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Similarity-based Learning Methods for the Semantic Web Claudia dAmato Dipartimento di Informatica Universit` a degli Studi di Bari Campus Universitario, Via Orabona 4, 70125 Bari, Italy Trento, 15 Ottobre 2007 Introduction &

  1. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Kernel Functions Similarity functions able to work with high dimensional feature spaces . Developed jointly with kernel methods : efficient learning algorithms realized for solving classification, regression and clustering problems in high dimensional feature spaces. Kernel machine : encapsulates the learning task kernel function : encapsulates the hypothesis language Introduced in the field pattern recognition Simplest goal: estimate a function using I/O training data able to correctly classify unseen examples ( x , y ) y is determined such that ( x , y ) is in some sense similar to the training examples. A similarity measure k is necessary C. d’Amato Similarity-based Learning Methods for the SW

  2. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals ...Kernel Functions... Possible Problem : overfitting for small sample sizes Intuition: a ”simple” (e.g., linear) function minimizing the error and that explains most of the data is preferable to a complex one (Occams razor). algorithms in feature spaces target a linear function for performing the learning task . Issue : not always possible to find a linear function C. d’Amato Similarity-based Learning Methods for the SW

  3. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals ...Kernel Functions Solution: mapping the initial feature space in a higher dimensional space where the learning problem can be solved by a linear function A kernel function performs such a mapping implicitly Any set that admits a positive definite kernel can be embedded into a linear space [Aronsza 1950] C. d’Amato Similarity-based Learning Methods for the SW

  4. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Similarity Measures based on Feature Matching Model Features can be of different types : binary, nominal, ordinal Tversky’s Similarity Measure : based on the notion of contrast model common features tend to increase the perceived similarity of two concepts feature differences tend to diminish perceived similarity feature commonalities increase perceived similarity more than feature differences can diminish it it is assumed that all features have the same importance Measures in propositional setting are not able to capture expressive relationships among data that typically characterize most complex languages. C. d’Amato Similarity-based Learning Methods for the SW

  5. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Relational Setting: Measures Based on Semantic Relations Also called Path distance measures [Bright,94] Measure the similarity value between single words ( elementary concepts ) concepts (words) are organized in a taxonomy using hypernym/hyponym and synoym links. the measure is a (weighted) count of the links in the path between two terms w.r.t. the most specific ancestor terms with a few links separating them are semantically similar terms with many links between them have less similar meanings link counts are weighted because different relationships have different implications for semantic similarity. C. d’Amato Similarity-based Learning Methods for the SW

  6. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Measures Based on Semantic Relations: WEAKNESS the similarity value is subjective due to the taxonomic ad-hoc representation the introduction of news term can change similarity values the similarity measures cannot be applied directly to the knowledge representation it needs of an intermediate step which is building the term taxonomy structure only ”linguistic” relations among terms are considered; there are not relations whose semantics models domain C. d’Amato Similarity-based Learning Methods for the SW

  7. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Measures Based on Information Content... Measure semantic similarity of concepts in an is-a taxonomy by the use of notion of Information Content (IC) [Resnik,99] Concepts similarity is given by the shared information The shared information is represented by a highly specific super-concept that subsumes both concepts Similarity value is given by the IC of the least common super-concept IC for a concept is determined considering the probability that an instance belongs to the concept C. d’Amato Similarity-based Learning Methods for the SW

  8. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals ...Measures Based on Information Content Use a criterion similar to those used in path distance measures , Differently from path distance measures , the use of probabilities avoids the unreliability of counting edge when changing in the hierarchy occur The considered relation among concepts is only is-a relation more semantically expressive relations cannot be considered C. d’Amato Similarity-based Learning Methods for the SW

  9. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Miscellaneous Approaches Propositionalization and Geometrical Models Path Distance and Feature Matching Approaches Feature Matching, Context-based and Information Content-based Approaches Geometrical models are largely used for their efficiency , but cab be applied only to propositional representations. Idea : focus the propositionalization problem Find a way for transforming a multi-relational representation into a propositional representation. Hence any method can be applied on the new representation rather than on the original one Hipothesis-driven distance [Sebag 1997]: a method for building a distance on first-order logic representation by recurring to the propositionalization is presented C. d’Amato Similarity-based Learning Methods for the SW

  10. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Relational Kernel Functions... Motivated by the necessity of solving real-world problems in an efficient way. Best known relational kernel function: the convolution kernel [Haussler 1999] Basic idea : the semantics of a composite object can be captured by a relation R between the object and its parts. The kernel is composed of kernels defined on different parts. Obtained by composing existing kernels by a certain sum over products , exploiting the closure properties of the class of positive definite functions. D � � k ( x , y ) = k d ( x d , y d ) (1) − → x ∈ R − 1 ( x ) , − → y ∈ R − 1 ( y ) d =1 C. d’Amato Similarity-based Learning Methods for the SW

  11. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals ...Relational Kernel Functions The term ”convolution kernel” refers to a class of kernels that can be formulated as shown in (1). Exploiting convolution kernel, string kernels, tree kernel, graph kernels etc.. have been defined. The advantage of convolution kernels is that they are very general and can be applied in several situations. Drawback: due to their generality, a significant amount of work is required to adapt convolution kernel to a specific problem Choosing R in real-world applications is a non-trivial task C. d’Amato Similarity-based Learning Methods for the SW

  12. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals Similarity Measures for Very Low Expressive DLs... Measures for complex concept descriptions [Borgida et al. 2005] A DL allowing only concept conjunction is considered (propositional DL) Feature Matching Approach : features are represented by atomic concepts An ordinary concept is the conjunction of its features Set intersection and difference corresponds to the LCS and concept difference Semantic Network Model and IC models The most specific ancestor is given by the LCS C. d’Amato Similarity-based Learning Methods for the SW

  13. Introduction & Motivation The Reference Representation Language Similarity Measures: Related Work Similarity Measures in Propositional Setting (Dis-)Similarity measures for DLs Similarity Measures in Relational Setting Applying Measures to Inductive Learning Methods Conclusions and Future Work Proposals ...Similarity Measures for Very Low Expressive DLs OPEN PROBLEMS in considering most expressive DLs: What is a feature in most expressive DLs? i.e. ( ≤ 3 R ) , ( ≤ 4 R ) and ( ≤ 9 R ) are three different features? or ( ≤ 3 R ) , ( ≤ 4 R ) are more similar w.r.t ( ≤ 9 R )? How to assess similarity in presence of role restrictions? i.e. ∀ R . ( ∀ R . A ) and ∀ R . A Key problem in network-based measures : how to assign a useful size for the various concepts in the description? IC-based model : how to compute the value p ( C ) for assessing the IC? C. d’Amato Similarity-based Learning Methods for the SW

  14. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Why New Measures Already defined similalrity/dissimilalrity measures cannot be directly applied to ontological knowledge They define similarity value between atomic concepts They are defined for representation less expressive than ontology representation They cannot exploit all the expressiveness of the ontological representation There are no measure for assessing similarity between individuals Defining new measures that are really semantic is necessary C. d’Amato Similarity-based Learning Methods for the SW

