In the context of the ℰℒ description logic, we define and study a new concept diference operator, called commonality subtraction operator (CSO), with respect to an acyclic definitional ontology , and noted ⊖ . CSO aims at removing from a concept description all common parts with another description , w.r.t. , which we call descriptional commonalities. Based on the proposed operator of tree subtraction (TSO), we give an algorithm to compute CSO along with its complexity. CSO fits well with existential restrictions and applies to any couple of concepts (, ), which makes it diferent from existing diference operators. We practically justify the definition of CSO by explaining our needs for such an operator in the context of a metrology resources management project.
wanted resources are built as ℰ ℒ concept descriptions. The semantically closest resources w.r.t. a user query are obtained by pairwisely comparing all resources w.r.t the query, in a semantic way using the ontology. This process produces a ranking of resources w.r.t the query based on the idea that the bigger the shared information between the query and the resources, the better. But instead of directly computing the shared information, we compute what is original in each resource w.r.t. the query. This means best resources are the ones which have the least original parts w.r.t. the query. Moreover, the process also computes what parts of the query are original w.r.t. each resource, which can be used to further refine the ranking if needed. The whole approach is based on a new diference operator for ℰ ℒ, namely the commonality subtraction operator (CSO, noted ⊖ ), which is the contribution presented in this paper. Example 1. In the metrology context, we may have the following ℰ ℒ ontology and resource and query descriptions (see figure 1): = { ≡ ⊓ ∃ℎ., ⊑ , ⊑ , ⊑ , ⊑ , ⊑ , ⊑ , ⊑ ℎ}, = ∃ . ⊓ ∃ℎ .( ⊓ ), and = ∃ . ⊓ ∃ℎ .. Intuitively, is defined as in which is included , is a kind of which is a kind of , as and . is a kind of , as , and is a kind of ℎ. describes a resource that is about a as instrument type, made of and . With , a user is looking for resources about as instrument type, made of . We then would like to have: (i) ⊖ = ∃ℎ . , which means shares all aspects of except the fact that is a resource about an instrument made of wood, and (ii) ⊖ = ∃ . ⊓ ∃ℎ .∃ℎ. , which means that shares with the fact that it describes resources made of , but does not share other aspects of (the instrument type and the inclusion inside the material).
The CSO ensures two important features: inverse subsumption criterion to define commonalities, and fine-grained diference. The inverse subsumption criterion states that a commonality between the minuend and the subtrahend exists when a part of the minuend subsumes the Δℐ It is assumed that: ℐ ⊆ Δℐ ∙ conjunctions do not contain ⊤ nor ℐ ⊆ Δℐ × Δℐ many times the same conjunct, and ℐ ∩ ℐ ∙ writing d=1 means the s are { ∈ Δℐ |∃ ∈ ℐ : not conjunctions themselves. (, ) ∈ ℐ } ℐ = ℐ ℐ ⊆ ℐ appears only once as the lhs of a definition. subtrahend. A contrario, in existing operators, the minuend is usually subsumed by (parts of) the subtrahend. This is justifed by our resource retrieval context where the minuend is seen as a query and the subtrahend may answer parts of it: we consider answering part of a query as corresponding to being subsumed by this part of the query. In example 1, considering ⊖ , ∃ℎ . is a commonality between and since it is a part of (once has been replaced by its definition ⊓ ∃ℎ.) and it subsumes . A contrario, ∃ℎ . is not a commonality with since it does not subsume . The fine-grained diference browses the tree structure implied by existential restrictions in order to precisely remove commonalities between the minuend and the subtrahend without modifying the remaining of the minuend. Going on with the same example, the CSO removes ∃ℎ . from since it is a commonality with , and it keeps ∃ℎ .∃ℎ., and ∃ . .
