Plan-Space Planning: Overcoming Search Limitations in Automated Planning, Slides of Computer Science

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Chapter 5

Plan-Space Planning

Lecture slides for

Automated Planning: Theory and Practice

Motivation

• Problem with state-space search
  • In some cases we may try many different orderings of the same actions before realizing there is no solution
• Least-commitment strategy: don’t commit to

orderings, instantiations, etc., until necessary

a b c b a

b a^ b a c b c (^) a c b

goal

dead end dead end

dead end dead end dead end dead end

Plan-Space Planning - Basic Idea

• Backward search from the goal
• Each node of the search space is a partial plan
  • A set of partially-instantiated actions
  • A set of constraints
• Make more and more refinements, until we have a solution
• Types of constraints:
  • precedence constraint : a must precede b
  • binding constraints :
    • inequality constraints, e.g., v 1 ≠ v 2 or v ≠ c
    • equality constraints (e.g., v 1 = v 2 or v = c ) and/or substitutions
  • causal link :
    • use action a to establish the precondition p needed by action b
• How to tell we have a solution: no more flaws in the plan
  • Will discuss flaws and how to resolve them

foo( x ) Precond: … Effects: p( x )

bar(y) Precond: ¬p( y ) Effects: …

baz( z ) Precond: p( z ) Effects: …

p( z )

x ≠ y

x = z

Flaws: 1. Open Goals

• Open goal:

• An action a has a precondition p that we haven’t

decided how to establish

• Resolving the flaw:

• Find an action b
  • (either already in the plan, or insert it)
• that can be used to establish p
  • can precede a and produce p
• Instantiate variables and/or

constrain variable bindings

• Create a causal link

foo( x ) Precond: … Effects: p( x )

baz( z ) Precond: p( z ) Effects: …

p( z )

foo( x ) Precond: … Effects: p( x )

baz( x ) Precond: p( x ) Effects: …

p( x )

The PSP Procedure

• PSP is both sound and complete
• It returns a partially ordered solution plan
  • Any total ordering of this plan will achieve the goals
  • Or could execute actions in parallel if the environment permits it

Example

• Similar (but not identical) to an example in Russell and Norvig’s Artificial Intelligence: A Modern Approach (1st edition)
• Operators:
  • Start Precond: none Effects: At(Home), sells(HWS,Drill), Sells(SM,Milk), Sells(SM,Banana)
  • Finish Precond: Have(Drill), Have(Milk), Have(Banana), At(Home)
  • Go( l,m ) Precond: At( l ) Effects: At( m ), ¬At( l )
  • Buy( p , s ) Precond: At( s ), Sells( s , p ) Effects: Have( p )

Start and Finish are dummy actions that we’ll use instead of the initial state and goal

Example (continued)

• The first three refinement steps
  • These are the only possible ways to establish the Have preconditions

At( s 1 ) At( s 2 )^ At( s 3 )

Buy(Drill, s 1 ) Buy(Milk, s 2 ) Buy(Bananas,^ s 2 )

Have(Drill)

Start

Buy(Drill, s 1 )

Finish

Have(Milk) (^) Have(Bananas)

Sells( s 1 , Drill) Sells( s 2 ,Milk)^ Sells( s 3 ,Bananas)

At(Home)

Why don’t we use Start to establish At(Home)?

Example (continued)

• Three more refinement steps
  • The only possible ways to establish the Sells preconditions

At(HWS) At(SM)^ At(SM)

Buy(Drill, s 1 ) (^) Buy(Milk, SM) Buy(Bananas, SM)

Have(Drill)

Start

Buy(Drill, HWS)

Finish

Have(Milk) (^) Have(Bananas)

Sells(HWS,Drill) Sells(SM,Milk)^ Sells(SM,Bananas)

At(Home)

Example (continued)

• Finally, a nondeterministic choice: how to resolve the threat to At( s 1 )?
  • Our choice: make Buy(Drill) precede Go(SM)
  • This also resolves the other two threats

At(SM)

Buy(Drill, s 1 )

Have(Drill)

Start

Finish

Have(Milk) (^) Have(Bananas)

Sells(SM,Milk) (^) Sells(SM,Bananas)

Go( l 2 , SM)

At(Home)

At( l 2 ) At( l 1 )

Go( l 1 ,HWS)

At(HWS) Sells(HWS,Drill) At(SM)

Buy(Drill, HWS) Buy(Milk, SM) Buy(Bananas, SM)

Example (continued)

• Nondeterministic choice: how to establish At( l 1 )?
  • We’ll do it from Start, with l 1 =Home

At(Home)

At(SM)

Buy(Drill, s 1 )

Have(Drill)

Start

Finish

Have(Milk) (^) Have(Bananas)

Sells(SM,Milk) (^) Sells(SM,Bananas)

Go( l 2 , SM)

Go(Home,HWS)

At(Home)

At( l 2 )

At(HWS) Sells(HWS,Drill) At(SM)

Buy(Drill, HWS) Buy(Milk, SM) Buy(Bananas, SM)

Example (continued)

• The only possible way to establish At(Home) for Finish
  • This creates a bunch of threats

Go( l 3 , Home)

At(Home)

At(SM)

Buy(Drill, s 1 )

Have(Drill)

Start

Finish

Have(Milk) (^) Have(Bananas)

Sells(SM,Milk) (^) Sells(SM,Bananas)

Go(HWS, SM)

Go(Home,HWS)

At(Home)

At(HWS)

At(HWS) Sells(HWS,Drill) At(SM)

Buy(Drill, HWS) Buy(Milk, SM) Buy(Bananas, SM)

Example (continued)

• To remove the threats to At(SM) and At(HWS), make them precede Go( l 3 ,Home)
  • This also removes the other threats

Go( l 3 , Home)

At(Home)

At(SM)

Buy(Drill, s 1 )

Have(Drill)

Start

Finish

Have(Milk) (^) Have(Bananas)

Sells(SM,Milk) (^) Sells(SM,Bananas)

Go(HWS, SM)

Go(Home,HWS)

At(Home)

At(HWS)

At(HWS) Sells(HWS,Drill) At(SM)

Buy(Drill, HWS) Buy(Milk, SM) Buy(Bananas, SM)

Discussion

• How to choose which flaw to resolve first and how to resolve it? - We’ll return to these questions in Chapter 10
• PSP doesn’t commit to orderings and instantiations until necessary - Avoids generating search trees like this one:
• Problem: how to prune infinitely long paths?
  • Loop detection is based on recognizing states we’ve seen before
  • In a partially ordered plan, we don’t know the states
• Can we prune if we see the same action more than once?
  • No. Sometimes we might need the same action several times in different states of the world (see next slide)

a b c b a

b a^ b a c b c (^) a c b

goal

• • •^ go(b,a)^ go(a,b)^ go(b,a)^ at(a)

Example

• 3-digit binary counter starts at 000, want to get to 110

s 0 = { d 3 =0, d 2 =0, d 1 =0} g = { d 3 =1, d 2 =1, d 1 =0}

• Operators to increment the counter by 1:

incr Precond: d 1 = Effects: d 1 =

incr Precond: d 2 =0, d 1 = Effects: d 2 =1, d 1 =

incr Precond: d 3 =0, d 2 =1, d 1 = Effects: d 3 =1, d 2 =0, d 1 =0 Docsity.com