Non-distributive description logic ⋆

. We deﬁne LE-ALC , a generalization of the description logic ALC based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE-ALC allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE-ALC , provide a tableaux algorithm for checking the consistency of LE-ALC knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE-ALC with acyclic TBoxes is in PTIME , while the complexity of the consistency checking of classical ALC with acyclic TBoxes is PSPACE -complete.


Introduction
Description Logic (DL) [2] is a class of logical formalisms, typically based on classical first-order logic, and widely used in Knowledge Representation and Reasoning to describe and reason about relevant concepts in a given application domain and their relationships.Since certain laws of classical logic fail in certain application domains, in recent years, there has been a growing interest in developing versions of description logics on weaker (non-classical) propositional bases.For instance, in [20], an intuitionistic version of the DL ALC has been introduced for resolving some inconsistencies arising from the classical law of excluded middle when applying ALC to legal domains.In [6,19], many-valued (fuzzy) description logics have been introduced to account for uncertainty and imprecision in processing information in the Semantic Web, and recently, frameworks of non-monotonic description logics have been introduced [14,18,15].One domain of application in which there is no consensus as to how classical logic should be applied is Formal Concept Analysis (FCA).In this setting, formal concepts arise from formal contexts P = (A, X, I), where A and X are sets (of objects and features respectively), and I ⊆ A × X.Specifically, formal concepts are represented as Galois-stable tuples (B, Y ) such that B ⊆ A and Y ⊆ X and B = {a ∈ A | ∀y(y ∈ Y ⇒ aIy)} and Y = {x ∈ X | ∀b(b ∈ B ⇒ bIx)}.The formal concepts arising from a formal context are naturally endowed with a partial order (the sub-concept/super-concept relation) as follows: This partial order is a complete lattice, which is in general non-distributive.The failure of distributivity in the lattice of formal concepts introduces a tension between classical logic and the natural logic of formal concepts in FCA.This failure motivated the introduction of lattice-based propositional (modal) logics as the (epistemic) logics of formal concepts [10,9].Complete relational semantics of these logics is given by enriched formal contexts (cf.Section 2.2), relational structures F = (P, R ✷ , R ✸ ) based on formal contexts.In this paper, we introduce LE-ALC, a lattice-based version of ALC which stands in the same relation to the lattice-based modal logic of formal concepts [12] as classical ALC stands in relation to classical modal logic: the language and semantics of LE-ALC is based on enriched formal contexts and their associated modal algebras.Thus, just like the language of ALC can be seen as a hybrid modal logic language interpreted on Kripke frames, the language of LE-ALC can be regarded as a hybrid modal logic language interpreted on enriched formal contexts.
FCA and DL are different and well known approaches in the formal representation of concepts (or categories).They have been used together for several purposes [1,4,17].Thus, providing a DL framework which allows us to describe formal contexts (possibly enriched, e.g. with additional relations on them) would be useful in relating these frameworks both at a theoretical and at a practical level.Proposals to connect FCA and DL have been made, in which concept lattices serve as models for DL concepts.Shilov and Han [21] interpret the positive fragment of ALC concepts over concept lattices and show that this interpretation is compatible with standard Kripke models for ALC.A similar approach is used by Wrum [22] in which complete semantics for the (full) Lambek calculus is defined on concept lattices.The approach of the present paper for defining and interpreting non-distributive description logic and modal logic in relation with concept lattices with operators differs from the approaches mentioned above in that it is based on duality-theoretic insights (cf.[10]).This allows us not only to show that the DL framework introduced in the present paper is consistent with the standard DL setting and its interpretation on Kripke models, but also to show that several properties of these logics and the meaning of their formulas can also be "lifted" from the classical (distributive) to non-distributive settings (cf.[7,12,8] for extended discussions).
