Lehmann, Jos

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     Edinburgh Research Explorer                                      A Case Study of Ontology Evolution in Atomic Physics as theBasis of the Open Structure Ontology Repair PlanCitation for published version:Lehmann, J 2009, 'A Case Study of Ontology Evolution in Atomic Physics as the Basis of the OpenStructure Ontology Repair Plan'. in Notes of the IJCAI-09 Workshop ARCOE-09.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Preprint (usually an early version)Published In:Notes of the IJCAI-09 Workshop ARCOE-09General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015A Case Study of Ontology Evolution in Atomic Physicsas the Basis of the Open Structure Ontology Repair Plan∗Jos LehmannSchool of Informatics, University of Edinburghjlehmann@inf.ed.ac.ukAbstractIn the GALILEO project a number of Ontology Re-pair Plans are being developed and implementedin higher-order logic. These plans resolve a con-tradiction between two or more ontologies that rep-resent the domain of physics. In this abstract, thetransition from Thomson’s plum pudding model ofthe atom to Rutherford’s planetary model is used asthe main inspiration of a new ontology repair plancalled Open Structure1 IntroductionIn the framework of the GALILEO project (Guided Analy-sis of Logical Inconsistencies Leads to Evolved Ontologies)a number of Ontology Repair Plans (ORPs) are being de-veloped and implemented in higher-order logic. For rea-sons of space, the interested reader is referred to [Bundy andChan, 2008] to find general indications on how the overallGALILEO approach, and thus what presented in this abstract,relates to analogues in the literature. It should be noted,though, that GALILEO’s future research objectives do in-clude a more precise positioning of the notion of ORPwith re-spect to existing formal-ontological and epistemological ap-proaches. ORPs resolve a contradiction between two or moreontologies that represent the domain of physics. In ORPs de-veloped thus far, one of the ontologies represents a physicaltheory – its theorems are interpreted as expectations or pre-dictions; a second ontology represents a sensory or experi-mental set-up for that theory – its theorems are interpreted asobservations. When the sensory ontology generates a theo-rem that contradicts a theorem of the theoretical ontology –i.e., when an observation contradicts an expectation – an ORPkicks in and amends the two ontologies. Such repair enablesthe theoretical ontology to make correct predictions and up-dates the experimental ontology with the new theory. ORPsmay act either as theory revision mechanisms or as signaturerevision mechanisms – in the latter case they bring about anevolution of the ontologies by changing their representationlanguage.The development of ORPs is inspired by cases in the history∗I’m grateful to Alan Bundy, Michael Chan and the ARCOE-09reviewers for comments on earlier drafts.of physics where the transition from an old theory to a newone was forced by a contradiction between observations andpredictions. For instance, the discovery of Latent Heat in-spired the ORP called Where is my stuff? (WMS). This planchanges the signature of the ontologies by splitting a functioninto a visible and an invisible part; such repair accounts forthe impossibility, at a given moment of the history of physics,to fully measure a given quantity (e.g. heat or presently mass,as Dark Matter has never been measured directly). The tran-sition from Boyle’ Law to the Ideal Gas Law inspired theORP called Inconstancy, which changes the signature of theontologies by adding an argument to a function; it may, forinstance, make a constant (e.g. the volume to pressure ra-tio of a gas) dependent on an quantity (e.g. temperature).The case of the ancient distinction between the Morning Starand the Evening Star, later identified by astronomers as thesame planet Venus, inspired the recently proposed ORP calledUnite. This plan operates as a theory revision mechanismwhich equates two functions.The present abstract provides an example of a modeling ex-ercise, from historical analysis to formalization. The tran-sition from Thomson’s plum pudding model of the atom toRutherford’s planetary model is used as source of inspirationof a new ORP called Open Structure (OP) and of its inverse,not presented here, called Close Structure (CS). OS and CSoperate as theory revision mechanisms, which delete one ofthe cases of a function, thus restructuring a given entity (e.g.an atom) by distributing a given quantity differently (e.g., thework excerted by an atom’s positive electric field on a particleorbiting around it)1. Section 2 presents part of the historicalanalysis that led to Open Structure and presents the formu-lae that may be