  15. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure between Concepts: Needs Necessity to have a measure really based on Semantics Considering [Tversky’77]: common features tend to increase the perceived similarity of two concepts feature differences tend to diminish perceived similarity feature commonalities increase perceived similarity more than feature differences can diminish it The proposed similarity measure is: C. d’Amato Similarity-based Learning Methods for the SW

  16. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure between Concepts Definition [d’Amato et al. @ CILC 2005]: Let L be the set of all concepts in ALC and let A be an A-Box with canonical interpretation I . The Semantic Similarity Measure s is a function s : L × L �→ [0 , 1] defined as follows: | C I | + | D I | − | I I | · max( | I I | | I I | | C I | , | I I | s ( C , D ) = | D I | ) where I = C ⊓ D and ( · ) I computes the concept extension wrt the interpretation I . C. d’Amato Similarity-based Learning Methods for the SW

  17. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure: Meaning If C ≡ D ( C ⊑ D and D ⊑ C )then s ( C , D ) = 1, i.e. the maximum value of the similarity is assigned. If C ⊓ D = ⊥ then s ( C , D ) = 0, i.e. the minimum similarity value is assigned because concepts are totally different. Otherwise s ( C , D ) ∈ ]0 , 1[. The similarity value is proportional to the overlapping amount of the concept extetions reduced by a quantity representing how the two concepts are near to the overlap. This means considering similarity not as an absolute value but as weighted w.r.t. a degree of non-similarity . C. d’Amato Similarity-based Learning Methods for the SW

  18. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure: Example... Primitive Concepts: N C = { Female , Male , Human } . Primitive Roles: N R = { HasChild , HasParent , HasGrandParent , HasUncle } . T = { Woman ≡ Human ⊓ Female; Man ≡ Human ⊓ Male Parent ≡ Human ⊓ ∃ HasChild.Human Mother ≡ Woman ⊓ Parent ∃ HasChild.Human Father ≡ Man ⊓ Parent Child ≡ Human ⊓ ∃ HasParent.Parent Grandparent ≡ Parent ⊓ ∃ HasChild.( ∃ HasChild.Human) Sibling ≡ Child ⊓ ∃ HasParent.( ∃ HasChild ≥ 2) Niece ≡ Human ⊓ ∃ HasGrandParent.Parent ⊔ ∃ HasUncle.Uncle Cousin ≡ Niece ⊓ ∃ HasUncle.( ∃ HasChild.Human) } . C. d’Amato Similarity-based Learning Methods for the SW

  19. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: Example... A = { Woman(Claudia), Woman(Tiziana), Father(Leonardo), Father(Antonio), Father(AntonioB), Mother(Maria), Mother(Giovanna), Child(Valentina), Sibling(Martina), Sibling(Vito), HasParent(Claudia,Giovanna), HasParent(Leonardo,AntonioB), HasParent(Martina,Maria), HasParent(Giovanna,Antonio), HasParent(Vito,AntonioB), HasParent(Tiziana,Giovanna), HasParent(Tiziana,Leonardo), HasParent(Valentina,Maria), HasParent(Maria,Antonio), HasSibling(Leonardo,Vito), HasSibling(Martina,Valentina), HasSibling(Giovanna,Maria), HasSibling(Vito,Leonardo), HasSibling(Tiziana,Claudia), HasSibling(Valentina,Martina), HasChild(Leonardo,Tiziana), HasChild(Antonio,Giovanna), HasChild(Antonio,Maria), HasChild(Giovanna,Tiziana), HasChild(Giovanna,Claudia), HasChild(AntonioB,Vito), HasChild(AntonioB,Leonardo), HasChild(Maria,Valentina), HasUncle(Martina,Giovanna), HasUncle(Valentina,Giovanna) } C. d’Amato Similarity-based Learning Methods for the SW

  20. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: Example | (Grandparent ⊓ Father) I | s (Grandparent , Father) = · | Granparent I | + | Father I | − | (Grandarent ⊓ Father) I | · max ( | (Grandparent ⊓ Father) I | , | (Grandparent ⊓ Father) I | ) = | Grandparent I | | Father I | 2 + 3 − 2 · max ( 2 2 2 , 2 = 3 ) = 0 . 67 C. d’Amato Similarity-based Learning Methods for the SW

  21. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure between Individuals Let c and d two individuals in a given A-Box. We can consider C ∗ = MSC ∗ ( c ) and D ∗ = MSC ∗ ( d ): s ( c , d ) := s ( C ∗ , D ∗ ) = s (MSC ∗ ( c ) , MSC ∗ ( d )) Analogously: ∀ a : s ( c , D ) := s (MSC ∗ ( c ) , D ) C. d’Amato Similarity-based Learning Methods for the SW

  22. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure: Conclusions... s is a Semantic Similarity measure It uses only semantic inference (Instance Checking) for determining similarity values It does not make use of the syntactic structure of the concept descriptions It does not add complexity besides of the complexity of used inference operator ( IChk that is PSPACE in ALC ) Dissimilarity Measure is defined using the set theory and reasoning operators It uses a numerical approach but it is applied to symbolic representations C. d’Amato Similarity-based Learning Methods for the SW

  23. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: Conclusions Experimental evaluations demonstrate that s works satisfying when it is applied between concepts s applied to individuals is often zero even in case of similar individuals The MSC ∗ is so specific that often covers only the considered individual and not similar individuals The new idea is to measure the similarity (dissimilarity) of the subconcepts that build the MSC ∗ concepts in order to find their similarity (dissimilarity) Intuition : Concepts defined by almost the same sub-concepts will be probably similar. C. d’Amato Similarity-based Learning Methods for the SW

  24. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ALC Normal Form D is in ALC normal form iff D ≡ ⊥ or D ≡ ⊤ or if D = D 1 ⊔ · · · ⊔ D n ( ∀ i = 1 , . . . , n , D i �≡ ⊥ ) with   � � � D i = A ⊓  ∀ R . val R ( D i ) ⊓ ∃ R . E  A ∈ prim( D i ) R ∈ N R E ∈ ex R ( D i ) where: prim( C ) set of all (negated) atoms occurring at C ’s top-level val R ( C ) conjunction C 1 ⊓ · · · ⊓ C n in the value restriction on R , if any (o.w. val R ( C ) = ⊤ ); ex R ( C ) set of concepts in the value restriction of the role R For any R , every sub-description in ex R ( D i ) and val R ( D i ) is in normal form. C. d’Amato Similarity-based Learning Methods for the SW

  25. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Overlap Function Definition [d’Amato et al. @ KCAP 2005 Workshop]: L = ALC / ≡ the set of all concepts in ALC normal form I canonical interpretation of A-Box A f : L × L �→ R + defined ∀ C = � n i =1 C i and D = � m j =1 D j in L ≡  ∞ C ≡ D   0 C ⊓ D ≡ ⊥  f ( C , D ) := f ⊔ ( C , D ) = max i = 1 , . . . , n f ⊓ ( C i , D j ) o.w.    j = 1 , . . . , m f ⊓ ( C i , D j ) := f P (prim( C i ) , prim( D j )) + f ∀ ( C i , D j ) + f ∃ ( C i , D j ) C. d’Amato Similarity-based Learning Methods for the SW

