Logics for Rough Concept Analysis

11/17/2018 ∙ by Giuseppe Greco, et al. ∙ 0

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a `nondistributive' (i.e. general lattice-based) setting.

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1 Introduction

This paper continues a line of investigation started in [9] and aimed at introducing sequent calculi for the logics of varieties of ‘rough algebras’, introduced and discussed in [1, 19]. The ‘rough algebras’ considered in the present paper are nondistributive (i.e. general lattice-based) generalizations of those of [19]; specifically, they are varieties of lattices expanded with normal modal operators, natural examples of which arise in connection with (certain subclasses of) rough formal contexts, introduced by Kent in [15] as the basic notion of Rough Concept Analysis (RCA), a synthesis of Rough Set Theory [18] and Formal Concept Analysis [7]. The core idea of Kent’s approach is to use a given indiscernibility relation on the objects of a formal context to generate -definable approximations and of the relation such that . The starting point of our approach is that and can be used to generate tuples of adjoint normal modal operators and . We identify conditions under which and are interior operators and and are closure operators. This provides the basic algebraic framework, which we axiomatically extend so as to define ‘nondistributive’ counterparts of the varieties introduced in [19].

From an algebraic perspective, it is interesting to observe that, unlike and , the modal operators and play the reverse roles they usually have in rough set theory: namely, , being an inflationary map, plays naturally the role of the closure operator providing the upper lax approximation of a given formal concept, and similarly , being a deflationary map, plays the role of the interior operator, providing the lower lax approximation of a given formal concept.

From a proof-theoretic perspective, these properties make it possible to introduce a modular generalization of the multi-type approach taken in [9] to endow the logics of ‘rough algebras’ with analytic calculi, so as to adapt it to a ‘nondistributive’ propositional base. For the sake of introducing the structural counterparts of the lattice connectives and (the reasons for which are explained below), our basic calculus does not have the display property, since the usual display rules for and are not sound in the general lattice setting. However, the cut elimination and subformula property for the calculi defined in Section 6 can be straightforwardly verified by appealing to the meta-theorem of [5]. Another interesting departure from the calculi of [9] concerns the counterparts of the IA3 condition, which in the present paper comes in two variants: the lower (strict), and the upper (lax). The inequality corresponding to the lower variant of IA3, which was analytic in the presence of distributivity, is not analytic inductive in the absence of distributivity (cf. [12, Definition 55]). However, the inequality corresponding to the upper variant of IA3 is analytic inductive, and hence can be captured in terms of an analytic structural rule.

2 Preliminaries

The purpose of this section, which is based on [3, Appendix] and [2] and [17, Sections 2.3 and 2.4], is to briefly recall the basic notions of the theory of enriched formal contexts (cf. Definition 2) while introducing the notation which will be used throughout the paper. For any relation , and any and , let

It can be easily verified that iff , that (resp. ) implies that (resp. ), and implies that and for all and .

Formal contexts, or polarities, are structures such that and are sets, and is a binary relation. Intuitively, formal contexts can be understood as abstract representations of databases [7], so that represents a collection of objects, as a collection of features, and for any object and feature , the tuple belongs to exactly when object has feature . In what follows, we use (resp. ) for elements of (resp. ), and (resp. ) for subsets of (resp. of ).

As is well known, for every formal context , the pair of maps

respectively defined by the assignments and , form a Galois connection and hence induce the closure operators and on and on respectively.111When (resp. ) we write for (resp.  for ). Moreover, the fixed points of these closure operators form complete sub--semilattices of and respectively, and hence are complete lattices which are dually isomorphic to each other via the restrictions of the maps and . This motivates the following

Definition 1

For every formal context , a formal concept of is a pair such that , , and and . The set is the extension of , which we will sometimes denote , and is the intension of , sometimes denoted . Let denote the set of the formal concepts of . Then the concept lattice of is the complete lattice

where for every ,

Then clearly, and , and the partial order underlying this lattice structure is defined as follows: for any ,

Theorem 2.1

(Birkhoff’s theorem, main theorem of FCA) Any complete lattice is isomorphic to the concept lattice of some formal context .

Definition 2

An enriched formal context is a tuple such that is a formal context, and and are -compatible relations, that is, (resp. ) and (resp. ) are Galois-stable for all and . The complex algebra of is

where is the concept lattice of , and and are unary operations on defined as follows: for every ,

Since and are -compatible, are well-defined.

Lemma 1

(cf. [17, Lemma 3]) For any enriched formal context , the algebra is a complete lattice expanded with normal modal operators such that is completely meet-preserving and is completely join-preserving.