After recalling notions about ℰ ℒ in section 2, we study CSO for ℰ ℒ (with an acyclic and definitional TBox) in section 3. In section 4, we relate CSO to other diference operators. At last, we conclude. When not given in the text body, full proofs of properties are given in appendix2. 2. Recalls about ℰ ℒ We assume to have two countably infinite sets: C for concept names and r for role names. From these, with the help of ℰ ℒ constructors (see table 1), ℰ ℒ concept descriptions can be built. From now on, unless stated otherwise, the term concept refers to the expression "ℰ ℒ concept description". A concept that is not a concept name nor ⊤ is called a compound concept. Given a concept , we can define its size.
Definition 1 (size [3]). Given a concept , its size noted size() is defined by induction on its structure: if ∈ C∪⊤ then size() = 1; if = 1⊓2, then size() = 1+size(1)+size(2); and if = ∃. then size() = 1 + size()
Concepts are given a model-theoretic semantics based on interpretations which are couples (Δℐ , .ℐ ) of, respectively, a universe of discourse and an interpretation function, see the third 2https://github.com/Myakko/DL2023-Appendix column of table 1. Axioms that relate concepts are of the following kinds: concept definitions of the form ≡ and primitive concept definitions of the form ⊑ . The size of an axiom is the sum of the sizes of the left and right hand sides of the axiom. An ℰ ℒ TBox, or just TBox, is a finite set of axioms. The size size( ) of a TBox is the sum of the sizes of its axioms. An interpretation (Δℐ , .ℐ ) is a model of a TBox if, for each axiom in , the condition given in the third column of table 1 is satisfied. A concept is subsumed by another concept w.r.t. a TBox , noted ⊑ (or |= ⊑ ), if ℐ ⊆ ℐ in every model of . When = ∅, we can note interchangeably ⊑ or ⊑∅.
A definitional TBox contains only concept definitions. A TBox containing primitive concept definitions can be made definitional in linear time w.r.t. size( ) [3] since (i) each primitive concept definition ⊑ can be transformed into the concept definition ≡ ⊓ , with a new concept name, and (ii) two primitive concept definitions ⊑ and ⊑ can be grouped into one ⊑ ⊓ . In the sequel, TBoxes are supposed to be definitional.
The signature of a TBox , noted sig , is the set of all concept names and roles that occur in . We note C = C ∩ sig , r = r ∩ sig , and ℰℒ the set of all concepts that can be built using elements of sig and ⊤. Concept names appearing as the left-hand side of a definition are called defined concepts and they define the set def ⊆ C . Defined concepts may only appear once as the left hand side of a concept definition. Other concept names are called primitive concepts. They define the set prpirmim. ⊆ C . The set of concepts built using only primitive concepts of and ⊤ is noted ℰℒ
Following definition 2.9 of [ 3], for , and ′ concept names, we say that directly uses in if there is in a primitive concept definition ⊑ , or a concept definition ≡ , such that occurs in . We say that uses if directly uses , or if there is a concept name ′ such that uses ′ and ′ directly uses . A TBox contains a cycle when some concept name uses itself. A TBox is acyclic if it contains no cycle. In the sequel, TBoxes are supposed to be acyclic, in addition to being definitional.
The complete expansion (a.k.a. unfolding) * of an acyclic definitional TBox [4, 5] rewrites every concept definition of into an equivalent one with only primitive concepts in its right hand side. Then, for a concept , * () is the complete expansion of w.r.t. . This process is EXPTIME in the sizes of and .
Example 2. We have the following acyclic definitional TBox: = { ≡ ⊓ , ≡ ⊓ ∃., ≡ , ≡ ∃ .}. Thus we have C = {, , , , , }, r = {}, prim = {, } and def = {, , , }. The complete expansion * is * = { ≡ ⊓ ⊓ ∃., ≡ ⊓ ∃., ≡ , ≡ ∃ .}.
In section 3.1, we define the CSO, first informally, by presenting the notions of characteristic branch and descriptional commonality, and then formally. In section 3.2, we present a new syntactical operator called the tree subtraction operator (TSO) and show how to use it to compute the CSO. We also give the main properties associated to both the TSO and the CSO, namely existence, unicity, and termination, soundness and complexity of the associated algorithms.