The main technical contribution of this paper is a tableaux algorithm for checking the consistency of LE-ALC ABoxes.We show that the algorithm is terminating, sound and complete.Interestingly, this algorithm has a polynomial time complexity, compared to the complexity of the consistency checking of classical ALC ABoxes which is PSPACE-complete.This means that the consistency checking problem for completely unravelled TBoxes (cf.Subsection 2.1) for our logic is in PTIME.The algorithm also constructs a model for the given ABox which is polynomial in size.Thus, it also implies that the corresponding hybrid modal logic has the finite model property.Structure of the paper.In Section 2, we give the necessary preliminaries on the DL ALC, lattice-based modal logics and their relational semantics.In Section 3, we introduce the syntax and the semantics of LE-ALC.In Section 4, we introduce a tableaux algorithm for checking the consistency of LE-ALC ABoxes and show that it is terminating, sound and complete.In Section 5, we conclude and discuss some future research directions.

Description logic ALC
Let C and R be disjoint sets of primitive or atomic concept names and role names.The set of concept descriptions over C and R are defined recursively as follows.
where A ∈ C and r ∈ R.An interpretation is a tuple I = (∆ I , • I ) s.t.∆ I is a non-empty set and • I maps every concept A ∈ C to a set A I ⊆ ∆ I , and every role name r ∈ R to a relation r I ⊆ ∆ I × ∆ I .This mapping extends to all concept descriptions as follows: Let S be a set of individual names disjoint from C and R, such that for every a in S, a I ∈ ∆ I .For any a, b ∈ S, any C ∈ C and r ∈ R, an expression of the form a : C (resp.(a, b) : r) is an ALC concept assertion (resp.role assertion).A finite set of ALC concept and role assertions is an ALC ABox.An assertion a : An ALC knowledge base is a tuple (A, T ), where A is an ALC ABox, and T is an ALC TBox.An interpretation I is a model for a knowledge base (A, T ) iff it satisfies all members of A and T .A knowledge base (A, T ) is consistent if there is a model for it.An ABox A (resp.TBox T ) is consistent if the knowledge base (A, ∅) (resp.(∅, T )) is consistent.
An ALC concept definition in T is an expression of the form A ≡ C where A is an atomic concept.We say that A directly uses B if there is a concept definition A ≡ C in T such that B occurs in C. We say that A uses B if A directly uses B, or if there is a concept name B ′ such that A uses B ′ and B ′ directly uses B. A finite set T of concept definitions is an acyclic TBox if 1. there is no concept in T that uses itself, 2. no concept occurs more than once on the left-hand side of a concept definition in T .A finite set T of concept definitions is a completely unravelled TBox if it is acyclic and no atomic concept occurring on the left-hand side of a concept definition in T is also present in some other concept definition on the right-side.Checking the consistency of a knowledge base is a key problem in description logics, usually solved via tableaux algorithms.In the ALC case, checking the consistency of any knowledge base is EXPTIME-complete while checking the consistency of a knowledge base with acyclic TBoxes is PSPACE-complete [2].

Basic normal non-distributive modal logic and its semantics
The logic introduced in this section is part of a family of lattice-based logics, sometimes referred to as LE-logics (cf.[11]), which have been studied in the context of a research program on the logical foundations of categorization theory [10,9,8,12].Let Prop be a (countable) set of atomic propositions.The language L is defined as follows: , where p ∈ Prop, and ✷ ∈ G and ✸ ∈ F for finite sets F and G of unary ✸-type (resp.✷-type) modal operators.The basic, or minimal normal L-logic is a set L of sequents ϕ ⊢ ψ, with ϕ, ψ ∈ L, containing the following axioms for every ✷ ∈ F and ✸ ∈ G: p ⊢ p ⊥ ⊢ p p ⊢ p ∨ q p ∧ q ⊢ p ⊤ ⊢ ✷⊤ ✷p ∧ ✷q ⊢ ✷(p ∧ q) p ⊢ ⊤ q ⊢ p ∨ q p ∧ q ⊢ q ✸⊥ ⊢ ⊥ ✸(p ∨ q) ⊢ ✸p ∨ ✸q and closed under the following inference rules: Note that unlike in classical modal logic, we cannot assume that ✷ and ✸ are inter-definable in LE-logics, hence we take all connectives as primitive.