  26. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Overlap Function / II f P (prim( C i ) , prim( D j )) := | (prim( C i )) I ∪ (prim( D j )) I | | ((prim( C i )) I ∪ (prim( D j )) I ) \ ((prim( C i )) I ∩ (prim( D j )) I ) | f P (prim( C i ) , prim( D j )) := ∞ if (prim( C i )) I = (prim( D j )) I � f ∀ ( C i , D j ) := f ⊔ (val R ( C i ) , val R ( D j )) R ∈ N R N � � p =1 ,..., M f ⊔ ( C k i , D p f ∃ ( C i , D j ) := max j ) R ∈ N R k =1 i ∈ ex R ( C i ) and D p where C k j ∈ ex R ( D j ) and wlog. N = | ex R ( C i ) | ≥ | ex R ( D j ) | = M , otherwise exchange N with M C. d’Amato Similarity-based Learning Methods for the SW

  27. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Dissimilarity Measure The dissimilarity measure d is a function d : L × L �→ [0 , 1] such that, for all C = � n i =1 C i and D = � m j =1 D j concept descriptions in ALC normal form:  0 f ( C , D ) = ∞  d ( C , D ) := 1 f ( C , D ) = 0 1 otherwise  f ( C , D ) where f is the function overlapping C. d’Amato Similarity-based Learning Methods for the SW

  28. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Discussion If C ≡ D (namely C ⊑ D e D ⊑ C) (semantic equivalence) d ( C , D ) = 0, rather d assigns the minimun value If C ⊓ D ≡ ⊥ then d ( C , D ) = 1, rather d assigns the maximum value because concepts involved are totally different Otherwise d ( C , D ) ∈ ]0 , 1[ rather dissimilarity is inversely proportional to the quantity of concept overlap , measured considering the entire definitions and their subconcepts. C. d’Amato Similarity-based Learning Methods for the SW

  29. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Dissimilarity Measure: example... C ≡ A 2 ⊓ ∃ R . B 1 ⊓ ∀ T . ( ∀ Q . ( A 4 ⊓ B 5 )) ⊔ A 1 D ≡ A 1 ⊓ B 2 ⊓ ∃ R . A 3 ⊓ ∃ R . B 2 ⊓ ∀ S . B 3 ⊓ ∀ T . ( B 6 ⊓ B 4 ) ⊔ B 2 where A i and B j are all primitive concepts. C 1 := A 2 ⊓ ∃ R . B 1 ⊓ ∀ T . ( ∀ Q . ( A 4 ⊓ B 5 )) D 1 := A 1 ⊓ B 2 ⊓ ∃ R . A 3 ⊓ ∃ R . B 2 ⊓ ∀ S . B 3 ⊓ ∀ T . ( B 6 ⊓ B 4 ) f ( C , D ) := f ⊔ ( C , D ) = max { f ⊓ ( C 1 , D 1 ) , f ⊓ ( C 1 , B 2 ) , f ⊓ ( A 1 , D 1 ) , f ⊓ ( A 1 , B 2 ) } C. d’Amato Similarity-based Learning Methods for the SW

  30. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Dissimilarity Measure: example... For brevity, we consider the computation of f ⊓ ( C 1 , D 1 ). f ⊓ ( C 1 , D 1 ) = f P (prim( C 1 ) , prim( D 1 )) + f ∀ ( C 1 , D 1 ) + f ∃ ( C 1 , D 1 ) Suppose that ( A 2 ) I � = ( A 1 ⊓ B 2 ) I . Then: f P ( C 1 , D 1 ) = f P (prim( C 1 ) , prim( D 1 )) = f P ( A 2 , A 1 ⊓ B 2 ) | I | = | I \ (( A 2 ) I ∩ ( A 1 ⊓ B 2 ) I ) | where I := ( A 2 ) I ∪ ( A 1 ⊓ B 2 ) I C. d’Amato Similarity-based Learning Methods for the SW

  31. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Dissimilarity Measure: example... In order to calculate f ∀ it is important to note that There are two different role at the same level T and S So the summation over the different roles is made by two terms. � f ∀ ( C 1 , D 1 ) = f ⊔ (val R ( C 1 ) , val R ( D 1 )) = R ∈ N R = f ⊔ (val T ( C 1 ) , val T ( D 1 )) + + f ⊔ (val S ( C 1 ) , val S ( D 1 )) = = f ⊔ ( ∀ Q . ( A 4 ⊓ B 5 ) , B 6 ⊓ B 4 ) + f ⊔ ( ⊤ , B 3 ) C. d’Amato Similarity-based Learning Methods for the SW

  32. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Dissimilarity Measure: example In order to calculate f ∃ it is important to note that There is only a single one role R so the first summation of its definition collapses in a single element N and M (numbers of existential concept descriptions w.r.t the same role ( R )) are N = 2 and M = 1 So we have to find the max value of a single element, that can be semplifyed. 2 � f ⊔ (ex R ( C 1 ) , ex R ( D k f ∃ ( C 1 , D 1 ) = 1 )) = k =1 = f ⊔ ( B 1 , A 3 ) + f ⊔ ( B 1 , B 2 ) C. d’Amato Similarity-based Learning Methods for the SW

  33. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Dissimilarity Measure: Conclusions Experimental evaluations demonstrate that d works satisfying both for concepts and individuals However , for complex descriptions (such as MSC ∗ ), deeply nested subconcepts could increase the dissimilarity value New idea: differentiate the weight of the subconcepts wrt their levels in the descriptions for determining the final dissimilarity value Solve the problem: how differences in concept structure might impact concept (dis-)similarity? i.e. considering the series dist ( B , B ⊓ A ) , dist ( B , B ⊓ ∀ R . A ) , dist ( B , B ⊓ ∀ R . ∀ R . A ) this should become smaller since more deeply nested restrictions ought to represent smaller differences .” [Borgida et al. 2005] C. d’Amato Similarity-based Learning Methods for the SW

  34. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs The weighted Dissimilarity Measure Overlap Function Definition [d’Amato et al. @ SWAP 2005]: L = ALC / ≡ the set of all concepts in ALC normal form I canonical interpretation of A-Box A f : L × L �→ R + defined ∀ C = � n i =1 C i and D = � m j =1 D j in L ≡  | ∆ | C ≡ D   C ⊓ D ≡ ⊥  0 f ( C , D ) := f ⊔ ( C , D ) = 1 + λ · max i = 1 , . . . , n f ⊓ ( C i , D j ) o.w.    j = 1 , . . . , m f ⊓ ( C i , D j ) := f P (prim( C i ) , prim( D j )) + f ∀ ( C i , D j ) + f ∃ ( C i , D j ) C. d’Amato Similarity-based Learning Methods for the SW

