DeepAI

BitSim: An Algebraic Similarity Measure for Description Logics Concepts

In this paper, we propose an algebraic similarity measure σBS (BS stands for BitSim) for assigning semantic similarity score to concept definitions in ALCH+ an expressive fragment of Description Logics (DL). We define an algebraic interpretation function, I_B, that maps a concept definition to a unique string (ω_B) called bit-code) over an alphabet Σ_B of 11 symbols belonging to L_B - the language over P B. IB has semantic correspondence with conventional model-theoretic interpretation of DL. We then define σ_BS on L_B. A detailed analysis of I_B and σ_BS has been given.

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1 Introduction

Semantic similarity measure serves as the foundation of knowledge discovery and management processes such as ontology matching, ontology alignment & mapping, ontology merging, etc [Shvaiko and Euzenat2013]. Ontological concept similarity can be based on different approaches: (i) string matching of concept labels (i.e. lexical similarity) [Stoilos et al.2005], (ii) external lexical resource/ontology based matching (i.e. lexico-semantic similarity) [Rada et al.1989a]

, (iii) graph-based matching using lexicons such as WordNet

[Stuckenschmidt2007] (i.e. structural similarity), (iv) property analysis (as in FCA-based similarity [Cimiano et al.2005]) or instance analysis (as in Jaccard similarity [Jaccard1998]) based matching over a large sample of concept instance occurrences (i.e. instance-driven similarity), (v) matching based on statistical analysis of attribute-value or distribution analysis within fixed context-windows of concepts over large corpora (i.e. statistical similarity) [Li and Clifton1994], and (vi) model-theoretic matching of formal concept descriptions (i.e. formal semantic similarity) [Alsubait et al.2014].

It can be argued that, in comparison to other approaches, formal semantic similarity measure modeling has not received equal research attention. Nevertheless, existing literature is significant, and can be broadly classified into two approaches: (i) Propositional Logics based

[Nienhuys-Cheng1998, Ramon and Bruynooghe1998], and (ii) Description Logics (DL) based [Alsubait et al.2014, Lehmann and Turhan2012, Stuckenschmidt2007, Fanizzi and d’Amato2006, Borgida et al.2005]. The former requires: (a) representation of ontologies (mostly in RDFS/OWL format) in First Order Predicate Logic, (b) a set of axioms (or domain knowledge, mostly as upper ontologies/thesaurus), and (c) a SAT solver that checks satisfiability (and hence, satisfiability) of disjointness of concept pairs. The latter approach, on the other hand, does not necessarily require any formal language transformation or satisfiability checker. In this paper we propose an algebraic similarity measure, called BitSim (), that can compute semantic similarity of pair of concepts defined in 111 : . The motivation behind is to formulate a formal semantic similarity measure that provides: (i) a platform for fast, scalable, and accurate semantic similarity computation of DL concepts, and (ii) a sound and complete correspondence with conventional semantic interpretation of DL. is algebraic, in the sense that it maps a given pair of concept codes (called bit-code), instead of concept DL definitions/axioms, to a positive real space. For this we define a novel algebraic interpretation function, called , that maps an definition to a unique string, called bit-code, () belonging to the language defined over a novel algebraic alphabet . We prove that has complete correspondence with . We also show that is highly adaptive to any kind of similarity measure that relies on set operation. As an example, we have shown how can be plugged into Jaccard similarity index. The contribution of the paper is as follows:

• : A novel algebraic semantic interpretation function for .

• Proof of mathematical correspondence of with semantic interpretation of .

• : A novel semantic similarity measure based on

• Comparative analysis of properties of with contemporary DL based similarity measures.

2 Related Work

DL based similarity measures, as described in the introduction, can be further sub-divided into: (i) taxonomic analysis, (ii) structural analysis [Tongphu and Suntisrivaraporn2014, Ontañón and Plaza2012, Joslyn et al.2008, Hariri et al.2006], (iii) language approximation [Stuckenschmidt2007, Tserendorj et al.2008, Groot et al.2005, Brandt et al.2002], and (iii) model-theoretic analysis [Distel et al.2014, Alsubait et al.2014, Lehmann and Turhan2012, Borgida et al.2005]. The most common approach for DL based similarity measure modeling adopts taxonomic analysis as proposed in [Rada et al.1989b, Resnik and others1999, Jiang and Conrath1997, Wu and Palmer1994, Lin1998]. These techniques can be further sub-divided, as mentioned in introduction, into graph-traversal approaches [Rada et al.1989b] and Information-Content approaches [Resnik and others1999]. However, these methods can work on a generalized ontology and hence, are not sensitive to DL definitions.

