L7_exAnswer and explanation Review Search vs. planning Situation calculus STRIPS operators

1. L9. Planning Agents • L7_exAnswer and explanation • Review • Search vs. planning • Situation calculus • STRIPS operators

2. L7_ex 解答例

3. Forward chaining example Let us add facts r1, r2, r3, f1, f2, f3 in turn into KB. r1. Buffalo(x) Pig(y)  Faster(x,y) r2.Pig(y)  Slug(z)  Faster(y,z) r3. Faster(x,y) Faster(y,z)  Faster(x,z) f1. Buffalo(Bob) [r1-c1, Bob/x, yes] f2. Pig(Pat) [r1-c2, Pat/y, yes] f4. Faster(Bob, Pat) f3. Slug(Steve) [r2-c2, Steve/z, yes] [r2, f2, f3, Pat/y, Steve/z, yes]  f5. Faster(Pat, Steve) [r3, f4, f5, Bob/x, Pat/y, Steve/z, yes]  f6. Faster(Bob, Steve)  x, y, z

4. Backward chaining example Bob is a buffalo | 1. Buffalo(Bob) --f1 Pat is a pig | 2. Pig(Pat) --f2 Buffaloes run faster than pigs| 3.  x, y Buffalo(x) Pig(y)  Faster(x,y) --r1 Goal: to prove Faster(Bob, Pat) Faster(x, y) r1 Buffalo(x)  Pig(y) R(2) – And Elimination Buffalo(x) Pig(y) R(8) – Universal Elimination R(8) – Universal Elimination {x/Bob} {y/Pat} Buffalo(Bob) Pig(Pat) {} {}

5. Search in problem solving a b c • Problem solution: • A path through the state space tree • State space search: • Search is a traversal of the tree • until the goals is reached. • State transitions is performed • by operators a b c

6. Search in problem solving Difficulty using standard search algorithms Standard search algorithms seems to fail miserably since the goal test is inadequate. What is “finish”?

7. Search in problem solving Planning problem in situation calculus • Consider the same task get milk, bananas, and a cordless drill • A planning problem is represented in situation calculus by logical sentences that describe the three main parts of a problem. • Initial state: • At(home, S0)  Have(Milk, S0)  Have(Banana, S0)  Have(Drill, S0) • Goal state: •  s At(home, s)  Have(Milk, s)  Have(Bananas, s)  Have(Drill, s) • Operators: • a, s Have(Milk, Result (a,s)) [ (a=Buy(Milk)) At(Supermarket, s) •  (Have(Milk, s) a Drop(Milk) )] • Plan: • p = [Go(Supermarket), Buy(Milk), Buy(Bananas), Go(HWS), …]

8. Search in problem solving Problem solving vs. planning • Representation of states • PS: Direct assignment of a symbol to each state • PL: Logic sentences • Representation of goals • PS: A goal state symbol • PL: Sentences that describe objective • Representation of actions • PS: Operators that transform one state symbol into another • PL: Addition/deletion of logic sentences describing world state • Representation of plans • PS: Path through state space • PL: Ordered or partially-orderer sequence of actions.

9. Search in problem solving Advantages of Planning Systems • Uniform language for describing states, goals, actions, • and their effects. • Ability to add actions to a plan whenever they are needed, • not just in an incremental sequence from some initial state. • Ability to capture the fact that most parts of the world are • independent of most other parts. • It performs better for complex worlds over standard search • algorithm since searching space becomes huge when there • are many initial states and operators in standard search • algorithms.

10. The Situation Calculus Wff - well formed formula • A goal can be described by a wff:  x On(x, B) • if we want to have a block on B • Planning: finding a set of actions to achieve a goal wff. • Situation Calculus (McCarthy, Hayes, 1969, Green 1969) • A Predicate Calculus formalization of states, actions, and their effects. • Sostate in figure can be described by: On(B, A)  On(A, C)  On(A, Fl)  Clear(B) we reify the state and include them as arguments • The atoms denotes relations over states.On(B, A, S0)  On(A, C, S0)  On(C, Fl, S0)  Clear(B, S0) • We can also have.x, y, s On(x, y, s)  (y = Fl) Clear(y, s) s Clear(Fl, s)

11. state action Representing actions • Reify the actions: denote an action by a symbol • actions are functions • move(B,A,Fl): move block B from block A to Fl • move (x,y,z) - action schema • do: A function constant, do denotes a function that maps actions and states into states • Express the effects of actions. • Example: (on, move) (expresses the effect of move on On) • positive effect axiom:

12. Effect axioms for (clear, move) move(x, y, z) matching? precondition are satisfied with B/x, A/y, S0/s, F1/z what was true in S0 remains true figure 21

13. Frame axioms A fluent is something that flows and changes across situations • Not everything true can be inferredSuch as, On(C,Fl) remains true but cannot be inferred • Actions have local effect • We need frame axioms for each action and each fluent that does not change as a result of the action • example: frame axioms for (move, on) • If a block is on another block and move is not relevant, it will stay the same. • Positive • Negative • Frame axioms for (move, clear)

14. STRIPS: describing goals and state STRIPS: STanford Research Institute Planning System • Basic approach in GPS (general Problem Solver): • Find a “difference” (Something in G that is not provable in S0) • Find a relevant operator f for reducing the difference • Achieve precondition of f; apply f; from resultant state, achieve G.

15. STRIPS planning • STRIPS uses logical formulas to represent the states • S0, G, P, etc: • Description of operator f:

16. A STRIPS planning example • On(B,A) • On(A,C) • On(C,F1) • Clear(B) • Clear(Fl) • The formula describes a set of world states • Planning search for a formula satisfying a goal description • On(A, C) • On(C, Fl) • On(B, Fl) • Clear(A) • Clear(B)

17. STRIPS Description of Operators • A STRIPS operator has 3 parts: • A set, PC - preconditions • A set D - the delete list • A setA - the add list • Usually described by Schema: Move(x,y,z) • PC: On(x,y) and On(Clear(x) and Clear(z) • D: Clear(z) , On(x,y) • A: On(x,z), Clear(y), Clear(F1) • A state S1 is created applying operator O by adding A and deleting D from S1.

18. Example: The move operator

19. Example1: The move operator G S0  f(P)->G S0->P    x/B, y/A, z/Fl x/B, y/A, z/Fl   On(x,y) Clear(x) Clear(z) f: move(x,y, z) add: On(x,z), Clear(y) del: On(x,y), Clear(z) On(x,z) Clear(x) Clear(y) P: f(P):

20. ABC Example1: The move operator + Clear(F2) G S0  f(P)->G S0->P    x/B, y/A, z/Fl x/B, y/A, z/Fl   On(x,y) Clear(x) Clear(z) f: move(x,y, z) add: On(x,z), Clear(y) del: On(x,y), Clear(z) On(x,z) Clear(x) Clear(y) P: f(P):

21. STRIPS algorithms are to search for operators!!! We will leave the discussion of STRIPS algorithms to next week.

22. ? Quiz： What differences are between standard search algorithms for problem solving and planning systems?