The following notation, notions and facts are from [8,12].For any binary relation T ⊆ U × V , and any U ′ ⊆ U and V ′ ⊆ V , we let T c denote the set-theoretic complement of T in U × V , and In what follows, we fix two sets A and X, and use a, b (resp.x, y) for elements of A (resp.X), and B, C, A j (resp.Y, W, X j ) for subsets of A (resp. of X).
A polarity or formal context (cf.[13]) is a tuple P = (A, X, I), where A and X are sets, and I ⊆ A × X is a binary relation.Intuitively, formal contexts can be understood as abstract representations of databases [13], so that A and X represent collections of objects and features, and for any object a and feature x, the tuple (a, x) belongs to I exactly when object a has feature x.
As is well known, for every formal context P = (A, X, I), the pair of maps (•) ↑ : P(A) → P(X) and (•) ↓ : P(X) → P(A), defined by the assignments B ↑ := I (1) [B] and )).It is well known (cf.[13]) that the sets B and Y are Galois-stable, and that the set of formal concepts of a polarity P, with the order defined by ), forms a complete lattice P + , namely the concept lattice of P.
For the language L defined above, an enriched formal L-context is a tuple ✸ [x] are Galois-stable in P. For each ✷ ∈ G and ✸ ∈ F , their associated relations R ✷ and R ✸ provide their corresponding semantic interpretations as operations [R ✷ ] and R ✸ on the concept lattice P + defined as follows: For any We refer to the algebra A valuation on such an F is a map V : Prop → P + .For each p ∈ Prop, we let

[[p]] := [[V (p)]] (resp. ([p]) := ([V (p)])) denote the extension (resp. intension) of the interpretation of p under V .
A model is a tuple M = (F, V ) where F = (P, R ✷ , R ✸ ) is an enriched formal context and V is a valuation on F. For every ϕ ∈ L, we let )) denote the extension (resp.intension) of the interpretation of ϕ under the homomorphic extension of V .The following 'forcing' relations can be recursively defined as follows: As to the interpretation of modal formulas, for every ✷ ∈ G and ✸ ∈ F : The definition above ensures that, for any L-formula ϕ, The interpretation of the propositional connectives ∨ and ∧ in the framework described above reproduces the standard notion of join and the meet of formal concepts used in FCA.The interpretation of the operators ✷ and ✸ is motivated by algebraic properties and duality theory for modal operators on lattices (cf.[12,Section 3] for an expanded discussion).In [8,Proposition 3.7], it is shown that the semantics of LE-logics is compatible with Kripke semantics for classical modal logic, and thus, LE-logics are indeed generalizations of classical modal logic.This interpretation is further justified in [8,Section 4] by noticing that, under the interpretations of the relation I as aIx iff "object a has feature x" and ✸ as aRx iff "there is evidence that object a has feature x", then, for any concept c, the extents of concepts ✷c and ✸c can be interpreted as "the set of objects which certainly belong to c" (upper approximation), and "the set of objects which possibly belong to c" (lower approximation) respectively.Thus, the interpretations of ✷ and ✸ have similar meaning in the LE-logic as in the classical modal logic.A similar justification regarding similarity of epistemic interpretations of ✷ in classical and lattice-based modal logics is discussed in [9].This transfer of meaning of modal axioms from classical modal logic to LE-logics has been investigated as a general phenomenon in [7, Section 4.3], [12].