Definition 3

For any formal context and any -compatible relations , the composition is defined as follows: for any and ,

3 Motivation: Kent’s Rough Concept Analysis

Below, we report on the basic definitions and constructions in Rough Concept Analysis [15], cast in the notational conventions of Section 2.

Rough formal contexts (abbreviated as Rfc) are tuples such that is a polarity (cf. Section 2), and is an equivalence relation (the indiscernibility relation between objects). For every we let . The relation induces two relations approximating , defined as follows: for every and ,

(1)

By definition, are -definable (i.e.  and for any ), and being reflexive immediately implies that

Lemma 2

For any Rfc , if and are defined as in (1), then

(2)

Intuitively, we can think of as the lax version of determined by , and as its strict version determined by . Following the methodology introduced in [4] and applied in [2, 3] to introduce a polarity-based semantics for the modal logics of formal concepts, under the assumption that and are -compatible (cf. Definition 2), the relations and can be used to define normal modal operators on defined as follows: for any ,

(3)
(4)

That is, the members of are exactly those objects that satisfy (possibly by proxy of some object equivalent to them) all features in the description of , while the members of are exactly those objects that not only satisfy all features in the description of , but that ‘force’ all their equivalents to also satisfy them. The assumption that implies that , hence is a sub-concept of . The assumption that implies that , hence is a super-concept of . Moreover, for any ,

(5)
(6)

That is, is the concept described by those features shared not only by each member of but also by their equivalents, while is the concept described by the common features of those members of which ‘force’ each of their equivalents to share them. The assumption that implies that , and hence is a sub-concept of . The assumption that implies that , and hence is a super-concept of . Summing up the discussion above, we have verified that the conditions and imply that the following sequents of the modal logic of formal concepts are valid on Kent’s basic structures:

(7)

where is interpreted as , as , as and as . Translated algebraically, these conditions say that and are deflationary, as interior operators are, and are inflationary, as closure operators are. Hence, it is natural to ask under which conditions they (i.e. their semantic interpretations) are indeed closure/interior operators. The next definition and lemma provide answers to this question.

Definition 4

An Rfc is amenable if , and (defined as in (1)) are -compatible.222The assumption that is -compatible does not follow from and being -compatible. Let for any polarity such that not all singleton sets of objects are Galois-stable. Hence is not -compatible. However, if , then are -compatible.

Lemma 3

For any amenable Rfc , if and and are defined as in (1), then

(8)
Proof

Let . To show that , let . By adjunction, this is equivalent to , which implies that , the last equality holding since is -compatible by assumption. Moreover, (cf. Lemma 2) implies that , which implies that , the last inclusion holding since is -compatible by assumption. Hence, . Suppose for contradiction that . By the -definability of , this is equivalent to . Hence , from which it follows that . Hence, , i.e. , against the assumption that .

Let . To show that , assume that . Since is -definable by construction, this is equivalent to . To show that , we need to show that for any and any . Let . Hence, by definition, for every . Since , this implies that for any , as required.

By the general theory developed in [4] and applied to enriched formal contexts in [17, Proposition 5], properties (8) guarantee that the following sequents of the modal logic of formal concepts are also valid on amenable Rfc’s:

(9)

Finally, again by [17, Proposition 5], the fact that by construction and (resp.  and ) are interpreted by operations defined in terms of the same relation guarantees the validity of the following sequents on amenable Rfc’s:

(10)

Axioms (7), (9) and (10) constitute the starting point and motivation for the proof-theoretic investigation of the logics associated to varieties of algebraic structures which can be understood as abstractions of amenable Rfc’s. We define these varieties in the next section.

4 Kent algebras

In the present section, we introduce basic Kent algebras (and the variety of abstract Kent algebras (aKa) to which they naturally belong), as algebraic generalizations of amenable Rfc’s, and then introduce some subvarieties of aKas in the style of [19].

Definition 5

A basic Kent algebra is a structure such that is a complete lattice, and are unary operations on such that for all ,

(11)

and for any ,

(12)
(13)

We let denote the class of basic Kent algebras.

From (11) it follows that, in basic Kent algebras, and are completely meet-preserving, and are completely join-preserving. For any amenable Rfc , if and are defined as in (1), then

where is the concept lattice of the formal context and are defined as in (3)–(6). The following proposition is an immediate consequence of [17, Proposition 5], using Lemmas 2 and 3, and the fact that and (resp.  and ) are defined using the same relation.

Proposition 1

If is an amenable Rfc, then is a basic Kent algebra.

The natural variety containing basic Kent algebras is defined as follows.

Definition 6

An abstract Kent algebra (aKa) is a structure such that is a lattice, and are unary operations on validating (11), (12) and (13). We let denote the class of abstract Kent algebras.

