12/5/18 Overview State Spaces & Partial-Order Planning • What is planning? AI Class 22 (Ch. 10 through 10.4.4 ) • Approaches to planning • GPS / STRIPS • Situation calculus formalism [revisited] • Partial-order planning Material from Dr. Marie desJardin, Some material adopted from notes by Andreas Geyer-Schulz and Chuck Dyer Planning Problem Planning Problem • put on right shoe • put on left shoe • put on pants • What is the planning problem? • What is the planning problem? • put on right sock • put on left sock • Find a sequence of actions that achieves a goal • Find a sequence of actions that achieves a goal • put on shirt when executed from an initial state . when executed from an initial state . • pants off • right shoe off • That is, given • That is, given • right sock off • A set of operators (possible actions) • A set of operators (possible actions) • right shoe off • An initial state description • An initial state description (etc) • A goal (description or conjunction of predicates) • A goal (description or conjunction of predicates) • pants on • Compute a sequence of operations: a plan . • Compute a sequence of operations: a plan . (etc) Typical Assumptions (1) Typical Assumptions • Atomic time : Each action is indivisible • Agent is the sole cause of change in the world • Can’t be interrupted halfway through putting on pants • Nobody else is putting on your socks • No concurrent actions allowed • Agent is omniscient: • Can’t put on socks at the same time • Has complete knowledge of the state of the world • Deterministic actions • Closed world assumption : • The result of actions are completely known – no uncertainty • Everything known-true about the world is in the state description • Anything not known-true is known-false 1
12/5/18 Blocks world Blocks World The blocks world consists of a table, set of blocks, and a • A micro-world robot gripper • Some domain constraints: Some domain constraints: • Only one block can be • Only one block on another block on another block • Any number of blocks on table • Hand can only hold one block • Any number of blocks can be on the table B Typical representation: ontable(a) handempty A C • The hand can only hold Meant to be a simple model! one block ontable(c) on(b,a) TABLE Try demo at: clear(b) clear(c) http://aispace.org/planning/ Typical BW planning problem Typical BW planning problem assertions atomic Initial state: Initial state: describing robot clear(a) clear(a) a state actions clear(b) clear(b) clear(c) clear(c) ontable(a) ontable(a) Plan: ontable(b) A C B ontable(b) A C B pickup(b) ontable(c) ontable(c) stack(b,c) handempty handempty pickup(a) Goal state: Goal state: stack(a,b) on(b,c) A on(b,c) A on(a,b) on(a,b) B B ontable(c) ontable(c) C C Major Approaches Planning vs. problem solving • GPS / STRIPS • Planning vs. problem solving: can often solve similar problems • Planning is more powerful and efficient because of the • Situation calculus representations and methods used • Partial order planning • States, goals, and actions are decomposed into sets of sentences (usually in first-order logic) • Hierarchical decomposition (HTN planning) • Search often proceeds through plan space rather than state • Planning with constraints (SATplan, Graphplan) space (though there are also state-space planners) • Sub-goals can be planned independently, reducing the • Reactive planning complexity of the planning problem 2
12/5/18 Yet Another BW planning problem Another BW planning problem Plan: Initial state: Initial state: unstack(c,b) clear(a) clear(c) A plan putdown(c) clear(b) ontable(a) pickup(a) unstack(b,a) clear(c) stack(a,b) on(b,a) putdown(b) ontable(a) unstack(a,b) backtracking { putdown(b) on(c,b) ontable(b) A C B A C B putdown(a) pickup(a) ontable(c) handempty pickup(b) stack(a,b) handempty stack(b,c) unstack(a,b) Goal: pickup(a) putdown(a) Goal: on(a,b) stack(a,b) pickup(b) on(a,b) A A on(b,c) stack(b,c) on(b,c) B B pickup(a) ontable(c) ontable(c) C C stack(a,b) “Get a quart of milk, a bunch of bananas Planning as Search and a variable-speed cordless drill.” Slightly more complex KB: • Can think of planning as a search problem • Actions: generate successor states • States: completely described & only used