LE Description logic
In this section, we introduce the non-classical DL LE-ALC, so that LE-ALC will be in same relation with LE-logic as ALC is with classical modal logic.This similarity extends to the models we will introduce for LE-ALC: in the same way as Kripke models of classical modal logic are used as models of ALC, enriched formal contexts, which provide complete semantics for LE-logic, will serve as models of LE-ALC.In this specific respect, LE-ALC can be seen as a generalization of the positive fragment (i.e. the fragment with no negations in concepts) of ALC in which we do not assume distributivity laws to hold for concepts.Consequently, the language of LE-ALC contains individuals of two types, usually interpreted as the objects and features of the given database or categorization.Let OBJ and FEAT be disjoint sets of individual names for objects and features.
The set R of the role names for LE-ALC is the union of three disjoint sets of relations: (1) the single relation While I is intended to be interpreted as the incidence relation of formal concepts, and encodes information on which objects have which features, the relations in R ✷ and R ✸ encode additional relationships between objects and features (cf.[8] for an extended discussion).
For any set C of atomic concept names, the language of LE-ALC concepts is: This language matches the language of LE-logic, and has an analogous intended interpretation on the complex algebras of enriched formal contexts (cf.Section 2.2).As usual, ∨ and ∧ are to be interpreted as the smallest common superconcept and the greatest common subconcept as in FCA.The constants ⊤ and ⊥ are to be interpreted as the largest and the smallest concept, respectively.We do not include ¬C as a valid concept in our language, since there is no canonical and natural way to interpret negations in non-distributive settings.
The concepts R ✸ C and [R ✷ ]C in LE-ALC are intended to be interpreted as the operations R ✸ and [R ✷ ] defined by the interpretations of their corresponding role names in enriched formal contexts, analogously to the way in which ∃r and ∀r in ALC are interpreted on Kripke frames.We do not use the symbols ∀r and ∃r in the context of LE-ALC because, as discussed in Section 2.2, the semantic clauses of modal operators in LE-logic use universal quantifiers, and hence using the same notation verbatim would be ambiguous or misleading.
TBox assertions in LE-ALC are of the shape C 1 ≡ C 2 , where C 1 and C 2 are concepts defined as above. 4The ABox assertions are of the form: aR ✷ x, xR ✸ a, aIx, a : C, x :: C, ¬α, where α is any of the first five ABox terms, and C is any concept in the language of LE-ALC.We refer to the terms of first three types as relational terms.The interpretations of the terms a : C and x :: C are: "object a is a member of concept C", and "feature x is in the description of concept C", respectively.
An interpretation for LE-ALC is a tuple I = (F, • I ), where F = (P, R ✷ , R ✸ ) is an enriched formal context, and • I maps: 1. individual names a ∈ OBJ (resp.x ∈ FEAT), to some a I ∈ A (resp.x I ∈ X); 2. relation names I, R ✷ and R ✸ to relations I I , R I ✷ and R I ✸ in F; 3. any atomic concept name D to D I ∈ F + , and other concepts as follows: ✷ ] and R I ✸ are defined as in Section 2.2.The satisfiability relation for an interpretation I is defined as follows: The framework of LE-ALC formally brings FCA and DL together in two important ways: (1) the concepts of LE-ALC are naturally interpreted as formal concepts in FCA; (2) the language of LE-ALC is designed to represent knowledge and reasoning in the setting of enriched formal contexts.

Tableaux algorithm for ABox of LE-ALC
In this section, we define a tableaux algorithm for checking the consistency of LE-ALC ABoxes.An LE-ALC ABox A contains a clash iff it contains both β and ¬β for some relational term β.The expansion rules below are designed so that the expansion of A will contain a clash iff A is inconsistent.The set sub(C) of sub-formulas of any LE-ALC concept C is defined as usual.
A concept C ′ occurs in A (in symbols: for some C such that one of the terms a : C, x :: C, ¬a : C, or ¬x :: C is in A. A constant b (resp.y) occurs in A (b ∈ A, or y ∈ A), iff some term containing b (resp.y) occurs in it.
