Language for planning problems• STRIPS: STanford Research Institute Problem Solver• world described by logical conditions• state as conjunction of positive literals

– propositional; e.g., Happy ^ Hungry to represent the state of the agent

– first-order ground and function-free terms; • e.g., At(Plane1, Verona) ^ At(Plane2,Malpensa)

• closed-world assumption; i.e., any not mentioned condition is false

• goal is a partially specified state– a state satisfies a goal if contains all the literals of the goal– e.g. state At(Plane1, Verona) ^ At(Plane2,Malpensa) satisfies goal

At(Plane2,Malpensa)

STRIPS actions (contd.)• Tidily arranged actions descriptions, restricted

language• Action schema:

– ACTION: specifies name and parameter list• Buy(x)

– PRECONDITION: conjunction of positive literals• At(p) ^ Sells(p, x)

– EFFECT: conjunction of literals (positive or negative)• Have(x)

– [Note: no information on how to execute the action!]• A complete set of STRIPS operators can be

translated into a set of successor-state axioms

Example: Air cargo transport• predicates In(·, ·),At(·, ·)• type predicates: Cargo(·), Plane(·),Airport(·)• Start

– At(C1, SFO) ^ At(C2, JFK) ^ At(P1, SFO) ^ At(P2, JFK) ^ Cargo(C1) ^ Cargo(C2) ^ Plane(P1) ^ Plane(P2) ^ Airport(JFK) ^ Airport(SFO)

• Goal– At(C1, JFK) ^ At(C2, SFO)

• Actions– Load(c, p, a)– Unload(c, p, a)– Fly(p, from, to)

Example: Air cargo transport• predicates In(·, ·),At(·, ·)• type predicates: Cargo(·), Plane(·),Airport(·)• Start

– At(C1, SFO) ^ At(C2, JFK) ^ At(P1, SFO) ^ At(P2, JFK) ^ Cargo(C1) ^ Cargo(C2) ^ Plane(P1) ^ Plane(P2) ^ Airport(JFK) ^ Airport(SFO)

• Goal– At(C1, JFK) ^ At(C2, SFO)

• Actions– Load(c, p, a)– Unload(c, p, a)– Fly(p, from, to)

Example: Air cargo transport• Actions

– Load(c, p, a)• PRE: At(c, a) ^ At(p, a) ^ Cargo(c) ^ Plane(p) ^

Airport(a)• EFF: ¬At(c, a) ^ In(c, p)

– Unload(c, p, a)• PRE: In(c, p) ^ At(p, a) ^ Cargo(c) ^ Plane(p) ^

Airport(a)• EFF: At(c, a) ^ ¬In(c, p)

– Fly(p, from, to)• PRE: At(p, from) ^ Plane(p) ^ Airport(from) ^

Airport(to)• EFF: ¬At(p, from) ^ At(p, to)

Example: Air cargo transport (contd.)

• a solution is– Load(C1, P1, SFO),– Fly(P1, SFO, JFK), – Unload(C1, P1, JFK),– Load(C2, P2, JFK), – Fly(P2, JFK, SFO), – Unload(C2, P2, SFO)

STRIPS Planning• STRIPS planning problem:

– find a sequence of actions that lead to a goal

– states and goals are defined by a conjunctions of literals

• State-space search– Forward search (goal progression) from the initial state

try to reach the goal

– Backward search (goal regression) from the goal and try to project it to the initial state

• Plan-space search– partial-order planning (POP) search the space of

partially build plans

State-space search• planning problem defines the search problem

– initial state is the start state

– goal test checks whether state satisfies the goal

– actions define the operators

– step cost is usually 1

• (a) forward or • (b) backward search

Forward search

• main search loop– select an action and unify precondition with the

state– if precondition is satisfied, apply the action

generating a new state– check whether the new state satisfies the goal

Forward search with the Shopping Example

• Start– At(Home) ^ Sells(SM,Milk) ^ Sells(SM,Banana) ^

Sells(HWS,Drill)^Loc(Home)^Loc(SM)^Loc(HWS)

• Buy(x)– PRE: At(store), Sells(store, x)– EFF: Have(x)

• Go(x, y)– PRE: At(x),Loc(y)– EFF: At(y),¬At(x)

• Goal– Have(Milk) ^ Have(Banana) ^ Have(Drill)

Forward search

• main search loop– select an action and unify precondition with the state

– if precondition is satisfied, apply the action generating a new state

– check whether the new state satisfies the goal

• state space is finite -> complete for complete search algorithms

• inefficient because of irrelevant actions -- needs good heuristics

Backward search• idea: select relevant actions only starting from the goal• action is

– relevant when it achieves one of the conjuncts (add-list)– consistent when doesn’t undo any desired literal (delete-list)

• select a relevant and consistent action and generate new state1. delete add-list2. add preconditions

• terminates with a state satisfied by the initial state• branching on relevant and consistent states

Backward search with the Shopping Example

• Start– At(Home) ^ Sells(SM,Milk) ^ Sells(SM,Banana) ^

Sells(HWS,Drill)^Loc(Home)^Loc(SM)^Loc(HWS)

• Buy(x)– PRE: At(store), Sells(store, x)– EFF: Have(x)

• Go(x, y)– PRE: At(x),Loc(y)– EFF: At(y),¬At(x)

• Goal– Have(Milk) ^ Have(Banana) ^ Have(Drill)

State-space vs. plan-space• forward and backward search explore linear sequences of

actions

• this is not necessary– consider the solution for the air cargo transport

• Load(C1, P1, SFO), Fly(P1, SFO, JFK),Unload(C1, P1, JFK),

• Load(C2, P2, JFK), Fly(P2, JFK, SFO),Unload(C2, P2, SFO)

– the only required ordering is among Load, Fly and Unload

– connected by causal effects; e.g., Load(C1, P1, SFO) achieve In(C1,P1), used by the precondition of Unload

– there’s no need to put Load(C2, P2, JFK) after Unload(C1, P1, JFK)

• partial-order plan: graph of actions including Start and Finish

Example

Partially ordered plans

• Partially ordered collection of steps with– Start step has the initial state description as its effect– Finish step has the goal description as its precondition– causal links from outcome of one step to precondition of another– temporal ordering between pairs of steps

• Open condition = precondition of a step not yet causally linked

• plan is complete iff every precondition is achieved• precondition is achieved iff it is the effect of an earlier step

and no possibly intervening step undoes it (conflict)• plans cannot contain cycles

Define the Start State and the Goal State

Define the Start State and the Goal State•Start

On(A,Table), On(B,Table), On(C,B), Clear(A), Clear(C)•End

•On(A,B), On(B,C), On(C,Table)

Define the Actions

Use either forward or backward search to build the plan for this problem.