for successor generation, heuristic fn. evaluation & goal testing • Goals: represented as a goal test and using a heuristic function • Plan representation: unbroken sequences of actions forward from initial states or backward from goal state Treating planning as a search problem isn’t very efficient! General Problem Solver Situation Calculus Planning • The General Problem Solver (GPS) system • Intuition: Represent the planning problem using • An early planner (Newell, Shaw, and Simon) first-order logic • Generate actions that reduce difference between current state • Situation calculus lets us reason about changes in the world and goal state • Use theorem proving to show (“prove”) that a sequence of actions will lead to a desired result, when applied to a world • Uses Means-Ends Analysis state / situation • Compare what is given or known with what is desired • Select a reasonable thing to do next • Use a table of differences to identify procedures to reduce differences • GPS is a state space planner • Operates on state space problems specified by an initial state, some goal states, and a set of operations 3
12/5/18 Situation Calculus Planning, cont. Situation Calculus Planning, cont. • Initial state : • Initial state : a logical sentence about (situation) S 0 At(Home, S 0 ) ∧ ¬ Have(Milk, S 0 ) ∧ ¬ Have(Bananas, S 0 ) ∧ ¬ Have(Drill, S 0 ) • Goal state: usually a conjunction of logical sentences • Goal state : ( ∃ s) At(Home,s) ∧ Have(Milk,s) ∧ Have(Bananas,s) ∧ Have(Drill,s) • Operators : descriptions of how the world changes as a • Operators: result of the agent’s actions: ∀ (a,s) Have(Milk,Result(a,s)) ⇔ � • Result( a,s ) names the situation resulting from executing ((a=Buy(Milk) ∧ At(Grocery,s)) ∨ (Have(Milk, s) ∧ a ≠ Drop(Milk))) action a in situation s . • Result(a,s) : situation after executing action a in situation s • Action sequences are also useful: ( ∀ s) Result’([ ],s) = s • Result’(l,s) : result of executing list of actions l starting in s ( ∀ a,p,s) Result’([a|p]s) = Result'(p,Result(a,s)) p=plan Situation Calculus, cont. Situation Calculus: Blocks World • Example situation calculus rule for blocks world: • Solution: a plan that when applied to the initial state • clear(X, Result(A,S)) ↔ � gives a situation satisfying the goal query : [clear(X, S) ∧ � ( ¬ (A=Stack(Y,X) ∨ A=Pickup(X)) � At(Home, Result'(p,S 0 )) ∨ (A=Stack(Y,X) ∧ ¬ (holding(Y,S)) � ∧ Have(Milk, Result'(p,S 0 )) ∨ (A=Pickup(X) ∧ ¬ (handempty(S) ∧ ontable(X,S) ∧ clear(X,S))))] � ∨ [A=Stack(X,Y) ∧ holding(X,S) ∧ clear(Y,S)] � ∧ Have(Bananas, Result'(p,S 0 )) ∨ [A=Unstack(Y,X) ∧ on(Y,X,S) ∧ clear(Y,S) ∧ handempty(S)] � ∧ Have(Drill, Result'(p,S 0 )) ∨ [A=Putdown(X) ∧ holding(X,S)] • Thus we would expect a plan (i.e., variable assignment • English translation: a block is clear if through unification) such as: (a) in the previous state it was clear AND we didn’t pick it up or stack something on it successfully, or p = [Go(Grocery), Buy(Milk), Buy(Bananas), Go(HardwareStore), � Wow. (b) we stacked it on something else successfully, or Buy(Drill), Go(Home)] (c) something was on it that we unstacked successfully, or (d) we were holding it and we put it down. Situation Calculus Planning: Analysis Basic Representations for Planning • Classic approach first used in the STRIPS planner circa 1970 • Fine in theory, but: • Problem solving (search) is exponential in the worst case • States represented as conjunction of ground literals • at(Home) ∧ ¬ have(Milk) ∧ ¬ have(bananas) ... • Resolution theorem proving only finds a proof (plan), not necessarily a good plan • Goals are conjunctions of literals, but may have variables* • at(?x) ∧ have(Milk) ∧ have(bananas) ... • So what can we do? • Don’t need to fully specify state • Restrict the language • Un-specified: either don’t-care or assumed-false • Blocks world is already pretty small… • Represent many cases in small storage • Use a special-purpose planner rather than general • Often only represent changes in state rather than entire situation theorem prover • Unlike theorem prover, not finding whether the goal is true , but whether there is a sequence of actions to attain it *generally assume ∃ 4
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