The tableaux algorithm below constructs a model (F, • I ) for every consistent A, where F = (P, R ✷ , R ✸ ) is such that, for any C ∈ A, some a C ∈ A and x C ∈ X exist such that, for any a ∈ A (resp.any We call a C and x C the classifying object and the classifying feature of C, respectively.To make our notation more easily readable, we will write a ✷C , x ✷C (resp. , where a C and x C are the classifying object and the classifying feature of C, respectively.Note that we can always assume w.l.o.g. that any consistent ABox A is satisfiable in a model with classifying objects and features (cf.Theorem 3).

Algorithm 1 tableaux algorithm for checking LE-ALC ABox consistency
Input: An LE-ALC ABox A. Output: whether A is inconsistent.1: if there is a clash in A then return "inconsistent".2: pick any applicable expansion rule R, apply R to A and proceed recursively.3: if no expansion rule is applicable return "consistent".
Below, we list the expansion rules.The commas in each rule are metalinguistic conjunctions, hence every tableau is non-branching.The basic rule and the logical rules for the connectives encode the semantics of the logical connectives in LE-ALC.The creation rule makes sure that, whenever successful, the algorithm outputs models with classifying object a C and feature x C for every concept C ∈ A. The adjunction rules and I-compatibility rules imply that every R ✷ ∈ R ✷ and R ✸ ∈ R ✸ are I-compatible.Appending and negative assertion rules encode the defining property of classifying objects and features of concepts.
It is easy to check that A has no LE-ALC model.The algorithm applies on A as follows (we only do the partial expansion to show that the clash exists): By applying the same process to b : x C1∧C2 :: Thus, there is a clash between ¬(bR ✷ y) and bR ✷ y in the expansion.
Example 2. Let A = {¬(bIy), y :: Note that no expansion rule is applicable anymore.It is clear that the tableau does not contain any clashes.Thus, this ABox has a model.By the procedure described in Section 4.2, this model is given by R

Termination of the tableaux algorithm
In this section, we show that Algorithm 1 always terminates for any finite LE-ALC ABox A. Since no rule branches, we only need to check that the number of new individuals added by the expansion rules is finite.Note that the only rules for adding new individuals are the creation and adjunction rules.The creation rules add one new object and feature for every concept C occurring in the expansion of A. Thus, it is enough to show that the number of individuals and new concepts added by applying adjunction rules is finite.To do so, we will show that any individual constant introduced by means of any adjunction rule will contain only finitely many modal operators applied to a constant occurring in A or added by the creation rule and any new concept added will contain finitely many ✷ and ✸ operators applied to a concept occurring in A.
The following Lemma bounds the length of concept and individual names appearing in a tableaux.Lemma 1.For any individual names b, and y, and concept C added during tableau expansion of A, Proof.We prove a stronger property of A, obtained from A after any finite number of expansion steps.Rules ∧ A , ∧ X , ∧ −1 A , ∧ −1 X : immediate from the definitions and the induction hypothesis.
I-compatibility rules: we only give a proof for the rule ✷y.The other proofs are analogous.We need to prove that item 2 holds for the newly added term bR ✷ y.By induction applied to the term bI✷y from item 1, we get ✷ ✷ and ✸ rules: we only give the proof for the ✷ case, as the proof for the ✸ rule is analogous.Using induction on b : [R ✷ ]C and y :: C, by items 4 and 5, we have Similarly, by items 4 and 5 we have Adjunction rules R ✷ and R ✸ : we only give the proof for R ✷ .The proof for the R ✸ case is analogous.We need to prove item 1 for the terms bIy and bI✷y.By the induction hypothesis applied to bR ✷ y, by item 2 we have Appending rules a C and x C : we only give the proof for x C .The proof for a C is analogous.We add a new term b : , by the induction hypothesis and using Equation (4) for x C , we immediately get the required conditions on ✷ D (C) and ✸ D (C).Moreover, by induction on term bIx C using item 1, we have Theorem 1 (Termination).For any ABox A, the tableaux algorithm 1 terminates in a finite number of steps which is polynomial in size(A).
