Planning

1. Planning • Points • Elements of a planning problem • Planning as resolution • Conditional plans • Actions as preconditions and effects • Goal regression

2. Elements of a planning problem • When is a problem a good application for planning? • It can be decomposed into subproblems in such a way that solving each subproblem (in some order) solves the whole problem. • It is conjunctive, that is, it can be expressed as a conjunction of conditions, each representing an elementary aspect if the world, the initial situation, the final situation. • Each subproblem can be reduced to facts about domain objects, and relations among such objects.

3. Elements of a planning problem (2) • States, state spaces. • Actions with preconditions and postconditions. • An action is a function from states to states. It makes local changes that affect few objects. • Actions are generic, facts are specific. • Actions are elementary: one-step plans. • A design decision: granularity of actions. • Conditional actions: If( C, A1, A2 ). • Frame axioms and the Closed World Assumption. • The timing of actions: is a real-time plan needed, or do we only need a sequence of actions with relative time?

4. Elements of a planning problem (3) • A plan is a sequence of actions that lead from an initial state to a goal state. • In a linear plan, the actions required to solve a subproblem precede the actions required to solve the next subproblem. • In a non-linear plan, actions for subproblems may be (or even may have to be) interleaved. • Optimality of plans: a simple criterion is the number of actions. • A plan may be executed either “in batch mode” (after the whole plan has been created), or with every action performed immediately (with feedback for more planning).

5. Elements of a planning problem (4) • A formal representation of plans may look, for example, like this: • Do( Action3, Do( Action2, Do( Action1, init ) ) ) • where init denotes an initial state. • There are two ways of implementing state changes (we must be able to undo such changes, because planning algorithms usually backtrack). • Explicit changes in the knowledge base. • Implicit changes, not in the knowledge base but only in the plan representation, recomputed as needed (this is costly, but undoing is easier).

6. Planning as resolution • The blocks world(find its description in any textbook ) • T( p, s ) means "predicate p is true in state s" • Elementary conditions: On, OnTbl and Clear. • Examples of actions: Stack, Unstack and NoOp. • T( OnTbl( x ), s ) T( Clear( x ), s )  T( Clear( y ), s )  x ≠ y  T( On( x, y ), Do( Stack( x, y ), s ) ) • T( On( x, y ), s )  T( Clear( x ), s ) T( OnTbl( x ), Do( Unstack( x, y ), s ) )  T( Clear( y ), Do( Unstack( x, y ), s ) ) • T( p, s )  T( p, Do( NoOp, s ) )

7. Planning as resolution (2) • On means “sitting on another block”. • OnTbl means “sitting on the table”. • Some facts about this worlds are always true.For example: • T( OnTbl( x ), s )    y T( On( x, y ), s) [=] • Frame axioms — two examples: • T( Clear( u ), s )  u ≠ y  T( Clear( u ), Do( Stack( x, y ), s ) ) • T( On( u, w ), s )  u ≠ x  T( On( u, w ), Do( Unstack( x, y ), s ) )

8. A S0 S1 C B C A B Planning as resolution (3) • An initial state • T( Clear( C ), S0 ) T( On( C, A ), S0 ) • T( Clear( B ), S0 ) • T( OnTbl( A ), S0 ) T( OnTbl( B ), S0 ) • Another (possibly final) state • T( Clear( A ), S1 ) T( On( A, B ), S1 ) • T( On( B, C ), S1 ) T( OnTbl( C ), S1 )

9. Plan construction by resolution(a very simple example) Planning as resolution (4) • We want to prove thatT( OnTbl( A ), t ) given S1. • We assume that we can do it in one action , so that state tshould look likeDo( , S1 ). • If this is not possible, we will try Do( , Do( , S1 ) ),Do( , Do( , Do( , S1 ) ) ), and so on. • Let’s negate the fact that we want proven. •  T( OnTbl( A ), Do( , S1 ) ) • Action  could be Unstack. That is because one of the clauses in its clausal-form definition is this: •  T( On( x, y ), s ),  T( Clear( x ), s ),T( OnTbl( x ), Do( Unstack( x, y ), s ) )

10. Plan construction by resolution(continued) Planning as resolution (5) •  T( OnTbl( A ), Do( , S1 ) ) • T( OnTbl( x ), Do( Unstack( x, y ), s ) ), T( On( x, y ), s ),  T( Clear( x ), s ) • This gives •  T( On( A, y ), S1 ),  T( Clear( A ), S1 ) • after assigning x  A, sS1, Unstack( A, y ). • This is given: T( On( A, B ), S1 ). • We resolve and get  T( Clear( A ), S1 ) with y  B. • This too is given: T( Clear( A ), S1 ). • so we produce the empty resolvent: done. • tDo( , S1 ) = Do( Unstack( A, B ), S1 )

11. Conditional plans • Let T( Clear( A ), S2 ) be the only given fact. • We can find a plan for the goal • T( OnTbl( A ), Do( , S2 ) ) • even though the problem is under-constrained. • Axioms for conditional actions • T( p, s )  T( q, Do( , s ) )  T( q, Do( If( p, ,  ), s ) ) • ¬ T( p, s )  T( q, Do( , s ) )  T( q, Do( If( p, ,  ), s ) ) • Again, we begin by assuming a one-step plan, but now it will be a conditional plan.