In structural analysis based approaches, a similarity measure is designed to capture the semantic equivalence of description trees of definitions of DL concept pairs. One way of achieving this is to calculate the degree of homomorphism between such trees, as proposed in [Tongphu and Suntisrivaraporn2014]. A refinement graph based anti-unification approach has been proposed in [Ontañón and Plaza2012] for computing instance similarity. Approximation based techniques aim at converting given DL expression to another lower expressive DL language on approximation. In [Stuckenschmidt2007] an upper and lower approximation interpretation for have been defined over a sub-vocabulary of . The sub-vocabulary can be formed by either removing concepts atoms in a given definition or by replacing them with structurally simpler concepts [Groot et al.2005]. Another technique, as proposed in [Noia et al.2004], is based on converting user query into a DL expression and try to classify the match to be either an exact match, or a potential match (i.e., match might happen if some concept atom and operators are added) or a partial match (where the user query and answer/description found in the knowledge base are in conflict).

One of the pioneer work on model-theoretic interpretation based similarity approach can be found in [Borgida et al.2005]. The work shows the inherent difficulty in measuring similarity of DL concepts using conventional taxonomic analysis based techniques. It then uses an Information-Content based approach to evaluate the similarity of two concept definitions. A work has been proposed by [Lehmann and Turhan2012] for similarity computation of concepts defined in

. In this work, a Jaccard Index

[Jaccard1998] based approach has been followed that compares common parents of a concept pair using a fuzzy connector (i.e. a similarity score aggregation function). A similar Jaccard Index based approach has been recently proposed in [Alsubait et al.2014]. Another recent approach has been proposed in [Distel et al.2014]. The work emphasizes the necessity of triangle inequality property of formal semantic similarity measure. It defines two versions of a relaxation function for computing dissimilarity of concepts defined in . However, it can be proved that triangle inequality is not always valid and hence, is not a necessary condition.

3 Preliminaries

3.1 ALCH+ - A Description Logics Fragment

We hereby define the semantic interpretation of (an extension of which also interprets current OWL 2 specification)222The syntax of follows conventional DL as defined in [Baader2003]. Let be an interpretation function, and be the universal domain. is defined as:

• (Attributive Language):

• Atomic concept:

• Role:

• Atomic Negation: =

• Top Concept:

• Bottom Concept:

• Conjunction:

• Value Restriction:

• Limited Existential Restriction:

• (Full Existential Restriction):

• (Concept Negation): =

• (Role Hierarchy):

• (Role Union):

• (Role Intersection):

3.2 Formal Similarity Measure

In this section we define the algebraic properties of as given in [Lehmann and Turhan2012]. Let is a DL concept in a given terminology (T-Box) .

Definition 1: A semantic similarity measure is a function defined as follows:
where

Properties of Similarity Measure: Arguably333* denotes that the property is not universally adopted as necessary condition. Also it cannot be proven to be valid in all types of algebraic spaces., should hold the following properties:

 Positiveness*:∀Ci,Cj∈T,σ(Ci,Cj)≥0 (1)
 Reflexive:∀Ci∈T,σ(Ci,Ci)=1 (2)
 Maximality:∀Ci,Cj,Ck∈T,σ(Ci,Ci)≥σ(Cj,Ck) (3)
 Symmetry*:∀Ci,Cj∈T,σ(Ci,Cj)=σ(Cj,Ci) (4)
 Equivalence Closure:∀Ci,Cj∈T,σ(Ci,Cj)=1⟺Ci≡Cj (5)
 Equivalence Invariance:∀Ci,Cj,Ck∈T,Ci≡Cj⟹σ(Ci,Ck)=σ(Cj,Ck) (6)
 Structural Dependency:∀Cin,Cjn,Ci,Cj∈T,∍Cin≡⨅k≤n (Ck⊓Ci);and,Cjn≡⨅k≤n (Ck⊓Cj);then,limx→∞σ(Cin,Cjn)=1 (7)
 Subsumption Preservation:∀Ci,Cj,Ck∈T, (8) Ci⊑Cj⊑Ck⟹σ(Ci,Cj)≥σ(Ci,Ck)
 Reverse Subsumption % Preservation:∀Ci,Cj,Ck∈T,Ci⊑Cj⊑Ck⟹σ(Cj,Ck)≥σ(Ci,Ck) (9)
 Strict Monotonicity:∀Ci,Cj,Ck∈T,∍if,∀Cl∈T,Ci⊑Cl;Cj⊑Cl⟹Ck⊑Clthen,σ(Ci,Ck)≥σ(Ci,Cj)and if,∃Cm∈T,∍Ci⊑Cm;Ck⊑Cm and Cj\nsqsubseteqCmthen,σ(Ci,Ck)>σ(Ci,Cj) (10)

It should be noted that the aforementioned necessary properties may not be sufficient and hence, detailed theoretical analysis has to be done on sufficiency.

