The situation calculus

1. K. V. S. Prasad Notes for TIN171/DIT410 (Friday, 26 March 2010) Based on Nils Nilsson, “Artificial Intelligence: a new synthesis”, Morgan Kaufmann Publishers, 1998 The situation calculus

2. The situation calculus • Features can leave out irrelevant properties • Important for goal condition • To get some block on B (don't care which) • “Exists x. (xB)” • Any predicate formula g can be a goal condition • Does state s satisfy g? • But state s is itself described by formulas • So try to prove g from those formulas • The situation calculus represents knowledge about both states and actions by predicate formulas

3. Blocks in the situation calculus • The state (BAC) could be written • On(B,A) and On(A,C) and On(C,T) and Clear(B) and ... • But explicitly including the state helps • E.g., (ML, last term) -> not (ML, this term) • “On(B,A,S0) and On(A,C,S0) and On(C,T,S0) and Clear(B,S0)” is a statement about S0 • All s. Clear(T,s) (The table is always clear) • All x, y, s. On(x,y,s) and not(y=T) -> not Clear(y,s) • We can prove “not Clear(A,S0)”

4. Actions in the situation calculus • States are terms in SC • So are actions: • “move(B,A,T)” is the term representing the action • Of moving B from A to the table • move(x,y,s) is a family of move actions • Instantiate the variables to get specific actions • “Do” takes (action, state) -> state • So now what is the effect of an action? • Knowledge about is encoded in the pred. calculus

5. Effects of actions • On(x,y,s) and Clear(x,s) and Clear(z,s) and not (x=z) -> On(x,z, do(move(x,y,z),s)) • In iconic notation, • (XY)(Z) –move(X,Y,Z)-->(Y)(XZ) • The above formula only tells you about (XZ) • Each effect axiom is like P -> Q, where P is the precondition for the action

6. Action effects, contd. • On(x,y,s) and Clear(x,s) and Clear(z,s) and not (x=z) -> not On(x,y, do(move(x,y,z),s)) • This negative effect axiom describes a proposition that became false after the action

7. Antecedent more than precondition • How does move cause Clear conditions? • On(x,y,s) and Clear(x,s) and Clear(z,s) and not (x=z) and not (y=z) -> Clear(y, do(move(x,y,z),s)) • On(x,y,s) and Clear(x,s) and Clear(z,s) and not (x=z) and not (T=z) -> not Clear(z, do(move(x,y,z),s)) • Here “not(y=z)” is not a precondition (the action can be done even if y=z) but the effect will not be given by the axiom above

8. Plans in the situation calculus • To go from S0=(BAC) to (AC)(B) • Try to prove “Exists s. On(B,T,s)” starting from the axioms describing S0 and the move axioms • You will succeed in proving “On(B,T,do(move(B,A,T),S0))” • The plan is “move(B,A,T)” and can be extracted from the proof • Programs from proofs is an idea that also crops up in type theory

9. Plans contd. • Let Ans(s) say that A is on B in state s • To get from S0=(BAC) to a state s where Ans(s) • Ans(do(move(A,C,B),do(move(B,A,T),S0))) • So the plan is move(B,A,T); move(A,C,B) • This used to be regarded as hopelessly hard even for small plans • But with fast SAT solvers these days ...

10. Problems with SC • Frame axioms • Actions typically have limited effects • Have to explicitly say rest stays unchanged • Both positive and negative literals can stay unchanged • Qualification axioms • Block not too heavy, not glued down, etc. • Ramification axioms • When a plane moves, everything in it moves • Mechanisms have been proposed for each