Alan Bundy University of Edinburgh

1. Alan Bundy University of Edinburgh Repair • Domain abstraction:Ball:spring • Energy of a spring: 8s:spring, t:time. TE(s,t) ::= PE(s,t) + KE(s,t) + EE(s,t) • Elastic potential energy: 8s:string, t:time. EE(s,t) ::= ((s).(Len(s,t)-NatLen(s))2)/(2.NatLen(s)) • Revised energy equations: Mass(Ball).G.Ht(Ball,0) = Mass(Ball).Vel(Ball,T-)2/2 = ((Ball) (Len(Ball,T)-NatLen(Ball))2)/(2.NatLen(Ball)) Aristotle vs Galileo • Aristotle conflated average and instantaneous velocity (Kuhn 1977). • Galileoet al discredited neo-Aristotelian physics. O ` Vel(Ball) = (Ht(Ball,T)-Ht(Ball,0))/T > 0 ²Vel(Ball) > 0 at t=0 • Distinguish: InVel(Ball) from AverVel(Ball). • Make both fluents: InVel(Ball,Mom), AverVel(Ball,Int). Paradox of the Bouncing Ball Paradox of Latent Heat • Latent heat: change of heat content without change of temperature. • Black discovered in 1761. • Before Black, heat and temperature conflated. • Wiser & Carey 1983 • Separation of conflated concepts necessary precursor to discovery. • Conflation of “morning star” and “evening star” into “Venus” in reverse direction. Andy deSessa’s (1983) Bouncing Ball:Where does energy go at moment of impact? Essential to idealize ball as deformable, eg spring. Representation as a Fluent • World infinitely rich: cannot model every aspect. • Fixed representations cannot cope with changing world and new challenges. • Representation needs to evolve under machine control. • Need to change signatures – not just beliefs. Towards a Theory of Ontology Repair or Truthfulness Considered Harmful Conclusion • Representation must be fluent in reasoning. • Not just belief revision but also signature revision. • Reasoning failure can motivate and direct reasoning refinement, • e.g. false theorem or inconsistency. • Towards formal theory. • Define most-general repair. • Avoid problems of truthfulness, semantic preservation, inconsistency creation and undefinedness. • Ontology Repair System • ORS Program:repairs faulty ontologies by analysing failed multi-agent plans. • Changes include abstraction and refinement of signatures, • e.g. adding arguments, changing predicates. • Allows agents with slightly different ontologies to communicate. • Physics as a Domain • Historical record of ontologies, their faults, diagnosis and successors. • Plenty of examples. • Personal background in Mecho Project. • Including some formalisations. • Subtle and profound issues. Examples Properties to be Avoided • Truthfulness: O `!(O) `() • Repair degenerates: Repair(O,,) ::= (O`Ʋ) )²() • Semantics preserving:² ! ²() • Repair degenerates: Repair(O,,) ::= (O `Ʋ) !(O) `() • Inconsistency creating: • O `?Æ(O) `? Theory Energy of a Particle • Energy Conservation • 8o:obj, t1:time, t2:time. TE(o,t1) = TE(o,t2) • Energy of particle: • 8p:part, t:time. TE(p,t) ::= PE(p,t) + KE(p,t) • Potential Energy: • 8o:obj, t:time. PE(o,t) ::= Mass(o).G.Ht(o,t) • Kinetic Energy: • 8o:obj, t:time. KE(o,t) ::= Mass(o).Vel(o,t)2/2 Formal Definitions • Repair(,) ::= O`Ʋ) ! ((O) `() Dz()) • InORSisHolds(Goal,Sit) • deliberately overloaded. • =?is special case, since² ?. • MGR(O,) ::= Repair(O,) Æ8': O  O. Repair(O, ') ! Th('(O)) ¾ Th((O)) • Most general repairs not unique. Bouncing Ball Snapshots • Initially:Ball : part Vel(Ball,0)=0, Ht(Ball,0)>0 • Just before contact: Vel(Ball,T-)>0, Ht(Ball,T-)=0 • At point of contact: Vel(Ball,T)=0, Ht(Ball,T)=0 • From energy equations: G.Ht(Ball,0) = Vel(Ball,T-)2 /2 = 0 • From which ? e.g. 0=Vel(Ball,T-)>0 Refinement Techniques • Predicate and Function: separating them. Vel(Ball)  InVel(Ball) + AverVel(Ball) • Domain: specialise types. Ball:obj  Ball:part • Propositional: add argument to predicate. InVel(Ball)  InVel(Ball,t) • Precondition: add rule condition [Heat(o,t) ::= Temp(o,t)]  [Solid(o) ! Heat(o,t) ::= Temp(o,t)] • Permutation: change argument order InVel(o,t)  InVel(t,o) Abstraction Techniques • Predicate & Function: merging them. MStar + EStar  Venus • Domain: generalise types. Ball:part  Ball:string • Propositional: drop arguments. AverVel(o,t)  AverVel(o) • Precondition: drop rule conditions [:Cold(o) ! Heat(o,t) ::= Temp(o,t)]  [Heat(o,t) ::= Temp(o,t)] Adapted from Walsh & Giunchiglia Predicate Abstraction and Refinement • Similar problems with truthfulness, inconsistency creation and undefinedness. • Also fix with semantic definitions. • Predicate Abstraction: Ax(a(O)) ::= {[p] | ([p1] 2 Ax(O)Ç[p2]) 2 Ax(O)Æ O ` :[p1Ç p2] } • Predicate Refinement: Ax(r(O)) ::= {[p1Ç p2] | [p] 2 Ax(O) ]} Propositional Abstraction & Refinement • Propositional abstraction is truthful. • Simple induction on proofs. • Propositional abstraction is inconsistency creating. P(a) 2 Ax(O) Æ: P(b) 2 Ax(O) !(O) ` P Æ: P • Propositional refinement is not totally defined. P 2 Ax(O) ! P(?) 2 Ax((O)) Unconflating Heat and Temperature • Effect of cold: 8o,c:obj,i:int. Adj(o,c,i) Æ Cold(c) ! Down( t. Heat(o,t),i) • Definition of Down: Down(f,i) ::= 8 t1,t2:mom. t12 i Æ t22 i Æ t1<t2! f(t1) > f(t2) • Inference: O ` Heat(H20,Start(Freeze)) > Heat(H20,End(Freeze)) • Observation: ²Heat(H20,Start(Freeze)) = Heat(H20,End(Freeze)) • Repair (function refinement): 8o:ob,t:mom. Solid(o,t) ! Heat(o,t) ::= Temp(o,t) 8o:ob,t:mom. Liquid(o,t) ! Heat(o,t) ::= Temp(o,t)+LHF(o) 8o:ob,t:mom. Gas(o,t) ! Heat(o,t) ::= Temp(o,t)+LHF(o) + LHV(o) • cf dark matter. Fix with Semantic Definitions • Propositional Abstraction: Ax(a(O)) ::= {[p(t)] | 9 t0. [p(t0, t)]2Ax(O) Æ O `:[9 x. p(x, t)]} • Propositional Refinement: Ax(r(O)) ::= {[9 x. p(x, t)] | [p(t)] 2 Ax(O)} • Blocks inconsistency: :P 2 Ax(a(O)) $:9 t0. : P(t0)2 Ax(O) Ç O`::9 x. P(x) $ O `9 x. P(x) since P(a) 2 Ax(O)