4 LB: Formal Language for Concept Coding

In this section we define the formal language on which the proposed algebraic interpretation function is defined. We first define bit (), the alphabet of , as follows:

Definition 2: A base alphabet () is an alphabet defined as: where

• 0 is the empty symbol. It is also called potential bit since it generates all other symbols (i.e. bits)

• 1 is base bit, called property bit, signifying the presence of a property at the string position that it holds.

Definition 3: An bit operator () is a set of operators on the base bits defined as: .

We now define a very important semantics for potential bit (i.e. 0) as follows:

 ¬B0=0 (1)

Definition 4: An derived alphabet () is an alphabet defined as: where

Based on the above definition the following observations can be made (using de Morgan’s law):

A further analysis shows that has a partial ordering (as shown in figure 1). It is interesting to note that .
Definition 5.1: An bit-alphabet () is defined as
: .

It is to be noted that satisfies commutativity and double negation over .

Definition 5.2: A quantifier-alphabet () is defined as
. The algebraic space of is defined as below:

Definition 5.3: A role-alphabet () is defined as
. The algebraic space of is defined as below:

Definition 6.1: A base bit-code () is defined as where is bit concatenation operator; ; ; .
Definition 6.2: A derived bit-code () is defined as = () () where is string concatenation operator.

It can be observed that the definition of is recursive. We leave the explanation and utility of the definition in section 5.3.
Definition 7: is defined as .

5 Encoding ALCH+ Concept

5.1 Motivation

The motivation behind BitSim ( ) is to develop a formal, efficient, and scalable matchmaking system that can be applied in DL based knowledge bases. Unlike other DL based similarity measures, was designed to satisfy all the necessary properties defined in section 3.2 with special emphasis on structural dependency and strict monotonicity.

At the same time, computation is over , rather than . This significantly improves the computational speed since essentially becomes a function over pairs (such bit-codes can be computed and stored off-line in the knowledge base). Since, computation is performed on pairs of bits holding the same position in , therefore we can chunk bit-codes in constant sizes and perform similarity over concept pairs on parallel computational platforms. This gives massive scalability to . Efficient optimization can be performed by caching similarity results of bit-code chunks that are frequently visited.

We will also show that, at a bit level, has a partial ordering . This allows application-oriented assignment of similarity scores to bit pairs at the lowest granularity. Also, is highly adaptive to all types of similarity measures that have set theoretic operations on DL concepts.

5.2 Encoding Atomic Concepts

Before we show that has complete correspondence with , we first provide the foundational axioms that helps us to encode atomic concepts in . For that we need to define the proposed algebraic interpretation function (also called bit-interpretation).

Definition 8: Bit-interpretation ( ) is a function as follows:
: ;

We hereby define , where is an arbitrary atomic concept in , using the following two axioms:

 Fundamental Axiom of Atomic % Concepts∀bAik∈ωAiB, bAik∈ΣBbase;ωAiB≡(Ai)IB. (1)
 Axiom of Significant % Property Bit∃bAik∈ωAiB,bAik=1 & bAik+1=0∗. (2)

where, k is the k-th position (in increasing order from right to left) in . In the second axiom holds the significant property bit.

We now show the method to encode inclusion axioms on atomic concepts using the following two axioms:

 Axiom of Property Bit % Inheritance∀Ai, Aj∈TALCH+,∍Ai⊑Aj,bAjk=1→bAik=1;∀bAjk∈ωAjB (3)
 Axiom of Atomic Bit-Code % UniquenessGiven,∀Ai,Aj∈TALCH+,∍Aj\nsqsubseteqAi∃bAik∈ωAiB,bAjk∈ωAjB,bik=1 & bjk≠1 (4)

The above four axioms ensure that a simple boolean intersection between any pair of and generates the bit-code of the least common subsumer (lcs) atomic concept. An example encoding instance has been illustrated in figure 2. For atomic roles we can use the same principle of encoding with an alphabet that corresponds to .

5.3 Encoding Derived Concepts

For encoding derived concepts, we cannot attain completeness using only. This is because, for a bounded number of atomic concepts (say, n) we need a mechanism to encode distinct and disjoint concepts, in the worst case. However, with only 11 bits in , we can only encode distinct concepts. It is because of this reason that we need to use . The method is to have nested encoding for bit operations over certain special bit pairs. These operations do not have direct mapping to the algebraic lattice shown in figure 1. Instead they map to intermediate and discrete compound bits (). which can be represented in terms of . We define as follows:

Definition 9: A compound bit is defined as:
= , where the algebra of is defined as shown in figure 3.

For any derived concept , we can state the following:

 Axiom of Binary Concept % OperationbCiωk⊕bCjωk≡(⟨∘B ωBderived∘⟩)Cik⊕(⟨∘B ωBderived∘⟩)Cjk≡(⟨∘B (ωCiBderived⊕ ωCjBderived) ∘⟩)k≡bCi ⊕ Cjωk (5)

Based on axiom 5 we can state that:

Lemma 1.

