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Dealing with Mathematical Relations in Web-Ontologies

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Dealing with Mathematical Relations in Web-Ontologies Dealing with Mathematical Relations in Web-Ontologies Muthukkaruppan Annamalai Leon Sterling Department of Computer Science & Software Department of Computer Science & Software Engineering Engineering The University of Melbourne The University of Melbourne Victoria 3010, Australia Victoria 3010, Australia [email protected] [email protected] ABSTRACT the Victorian Partnership for Advanced Computing [21] to The growing use of agent systems and the widespread pen- investigate whether ontologies would be useful for EHEP col- etration of the Internet has opened up new possibilities for laboration, in particular in the Belle [4] collaboration. The scientific collaboration. We have been investigating the role research is founded on the idea that suitable web-ontologies for agent systems to aid with collaboration among Exper- be developed and reused to facilitate this scientific commu- imental High-Energy Physics (EHEP) physicists. A neces- nity to produce and share information effectively on the se- sary component is an agreed ontology, which must include mantic web. While the final verdict has not been reached, complex mathematical relations involving such quantities as it is clear that our project extends existing capabilities of the energy and momentum of elementary physics particles. web-ontology specification languages and tools. We claim that the current web-ontology specification lan- guages are not sufficiently expressive to be useful for explicit One concerning issue is as to how to define ‘function-concepts’ representation of mathematical expressions. We adapt some in the EHEP ontology, that is concepts equated with math- previous work on representing mathematical expressions to ematical expressions involving such quantities as the energy produce a set of mathematical representational primitives and momentum of elementary physics particles. A function- and supporting definitions that will allow knowledge shar- concept is a manifestation of an n-ary mathematical relation ing in agent systems. The paper sketches out a scheme or function binding a set of quantifiable terms defined in the for dealing with mathematical relations in scientific domain ontology. The current web-ontology specification languages, web-ontologies, illustrated with examples arising from our such as DAML [17] and OIL [10] are founded on Description interactions with the EHEP physicists. Logic [2] and can only express unary and binary relations. They offer no representational features for expressing func- tions. We ask, “How do we facilitate the semantic recogni- 1. INTRODUCTION tion of function-concepts in web-ontology?” The answer to The growing use of agent systems and the widespread pen- this question led us to cognise the extensions required for ad- etration of the Internet has opened up new possibilities and ditional expressivity in web-ontology. This paper shows the created new challenges for scientific collaboration. On one approach we have taken in order to ensure that we could de- hand, it is possible to make large amounts of scientific anal- scribe mathematical relation underlying a function-concept yses of Experimental High-Energy Physics (EHEP) experi- in the ontology. ments available to scientists around the world [3, 4, 7]. On the other hand different scientific groups, even within a sin- This paper is organised as follows. In Section 2, we provide gle collaboration, utilise different calculation methods, and a glimpse of the function-concepts in the EHEP domain. In it is sometimes difficult to know how to interpret particu- Section 3, we describe the principles of dealing with math- lar analyses. It is assumed that practitioners in this domain ematical relations in web-ontology and our approach to the possess the necessary background knowledge to interpret the problem is elaborated in Section 4. Finally, Section 5 con- intended meaning of the appropriated jargon in the domain cludes the contribution of this paper. of discourse. Unfortunately, application developers, new- comers to this field, and software agents lacking in relevant expertise are not capable of making a similar kind of inter- 2. EVENT-VARIABLES IN THE EHEP ON- pretation. Knowledge models, or ontologies built to express TOLOGY specific facts about a domain can serve as the basis for un- Mathematics plays a significant role in the EHEP experi- derstanding the discourse in that domain [5]. mental analysis – from the time sensor data is captured up to statistical analysis and systematic error calculation. In The notion of ontology as specification of a partial account this paper however, we limit our scope of discussion to the of shared conceptualisation [13, 16] is adopted in this pa- treatment of mathematics in the vital event selection phase. per, that is an ontology defines a set of representational vocabulary for specific classes of objects and the describable The superfluous event data captured by detectors