Artificial Intelligence 2004 Planning: Situation Calculus

Transcription

1 Artificial Intelligence 2004 Planning: Situation Calculus Review STRIPS POP Hierarchical Planning Situation Calculus (John McCarthy) situations actions axioms Review Planning 1 STRIPS (Nils J. Nilsson) actions specified by preconditions and effects stated as formulae in (restricted) First-Order Predicate Logic planning as search in space of (world) states plan is sequence of actions from start to goal state Partial Order Planning planning through plan refinement parallel expansion to satisfy preconditions causal links (effect of a used in precondition of a') threats (effect of a negates precondition of a'; a'<a)

2 Review Planning 2 Plan Decomposition / Hierarchical Planning hierarchical organisation of 'actions' complex and less complex (or: abstract) actions lowest level reflects directly executable actions planning starts with complex action on top plan constructed through action decomposition substitute complex action with plan of less complex actions (pre-defined plan schemata; or learning of plans/plan abstraction, see ABSTRIPS) overall plan must generate effect of complex action Situation Calculus - Overview Situation Calculus (John McCarthy) models actions and events in First-Order Predicate Logic situation as additional parameter for some formulae (fluents) allows to specify change due to events action applied in situation: Result-function effect (changes) and frame (remain) of an action specified through axioms planning as theorem-proving

3 Situations A situation corresponds to a World State. Situations are denoted through constants and variables s, s' (reification) Situations are described through FOPL formulae. Actions transform situations. Situations - Blocks World Example Situation s 0 s 0 = {on(a,b),on(b,fl),clear(a),clear(fl)} on(a,b,s 0 ),on(b,fl,s 0 ),clear(a,s 0 ), clear(fl,s 0 ) Action: move (A, B, Fl) Situation s 1 A A B B s 1 = {on(a,fl), on(b,fl), clear(a),clear(b),clear(fl)} on(a,f,s 1 ),on(b,fl,s 1 ),clear(a,s 1 ),clear(b,s 1 ),clear(fl,s 1 )

4 Actions Actions are written like functions with their name and parameter list. They can also be referred to by variables (reification). Actions transform situations. The performance of an action in a situation is denoted through the result (do) function. The performance of an action a in a situation s yields a new situation s'. Result Function Result (or: do) is a function from actions and situations into situations. Example s' = Result (move (x, y, z), s) specifies a new situation s' which is the result of performing a move-action in situation s. General s = Result (a, s) for action a and situations s, s

5 Result Function - Example situation s = {on(a,b), on(b,fl), clear(c)} action a = move (A,B,C) apply action a in situation s Result (move (A,B,C), s) = s' s' = {on(a,c), on(b,fl), clear (B)} Fluents Formulas affected by actions have situations as parameters. Predicates and functions in these formulae which change due to actions are called fluents. Integrate situation parameter into these fluent formulas. on(a,b,s), on(b,fl,s), clear(c,s) Note: Block(A), Block(B),... without s

6 Situations in Formulas situation s action a on(a,b,s), on(b,fl,s), clear(c,s) move (A,B,C) apply action a in situation s Result (a,s) Result (move (A,B,C), s) = s' situation s' on(a,c,s'), on(b,fl,s'), clear(b,s') Now the Calculus

7 Situation Calculus Effect axioms describe how an action changes a situation when the action is performed. Frame axioms describe what remains unchanged between situations. Successor-state axioms combine effect and frame axioms for an action. action move (x, y, z) effect-axiom: Effect Axiom on (x, y, s) clear (z, s) x z on (x, z, Result (move (x, y, z), s)) Explanation: If the left side (condition) of the axiom holds, then the action can be performed, and the right side (consequence) follows. The consequence states what is true in the resulting situation, here: on(x,z)

8 action move (x, y, z) Frame Axiom Frame Axiom - example : on (x, y, s) x u on (x, y, Result (move (u, v, z), s)) Explanation: A Frame Axiom states what remains true or unaffected when an action is performed. In the example here: the blocks x, y which are not moved remain where they are, on (x, y) Action Description - Axioms Axioms specify what changes and what remains. Consider every combination of action and fluent. effect-axioms specify effects, what changes positive effects state what becomes true negative effects state what becomes false frame-axioms specify frame, what remains positive effects state what remains true negative effects state what remains false In addition, general axioms specify general laws or rules of the domain.

9 Action move (x, y, z) Effect Axioms effect-axioms specify change (for pair move-on) positive effect on (x, y, s) clear (x, s) clear (z, s) y z on (x, z, Result (move (x, y, z), s)) If x is on y and both x and z are clear, then the moveaction can be performed and the result is that x is on z. negative effect on (x, y, s) clear (x, s) clear (z, s) y z on (x, y, Result (move (x, y, z), s)) If x is on y and both x and z are clear, then the moveaction can be performed and the result is that x is not anymore on y. Frame Axioms Action move (x, y, z) frame-axioms specify frame, i.e. what remains positive frame on (x, y, s) x u on (x, y, Result (move (u, v, z), s)) If a block x is on another block y, and x is not moved, then it stays on y. negative frame on (x, y, s) (x u y z) on (x, y, Result (move (u, v, z), s)) If a block x is not on another block y, and x is not moved, nor is something put on y, then x will still not be on y after the move.

10 Situation Calculus Successor-State Axioms successor-state-axioms combine frame and effect axioms; specified for each fluent general structure predicate expression true in follow state the action made it true; it was true and the action did not make it false. Successor-State Axiom - Example Successor-state axiom for on(x,y,s) x,y,s,a: on (x, y, Result(a,s)) [a = move(x,z,y) clear(x,s) clear(z,s) y z ] [on(x,y,s) a move(x,z,y) ] First part specifies action achieving on(x,y) Second part specifies actions not deleting on(x,y)

11 General axioms Situation Calculus - General Axioms Formulas which are true in all situations or states. Example: x, y, s: on (x, y, s) (y=table) clear (y, s) For all situations s and all objects x and y: if something is on object y in s, and y is not the table, then y is not clear in s. s: clear (Table, s) The table (or floor) is always clear. Situation Calculus - Problems Frame-Problem specify everything which remains stable Leads to too many conditions which would have to be explicitly stated for any state transformation. Computationally very expensive. Approach: successor-state axioms; STRIPS Qualification-Problem specify everything which is precondition to an action Difficult to include every precondition which could prevent (if not fulfilled) the action to be performed). Approach: non-monotonic reasoning with standard preconditions and effects as defaults.

12 Situation Calculus - Problems Ramification-Problem derived formulae conflict between change and frame Some axioms state conclusions about fluents indirectly affected by actions. This can contradict frame-axioms. Example: An agent grabs an object and holds it. When the agent moves, the object moves too, though not explicitly stated. Rule 1: every object stays where it is unless it is moved. Rule 2: if an object is attached to another object and one of the objects moves, the other object moves too. Approach: Integrate TMS for derived formulae. Situation Calculus and Planning Planning starts with a specified start situation and the specification of a goal situation. Planning comprises of finding a proof which infers the goal situation from the start situation using successor-state and other axioms. A Plan is a sequence of actions which specifies a sequence of transformations of situations from the initial situation to the final situation.

13 Additional References Nils J. Nilsson: Artificial Intelligence A New Synthesis. Morgan Kaufmann, San Francisco, 1998.