  35. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Looking toward Information Content: Motivation The use of Information Content is presented as the most effective way for measuring complex concept descriptions [Borgida et al. 2005] The necessity of considering concepts in normal form for computing their (dis-)similarity is argued [Borgida et al. 2005] confirmation of the used approach in the previous measure A dissimilarity measure for complex descriptions grounded on IC has been defined ALC concepts in normal form based on the structure and semantics of the concepts. elicits the underlying semantics , by querying the KB for assessing the IC of concept descriptions w.r.t. the KB extension for considering individuals C. d’Amato Similarity-based Learning Methods for the SW

  36. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Information Content: Defintion A measure of concept (dis)similarity can be derived from the notion of Information Content (IC) IC depends on the probability of an individual to belong to a certain concept IC ( C ) = − log pr ( C ) In order to approximate the probability for a concept C , it is possible to recur to its extension wrt the considered ABox. pr ( C ) = | C I | / | ∆ I | A function for measuring the IC variation between concepts is defined C. d’Amato Similarity-based Learning Methods for the SW

  37. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Function Definition /I [d’Amato et al. @ SAC 2006] L = ALC / ≡ the set of all concepts in ALC normal form I canonical interpretation of A-Box A f : L × L �→ R + defined ∀ C = � n i =1 C i and D = � m j =1 D j in L ≡  0 C ≡ D   ∞ C ⊓ D ≡ ⊥  f ( C , D ) := f ⊔ ( C , D ) = max i = 1 , . . . , n f ⊓ ( C i , D j ) o.w.    j = 1 , . . . , m f ⊓ ( C i , D j ) := f P (prim( C i ) , prim( D j )) + f ∀ ( C i , D j ) + f ∃ ( C i , D j ) C. d’Amato Similarity-based Learning Methods for the SW

  38. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Function Definition / II  ∞ if prim( C i ) ⊓ prim( D j ) ≡ ⊥   f P (prim( C i ) , prim( D j )) := IC (prim( C i ) ⊓ prim( D j ))+1 o.w.   IC ( LCS (prim( C i ) , prim( D j )))+1 � f ∀ ( C i , D j ) := f ⊔ (val R ( C i ) , val R ( D j )) R ∈ N R N i , D p � � p =1 ,..., M f ⊔ ( C k f ∃ ( C i , D j ) := max j ) R ∈ N R k =1 i ∈ ex R ( C i ) and D p where C k j ∈ ex R ( D j ) and wlog. N = | ex R ( C i ) | ≥ | ex R ( D j ) | = M , otherwise exchange N with M C. d’Amato Similarity-based Learning Methods for the SW

  39. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Dissimilarity Measure: Definition The dissimilarity measure d is a function d : L × L �→ [0 , 1] such that, for all C = � n i =1 C i and D = � m j =1 D j concept descriptions in ALC normal form:  0 f ( C , D ) = 0  f ( C , D ) = ∞ d ( C , D ) := 1 1 1 − otherwise  f ( C , D ) where f is the function defined previously C. d’Amato Similarity-based Learning Methods for the SW

  40. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Discussion d ( C , D ) = 0 iff IC=0 iff C ≡ D (semantic equivalence) rather d assigns the minimun value d ( C , D ) = 1 iff IC → ∞ iff C ⊓ D ≡ ⊥ , rather d assigns the maximum value because concepts involved are totally different Otherwise d ( C , D ) ∈ ]0 , 1[ rather d tends to 0 if IC tends to 0; d tends to 1 if IC tends to infinity C. d’Amato Similarity-based Learning Methods for the SW

  41. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ALN Normal Form C is in ALN normal form iff C ≡ ⊥ or C ≡ ⊤ or if � � C = P ⊓ ( ∀ R . C R ⊓ ≥ n . R ⊓ ≤ m . R ) P ∈ prim( C ) R ∈ N R where: C R = val R ( C ), n =min R ( C ) and m = max R ( C ) prim( C ) set of all (negated) atoms occurring at C ’s top-level val R ( C ) conjunction C 1 ⊓ · · · ⊓ C n in the value restriction on R , if any (o.w. val R ( C ) = ⊤ ); min R ( C ) = max { n ∈ N | C ⊑ ( ≥ n . R ) } (always finite number); max R ( C ) = min { n ∈ N | C ⊑ ( ≤ n . R ) } (if unlimited max R ( C ) = ∞ ) C. d’Amato Similarity-based Learning Methods for the SW For any R , every sub-description in val R ( C ) is in normal form.

  42. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Measure Definition / I [Fanizzi et. al @ CILC 2006] L = ALN / ≡ the set of all concepts in ALN normal form I canonical interpretation of A A-Box s : L × L �→ [0 , 1] defined ∀ C , D ∈ L : s ( C , D ) := λ [ s P (prim( C ) , prim( D )) + 1 1 � + s (val R ( C ) , val R ( D )) + | N R | · | N R | R ∈ N R � · s N ((min R ( C ) , max R ( C )) , (min R ( D ) , max R ( D )))] R ∈ N R where λ ∈ ]0 , 1] (let λ = 1 / 3), C. d’Amato Similarity-based Learning Methods for the SW

  43. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Measure Defintion / II P C ∈ prim( C ) P I Q D ∈ prim( D ) Q I | � C ∩ � D | s P (prim( C ) , prim( D )) := P C ∈ prim( C ) P I Q D ∈ prim( D ) Q I | � C ∪ � D | s N (( m C , M C ) , ( m D , M D )) := min( M C , M D ) − max( m C , m D ) + 1 max( M C , M D ) − min( m C , m D ) + 1 s N (( m C , M C ) , ( m D , M D )) := 0 if min( M C , M D ) > max( m C , m D ) C. d’Amato Similarity-based Learning Methods for the SW

  44. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Similarity Measure: example... Let A be the considered ABox Person(Meg) , ¬ Male(Meg) , hasChild(Meg,Bob) , hasChild(Meg,Pat) , Person(Bob) , Male(Bob) , hasChild(Bob,Ann) , Person(Pat) , Male(Pat) , hasChild(Pat,Gwen) , Person(Gwen) , ¬ Male(Gwen) , Person(Ann) , ¬ Male(Ann) , hasChild(Ann,Sue) , marriedTo(Ann,Tom) , Person(Sue) , ¬ Male(Sue) , Person(Tom) , Male(Tom) and let C and D be two descriptions in ALN normal form: C ≡ Person ⊓ ∀ marriedTo . Person ⊓ ≤ 1 . hasChild D ≡ Male ⊓ ∀ marriedTo . (Person ⊓ ¬ Male) ⊓ ≤ 2 . hasChild C. d’Amato Similarity-based Learning Methods for the SW

  45. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: example... In order to compute s ( C , D ) let us consider: Let be λ := 1 3 N R = { hasChild, marriedTo } → | N R | = 2  1  s P (prim( C ) , prim( D )) + 1 � s ( C , D ) := s (val R ( C ) , val R ( D )) + 3 2 R ∈ N R  1 � + s N ((min R ( C ) , max R ( C )) , (min R ( D ) , max R ( D )))  2 R ∈ N R C. d’Amato Similarity-based Learning Methods for the SW