is system- relationships that exist among them in the modelled world atically filtered to suppress much of the background events, of a shared domain. while preserving the vital signal events. Parametric restric- tions on the event selection variables or ‘cuts’ are utilised to In 2002, we were involved in a project [1, 8] supported by sift the signal events from background events. Loose cuts (or skimming), followed by more decisive topological, kinematic the EngMath ontology [14], an extensive past attempt to and geometric cuts is aimed to produce a set of desired event capture the semantics associated with mathematical expres- data, fit for justification of the empirical findings. sions in engineering models. EngMath is a declarative first order KIF [11] axiomatisation, which is supported by sets A category of event selection variables is defined in the of theories for describing physical quantities, mathematical EHEP ontology. These event-variables are identified based object such as scalar, vector, tensor and, functions and op- on their use and need to specify the event selection or back- erations associated with them. ground suppression cuts. A set of competency questions [15] drawn-up while the ontology being built guides the concep- To a lesser degree, general-purpose ontologies like CYC [9] tualisation of these required event-variables. A typical com- and SUMO [20] also attempt to declaratively capture the petency question is: “What are the kinematic selection cri- semantics of ‘evaluatable’ function. However, such function teria applied in an analysis?” The ensuing answer would is categorised as unary, binary, ternary, quaternary and con- list the selection criteria enforced, such as: “Beam con- tinuous types; thereby placing a limit on the number of its 2 strained mass Mbc > 5.2 GeV/c , Track transverse momen- argument. Perhaps, this restrain is imposed in order to refer tum PT > 100 MeV/c, Energy difference ∆E < 0.2 GeV , to a function’s argument based on its order in the ‘argument Likelihood of electron over kaon Le/K > 0.95”. The ontol- list’. In our case, a function argument must be able to de- ogy must define the necessary vocabulary to represent the note a collection of class instances, which SUMO does not competency questions that arose and the answers that were allow. In EngMath, such collection is represented as tensor generated. or vector. Some event-variables are constants, while others are func- Although, the EngMath approach of providing rigorous de- tions. In the above illustrative example, Le/K is a con- scriptions appears to faithfully represent mathematical ex- stant; whereas, Mbc is a function event-variable. Mbc = pressions in instantiated models, it is difficult to understand q 2 2 and apply to working systems. The theories represented E − P , where Ebeam is the energy of the beam and beam 3B in declarative style, as axioms (KIF sentences) are hard to P is the 3-Momentum of B particle. E is a constant. 3B beam read and understand, more so by physicists who are not well The 3-Momentum P = pP 2 + P 2 + P 2 is a function. In 3 x y z versed with ontology. Furthermore, it is not feasible to port here, P , P and P are constant event-variables that de- x y z this ontology on the web because web-ontology languages do note the individualised track momentum along the x, y and not provide for definitions of arbitrary n-ary relations and z axes, respectively. functions, and axioms that make up EngMath. Note that a function event-variable is expressed algebraically Since we aim to build suitable EHEP ontologies for the se- in terms of constant event-variables. A simple function mantic web, we parted from the EngMath approach from the event-variable accepts only constants as parameters. Ex- outset. The contrast between EngMath and our modelling amples are 3-Momentum, P and Transverse Momentum, 3 principles are described below. PT that describe the momentum of a track. The function p 2 2 P3 is given above, while PT = Px + Py . I. Web-ontologies are serialised in XML [22], a mark up On the other hand, a higher-order function event-variable language intended to encode metadata concerning web also admits other functions as parameters. The M is a bc document. higher-order function. One of its parameter is P3, a func- tion event-variable. Another example is the Fox-Wolfram II. The domain concepts are assembled in a hierarchy P P Moment-0 event-variable, H0 = j k(P3j × P3k ), whose and their distinguishing properties and relationships parameter is a set of 3-Momentum of tracks. The indices j among them are specified, using a frame-like syntax. and k enumerate the tracks in an event. In light of this fact, we have conceived the concepts using notions like class, subclass, range-relation and Sometimes, a function requires weighted parameters. An ex- cardinality. (See the examples in next section) P ample is the Fisher Discriminant event-variable F = i(αi× Ri), which combines a set of correlated event-variables R1, III. A mathematical relation is closed on a function-concept R2, etceteras to form a single variable.
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