Proof.New individuals are added to the tableau only in the following ways: (1) individuals of the form a C or x C can be added by creation rules; (2) individuals of the form ✷y, y, ✸b, and b can be added through the expansion rules for bR ✷ x and yR ✸ a.
New concepts can only be added by the appending through some constant that was already added.Note that no new propositional connective is added by any of the rules.Thus, the only concept that can appear are added by the application of ✷ and ✸ operators to concepts already appearing in A. By Equation (2) of Lemma 1, the maximum number of ✷ or ✸ connectives appearing in any concept added is bounded by ✷ D (A) + 1 and ✸ D (A) + 1, respectively.Also, by Equations ( 3) and ( 4 Since the tableaux algorithm for LE-ALC does not involve any branching, the above theorem implies that the time complexity of checking the consistency of an LE-ALC ABox A using the tableaux algorithm is P oly(size(A)).

Soundness of the tableaux algorithm
For any consistent ABox A, we let its completion A be its maximal expansion (which exists due to termination) after post-processing.If there is no clash in A, we construct a model (F, • I ) where A and X are the sets of names of objects and features occurring in the expansion, and for any a ∈ A, x ∈ X, and any role names R ✷ ∈ R ✷ , R ✸ ∈ R ✸ we have aIx, aR ✷ x, xR ✸ a iff such relational terms explicitly occur in A. We also add a new element x ⊥ (resp.a ⊤ ) to X (resp.A) such that it is not related to any element of A (resp.X) by any relation.Let F = (A, X, I, R ✷ , R ✸ ) be the relational structure obtained in this manner.We define an interpretation I on it as follows.For any object name a, and feature name x, we let a I := a and x I := x.For any atomic concept D, we define D I = (x D ↓ , a D ↑ ).Next, we show that I is a valid interpretation for LE-ALC.To this end, we need to show that F is an enriched formal context, i.e. that all R ✷ and R ✸ are I-compatible, and that D I is a concept in the concept lattice P + of P = (A, X, I).The latter condition is shown in the next lemma, and the former in the subsequent one.
Proof.We prove that R (0) ✷ [y] is Galois-stable for every y appearing in A. All the remaining proofs are similar.Consider the case where ✷y appears in the tableaux expansion: if bI✷y (resp.bR ✷ y) is in A, then bR ✷ y (resp.bI✷y) is added by the rule ✷y (resp.R ✷ ).
In the case where ✷y does not appear in A, then there is no term of the form bR ✷ y in A. Therefore, we have that R (0) is Galois-stable, because we have a feature x ⊥ which is not related to any of the objects.
From the lemmas above, it immediately follows that the tuple M = (F, • I ), with F and • I defined at the beginning of the present section, is a model for LE-ALC.The following lemma states that the interpretation of any concept C in the model M is completely determined by the terms of the form bIx C and a C Iy occurring in the tableau expansion.
Proof.The proof is by simultaneous induction on the complexity of C. The base case is obvious by the construction of M .For the induction step, we distinguish four cases.
and moreover for every ✷ ∈ G and ✸ ∈ F : 1.There exists a C ∈ A ′ and x C ∈ X ′ such that: . For every individual b in A there exist ✸b and b in A ′ such that: . For every individual y in X there exist ✷y and y in X ′ such that: Proof.Fix ✷ ∈ G and ✸ ∈ F .Let M ′ be defined as follows.For every concept C, we add new elements a C and x C to A and X (respectively) to obtain the sets A ′ and X ′ .For any J ∈ {I, R ✷ }, any a ∈ A ′ and x ∈ X ′ , we set aJ ′ x iff one of the following holds: 1. a ∈ A, x ∈ X, and aJx; Theorem 3 (Completeness).Let A be a consistent ABox and A ′ be obtained via the application of any expansion rule or post-processing applied to A. Then A ′ is also consistent.