  46. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: example... In order to compute s P let us note that: prim(C) = Person prim(D) = Male s P ( { Person } , { Male } ) = = |{ Meg , Bob , Pat , Gwen , Ann , Sue , Tom }∩{ Bob , Pat , Tom }| |{ Meg , Bob , Pat , Gwen , Ann , Sue , Tom }∪{ Bob , Pat , Tom }| = |{ Bob , Pat , Tom }| = |{ Meg , Bob , Pat , Gwen , Ann , Sue , Tom }| = 3 / 7 C. d’Amato Similarity-based Learning Methods for the SW

  47. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: example... To compute s for value restrictions, it is important to note that N R = { hasChild , marriedTo } val hasChild ( C ) = ⊤ val marriedTo ( C ) = Person and val marriedTo ( D ) = Person ⊓ ¬ Male and val hasChild ( D ) = ⊤ s (Person , Person ⊓ ¬ Male) + s ( ⊤ , ⊤ ) = = 1 3 · ( s P (Person , Person ⊓ ¬ Male) + 1 2 · (1 + 1) + 1 2 · (1 + 1))+ + 1 3 · (1 + 1 + 1) = 1 3 · ( 4 7 + 1 + 1) + 1 = 13 7 C. d’Amato Similarity-based Learning Methods for the SW

  48. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Similarity Measure: example To compute s for number restrictions it is important to note that N R = { hasChild , marriedTo } min( M C , M D ) > max( m C , m D ) min marriedTo ( C ) = 0; max marriedTo ( C ) = | ∆ | + 1 = 7 + 1 = 8 min hasChild ( C ) = 0; max hasChild ( C ) = 1 min marriedTo ( D ) = 0; max marriedTo ( D ) = | ∆ | + 1 = 7 + 1 = 8 min hasChild ( D ) = 0; max hasChild ( D ) = 2 s N ( ( m hasChild ( C ) , M hasChild ( C )) , ( m hasChild ( D ) , M hasChild ( D ))) + + s N (( m marriedTo ( C ) , M marriedTo ( C )) , ( m marriedTo ( D ) , M marriedTo ( D ))) = = min ( M hasChild ( C ) , M hasChild ( D )) − max ( m hasChild ( C ) , m hasChild ( D ))+1 max ( M hasChild ( C ) , M hasChild ( D ) − min ( m hasChild ( C ) , m hasChild ( D ))+1) + 1 = = min (1 , 2) − max (0 , 0)+1 max (1 , 2) − min (0 , 0)+1) + 1 = 2 3 + 1 = 5 3 C. d’Amato Similarity-based Learning Methods for the SW

  49. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Relational Kernel Function: Motivation Kernel functions jointly with a kernel method. Advangate: 1) efficency; 2) the learning algorithm and the kernel are almost completely independent. An efficient algorithm for attribute-value instance spaces can be converted into one suitable for structured spaces by merely replacing the kernel function . A kernel function for ALC normal form concept descriptions has been defined. Based both on the syntactic structure (exploiting the convolution kernel [Haussler 1999] and on the semantics , derived from the ABox. C. d’Amato Similarity-based Learning Methods for the SW

  50. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Kernel Defintion/I [Fanizzi et al. @ ISMIS 2006] Given the space X of ALC normal form concept descriptions, D 1 = � n i and D 2 = � m i =1 C 1 j =1 C 2 j in X , and an interpretation I , the ALC kernel based on I is the function k I : X × X �→ R inductively defined as follows. disjunctive descriptions : k I ( D 1 , D 2 ) = λ � n � m j =1 k I ( C 1 i , C 2 j ) with λ ∈ ]0 , 1] i =1 conjunctive descriptions : k I ( C 1 , C 2 ) � � k I (val R ( C 1 ) , val R ( C 2 )) · k I ( P 1 , P 2 ) · = P 1 ∈ prim( C 1 ) R ∈ N R P 2 ∈ prim( C 2 ) � � k I ( C 1 i , C 2 · j ) R ∈ N R C 1 i ∈ ex R ( C 1 ) C 2 j ∈ ex R ( C 2 ) C. d’Amato Similarity-based Learning Methods for the SW

  51. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Kernel Definition/II primitive concepts : k I ( P 1 , P 2 ) = k set ( P I 1 , P I = | P I 1 ∩ P I 2 | 2 ) | ∆ I | | ∆ I | where k set is the kernel for set structures [Gaertner 2004] . This case includes also the negation of primitive concepts using set difference: ( ¬ P ) I = ∆ I \ P I C. d’Amato Similarity-based Learning Methods for the SW

  52. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Computing the kernel fucntion: Example... Considered concept descriptions: C ≡ ( P 1 ⊓ P 2 ) ⊔ ( ∃ R . P 3 ⊓ ∀ R . ( P 1 ⊓ ¬ P 2 )) D ≡ P 3 ⊔ ( ∃ R . ∀ R . P 2 ⊓ ∃ R . ¬ P 1 ) Supposing: 3 = { a , b , d } , ∆ I = { a , b , c , d , e } P I 1 = { a , b , c } , P I 2 = { b , c } , P I Disjunctive level: 2 2 � � k I ( C , D ) = λ k I ( C i , D j ) = i =1 j =1 = λ · ( k I ( C 1 , D 1 ) + k I ( C 1 , D 2 ) + k I ( C 2 , D 1 ) + k I ( C 2 , D 2 )) C 1 ≡ P 1 ⊓ P 2 , C 2 ≡ ∃ R . P 3 ⊓ ∀ R . ( P 1 ⊓ ¬ P 2 ), where D 1 ≡ P 3 , D 2 ≡ ∃ R . ∀ R . P 2 ⊓ ∃ R . ¬ P 1 . C. d’Amato Similarity-based Learning Methods for the SW

  53. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Computing the kernel fucntion: Example... The kernel for the conjunctive level has to be compute for every couple C i , D j � � k I ( P C 1 , P D k I ( C 1 , D 1 ) = 1 ) · k I ( ⊤ , ⊤ ) · k I ( ⊤ , ⊤ ) = P C P D 1 ∈ prim( C 1 ) 1 ∈ prim( D 1 ) k I ( P 1 , P 3 ) · k I ( P 2 , P 3 ) · 1 · 1 = = |{ a , b , c } ∩ { a , b , d }| · |{ b , c } ∩ { a , b , d }| = 2 5 · 1 5 = 2 = a , b , c , d , e a , b , c , d , e 25 No contribution comes from value and existential restrictions: the factors amount to 1 since val R ( C 1 ) = val R ( D 1 )) = ⊤ and ex R ( C 1 ) = ex R ( D 1 ) = ∅ which make those equivalent to ⊤ too. C. d’Amato Similarity-based Learning Methods for the SW