12. Conditional plans (2) • ¬ T( OnTbl( A ), Do( , S2 ) ) • Assume If( p, ,  ). • ¬ T( OnTbl( A ), Do( If( p, ,  ), S2 ) ) • The first axiom as a clause: [+] • T( q, Do( If( p, ,  ), s ) ),¬ T( p, s ), ¬ T( q, Do( , s ) ) • Resolve with q OnTbl( A ), sS2 and get • ¬ T( p, S2 ), ¬ T( OnTbl( A ), Do( , S2 ) ) • We need again that axiom for Unstack: • T( OnTbl( x ), Do( Unstack( x, y ), s ) ), T( On( x, y ), s ),  T( Clear( x ), s )

13. Conditional plans (3) • ¬ T( p, S2 ), ¬ T( OnTbl( A ), Do( , S2 ) ) • T( OnTbl( x ), Do( Unstack( x, y ), s ) ), T( On( x, y ), s ),  T( Clear( x ), s ) • With x  A, Unstack( A, y ), sS2 we get • ¬ T( p, S2 ),  T( On( A, y ), S2 ),  T( Clear( A ), S2 ) • We resolve this with our only fact: • ¬ T( p, S2 ),  T( On( A, y ), S2 ) • ... and this is it. We need something new to move on. • One possibility is factoring: look for matching literals on the resolvent. Here we can assign p On( A, y ) and reduce the resolvent to  T( On( A, y ), S2 ).

14. Conditional plans (4) • We can conclude the proof if we assume • T( On( A, y ), S2 ) • Here, the conditional nature of the plan helps.We combine everything we have found so far.Our one-action plan looks like this: • Do( If( On( A, y ), Unstack( A, y ),  ), S2 ) • To also determine , we return to the point marked [+] and look at the second axiom for conditional actions: • T( q, Do( If( p, ,  ), s ) ), T( p, s ), ¬ T( q, Do( , s ) ) • ¬ T( OnTbl( A ), Do( , S2 ) ) • Resolve with q OnTbl( A ), sS2 and get • T( p, S2 ), ¬ T( OnTbl( A ), Do( , S2 ) )

15. Conditional plans (5) • T( p, S2 ), ¬ T( OnTbl( A ), Do( , S2 ) ) • The clausal form of action NoOp: • ¬ T( p, s ), T( p, Do( NoOp, s ) ) • Assign p OnTbl( A ),   NoOp and get this: • T( p, S2 ), ¬ T( OnTbl( A ), S2 ) • It is time to recall that useful fact [=] about OnTbl: • T( OnTbl( x ), s )    y T( On( x, y ), s) • So, we can replace one side with the other: • T( p, S2 ),  y T( On( A, y ), S2) • Skolemize ( is the Skolem constant): • T( p, S2 ), T( On( A,  ), S2)

16. Conditional plans (6) • T( p, S2 ), T( On( A,  ), S2) • We perform factorization, assigning p On( A,  ): • T( On( A,  ), S2) • To conclude the second part of the proof, we assume •  T( On( A,  ), S2 ) • Now, our one-action plan is either of these: • Do( If( On( A, y ), Unstack( A, y ),  ), S2 ) • Do( If( On( A,  ), , NoOp ), S2 ) • We can finally put these partial plans together: • Do( If( On( A,  ), Unstack( A,  ), NoOp ), S2 )

17. Another representation of actions • Preconditions of an action + • facts added (made true) by this action + • facts deleted (made false) by this action. • Pre( Unstack( x, y ) ) = { On( x, y ), Clear( x ) } • Add( Unstack( x, y ) ) = { OnTbl( x ), Clear( y ) } • Del( Unstack( x, y ) ) = { On( x, y ) } • The Closed World Assumption serves as the frame axiom: • facts not on the Add list and on the Del listare by default preserved. • A state is represented as a set of facts, for example: • { OnTbl( A ), OnTbl( B ) }

18. Goal regression • Reduce one of the goals to its subgoal or subgoals. • This means working backward from a state to a state by looking at the preconditions and effects of some action. • Let Regr( q,  ) be a state achieved by regression from q through  (“undoing” the effects of ). •  cannot delete a fact that must be true in q: • q  Del(  ) = Ø • If  has this property, we can define: • Regr( q,  ) = Pre(  )  (q — Add(  ) ) • So: keep the preconditions of , undo the effects of .

19. Goal regression (2) • Example • If we begin with • { OnTbl( A ), OnTbl( B ) } • we can “recreate” • { Clear( A ), On( A, B ), OnTbl( B ) } • by regression through Unstack( A, B ). • Protection violation • Actions for a subproblem must not undo the effects of earlier actions for already solved subproblems.

20. A planning strategy for conjunctive problems • Going left to right, make a plan for each conjunct and freeze the effects (put the added or preserved goals on a list of protected goals). • This simple method is weak. It will not work even for the following trivial problem. • SinitialOn( C, A ) OnTbl( A ) OnTbl( B ) • SfinalOn( C, A ) On( A, B ) • On( C, A ) is true in Sinitial and must be protected. But: it must be temporarily destroyed to achieve On( A, B ). Goal protection makes planning impossible. • Here, rearrangement helps: On( A, B ) On( C, A ) “works”. In general non-linear planning is necessary. • One such a smarter planner is Warplan.