From (11) it follows that, in aKas, and are finitely meet-preserving, and are finitely join-preserving.

Lemma 4

For any aKa and every ,

(14)
(15)
(16)
(17)
Proof

The inequalities in (15) are straightforward consequences of (11). The inequalities in (14) and (16) follow from (12) and (15), using the transitivity of the order. The inequalities in (17) follow from those in (13) using (11).

Conditions (17) define the‘Kent algebra’ counterparts of topological quasi Boolean algebras 5 (tqBa5) [19]. In the next definition, we introduce ‘Kent algebra’ counterparts of some other varieties considered in [19], and also varieties characterized by interaction axioms between lax and strict connectives which follow the pattern of the 5-axioms in rough algebras.

Definition 7

An aKa as above is an aKa5’ if for any ,

(18)

is a K-IA3 if for any ,

(19)

and is a K-IA3 if for any ,

(20)

Interestingly, the third and fourth inequality in (18) are not analytic inductive (cf. [12, Definition 55]); however, they are equivalent to analytic inductive inequalities in the multi-type language of the heterogeneous algebras discussed in the next section.

5 Multi-type presentation of Kent algebras

Similarly to what holds for rough algebras (cf. [9, Section 3]), since the modal operations of any aKa are either interior operators or closure operators, each of them factorizes into a pair of adjoint normal modal operators which are retractions or co-retractions, as illustrated in the following table:

       

where , , , and , and such that for all , , , , ,

(21)

Again similarly to what observed in [9], the lattice structure of can be exported to each of the sets and via the corresponding pair of modal operators as follows.

Definition 8

For any aKa , the strict interior kernel and the strict closure kernel are such that, for all , and all ,

The lax interior kernel and the lax closure kernel are such that, for all , and all ,

Similarly to what observed in [9], it is easy to verify that the algebras defined above are lattices, and the operations indicated with a circle (either black or white) are lattice homomorphisms (i.e. are both normal box-type and normal diamond-type operators). The construction above justifies the following definition of class of heterogeneous algebras equivalent to aKas:

Definition 9

A heterogeneous aKa (haKa) is a tuple

such that:

  • are bounded lattices;

  • , , , are lattice homomorphisms;

  •         ;

  • 333Condition H3 implies that and are -hemimorphisms and and are -hemimorphisms; condition H4 implies that the black connectives are surjective and the white ones are injective.

The haKas corresponding to the varieties of Definition 7 are defined as follows:

Algebra Acronym Conditions
heterogeneous aKa5’ haKa5’
heterogeneous K-IA3 hK-IA3 and imply
heterogeneous K-IA3 hK-IA3 and imply

Notice that the inequalities defining haKa5’ are all analytic inductive. A heterogeneous algebra is perfect if every lattice in the signature of is perfect (cf. [4, Definition 1.8]), and every homomorphism (resp. hemimorphism) in the signature of is a complete homomorphism (resp. hemimorphism).

Similarly to what discussed in [9, Section 3], one can readily show that the classes of haKas defined above correspond to the varieties defined in Section 4. That is, for any aKa one can define its corresponding haKa using the factorizations described at the beginning of the present section and Definition 8, and conversely, given a haKa , one can define its corresponding aKa by endowing its first domain with modal operations defined by taking the appropriate compositions of pairs of heterogeneous maps of . Then, for every aKa, aKa5’, K-IA3, K-IA3, letting denote its corresponding class of heterogeneous algebras, the following holds:

Proposition 2
  1. If , then ;

  2. If , then ;

  3. The isomorphisms of the previous item restrict to perfect members of and .

  4. If , then and if , then .

6 Multi-type calculi for the logics of Kent algebras

In the present section, we introduce the multi-type calculi associated with each class of algebras . The language of these logics matches the language of haKas, and is built up from structural and operational (i.e. logical) connectives. Each structural connective is denoted by decorating its corresponding logical connective with (resp.  or ). In what follows, we will adopt the convention that unary connectives bind more strongly than binary ones.


general lattice


strict-interior kernel  
 lax-interior kernel  
 

 


strict-closure kernel  
 lax-closure kernel  
 

 

  • Interpretation of structural connectives as their logical counterparts444 The connectives which appear in a grey cell in the synoptic tables will only be included in the present language at the structural level.

  1. structural and operational pure -type connectives:

     structural operations
     logical operations
  2. structural and operational pure -type and -type connectives:

     structural operations
     logical operations
  3. structural and operational pure -type and -type connectives:

     structural operations
     logical operations
  4. structural and operational multi-type strict connectives:

     types
     structural operations
     logical operations
  5. structural and operational multi-type lax connectives:

     types