  54. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Computing the kernel fucntion: Example... The conjunctive kernel for C 1 and D 2 has to be computed. Note that there are no universal restrictions and N R = { R } ⇒ | N R | = 1 this means that all products on varying R ∈ N R can be simplified. Empty prim is equivalent to ⊤ . � k I ( C 1 , D 2 ) = [ k I ( P 1 , ⊤ ) · k I ( P 2 , ⊤ )] · k I ( ⊤ , ⊤ ) · k I ( E C , E D ) E C ∈ ex R ( C 1 ) E D ∈ ex R ( D 2 ) = (3 · 2) · 1 · [ k I ( ⊤ , ∀ R . P 2 ) + k I ( ⊤ , ¬ P 1 )] = � k I ( C ′ , D ′ ) + 2] = = 6 · [ λ C ′ ∈{⊤} D ′ ∈{∀ R . P 2 } = 6 · [ λ · (1 · k I ( ⊤ , P 2 ) · 1) + 2] = 6 · [ λ · ( λ · 1 · 2 / 5 · 1) + 2] = 12( λ 2 / 5 + 1) = C. d’Amato Similarity-based Learning Methods for the SW

  55. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Computing the kernel fucntion: Example... � k I ( ⊤ , P 3 ) · k I (val R ( C 2 ) , ⊤ ) · k I ( C 2 , D 1 ) = k I ( E C , E D ) = E C ∈ ex R ( C 2 ) E D ∈ ex R ( D 1 ) 3 / 5 · k I ( P 1 ⊓ ¬ P 2 , ⊤ ) · k I ( P 3 , ⊤ ) = = = 3 / 5 · [ λ ( k I ( P 1 , ⊤ ) · k I ( ¬ P 2 , ⊤ ))] · 3 / 5 = 3 / 5 · [ λ (3 / 5 · 3 / 5)] · 3 / 5 = 81 λ/ 625 = Note that the absence of the prim set is equivalent to ⊤ and, since one of the sub-concepts has no existential restriction the product gives no contribution. C. d’Amato Similarity-based Learning Methods for the SW

  56. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs ...Computing the kernel fucntion: Example Finally, the kernel function on the last couple of disjuncts � k I ( C ′′ , D ′′ ) = k I ( ⊤ , ⊤ ) · k I ( P 1 ⊓ ¬ P 2 , ⊤ ) · k I ( C 2 , D 2 ) = C ′′ ∈{ P 3 } D ′′ ∈{∀ R . P 2 , ¬ P 1 } 1 · 9 λ/ 25 · [( k I ( P 3 , ∀ R . P 2 ) + k I ( P 3 , ¬ P 1 )] = = = 9 λ/ 25 · [ λ · k I ( P 3 , ⊤ ) · k I ( ⊤ , P 2 ) · k I ( ⊤ , ⊤ ) + 1 / 5] = 9 λ/ 25 · [ λ · 3 / 5 · 2 λ/ 5 · 1 + 1 / 5] = = 9 λ/ 25 · [6 λ 2 / 25 + 1 / 5] = By collecting the four intermediate results, the value for the computed kernel function on C and D can be computed: k I ( C , D ) = 2 / 25 + 12( λ 2 / 5 + 1) + 81 λ/ 625 + 9 λ/ 25 · [6 λ 2 / 25 + 1 / 5] C. d’Amato Similarity-based Learning Methods for the SW

  57. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Kernel function: Discussion The kernel function can be extended to the case of individuals/concept The kernel is valid The function is symmetric The function is closed under multiplication and sum of valid kernel (kernel set). Being the kernel valid, and induced distance measure (metric) can be obtained [Haussler 1999] � k I ( C , C ) − 2 k I ( C , D ) + k I ( D , D ) d I ( C , D ) = C. d’Amato Similarity-based Learning Methods for the SW

  58. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Semi-Distance Measure: Motivations Most of the presented measures are grounded on concept structures ⇒ hardly scalable w.r.t. most expressive DLs IDEA : on a semantic level, similar individuals should behave similarly w.r.t. the same concepts Following HDD [Sebag 1997] : individuals can be compared on the grounds of their behavior w.r.t. a given set of hypotheses F = { F 1 , F 2 , . . . , F m } , that is a collection of (primitive or defined) concept descriptions F stands as a group of discriminating features expressed in the considered language As such, the new measure totally depends on semantic aspects of the individuals in the KB C. d’Amato Similarity-based Learning Methods for the SW

  59. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Semantic Semi-Dinstance Measure: Definition [Fanizzi et al. @ DL 2007] Let K = �T , A� be a KB and let Ind( A ) be the set of the individuals in A . Given sets of concept descriptions F = { F 1 , F 2 , . . . , F m } in T , a family of semi-distance functions d F p : Ind( A ) × Ind( A ) �→ R is defined as follows: � m � 1 / p p ( a , b ) := 1 d F � | π i ( a ) − π i ( b ) | p ∀ a , b ∈ Ind( A ) m i =1 where p > 0 and ∀ i ∈ { 1 , . . . , m } the projection function π i is defined by:  1 F i ( a ) ∈ A ( K | = F i ( a ))  ∀ a ∈ Ind( A ) ¬ F i ( a ) ∈ A ( K | = ¬ F i ( a )) π i ( a ) = 0 1 otherwise  2 C. d’Amato Similarity-based Learning Methods for the SW

  60. A Semantic Similarity Measure for ALC Introduction & Motivation A Dissimilarity Measure for ALC The Reference Representation Language Weighted Dissimilarity Measure for ALC Similarity Measures: Related Work A Dissimilarity Measure for ALC using Information Content (Dis-)Similarity measures for DLs A Similarity Measure for ALN Applying Measures to Inductive Learning Methods A Relational Kernel Function for ALC Conclusions and Future Work Proposals A Semantic Semi-Distance Measure for Any DLs Semi-Distance Measure: Discussion More similar the considered individuals are , more similar the project function values are ⇒ d F p ≃ 0 More different the considered individuals are , more different the projection values are ⇒ the value of d F p will increase The measure complexity mainly depends from the complexity of the Instance Checking operator for the chosen DL Compl ( d F p ) = | F | · 2 · Compl (IChk) Optimal discriminating feature set could be learned C. d’Amato Similarity-based Learning Methods for the SW

  61. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Goals for using Inductive Learning Methods in the SW Instance-base classifier for Semi-automatize the A-Box population task Induce new knowledge not logically derivable Improve concept retrieval and query answearing inference service Realized algorithms Relational K-NN Relational kernel embedded in a SVM Unsupervised learning methods for Improve service discovery task Exploiting (dis-)similarity measures for improving the ranking of the retrieved services C. d’Amato Similarity-based Learning Methods for the SW

  62. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Classical K-NN algorithm... C. d’Amato Similarity-based Learning Methods for the SW

  63. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals ...Classical K-NN algorithm... C. d’Amato Similarity-based Learning Methods for the SW

  64. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals ...Classical K-NN algorithm Generally applied to feature vector representation In classification phase it is assumed that each training and test example belong to a single class, so classes are considered to be disjoint An implicit Closed World Assumption is made C. d’Amato Similarity-based Learning Methods for the SW

  65. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Difficulties in applying K-NN to Ontological Knowledge To apply K-NN for classifying individual asserted in an ontological knowledge base 1 It has to find a way for applying K-NN to a most complex and expressive knowledge representation 2 It is not possible to assume disjointness of classes. Individuals in an ontology can belong to more than one class (concept). 3 The classification process has to cope with the Open World Assumption charactering Semantic Web area C. d’Amato Similarity-based Learning Methods for the SW

