Ontology Applying logic to the real world

D Goforth - COSC 4117, fall Real world knowledge general knowledge / common sense reasoning domain expertise / specific knowlege problem / facts example: understand a news report language and world knowledge; specific situation and terminology  interpret story

D Goforth - COSC 4117, fall Ontology: structure of knowledge  E.g. java programming ontology – object-oriented design: class/object inheritance and interfaces part-of hierarchy API message-passing sequential execution, threads

D Goforth - COSC 4117, fall Ontology: structure of knowledge  general knowledge: what top-level structure? Anything (like class Object) AbstractObject (eternal) GeneralizedEvent (time-limited existence)

D Goforth - COSC 4117, fall Upper Ontology

General knowledge Domain knowledge Problem facts and questions Ontology in Knowledge base “Abstract objects” “Generalized events” Categories (ontology) Logic (sentences)

D Goforth - COSC 4117, fall Representing Knowledge  how ‘deep’? shallow – as predicate:  Terrier(x) deep ‘reification’ category with meaning structure:  Terriers ⊆ Dogs, Dogs ⊆ Mammals  Member (x, Terriers)

D Goforth - COSC 4117, fall Categories  like set theory – easy to reason with in FOL subcategories / subsets categories of categories intersections, unions, disjoint sets, partitions

D Goforth - COSC 4117, fall Reasoning about categories  disjoint subcategories – no common objects NO: x  Students, x  Employed YES:x  Mazdas, x  Mercedes x  Mazdas ⇒ ~( x  Mercedes) x  Mercedes ⇒ ~( x  Mazdas)

D Goforth - COSC 4117, fall Reasoning about categories  exhaustive decomposition – all objects of a category belong to at least one of the subcategories Namedstreets  Cityroutes Numberedroads  Cityroutes x  Cityroutes ⇒ (x  Namedstreets)  (x  Numberedroads) (could be both named and numbered)

D Goforth - COSC 4117, fall Reasoning about categories  Partitioning a category subcategories are disjoint subcategories form exhaustive decomposition e.g., Players are teammates or opponents: Players = Teammates  Opponents Teammates  Opponents = {}

D Goforth - COSC 4117, fall Physical objects  properties things – a pile of sand  measurable / quantities vs stuff - sand  intrinsic qualities PhysicalObjects StuffThings

D Goforth - COSC 4117, fall Situation calculus  in specific domain (TimedEvents vs AbstractObjects) some objects are ‘fluent’  functions and properties can change over time (position, orientation, etc) some objects are ‘eternal’  existence and properties remain fixed during period of reasoning (more efficient) (recall wumpus world example)

D Goforth - COSC 4117, fall GeneralizedEvents – the time problem  objects in this hierarchy have time property physical objects events processes intervals  ‘fluent’ (“fleeting”) vs ‘eternal’ abstract objects

D Goforth - COSC 4117, fall Situation calculus universe is defined as sequence of ‘situations’  ‘actions’ are like inferences: preconditions – required facts in current situation effects – facts that are true in subsequent situation if action is applied situation S 3 preconditioneffect action situation S 2 situation S 1 situation S 0

D Goforth - COSC 4117, fall Situation calculus - example  blocks world tabletop and three blocks actions and situations AB C eternal Table(x) Block(x) fluent On(x,y,s) ClearTop(x,s) s is situation variable objects/terms in FOL

D Goforth - COSC 4117, fall Situation calculus - example  actions are functions (objects)  situations are objects AB C  PutOn(x,y) preconditions:  ClearTop(x,s)  ClearTop(y,s) V Table(y) effect:  On(x,y,Result(PutOn(x,y),s))

D Goforth - COSC 4117, fall Situation calculus - example  each situation is a function of the previous one – Result function AB C  Result(a,s) preconditions:  action a can be applied at s effect:  Result is next situation after a is applied at s

D Goforth - COSC 4117, fall T Situation calculus - example  e.g. KB: function PutOn(x,y) ∀ x (~ ∃ y On(y,x,s)) ⇒ ClearTop(x,s) ∀ x,y,s ClearTop(x,s) ^ (ClearTop(y,s) v Table(y) =>On(x,y,Result(PutOn(x,y),s)) constants: A, B, C, T, S 0, S 1, S 2,… Table(T), Block(A), Block(B), Block(C) On(A,B,S 0 ), On(B,C,S 0 ), On(C,T,S 0 ) A B C S0S0

D Goforth - COSC 4117, fall T Situation calculus - example  action: PutOn(A,T) preconditions: ClearTop(A,S 0 ),Table(T) effect: On(A,T,Result(PutOn(A,T),S 0 )) BUT… what happens to other fluents? some propagated, some not A B C S1S1 A

D Goforth - COSC 4117, fall T Situation calculus - example  ‘On’ axiom: ∀ x,y,z,a,s On(x,y,Result(a,s))  [ClearTop(x,s)^(ClearTop(y,s)vTable(y))^ a= PutOn(x,y) v [ On(x,y,s)^~(a=PutOn(x,z))] C S1S1 A A B