For any binary operation between and , the length of both the operands, and , must be same; where k: k-th position of in .

Lemma 2.

For any binary operation between and , the length of the resultant has growth = , where and n is length of operand .

If , then

If , then

If , then

Lemma 6.

If , then

We now postulate the following axioms for operations over derived concepts:

 Axiom of Concept Negation(¬Ci)IB≡{¬bCik}≡{b¬Cik} (6)
 Axiom of Binary Concept % Operation(Ci⊕Cj)IB≡(Ci)IB⊕(Cj)IB≡{bCik}⊕B{bCjk}≡{bCi ⊕ Cjk};where ⊕B≡⊕IB;⊕∈{⊔,⊓} (7)
 Axiom of Universal Role % Restriction(∀Rj.Ci)IB≡||⟨∘Bb∀k∘⟩||||⟨∘B{bRjRk}∘⟩||∘B{bCik} (8)
 Axiom of Full Existential % Role Restriction(∃Rj.Ci)IB≡||⟨∘Bb∃k∘⟩||||⟨∘B{bRjRk}∘⟩||∘B{bCik} (9)

It is to be noted that for the above axioms can be both a simple and a compound bit. We now provide a proof for showing the mathematical correspondence between ( ) and ( ).

Lemma 7.

k: k-th position in

Proof.

Proof can be derived from the lattice structure of (see figure 1.) ∎

Following the above lemma we can state that:

Proof.

, if ; else converse over . ∎

Proof.

From axiom 7 and 8, we can show that is unique. This is because is unique. In other words, the algebra has complete correspondence with . ∎

, is unique.

Proof.

Follows from axiom 4 and lemma 1 - 6. ∎

Theorem 5.

has complete correspondence with

Proof.

The proof follows from theorem 1 - 3. ∎

6 BitSim Similarity Measure

6.1 Outline

In this section we provide a generic definition for BitSim. We first define (i.e. similarity at a bit level) as follows:

Definition 10: : ; where .

It is to be noted that can be as well. One can see that has a total order (see figure 4). In order to compute similarity at a bit-code level, we define an aggregation function called . There are two parameters that should influence the value output of : (i) and (ii) code-generativity (). We define code-generativity as follows:

Definition 11: Code-generativity () of any of a concept is the total number of distinct and disjoint concepts that are covered by the .

As an example, ; (i.e. ). We now define the similarity measure at a bit-code level (we call it as follows:

Definition 12: : { } .

We now postulate the following axioms:

 Axiom of 0-bit Similarity∀k;σBS(0k,0k) is ignored. (10)
 Axiom of ⊤− % bit Similarity∀a∈ΣB;σBS(⊤,a)=1 if a=⊤; else undefined. (11)
 Axiom of ⊥− % bit Similarity∀a∈ΣB;σBS(⊥,a)=1 if a=⊥; else undefined. (12)

6.2 ^σBS: Property Analysis

In this section we show that follows the necessary conditions: (i) reflexive, (ii) maximality, (iii) equivalence closure, (iv) equivalence invariance, (v) structural dependency, (vi) subsumption preservation, (vii) reverse subsumption preservation, and (viii) strict monotonicity. The first two properties trivially hold true. The following theorems show that the rest of the properties hold:

Theorem 6.

Equivalence closure and invariance holds true for .

Proof.

Follows from theorem 1 and theorem 4. ∎

Theorem 7.

Structural dependency holds true for .

Proof.

Under the condition of structural dependency, for two concepts and the that is generated for them will have a length, say l, that grows exponentially as the number of inner intersections in the definition of both and tends to . Therefore, except for the word length of the invariant concepts in the definition of and , all of and will be exact. Hence, ()

Theorem 8.

Strict monotonicity holds true for .

Proof.

When more than one concepts (say, ) subsume two (say, ) out of three arbitrary concepts (say, ), while only one (say, ) subsumes all three, then () (). This is because, since is property based measure, will inherit more common bits than . ∎

We now show how can be adapted to a third-party similarity measure as follows:

Definition 13: - ()

7 Discussion

As can be seen, can be adapted as an alternate paradigm for DL subsumption reasoning. Since can be mapped to a boolean space one can perform bit operations at high speed and that too on a distributed and parallel platform. At the same time, various caching techniques can be applied efficiently. In the future we will be exploring these research prospects and other possibilities such as probabilistic reasoning on , A-Box reasoning, and reasoning over higher expressive DL.

8 Conclusion

In this paper we have proposed BitSim ( ) - an algebraic similarity measure for concept definitions in . We show that satisfies all the necessary algebraic properties recommended for a formal similarity measure. Being based on , is highly sensitive to standard DL interpretation. Furthermore, is highly adaptive to any similarity measure that uses set theoretic operations.

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