The CSO operator ⊖ is intended to remove from the minuend all concept parts shared with the subtrahend w.r.t. some TBox . We call these shared parts descriptional commonalities (or commonalities for short) from to . Syntactically, we want commonalities to be removable from the minuend without impacting its other parts. So they have to be atomic defined as a characteristic branch of w.r.t. that subsumes : in some sense. Since ℰ ℒ concepts have a tree structure (see [6]), we capture the notion of atomic parts of a concept as its branches in its tree structure. We define the notion of branch with the ones of subdescription and width of a concept. Semantically, being a commonality from to means being a part of linked to : we propose a commonality from to to be • A characteristic branch of w.r.t. is a primitive branch of or of a concept equivalent to (w.r.t. ) such that it cannot be syntactically removed from without changing its semantics. Fine-grained diference (cf. introduction) is achieved by working at the level of characteristic branches. • Imposing being subsumed by a characteristic branch of expresses the fact that commonalities from to are parts of to which answers (by being subsumed by them). This is how the inverse subsumption criterion is implemented (cf. introduction). Then, ⊖ is defined as the minimal concept that subsumes such that there are no commonalities from to (meaning all commonalities from to have been removed).
We now formalize these notions. First, the width of a concept is the maximum number of conjuncts composing any conjunction occuring in .
Definition 2 (width of a concept). The width of , noted wid(), is defined as follows: wid() = wid() if = ∃. wid() = 1 if ∈ C ∪ {⊤} wid() = (, {wid(), 1 ≤ ≤ }) if = d =1 , ≥ 2
A subdescription of a concept is obtained by removing zero or many conjuncts anywhere in , provided it remains a syntactically correct concept3. The following definition formalizes this idea. It is equivalent to the definition given in [7] (restricted to ℰ ℒ). subd , is set to {} if ∈ C ∪ {⊤}, or to {∃. | ∈ subd} if = ∃., or to Definition 3 (subdescription of a concept). With ≥ 2, the set of subdescriptions of , noted ︁( d =1 ︁) )︂ if = d =1 .
⋃︀ ⊆{ |1≤ ≤ } with ||≠=0 ︂( ⟨1,...,⟩∈ ∏︀ subd ⋃︀
∈ We note subdpr,im = subd ∩ ℰℒ primitive concepts or ⊤ only (w.r.t. ).
Example 3. Let = ⊓ ∃.( ⊓ ). We have: subd = { ⊓ ∃.( ⊓ ), ∃.( ⊓ ), ⊓ ∃., ⊓ ∃., ∃., ∃., } 3The notion of subdescription is close to but not the same as the one of subconcept defined in [ 3]. Informally, a subconcept is any conjunction taken in the original concept from which zero or many conjuncts have been removed (keeping at least one).
prim the set of subdescriptions of where concept names are
A branch is essentially a concept having a width of at most 1. bbrrprim=={{∈ ∈ℰℒℰpℒ|riwmid() = 1}
| wid() = 1} bbrrpr,im=={{∈ ∈susbudbdp|r,wimid|(wi)d(=)1 = 1}
} Definition 4 (branch). Let be a TBox and ∈ ℰℒ. The set of branches over is noted br . The set of primitive branches over is noted brprim. The set of branches of is noted br . And the set of primitive branches of is noted brpr,im. These sets are defined as follows: ⊖ =
⊤ Example 4. Using TBox from example 2, let be the following concept: = ⊓ ∃1.(∃2.∃3.⊤ ⊓ ⊓ ∃4.). Then we have: br = {, ∃1.∃2.∃3.⊤, ∃1., ∃1.∃4.} and brpr,im = {∃1.∃2.∃3.⊤, ∃1.∃4.}.
