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
Alan Bundy
University of Edinburgh
Ontology Repair System
ORS Program: repairs faulty ontologies by analysing failed multiagent plans.
 Changes include abstraction and refinement of signatures,
e.g. adding arguments, changing predicates.
Physics as a Domain
Allows agents with slightly different ontologies to communicate.  Historical record of ontologies, their faults, diagnosis and successors.
 Plenty of examples.
 Personal background in Mecho Project.
 Including some formalisations.
Examples
 Subtle and profound issues.
Paradox of the Bouncing Ball
Andy deSessa’s (1983)
Bouncing Ball: Where
does energy go at
moment of impact?
Essential to idealize ball as
deformable, eg spring.
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
 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.
Properties to be Avoided
Theory
Energy of a Particle
Energy Conservation
8 o:obj, t1:time, t2:time. TE(o,t1) = TE(o,t2)
Energy of particle:
8 p:part, t:time. TE(p,t) ::= PE(p,t) + KE(p,t)
Potential Energy:
8 o:obj, t:time. PE(o,t) ::= Mass(o).G.Ht(o,t)
Kinetic Energy:
8 o:obj, t:time. KE(o,t) ::= Mass(o).Vel(o,t)2/2
Repair
Conclusion
Formal Definitions
Truthfulness: O `  ! (O) ` ()
 Repair degenerates:
Repair(O,,) ::= (O` Æ ²) ) ²()
Semantics preserving: ² ! ²()
Repair degenerates:
Repair(O,,) ::= (O ` Æ ²) ! (O) ` ()
Inconsistency creating:
O ` ? Æ (O) ` ?
Repair(,) ::= O` Æ ²) ! ((O) ` () Ç ²())
In ORS  is Holds(Goal,Sit)
  deliberately overloaded.
 =? is special case, since ² ?.
MGR(O,) ::= Repair(O,) Æ 8 ': O  O. Repair(O, ') ! Th('(O)) ¾ Th((O))
Most general repairs not unique.
Abstraction Techniques
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)]
 Predicate & Function: merging them.
Domain abstraction: Ball:spring
MStar + EStar  Venus
 Energy of a spring:
 Domain: generalise types.
8 s:spring, t:time. TE(s,t) ::= PE(s,t) + KE(s,t) +
Ball:part  Ball:string
EE(s,t)
 Propositional: drop arguments.
 Elastic potential energy:
AverVel(o,t)  AverVel(o)
8 s:string, t:time. EE(s,t) ::=
 Precondition: drop rule conditions
2
((s).(Len(s,t)-NatLen(s)) )/(2.NatLen(s))
[:Cold(o) ! Heat(o,t) ::= Temp(o,t)]
 Revised energy equations:
 [Heat(o,t) ::= Temp(o,t)] Permutation: change argument order
InVel(o,t)  InVel(t,o)
Mass(Ball).G.Ht(Ball,0) = Mass(Ball).Vel(Ball,T-)2/2 =
Aristotle vs Galileo
Adapted from Walsh & Giunchiglia
((Ball) (Len(Ball,T)-NatLen(Ball))2)/(2.NatLen(Ball))
Aristotle conflated average and instantaneous velocity (Kuhn 1977).
Predicate Abstraction and Refinement
Propositional Abstraction & Refinement
Galileo et al discredited neo-Aristotelian physics.
 Similar problems with truthfulness, inconsistency creation and undefinedness.
Propositional abstraction is truthful.
O ` Vel(Ball) = (Ht(Ball,T)-Ht(Ball,0))/T > 0
Also fix with semantic definitions.
 Simple induction on proofs.
² Vel(Ball) > 0 at t=0
 Predicate Abstraction:
 Propositional abstraction is inconsistency creating.
Distinguish: InVel(Ball) from AverVel(Ball).
Ax(a(O)) ::= {[p] | ([p1] 2 Ax(O) Ç [p2]) 2 Ax(O) Æ O ` :[p1 Ç p2] }
P(a) 2 Ax(O) Æ : P(b) 2 Ax(O) ! (O) ` P Æ : P
Make both fluents:
 Predicate Refinement:
 Propositional refinement is not totally defined.
InVel(Ball,Mom), AverVel(Ball,Int).
Ax(r(O)) ::= {[p1 Ç p2] | [p] 2 Ax(O) ]}
Unconflating Heat and Temperature
P 2 Ax(O) ! P(?) 2 Ax((O))
Effect of cold:
8 o,c:obj,i:int. Adj(o,c,i) Æ Cold(c) ! Down( t. Heat(o,t),i)
Definition of Down:
Fix with Semantic Definitions
Paradox of Latent Heat
Down(f,i) ::= 8 t1,t2:mom. t1 2 i Æ t2 2 i Æ t1<t2 ! f(t1) > f(t2)
Propositional Abstraction:
Latent heat: change of heat content without change of Inference:
Ax(a(O)) ::= {[p(t)] | 9 t0. [p(t0, t)]2Ax(O) Æ O `:[9 x. p(x, t)]}
temperature.
O ` Heat(H20,Start(Freeze)) > Heat(H20,End(Freeze))
 Propositional Refinement:
 Black discovered in 1761.
Observation:
Ax(r(O)) ::= {[9 x. p(x, t)] | [p(t)] 2 Ax(O)}
 Before Black, heat and temperature conflated.
² Heat(H20,Start(Freeze)) = Heat(H20,End(Freeze))
Blocks inconsistency:
Wiser & Carey 1983
Repair (function refinement):
: P 2 Ax(a(O)) $ :9 t0. : P(t0)2 Ax(O) Ç O` :: 9 x. P(x) $ O ` 9 x. P(x) since
 Separation of conflated concepts necessary precursor
8 o:ob,t:mom. Solid(o,t) ! Heat(o,t) ::= Temp(o,t)
P(a) 2 Ax(O)
to discovery.
8 o:ob,t:mom. Liquid(o,t) ! Heat(o,t) ::= Temp(o,t)+LHF(o)
 Conflation of “morning star” and “evening star” into
8 o:ob,t:mom. Gas(o,t) ! Heat(o,t) ::= Temp(o,t)+LHF(o) + LHV(o)
“Venus” in reverse direction.
cf dark matter.