Proof.If A is consistent, by Lemma 5, a model M ′ of A exists which satisfies (5), ( 6) and (7).The statement follows from the fact that any term added by any expansion rule or in post-processing is satisfied by M ′ where we interpret a C , x C , b, ✸b, ✷y, y as in Lemma 5.
Remark 1.The algorithm can easily be extended to acyclic TBoxes, via the unravelling technique (cf.[3] for details).Notice that in the presence of TBoxes that are not completely unravelled (cf.Subsection 2.1), polynomial-time complexity for the consistency check procedure is not necessarily preserved.

Conclusion and future work
In this paper, we define a two-sorted non-distributive description logic LE-ALC to describe and reason about formal concepts arising from (enriched) formal contexts from FCA.We describe ABox and TBox terms for the logic and define a tableaux algorithm for it.This tableaux algorithm decides the consistency of ABoxes and acyclic TBoxes, and provides a procedure to construct a model when the input is consistent.We show that this algorithm is computationally more efficient than the tableaux algorithm for ALC.
This work can be extended in several interesting directions.
Dealing with cyclic TBoxes and RBox axioms.In this paper, we introduced a tableaux algorithm only for knowledge bases with acyclic TBoxes.We conjecture that the following statement holds of general (i.e.possibly cyclic) TBoxes.
Developing such an algorithm is a research direction we are currently pursuing.Another aspect we intend to develop in future work concerns giving a complete axiomatization for LE-ALC.RBox axioms are used in description logics to describe the relationship between different relations in knowledge bases and the properties of these relations such as reflexivity, symmetry, and transitivity.It would be interesting to see if it is possible to obtain necessary and/or sufficient conditions on the shape of RBox axioms for which a tableaux algorithm can be obtained.This has an interesting relationship with the problem in LElogic of providing computationally efficient proof systems for various extensions of LE-logic in a modular manner [16,5].
Generalizing to other semantic frameworks.
The non-distributive DL introduced in this paper is semantically motivated by a relational semantics for LE-logics which establishes a link with FCA.A different semantics for the same logic, referred to as graph-based semantics [12], provides another interpretation of the same logic as a logic suitable for evidential and hyper-constructivist reasoning.In the future, we intend to develop description logics for reasoning in the framework of graph-based semantics, to appropriately model evidential and hyper-constructivist settings.
Generalizing to more expressive description logics.The DL LE-ALC is the non-distributive counterpart of ALC.A natural direction for further research is to explore the non-distributive counterparts of extensions of ALC such as ALCI and ALCIN .
Description logic and Formal Concept Analysis.The relationship between FCA and DL has been studied and used in several applications [1,4,17].The framework of LE-ALC formally brings FCA and DL together, both because its concepts are naturally interpreted as formal concepts in FCA, and because its language is designed to represent knowledge and reasoning in enriched formal contexts.Thus, these results pave the way to the possibility of establishing a closer and more formally explicit connection between FCA and DL, and of using this connection in theory and applications.
form a Galois connection, and hence induce the closure operators (•) ↑↓ and (•) ↓↑ on P(A) and on P(X) respectively.The fixed points of (•) ↑↓ and (•) ↓↑ are the Galois-stable sets.A formal concept of a polarity P = (A, X, I) is a tuple c = (B, Y ) such that B ⊆ A and Y ⊆ X, and B = Y ↓ and Y = B ↑ .The subset B (resp.Y ) is the extension (resp.the intension) of c and is denoted by [[c]] (resp.([c] 2. I |= a : C iff a I ∈ [[C I ]] and I |= x :: C iff x I ∈ ([C I ]). 3. I |= aIx (resp.aR ✷ x, xR ✸ a) iff a I I I x I (resp.a I R I ✷ x I , x I R I ✸ a I ). 4. I |= ¬α, where α is any ABox term, iff I |= α.An interpretation I is a model for an LE-ALC knowledge base (A, T ) if I |= A and I |= T .An LE-ALC knowledge base (A, T ) (resp.TBox T , resp.ABox A) is said to be inconsistent if it has no model.