A characteristic branch of w.r.t. is a primitive branch of or of a concept equivalent to such that it cannot be syntactically removed from without changing its semantics. Definition 5 (characteristic branch). Let be a TBox and ∈ ℰℒ. The set char of characteristic branches of w.r.t. is defined as follows: char = { ∈ brprim | ∃′ ∈ ℰℒ such that
︀( ≡ ′ and ∈ br′ and ∀′′ ∈ ℰℒ (br′′ = br′ ∖ {}) → (′′ ̸≡ ))︀ } commonalities from to are defined as characteristic branches of that subsume . Definition 6 (descriptional commonality). Let be a TBox and (, ) ∈ (ℰℒ)2. The set dcom, of descriptional commonalities from to is defined as follows: dcom, = { ∈ char | ⊑ }
At last, ⊖ is defined as the minimal concept (w.r.t. ⊑ ) that subsumes with no commonalities from to (unicity is shown in proposition 2). When all characteristic branches are commonalities (and thus must all be removed from ), the result is ⊤. Definition 7 (CSO). Let be a TBox and (, ) ∈ (ℰℒ)2. The binary operator ⊖ , called commonality subtraction operator (CSO) for , is defined as follows: {︃ ⊑ { ∈ ℰℒ | ⊑ and dcom, = ∅} if char ̸⊆ dcom, if char ⊆ dcom, Example 5. Table 2 shows two examples of CSO, with from example 2: ⊖ and ⊖ where = ⊓ ⊓ ∃.⊤ ⊓ ∃.⊤ and = ⊓ ∃. ⊓ ∃.. In both cases, the corresponding sets of characteristic branches and descriptional commonalities are given.
In order to compute ⊖ , we propose an approach based on a syntactical diference operator that operates on branches of expansions of and . This operator is not the classical set diference of the branch sets since it takes into account subsumption relationships between
= { ≡ ⊓ , ≡ ⊓ ∃., ≡ , ≡ ∃ .} = ⊓ ⊓ ∃.⊤ ⊓ ∃.⊤ = ⊓ ∃. ⊓ ∃. * () = ⊓ ⊓ ∃. ⊓ ∃. ⊓ ∃.⊤ ⊓ ∃.⊤ * () = ⊓ ∃. ⊓ ∃. char = {, , ∃., ∃., ∃.⊤} char = {, ∃., ∃.} dcom, = {, ∃., ∃.⊤} and thus ⊖ = ⊓ ∃. dcom, = {, ∃.} and thus ⊖ = ∃. branches, e.g. those involving the concept ⊤. Moreover it does not change the original tree structure of the minuend. We call this operator the tree subtraction operator (TSO), noted . Informally, , to be read " deprived of ", is intended to be the minimal concept w.r.t. ⊑ that subsumes such that all branches in br that subsume a branch in br have been removed from br . That means we remove from br branches that are also in br, but also branches of br that end by ⊤ when there is in br a branch beginning with the same existential restrictions. If all branches of are removed, then the result is set to be ⊤.
Definition 8 (TSO). Let and be two concepts. The binary operator , called tree subtraction operator (TSO), is defined as follows:
{︃ ⊑{ | ⊑ and br = br} if br ̸= ∅ =
⊤ if br = ∅ with br = br ∖ (︀ br ∪ { = ∃1.∃2...∃.⊤ ∈ br , ≥ 0 | ∃′ = ∃1...∃.∃+1...∃+. ∈ br, ≥ 0 )︀ } and any concept name or ⊤ (with the exception that cannot be ⊤ when = 0). Example 6. Following table 2, with = * () and = * (), we have: br ∖ (︀ br ∪ { = ∃1.∃2...∃.⊤ ∈ br, ≥ 0 |
∃′ = ∃1...∃.∃+1...∃+. ∈ br, ≥ 0})︀ = {, ∃.}.
Thus ⊖ = = ⊓ ∃..
It is not dificult to show that definition 8 ensures keeps the tree structure of , i.e. is a subdescription of . We can illustrate this with non equivalent concepts having the same set of branches, like 1 = ∃. ⊓ ∃. and 2 = ∃.( ⊓ ) which set of branches is br1 = br2 = {∃., ∃.}. Suppose we remove from 1 (resp. 2) a concept that has no commonality with 1 (resp. 2), then the result is 1 and not 2 (resp. 2 and not 1), thus keeping the initial tree structure.
