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Chapter 17

Planning Based on Model Checking

Lecture slides for

Automated Planning: Theory and Practice

Motivation

• Actions with multiple possible

outcomes

• Action failures
  • e.g., gripper drops its load
• Exogenous events
  • e.g., road closed
• Nondeterministic systems are like

Markov Decision Processes (MDPs), but without probabilities attached to the outcomes

• Useful if accurate probabilities aren’t available, or if probability calculations would introduce inaccuracies

a

c b

grasp(c)

a

c

b Intended outcome

a b Unintended outcome

Goal

Start

2

Example

• Robot r1 starts at location l
• Objective is to get r1 to location l
• π 1 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4))}
• π 2 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4)), (s5, move(r1,l3,l4))}
• π 3 = {(s1, move(r1,l1,l4))}

Goal

Start

2

Example

• Robot r1 starts at location l
• Objective is to get r1 to location l
• π 1 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4))}
• π 2 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4)), (s5, move(r1,l3,l4))}
• π 3 = {(s1, move(r1,l1,l4))}

Goal

Start

2

Execution Structures

• Execution structure for a policy π: - The graph of all of π’s execution paths
• Notation: Σπ = ( Q , T )
  • Q ⊆ S
  • T ⊆ S × S
• π 1 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4))}
• π 2 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4)), (s5, move(r1,l3,l4))}
• π 3 = {(s1, move(r1,l1,l4))}

s

s

s

s

s

Execution Structures

• Execution structure for a policy π: - The graph of all of π’s execution paths
• Notation: Σπ = ( Q , T )
  • Q ⊆ S
  • T ⊆ S × S
• π 1 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4))}
• π 2 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3, move(r1,l3,l4)), (s5, move(r1,l3,l4))}
• π 3 = {(s1, move(r1,l1,l4))}

s

s

s

s

s

Execution Structures

• Execution structure

for a policy π:

• The graph of all of π’s execution paths
• Notation: Σπ = ( Q , T )
• Q ⊆ S
• T ⊆ S × S
• π 1 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3,

move(r1,l3,l4))}

• π 2 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3,

move(r1,l3,l4)), (s5, move(r1,l3,l4))}

• π 3 = {(s1, move(r1,l1,l4))}

s

s

s

s

s

Goal

Start

2

Execution Structures

• Execution structure

for a policy π:

• The graph of all of π’s execution paths
• Notation: Σπ = ( Q , T )
• Q ⊆ S
• T ⊆ S × S
• π 1 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3,

move(r1,l3,l4))}

• π 2 = {(s1, move(r1,l1,l2)), (s2, move(r1,l2,l3)), (s3,

move(r1,l3,l4)), (s5, move(r1,l3,l4))}

• π 3 = {(s1, move(r1,l1,l4))}

s1 (^) s

Types of Solutions

• Weak solution : at least one execution path

reaches a goal

• Strong solution : every execution path reaches a

goal

s s

s a0 Goal a

a s

a

s s

s a0 Goal a

a s

s s

s a0 a1 Goal

a s

Goal

a

Finding Strong Solutions

• Backward breadth-first search

• StrongPreImg( S )

= {( s,a ) : γ( s,a ) ≠ ∅, γ( s,a ) ⊆ S }

– all state-action pairs for which

all of the successors are in S

• PruneStates(π, S )

= {( s,a ) ∈ π : s ∉ S }

– S is the set of states we’ve

already solved

– keep only the state-action

pairs for other states

Start

Example

π = failure

S π ' = ∅

S g ∪ S π ' = {s4}

π '' ← PreImage =

{(s3,move(r1,l3,l

(s5,move(r1,l5,l

Goal

s

s

s

2

Start

Example

π = failure

S π ' = ∅

S g ∪ S π ' = {s4}

π '' ← PreImage =

{(s3,move(r1,l3,l4)

(s5,move(r1,l5,l4))

Goal

s

s

s

2

Example

π ' = {(s3,move(r1,l3,l4)),

(s5,move(r1,l5,l4))}

S π ' = {s3,s5}

S g ∪ S π ' = {s3,s4,s5}

PreImage ←

{(s2,move(r1,l2,l3)),

(s3,move(r1,l3,l4)),

(s5,move(r1,l5,l4)),

(s3,move(r1,l4,l3)),

(s5,move(r1,l4,l5))}

π '' ← {(s2,move(r1,l2,l3))}

Goal

Start

s

s

s

s

2

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Example

π = {(s3,move(r1,l3,l4)),

(s5,move(r1,l5,l4))}

π ' = {(s2,move(r1,l2,l3)),

(s3,move(r1,l3,l4)),

(s5,move(r1,l5,l4))}

S π ' = {s2,s3,s5}

S g ∪ S π ' = {s2,s3,s4,s5}

Goal

Start

s

s

s

s

2