  66. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Choices for applying K-NN to Ontological Knowledge 1 To have similarity and dissimilarity measures applicable to ontological knowledge allows applying K-NN to this kind of knowledge representation 2 A new classification procedure is adopted, decomposing the multi-class classification problem into smaller binary classification problems (one per target concept). For each individual to classify w.r.t each class (concept), classification returns { -1,+1 } 3 A third value 0 representing unknown information is added in the classification results { -1,0,+1 } 4 Hence a majority voting criterion is applied C. d’Amato Similarity-based Learning Methods for the SW

  67. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Realized K-NN Algorithm... [d’Amato et al. @ URSW Workshop at ISWC 2006] Main Idea: similar individuals, by analogy, should likely belong to similar concepts for every ontology, all individuals are classified to be instances of one or more concepts of the considered ontology For each individual in the ontology MSC is computed MSC list represents the set of training examples C. d’Amato Similarity-based Learning Methods for the SW

  68. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals ...Realized K-NN Algorithm Each example is classified applying the k-NN method for DLs, adopting the leave-one-out cross validation procedure. k ˆ � h j ( x q ) := argmax ω i · δ ( v , h j ( x i )) ∀ j ∈ { 1 , . . . , s } (2) v ∈ V i =1 where  +1 C j ( x ) ∈ A  h j ( x ) = 0 C j ( x ) �∈ A − 1 ¬ C j ( x ) ∈ A  C. d’Amato Similarity-based Learning Methods for the SW

  69. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experimentation Setting ontology DL SOF ( D ) FSM S.-W.-M. ALCOF ( D ) ALCN Family ALCIF Financial ontology #concepts #obj. prop #data prop #individuals FSM 20 10 7 37 19 9 1 115 S.-W.-M. Family 14 5 0 39 60 17 0 652 Financial C. d’Amato Similarity-based Learning Methods for the SW

  70. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Measures for Evaluating Experiments Performance evaluated by comparing the procedure responses to those returned by a standard reasoner (Pellet) Predictive Accuracy: measures the number of correctly classified individuals w.r.t. overall number of individuals. Omission Error Rate: measures the amount of unlabelled individuals C ( x q ) = 0 with respect to a certain concept C j while they are instances of C j in the KB. Commission Error Rate: measures the amount of individuals labelled as instances of the negation of the target concept C j , while they belong to C j or vice-versa. Induction Rate: measures the amount of individuals that were found to belong to a concept or its negation, while this information is not derivable from the KB. C. d’Amato Similarity-based Learning Methods for the SW

  71. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experimentation Evaluation Results (average ± std-dev.) using the measure based on overlap. Match Commission Omission Induction Rate Rate Rate Rate family .654 ± .174 .000 ± .000 .231 ± .173 .115 ± .107 fsm .974 ± .044 .026 ± .044 .000 ± .000 .000 ± .000 S.-W.-M. .820 ± .241 .000 ± .000 .064 ± .111 .116 ± .246 Financial .807 ± .091 .024 ± .076 .000 ± .001 .169 ± .076 Results (average ± std-dev.) using the measure based in IC Match Commission Omission Induction family .608 ± .230 .000 ± .000 .330 ± .216 .062 ± .217 fsm .899 ± .178 .096 ± .179 .000 ± .000 .005 ± .024 S.-W.-M. .820 ± .241 .000 ± .000 .064 ± .111 .116 ± .246 Financial .807 ± .091 .024 ± .076 .000 ± .001 .169 ± .046 C. d’Amato Similarity-based Learning Methods for the SW

  72. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experimentation: Discussion... For every ontology, the commission error is almost null ; the classifier almost never mades critical mistakes FSM Ontology : the classifier always assigns individuals to the correct concepts; it is never capable to induce new knowledge Because individuals are all instances of a single concept and are involved in a few roles, so MSCs are very similar and so the amount of information they convey is very low C. d’Amato Similarity-based Learning Methods for the SW

  73. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals ...Experimentation: Discussion... SURFACE-WATER-MODEL and FINANCIAL Ontology The classifier always assigns individuals to the correct concepts Because most of individuals are instances of a single concept Induction rate is not null so new knowledge is induced . This is mainly due to some concepts that are declared to be mutually disjoint some individuals are involved in relations C. d’Amato Similarity-based Learning Methods for the SW

  74. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals ...Experimentation: Discussion FAMILY Ontology Predictive Accuracy is not so high and Omission Error not null Because instances are more irregularly spread over the classes, so computed MSCs are often very different provoking sometimes incorrect classifications (weakness on K-NN algorithm) No Commission Error (but only omission error) The Classifier is able of induce new knowledge that is not derivable C. d’Amato Similarity-based Learning Methods for the SW

  75. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Comparing the Measures The measure based on IC poorly classifies concepts that have less information in the ontology The measure based on IC is less able , w.r.t. the measure based on overlap, to classify concepts correctly, when they have few information (instance and object properties involved); Comparable behavior when enough information is available Inducted knowledge can be used for semi-automatize ABox population improving concept retrieval C. d’Amato Similarity-based Learning Methods for the SW

  76. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experiments: Querying the KB exploiting relational K-NN Setting 15 queries randomly generated by conjunctions/disjunctions of primitive or defined concepts of each ontology. Classification of all individuals in each ontology w.r.t the query concept Performance evaluated by comparing the procedure responses to those returned by a standard reasoner (Pellet) employed as a baseline. The Semi-distance measure has been used All concepts in ontology have been employed as feature set F C. d’Amato Similarity-based Learning Methods for the SW

  77. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Ontologies employed in the experiments ontology DL SOF ( D ) FSM ALCOF ( D ) S.-W.-M. ALCIF ( D ) Science SHIF ( D ) NTN ALCIF Financial ontology #concepts #obj. prop #data prop #individuals 20 10 7 37 FSM 19 9 1 115 S.-W.-M. 74 70 40 331 Science 47 27 8 676 NTN 60 17 0 652 Financial C. d’Amato Similarity-based Learning Methods for the SW

  78. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experimentation: Resuls Results (average ± std-dev.) using the semi-distance semantic measure match commission omission induction rate rate rate rate 97.7 ± 3.00 2.30 ± 3.00 0.00 ± 0.00 0.00 ± 0.00 FSM 99.9 ± 0.20 0.00 ± 0.00 0.10 ± 0.20 0.00 ± 0.00 S.-W.-M. 99.8 ± 0.50 0.00 ± 0.00 0.20 ± 0.10 0.00 ± 0.00 Science 90.4 ± 24.6 9.40 ± 24.5 0.10 ± 0.10 0.10 ± 0.20 Financial 99.9 ± 0.10 0.00 ± 7.60 0.10 ± 0.00 0.00 ± 0.10 NTN C. d’Amato Similarity-based Learning Methods for the SW