TSO is implemented by algorithm 1. Its principle is to traverse at the same time the tree structures of both and , removing from branches that subsume, w.r.t. the empty TBox, a branch of . Properties of the TSO and algorithm 1 (unicity, termination, soundness, complexity) are given in proposition 1.
Proposition 1 (Properties of TSO and algorithm 1). Let and be two concepts. We have: . always exists and is unique.
. Algorithm 1 terminates and produces a unique result. Algorithm 2 cso( , , ) Require:
∙∙ (a,na)c∈yc(licℰaℒn)d2 definitional ℰ ℒ TBox Ensure: ⊖ (cf. def. 7) 1: return tso( * (), * ()) . = tso(, ) (soundness).
. Computing tso(, ) is in PTIME in the sizes of and .
Sketch of proof. . Existence comes from the definition of br which always corresponds to at least one concept, or ⊤ when it is empty. Unicity trivially comes from the minimality w.r.t. ⊑.
. We show by induction on the sizes of and that tso(, ) always terminates and generates an output which size is strictly less than size() + size().
. Soundness is showed in 3 steps: (i) from the characterization of subumption in ℰ ℒ without any TBox given in [6], we derive a characterization of subsumption in ℰ ℒ in terms of subdescriptions , (ii) then is derived a characterization of TSO in terms of subdescriptions, and (iii) at last a proof by induction of soundness is given, using the characterization obtained at step (ii).
. Tractability is showed by finding a worst case (when and are conjunctions of concepts names, without any existential restriction) and studying the complexity in this case.
Now, we can use TSO to compute CSO: ⊖ is obtained by computing * () * (), cf. algorithm 2. Proposition 2 shows properties associated to CSO, namely existence, uniqueness, and soundness and complexity of algorithm 2 (termination is trivially implied by terminations of the complete expansion and algorithm 1).
Proposition 2. Let be a TBox and (, ) ∈ (ℰℒ)2. There is: . ⊖ exists and is unique (up to ≡ ). . ⊖ = * () * () = cso( , , ) (soundness).
. Computing cso( , , ) is in EXPTIME in the sizes of , and and in PTIME in the sizes of * () and * ().
Sketch of proof. . Existence and unicity of CSO easily come from definition 7.
. Soundness of cso( , , ) is grounded on (i) the characterization of subsumption in terms of subdescriptions already used in proof of proposition 1, and on (ii) a lemma stating br is the set of branches of that do not subsume .
. The result easily comes from complexity of the complete expansion and algorithm 1.
As far as we know, five diference operators have been defined for DLs before CSO, in [ 8, 7, 9, 10, 11]. In the sequel, we respectively note them ⊖ , ⊖ , ⊖ , ⊖ and ⊖ (and ⊖ for CSO). We do not consider the contraction operator defined in [ 12], since it appears to be more a matchmaking operator rather than a diference one.
We first begin by illustrating how these diference operators behave. Let’s take the following two descriptions: 1 ≡ ⊓ ⊓ ∃.( ⊓ ) and 2 ≡ ⊓ ⊓ ∃.. In table 3, we give [8] ⊖ finds the maximal concept w.r.t. ⊑ that added to by conjunction gives a concept equivalent to . [7] ⊖ finds the minimal concepts w.r.t. the subdescription order such that, added to by conjunction, they give a concept equivalent to ⊓ . [9] ⊖ removes the minimal con- S juncts of , w.r.t. a syntactical total order on ℰ ℒ concepts, that are subsumed by conjuncts of (one removed conjunct for one conjunct of ). [10] ⊖ removes conjuncts of that are subsumed by the smallest conjunct of , w.r.t. a syntactical total order. Recursive process inside existential restrictions. [11] Extending the notion of subdescrip- S tion to contain concepts obtained after replacing a concept name by ⊤, ⊖ finds the single subdescription (full meet mode) or the many subdescriptions (maxichoice mode) of that are minimal w.r.t. ⊑ such that ̸⊑ . [This paper] ⊖ removes descrip- S tional commonalities from to , i.e. the characteristic branches of that subsume .