subject to ontological evolution. Section 3presents Open Structure and its application to the case study.Section 4 sets further research objectives.2 From Thomson’s to Rutherford’s AtomThomson’s Atom Around 1904 Thomson believed that theatom was a uniform sphere of positive charge, as representedin Fig. 1 (left), with electrons rotating and oscillating in it.Negative charges were placed on concentric rings or at the1As noted by Alan Bundy during a recent private exchange, ingeneral mathematical terms it may be said that OS and CS let a non-monotonic function evolve into a monotonic one.center. The number of (electrons on) the rings would in-crease monotonically while the number of charges at the cen-ter would periodically increase and decrease [Baily, 2008].Figure 1: Left: Thomson’s atom (11 electron atom and 15electron atom). Right: Rutherford’s atom (11 electron atom15 electron atom). Black dots and circumferences representnegative charges and their orbits; red circles represent posi-tive charge (more intense where darker).Scattering Experiment Together with Geiger and Mars-den, around 1911 Rutherford learnt how to control a heavyemission named α-particle, i.e. doubly positively charged he-lium atoms (He2+). Using the apparatus shown in Fig. 22, astream of alpha particles originating from source R was shotat a thin foil of gold atoms (F). Scintillations, which weredue to the particles scattering against a rotating screen (S),were observed through a microscope (M) mounted behindthe screen. The chamber was evacuated and could be rotatedaround the foil.Based on Thomson’s model, Rutherford expected the sphereFigure 2: Rutherford’s scattering apparatusof positive charge and the electrons in it to offer virtually noresistance to the passage of α-particles. Given their highspeed, their momentum would overcome the repulsion ex-certed by the positively charged sphere. Thus the particleswould be deflected slightly (just like, on left side of Fig. 3,the trajectories with impact parameter around ± b′) or gostraight through (like virtually all other trajectories). Also,since α-particles are 7000 times heavier than electrons, theirtrajectory would never be curved by electrons (that is why nonegative charges are represented in Fig. 3). Contrary to theexpectations, three degrees of scattering were observed:2Originally in [Geiger and Marsden, 1913], downloaded from:galileo.phys.virginia.edu/classes/252/Rutherford /Scattering/Rutherford Scattering.html• most α-particles passed straight through the gold foilwithout any or with very little deflection (like most par-ticles with impact parameters between ± b and ± b′ onright side of Fig. 3);• some α-particles were deflected through large angles(i.e. particles with impact paraments around ± b);• a few α-particles rebounded completely (i.e. particleswith impact parameter around 0).Figure 3: Left: expected scattering. Right: observed scatter-ing. Small red spots represent α-particles; red lines representparticle’s paths; half-dashed thin red lines represent ideal un-deflected paths; b’s and −b’s are impact parameters, i.e. per-pendicular distances between a particle’s velocity vector andthe centre of the target atom (b = 0 for third particle); r’sare distances between a point of the atom’s electric field andthe atom’s center; R is the atom’s radius; th’s are scatteringangles.Rutherford’s Atom Rutherford analyzed the results of theexperiment in terms of Newtonian mechanics and obtained aformula to calculate the differential scattering cross-sectionof the α-particles, i.e. the probability, given the impact pa-rameter, of an α-particle to be deflected through a given an-gle. This made it possible to derive the following conclusions:1. Since most of the α-particles were not deflected, there isa lot of empty space in an atom.2. Since some of the α-particles were deflected, there is acentre of positive charge in an atom.3. Since very few α-particles rebounded, the nucleus isvery dense and hard.4. Since 1 to 3, the structure of an atom is comparable tothe Solar System’s structure, with the nucleus formingthe main mass and the electrons revolving around it.Evolution between the Two Atoms As shown in [Zoli,1998], the differences between the scattering of α-particlesin Thomson’s and Rutherford’s atoms may be expressed invarious ways. One way is to define two distinct scattering po-tential functions (like 1 below for Thomson and 2 for Ruther-ford) to calculate the amount of work excerted by the elec-tric fields of the two atoms when deflecting incident particlesplaced at a distance r from their centers. Note that both po-tentials are central but, while 1 is both non-Coulombic (i.e.for values of r lower than the atom’s radiusR, the potential isdirectly proportional to r) and Coulombic (i.e. for values ofr