Describing State Change

1. Describing State Change Predicate logic is stateless: if p(a) can once be deduced from a set of axioms then it can always be deduced from those axioms. Modelling the real world, however, requires us to talk about state change. We look at one simple (but common) way to model state in a stateless logic.

2. Blocks World Example Op1 Opn trans(Block, From, To) …….. a c b a b c t1 t2 t3 t1 t2 t3

3. State Change as Recursion poss(s0). poss(do(Op, S)) :- poss(S), pact(do(Op, S)). do(Op2 , do(Op1, s0 ) ) do( Op1, s0 ) s0 c a b t1 t2 t3

4. Possible Actions pact(do(trans(X, Y, Z), S)) :- holds(clr(Z), S), holds(clr(X), S), holds(on(X, Y), S), \+(X = Z), \+(Z = Y), \+(X = Y). trans(c, a, t3) clr(t3) clr(c) c on(c,a) c a b a b t1 t2 t3 t1 t2 t3

5. Initial Conditions holds(on(c,a), s0). holds(on(a,t1), s0). holds(on(b,t2), s0). holds(clr(c), s0). holds(clr(b), s0). holds(clr(t3), s0). c a b t1 t2 t3

6. Transition Effects on State holds(clr(Y), do(trans(_, Y, _), _)). holds(on(X, Z), do(trans(X, _, Z), _)). trans(c, a, t3) clr(a) c c a b a b on(c,t3) t1 t2 t3 t1 t2 t3

7. Frame Axiom holds(C, do(trans(X, Y, Z), S)) :- holds(C, S), \+(C = clr(Z)), \+(C = on(X, Y)). clr(t3) trans(c, a, t3) on(c,a) on(a,t1) on(a,t1) on(b,t2) on(b,t2) c clr(c) clr(c) c a b a b t1 t2 t3 clr(b) clr(b) t1 t2 t3

8. Goal State goal :- poss(S), holds(on(a,b), S), holds(clr(a), S), holds(on(b,c), S), holds(on(c,t3), S), holds(clr(t1), S), holds(clr(t2), S). a b c t1 t2 t3

9. Plan poss(do(trans(a,t1,b), do(trans(b,t2,c), do(trans(c,a,t3), s0)))) trans(c, a, t3) trans(b, t2, c) trans(a, t1, b) a b c b c a b c a a b c t1 t2 t3 t1 t2 t3 t1 t2 t3 t1 t2 t3

10. Subsumption Based Systems • Subsumption based inference is used in programming systems that employ types, classes, description logics and the like. • We look at how to infer subsumption in two ways: • Through term structure; • Through translation to FOPC • There are other ways (e.g. by writing a type unification system) but we don’t cover those.

11. Term Encoding For Subsumption animal vehicle mammal reptile car bus dog Type described as a structured term e.g. animal(mammal(dog(X))) | ?- animal(X) = animal(mammal(Y)). X = mammal(Y) | ?- animal(X) = vehicle(Y). no | ?- animal(mammal(dog(rover))) = vehicle(car(rover)). no

12. Vocabulary Describing Classes “Mammals are animals” “Mammals are warm blooded and lactating” subclass(mammal, animal) subclass(intersection(warm_blooded, lactating), mammal) domain(biomass,animal) range(biomass,number) etc… “Biomass is a function over animals” “Biomass is a function returning a number”

13. Translator to FOPC th(A, X, T) :- atom(A), T =.. [A,X]. th(intersection(C,D), X, (T1,T2)) :- th(C, X, T1), th(D, X, T2). th(all(R,C), X, (T1 :- T2)) :- th(C, Y, T1), T2 =.. [R,X,Y]. tb(A, X, T) :- atom(A), T =.. [A,X]. tb(intersection(C,D), X, (T1,T2)) :- tb(C, X, T1), tb(D, X, T2). tb(union(C,D), X, (T1 ; T2)) :- tb(C, X, T1), tb(D, X, T2). tb(some(R,C), X, (T1,T2)) :- tb(C, Y, T2), T1 =.. [R,X,Y]. t(subclass(C,D), (T1 :- T2)) :- th(D, Y, T1), tb(C, Y, T2). t(equivalent(C,D), (T1,T2)) :- t(subclass(C,D), T1), t(subclass(D,C), T2). t(range(R,D), (T1 :- T2)) :- th(D, Y, T1), T2 =.. [R,_,Y]. t(domain(R,D), (T1 :- T2)) :- th(D, X, T1), T2 =.. [R,X,_]. t(X : A, T) :- th(A, X, T). t((X1,X2):R, T) :- atom(R), T =.. [R,X1,X2].

14. Applying FOPC Translator | ?- t(subclass(mammal, animal), S). S = (animal(X) :- mammal(X)) | ?- t(subclass(intersection(warm_blooded, lactating), mammal), R). R = (mammal(X) :- warm_blooded(X), lactating(X)) | ?- t(domain(biomass,animal), D). D = (animal(X) :- biomass(X,_)) | ?- t(range(biomass,number), R). R = (number(X) :- biomass(_,X))