1 Ab: 1 X
For any C ∈ A create aC : C, xC :: C b : C, y :: C I bIy Rules for the logical connectives I-compatibility rules b : C1 ∧ C2 ∧A b : C1, b : C2 y :: C1 ∨ C2 ∨X y :: C1, y :: the lattice connectivesb : C1, b : C2, C1 ∧ C2 ∈ A ∧ −C1 ∧ C2 y :: C1, y :: C2, C1 ∨ C2 ∈ A ∨ −In the adjunction rules the individuals b, ✸b, ✷y, and y are new and unique for each relation R ✷ and R ✸ , except for ✸a C = a ✸C and ✷x C = x ✷C .Side conditions for rules ∧ −1 A , and ∨ −1 X ensure we do not add new joins or meets to concepts.It is easy to check that the following rules are derivable in the calculus.b : C 1 ∨ C 2 , y :: C 1 , y :: C 2 ∨A bIy y :: C 1 ∧ C 2 , b : C 1 , b : C 2 ∧X bIy b : C adj✷ b : [R ✷ ]C y :: C adj✸ y :: R ✸ C we add the terms b : C 1 and b : C 2 to the tableau.Then the further tableau expansion is as follows: Rule Premises Added terms ∧ X

Lemma 2 .
x ↓↑ D = a ↑ D and a ↑↓ D = x ↓ D for any D ∈ C. Proof.For any atomic concept D, we have a D Ix D by the creation and appending rules.Therefore, I (0) [I (1) [a D ]] ⊆ I (0) [x D ]. Suppose a D Iy and bIx D are in A, then by the appending and basic rule we get bIy ∈ A. Therefore, I (0) [x D ] ⊆ I (0) [I (1) [a D ]].Hence the statement is proved.

Lemma 4 .
Let M = (F, • I ) be the model defined by the construction above.Then for any concept C and individuals b, y occurring in A, added which by the creation and appending rule implies bIx C1∧C2 ∈ A. Conversely, suppose bIx C1∧C2 ∈ A. By the appending rule we have b : C 1 ∧ C 2 ∈ A. By rule ∧ A , we have b : C 1 , b : C 2 ∈ A. By the creation and basic rule we have bIx C1 , bIx C2 ∈ A. By induction hypothesis, this implies b ∈ [[C 1 ]], and b ∈ [[C 2 ]].By definition, this implies 2. x ∈ X, and a = a C for some concept C, and bJx for all b ∈ [[C I ]]; 3. a ∈ A, and x = x C for some concept C, and aJy for all y ∈ ([C I ]); 4. a = a C1 and x = x C2 for some C 1 , C 2 , and bJy for all b ∈ [[C I 1 ]], and y ∈ ([C I 2 ]).We set xR ′ ✸ a iff one of the following holds: 1. a ∈ A, x ∈ X, and xR ✸ a; 2. x ∈ X, and a = a C for some concept C, and xR ✸ b for all b ∈ [[C I ]]; 3. a ∈ A, and x = x C for some concept C, and yR ✸ a for all y ∈ ([C I ]); 4. a = a C1 and x = x C2 for some C 1 , C 2 , and yR ✸ b for all b ∈ [[C I 1 ]], y ∈ ([C I 2 ]).For any b ∈ A, y ∈ X, let b = a (cl(b)) , ✸b = a ✸(cl(b)) , y = x (cl(y)) , and ✷y = x ✷(cl(y)) , where cl(b) (resp.cl(y)) is the smallest concept generated by b (resp.y), and the operations and are the adjoints of operations ✷ and ✸, respectively.For any C, let C I ′ = (I ′(0) [x C ], I ′(1) [a C ]). Then M ′ is as required.