higher than R, the potential is inversely proportional to r),2 is only Coulombic (R needs not to be considered).V (r)T =8><>:QAQB4pi01rR ≤ rQAQB4pi012R3(3R2 − r2) 0 ≤ r ≤ R(1)V (r)R =QAQB4pi01r(2)where QA is the charge of incident particle, QB is the charge ofthe target atom, 1/4pi0 is the Coulomb constant, r is the distancebetween the incident particle and the centre of the target atom, R isthe radius of the target atom.3 Open Structure Ontology Repair PlanIn this section the evolution from V (r)T to V (r)R is modeledbyOpen Structure (3 below). The idea of the plan is to evolvea function that has values that are partly directly proportionaland partly inversely proportional to its argument into a func-tion that is only inversely proportional to its argument. In thisway the physical structure of the entity described by the func-tion loses an internal and/or external boundary.Just like all ORPs, OS models the function that is subject toevolution as stuff . V is stuff . Just like V ranges over thetype dis of distances r’s, stuff ranges over a type δ. Thedistance R plays the role of cut-off point of V . So stuff ’sdomain contains a cut-off point, cop. As the relation betweenall other quantities in V is constant, these are modeled asK.Two cases of contradiction between Ot and Os trigger OS:1. In Ot, for all arguments below the cut-off point, thevalue of stuff is directly proportional to the argument,while for all other arguments the value of stuff is in-versely proportional to the argument. In contrast withthis, in Os the value of stuff is always inversely propor-tional to its argument. In this case, the solution of thecontradiction by OS results in the structure of the en-tity described by the function to lose a physical internalboundary (where the function is at its maximum).2. In Ot, for all arguments below the cut-off point, thevalue of stuff is inversely proportional to the argumentwhile, for all other arguments, the value of stuff is di-rectly proportional to the argument. Os is the same asin the first case above. Here, after applying OS, thestructure of the entity described by the function loses aphysical internal boundary (where the function is at itsminimum) as well as an external boundary (where thefunction is at its maximum).In all cases the contradiction is repaired according to whatdictated by Os (4 through 6 below). This is also true forthe inverse of OS, Closed Structure, but here Os forces theacquisition of an external boundary.OS may be formlized and applied to the case study as below.TriggerOt ` d4 > d3 ≥ cop ≥ d2 > d1 ∧ (3)((stuff (d2) > stuff (d1) ∧ stuff (d3) > stuff (d4)) ∨(stuff (d1) > stuff (d2) ∧ stuff (d4) > stuff (d3))).Os ` ∀d, d′ : δ. d′ > d→ stuff (d) > stuff (d′).Open Structureν(stuff ) ::= λd : δ. K/d. (4)Create New AxiomsAx(ν(Ot)) ::= Ax(Ot) \ (5){stuff ::= λd : δ. (cop > d ∧Kd) ∨K/d} ∪{ν(stuff ) ::= λd : δ. K/d}.Ax(ν(Os)) ::= Ax(Os) \ (6){stuff ::= λcop, d : δ. (cop > d ∧Kd) ∨K/d} ∪{ν(stuff ) ::= λd : δ. K/d}.The following substitutions in the atom case study wouldyield the repair below:V/stuff , di/ri, cop/R,K/QAQB4pi0ffAx(ν(Ot)) ::= Ax(Ot) \{V ::= λr : dis. (R > r ∧Kr) ∨K/r} ∪{ν(V ) ::= λr : dis. K/r}.Ax(ν(Os)) ::= Ax(Os) \{V ::= λr : dis. (R > r ∧Kr) ∨K/r} ∪{ν(V ) ::= λr : dis. K/r}.4 ConclusionThe transition from Thomson’s plum pudding model of theatom to Rutherford’s planetary model was used as the basisof a new ontology repair plan called Open Structure and of itsinverse, Close Structure. Future research will focus on:• the interpretation of the second case of OS: what are theexamples of the physical structure represented by Ot inthe OS’s second case?;• the interpretation of CS, OS’s inverse: what are the ex-amples of the physical structure represented by Os inCS?• the alternative to OS and CS: what would be the advan-tages of modeling the evolution between the two atomsin terms of their different scattering angle functions (i.e.by comparing alternative trajectories of particles thathave the same impact paramater)?References[Baily, 2008] C. Baily. Atomic modeling in the early 20thcentury: 1904-1913. Technical report, University of Col-orado, Boulder - Department of Physics, 2008.[Bundy and Chan, 2008] A. Bundy and M. Chan. Towardsontology evolution in physics. W. Hodges, editor, Procs.Wollic 2008. LNCS Springer-Verlag, 2008.[Geiger and Marsden, 1913] H. Geiger and E. Marsden. Thelaws of deflection of alpha particles through large angles.Philosophical Magazine (6), 25, 507, 1913.[Zoli, 1998] M. Zoli. Alpha-particle scattering in the thom-son and rutherford atomic models. European Journal ofPhysics 19 187-193, 19:187–193, 1998.

A Case Study of Ontology Evolution in Atomic Physics as the Basis of the Open Structure Ontology Repair Plan

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