  79. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experimentation: Discussion Very low commission error: almost never the classifier makes critical mistakes Very high match rate 95%(more than the previous measures 80%) ⇒ Highly comparable with the reasoner Very low induction rate ⇒ Less able (w.r.t. previous measures) to induce new knowledge Lower match rate for Financial ontology as data are not enough sparse The usage of all concepts for the set F made the measure accurate , which is the reason why the procedure resulted conservative as regards inducing new assertions. C. d’Amato Similarity-based Learning Methods for the SW

  80. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Testing the Effect of the Variation of F on the Measure Espected result : with an increasing number of considered hypotheses for F , the accuracy of the measure would increase accordingly. Considered ontology: Financial as is is the most populated Experiment repeated with an increasing percentage of concepts randomly selected for F from the ontology. Results confirm the hypothesis Similar results for the other ontologies C. d’Amato Similarity-based Learning Methods for the SW

  81. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experimentation: Results % of concepts match commission omission Induction 20% 79.1 20.7 0.00 0.20 40% 96.1 03.9 0.00 0.00 50% 97.2 02.8 0.00 0.00 70% 97.4 02.6 0.00 0.00 100% 98.0 02.0 0.00 0.00 C. d’Amato Similarity-based Learning Methods for the SW

  82. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals SVM and Relational Kernel Function for the SW A SMV is a classifier that, by means of kernel function, implicitly maps the training data into a higher dimensional feature space where they can be classified using a linear classifier A SVM from the LIBSVM library has been considered Learning Problem : Given an ontology, classify all its individuals w.r.t. all concepts in the ontology [Fanizzi et al. @ KES 2007] Problems to solve: 1) Implicit CWA; 2) Assumption of class disjointness Solutions: Decomposing the classification problem is a set of ternary classification problems { +1 , 0 , − 1 } , for each concept of the ontology C. d’Amato Similarity-based Learning Methods for the SW

  83. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Ontologies employed in the experiments ontology DL People ALCHIN ( D ) University ALC family ALCF FSM SOF ( D ) ALCOF ( D ) S.-W.-M. ALCIF ( D ) Science SHIF ( D ) NTN ALCF ( D ) Newspaper ALCIO ( D ) Wines ontology #concepts #obj. prop #data prop #individuals 60 14 1 21 People University 13 4 0 19 14 5 0 39 family FSM 20 10 7 37 19 9 1 115 S.-W.-M. Science 74 70 40 331 47 27 8 676 NTN Newspaper 29 28 25 72 112 9 10 188 Wines C. d’Amato Similarity-based Learning Methods for the SW

  84. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experiment: Results match rate ind. rate omis.err.rate comm.err.rate Ontoly avg. 0.866 0.054 0.08 0.00 People range 0.66 - 0.99 0.00 - 0.32 0.00 - 0.22 0.00 - 0.03 avg. 0.789 0.114 0.018 0.079 University range 0.63 - 1.00 0.00 - 0.21 0.00 - 0.21 0.00 - 0.26 avg. 0.917 0.007 0.00 0.076 fsm range 0.70 - 1.00 0.00 - 0.10 0.00 - 0.00 0.00 - 0.30 avg. 0.619 0.032 0.349 0.00 Family range 0.39 - 0.89 0.00 - 0.41 0.00 - 0.62 0.00 - 0.00 avg. 0.903 0.00 0.097 0.00 NewsPaper range 0.74 - 0.99 0.00 - 0.00 0.02 - 0.26 0.00 - 0.00 avg. 0.956 0.004 0.04 0.00 Wines range 0.65 - 1.00 0.00 - 0.27 0.01 - 0.34 0.00 - 0.00 avg. 0.942 0.007 0.051 0.00 Science range 0.80 - 1.00 0.00 - 0.04 0.00 - 0.20 0.00 - 0.00 avg. 0.871 0.067 0.062 0.00 S.-W.-M. range 0.57 - 0.98 0.00 - 0.42 0.00 - 0.40 0.00 - 0.00 avg. 0.925 0.026 0.048 0.001 N.T.N. range 0.66 - 0.99 0.00 - 0.32 0.00 - 0.22 0.00 - 0.03 C. d’Amato Similarity-based Learning Methods for the SW

  85. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Experiments: Discussion High matching rate Induction Rate not null ⇒ new knowledge is induced For every ontology, the commission error is quite low ⇒ the classifier does not make critical mistakes Not null for University and FSM ontologies ⇒ They have the lowest number of individuals There is not enough information for separating the feature space producing a correct classification In general the match rate increases with the increase of the number of individuals in the ontology Consequently the commission error rate decreases Similar results by using the classifier for querying the KB C. d’Amato Similarity-based Learning Methods for the SW

  86. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Why the Attention to Modeling Service Descriptions WS Technology has allowed uniform access via Web standards to software components residing on various platforms and written in different programming languages WS major limitation: their retrieval and composition still require manual effort Solution : augment WS with a semantic description of their functionality ⇒ SWS Choice: DLs as representation language, because : DLs are endowed by a formal semantics ⇒ guarantee expressive service descriptions and precise semantics definition DLs are the theoretical foundation of OWL ⇒ ensure compatibility with existing ontology standards Service discovery can be performed exploiting standard and non-standard DL inferences C. d’Amato Similarity-based Learning Methods for the SW

  87. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals DLs-based Service Descriptions [Grimm et al. 2004] A service description is expressed by a set of DL-axioms D = { S , φ 1 , φ 2 , ..., φ n } , where the axioms φ i impose restrictions on an atomic concept S, which represents the service to be performed D r = { S r ≡ Company ⊓ ∃ payment.EPayment ⊓ ∃ to. { bari } ⊓ ⊓ ∃ from. { cologne,hahn } ⊓ ≤ 1 hasAlliance ⊓ ⊓ ∀ hasFidelityCard. { milesAndMore } ; { cologne,hahn } ⊑ ∃ from − . S r } KB = { cologne:Germany, hahn:Germany, bari:Italy, milesAndMore:Card } C. d’Amato Similarity-based Learning Methods for the SW

  88. Introduction & Motivation K-Nearest Neighbor Algorithm for the SW The Reference Representation Language SVM and Relational Kernel Function for the SW Similarity Measures: Related Work DLs-based Service Descriptions by the use of Constraint Hardness (Dis-)Similarity measures for DLs Unsupervised Learning for Improving Service Discovery Applying Measures to Inductive Learning Methods Ranking Service Descriptions Conclusions and Future Work Proposals Introducing Constraint Hardness [d’Amato et al. @ Sem4WS Workshop at BPM 2006] In real scenarios a service request is characterized by some needs that must be satisfied and others that may be satisfied HC represent necessary and sufficient conditions for selecting requested service instances SC represent only necessary conditions. Definition Let D HC = { S HC , σ HC , ..., σ HC } be the set of HC for a requested r r 1 n service description D r and let D SC = { S SC , σ SC 1 , ..., σ SC m } be the r r set of SC for D r . The complete description of D r is given by D r = { S r ≡ S HC ⊔ S SC , σ HC , ..., σ HC , σ SC 1 , ..., σ SC m } . r r n 1 C. d’Amato Similarity-based Learning Methods for the SW

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