Logics for Rough Concept Analysis

Abstract

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.

The research of the fourth author is supported by the NWO Vidi grant 016.138.314, the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded in 2013.

A Properties

Throughout this section, we let $\mathsf {K}\in \{aKa, aKa5', K\text {-}IA3_{\ell }\}$, and $\mathsf {HK}$ the class of heterogeneous algebras corresponding to $\mathsf {K}$. Further, we let $\text {D.K}$ denote the multi-type calculus for the logic $\text {H.K}$ canonically associated with $\mathsf {K}$.

1.1 A.1 Soundness for Perfect $\mathsf {HK}$ Algebras

The verification of the soundness of the rules of $\text {D.K}$ w.r.t. the semantics of perfect elements of $\mathsf {HK}$ (see Definition 9) is analogous to that of many other multi-type calculi (cf. [6, 11,12,13,14, 16, 23]). Here we only discuss the soundness of the rule $k\text {-}ia3_{\ell }$. By definition, the following quasi-inequality is valid on every K-IA3$_{\ell }$:

$$\begin{aligned} \Box _{\ell } a\le \Box _{\ell } b \text { and } \Diamond _{\ell } a\le \Diamond _{\ell } b\text { imply } a\le b. \end{aligned}$$

This quasi-inequality equivalently translates into the multi-type language as follows:

$$\begin{aligned} \Box _C\, {\bullet _C}\, a\le \Box _C\, {\bullet _C}\, b \text { and } \Diamond _I\, {\bullet _I}\, a\le \Diamond _I\, {\bullet _I}\, b\text { imply } a\le b. \end{aligned}$$

By adjunction, the quasi-inequality above can be equivalently rewritten as follows:

$$\begin{aligned} \Diamond _C\, {\bullet _C}\, \Box _C\, {\bullet _C}\, a\le b \text { and } a\le \Box _I\, {\bullet _I}\, \Diamond _I\, {\bullet _I}\, b\text { imply } a\le b, \end{aligned}$$

which, thanks to a well known property of adjoint maps, simplifies as:

$$\begin{aligned} \Diamond _C\, {\bullet _C}\, a\le b \text { and } a\le \Box _I\, {\bullet _I}\, b\text { imply } a\le b. \end{aligned}$$

Hence, the quasi-inequality above is equivalent to the following inequality:

$$\begin{aligned} a\wedge \Box _I\, {\bullet _I}\, b\le \Diamond _C\, {\bullet _C}\, a\vee b. \end{aligned}$$

The inequality above is analytic inductive (cf. [7, Definition 55]), and hence running ALBA on this inequality produces:

The last quasi-inequality above is the semantic translation of the rule $k\text {-}ia3_{\ell }$:

which we then proved to be sound on every perfect heterogeneous K-IA3$_{\ell }$, by the soundness of the ALBA steps. Likewise, the defining condition of K-IA3$_{\ell }$ translates into the inequality

which, however, is not analytic inductive, and hence it cannot be transformed into an analytic rule via ALBA.

1.2 A.2 Completeness

Let $A^\tau \vdash B^\tau$ be the translation of any sequent $A\vdash B$ in the language of $\text {H.K}$ into the language of $\text {D.K}$ induced by the correspondence between $\mathsf {K}$ and $\mathsf {HK}$ described in Sect. 5.

Proposition 3

For every $\text {H.K}$-derivable sequent $A {\ \vdash \ }B$, the sequent $A^{\tau } {\ \vdash \ }B^{\tau }$ is derivable in $\text {D.K}$.

Below we provide the multi-type translations of the single-type sequents corresponding to inequalities (11). All of them are derivable in D.AKA by logical introduction rules, display rules, and the rules $\,\check{\Box }\, \,\tilde{{\bullet }}\,$ and .

Below we provide the multi-type translations of the single-type sequents corresponding to inequalities (12) and (13), respectively. All of them are derivable in D.AKA by logical introduction rules and display rules.

Below we provide the multi-type translation of the single-type sequents corresponding to inequalities (18). All of them are derivable in D.AKA5’.

Below we provide the multi-type translations of the single-type rules corresponding to quasi-inequality (20), respectively.

Below, we derive (20). Firstly, $A \wedge \Box _C\, {\bullet _C}\, B {\ \vdash \ }\Diamond _I\, {\bullet _I}\, A \vee B$ is derivable via $k\text {-}ia3_\ell$ by means of the following derivation $\mathcal {D}$:

Assuming $\Diamond _I\, {\bullet _I}\, A {\ \vdash \ }\Diamond _I\, {\bullet _I}\, B$ and $\Box _C\, {\bullet _C}\, A {\ \vdash \ }\Box _C\, {\bullet _C}\, B$, we derive $A{\ \vdash \ }B$ via cut as follows:

1.3 A.3 Conservativity

To argue that $\text {D.K}$ is conservative w.r.t. $\text {H.K}$, we follow the standard proof strategy discussed in [7, 9]. We need to show that, for all formulas A and B in the language of $\text {H.K}$, if $A^\tau \vdash B^\tau$ is a $\text {D.K}$-derivable sequent, then $A \vdash B$ is derivable in $\text {H.K}$. This claim can be proved using the following facts: (a) The rules of $\text {D.K}$ are sound w.r.t. perfect members of $\mathsf {HK}$ (cf. Sect. A.1); (b) $\text {H.K}$ is complete w.r.t. the class of perfect algebras in $\mathsf {K}$; (c) A perfect element of $\mathsf {K}$ is equivalently presented as a perfect member of $\mathsf {HK}$ so that the semantic consequence relations arising from each type of structures preserve and reflect the translation. Let A, B be as above. If $A^\tau \vdash B^\tau$ is $\text {D.K}$-derivable, then by (a), $\models _{\mathbb {HK}} A^\tau \vdash B^\tau$. By (c), this implies that $\models _{\mathsf {K}} A\vdash B$, where $\models _{\mathsf {K}}$ denotes the semantic consequence relation arising from the perfect members of class $\mathsf {K}$. By (b), this implies that $A\vdash B$ is derivable in $\text {H.K}$, as required.

1.4 A.4 Cut Elimination and Subformula Property

Cut elimination and subformula property for each $\text {D.K}$ are obtained by verifying the assumptions of [5, Theorem 4.1]. All of them except $\mathrm {C}'_8$ are readily satisfied by inspecting the rules. Condition $\mathrm {C}'_8$ requires to check that reduction steps can be performed for every application of cut in which both cut-formulas are principal, which either remove the original cut altogether or replace it by one or more cuts on formulas of strictly lower complexity. In what follows, we only show $\mathrm {C}'_8$ for some heterogeneous connectives.

The remaining cases are analogous.