑ and ↓ are often referred to as Shefer functions as each one of them can generate the class of all Boolean functions by taking compositions and variable substitutions.

Observe that the constant tuples 0 and 1 belong to every ( ∈ {1, 2}), and thus every such is invariant under I, i.e., I ⊆ . Hence, for every , ∈ {1, 2}, Pol( , ′ ) is stable under right composition with I. This leads us to considering the following notion.

A function is said to be a -minor of a function if ∈ . Recall that in the particular case when = {0,1}, is called a minor of . The functions and are said to be equivalent, denoted by ≡ , if is a minor of and is a minor of . For further background on these notions and variants see, e.g., [ 43, 44, 37 ].

Since Pol( , ′ ) is stable under right composition with I, this means that if an I-minor of a function does not belong to Pol( , ′ ), then neither does . This observation constitutes a main tool in describing the sets of the form AP( , ) = Pol( , ′ ).

Proposition 10. Let L denote the clone of afine functions. We have AP( 2, 2) = AP( 2, 1) = AP( 1, 2) = AP( 1, 1) = L.

Proof. We make use of the fact that AP(, ) = Pol(, ′). Since 1 = 1′, it follows immediately that Pol( 1, 1′) = Pol 1 is a clone, and it is well known that Pol 1 = L.

To prove AP( 2, 2) = Pol( 2, 2′) = L, we make use of main tool given above. Since ( 2, 2′) is a relaxation of ( 1, 1′), Pol( 2, 2′) ⊇ Pol( 1, 1′) = L. Furthermore, • ∧, ∨ ∈/ Pol( 2, 2′) because • ↑, ↓ ∈/ Pol( 2, 2′) because = ⎜⎜⎝ 10⎟⎟⎠ ∈/ 2′, and and = ⎜⎜⎝ 01⎟⎟⎠ ∈/ 2′. • ↛, → ∈/ Pol( 2, 2′) because and

⎛ 1⎞ = ⎜⎜⎝ 11⎟⎟⎠ 0 ∈/ 2′.

The result now follows by observing that any function outside of L has an I-minor in {∧, ∨, ↑, ↓, ↛, →}. (For further details, see [ 31 ].) By similar arguments, it also follows that AP( 2, 1) = AP( 1, 2) = AP( 1, 1) = L. 5. Conclusion and Perspectives In this paper we survey a general Galois framework for studying analogical classifiers that does not depend on the underlying domains nor the formal models of analogy considered. We also illustrate its usefulness by explicitly describing sets of analogical classifiers with respect to two classical models of analogy. As future work, we intend to further explore diferent formal models of analogy that may be obtained by considering diferent algebraic signatures.

Acknowledgments

The authors wish to thank Esteban Marquer, Pierre-Alexandre Murena and the reviewers for the useful suggestions for improving this manuscript.

The research work by the first named author was partially supported by TAILOR, a project funded by EU Horizon 2020 research and innovation program under GA No 952215, and the Inria Project Lab “Hybrid Approaches for Interpretable AI” (HyAIAI).

The research work by the second named author was partially funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects PTDC/MAT-PUR/31174/2017, UIDB/00297/2020, and UIDP/00297/2020 (Center for Mathematics and Applications).

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