○ 1 C C S
M
T ○ 2 M B B M ○ 3 M − − − ○ 4 N N N Y
Y B
M
Y
Remarks and complexity ∙ Defined for any DL ℒ. ∙ Not defined if ̸⊑ . ∙ Unstudied with a TBox. ∙ Complexity is not given. ∙ is in ℒ, is in ℒℰ . ∙ Unstudied with a TBox. ∙ PTIME in sizes of and given an oracle for subsumption. ∙ Defined for ℰ ℒ. ∙ is acyclic and definitional ∙ EXPTIME in the sizes of , and and PTIME in the sizes of * () and * (). ∙ Defined for ℰ ℒ. ∙ is acyclic and definitional. ∙ ⊖ = if ̸⊑ . ∙ Complexity is not given. ∙ Defined for ℰ ℒ. ∙ is acyclic and definitional. ∙ ⊖ = if ̸⊑ . ∙ Complexity is not given. ∙ Defined for ℰ ℒ. ∙ is acyclic and definitional. ∙ EXPTIME in the sizes of , and and PTIME in the sizes of * () and * (). the result of 1 ⊖ 2 and 2 ⊖ 1, for ⊖ replaced by each operator. Note that we suppose to have an empty TBox , for a sake of simplicity. We can see that each of the six operators give diferent results when considering both 1 ⊖ 2 and 2 ⊖ 1.
Second, we propose to classify these operators in table 4 according to 4 dimensions (precise definitions of operators are recalled in appendix 4. ○ 1 is their type: S for subtraction-based (something is removed from the minuend) or C for completion-based (something is added to the subtrahend). ○ 2 is the definition type of the diference: M for semantical or B for both semantical and syntactical. ○ 3 is the optimization (min or max) criterion type to choose the best result: T for syntactical, M for semantical, or − if non relevant. ○ 4 is the fine grained
diference property i.e. the ability to remove precise subdescriptions of the minuend inside nested existential restrictions without removing unnecessary ones: Y for yes and N for no.
Tables 3 and 4 show that CSO is original w.r.t. existing operators. More precisely: • ⊖ and ⊖ are completion-based operators, they do not achieve fine-grained diference, and ⊖ does not ensure unicity. • ⊖ is a subtraction operation, however it is not a fine-grained diference. Moreover the notion of commonality is based on subsumption and not inverse subsumption. • ⊖ and ⊖ are a fine-grained subtraction operators, as is CSO. However, the diferences with CSO are the following ones: (i) for both operators, there can be subtraction only when the minuend is subsumed by the subtrahend, which is a restriction CSO does not have; (ii) in ⊖ , the notion of common part is not based on inverse subsumption; and (iii) ⊖ is sometimes too much fine-grained, which may lead it not to remove existential restrictions in cases where CSO does, e.g. ∃. ⊖ ∃. = ∃.⊤, while ∃. ⊖ ∃. = ⊤.
We propose a diference operator for ℰ ℒ w.r.t. an acyclic and definitional TBox, named CSO. It is based on the TSO, a operator to achieve a syntactical tree diference between two concepts. We propose a tractable algorithm to compute TSO and thus CSO (in the sizes of the complete expansion of the inputs w.r.t. the TBox), and show that CSO is an original diference operator w.r.t existing ones. Also, an implementation of these operators has been done in the Ruby programming language for integration into the metrology platform. Performance tests are being achieved.
Even if CSO is intended to be used in a matchmaking process, we plan to study how it instanciates properties that are defined for diference operators in the AGM approach of agent belief revision [13], namely preservation, success, inclusion, vacuity, recovery, failure, fullness and relevance. This would provide a more precise insight on CSO w.r.t. existing operators and help decide its potential interest in the AGM framework. Besides, we would like to extend this work to the case where TBoxes can be general and cyclic, for which we do not know any DL diference operator yet.
This work has been fully supported by the European Regional Development Fund5.
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