Planning and Optimization C4. Delete Relaxation: Relaxed Task Graphs Gabriele R¨oger and Thomas Keller

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Planning and Optimization

C4. Delete Relaxation: Relaxed Task Graphs

Gabriele R¨oger and Thomas Keller

Universit¨at Basel

October 22, 2018

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Relaxed Task Graphs Construction Reachability Analysis Remarks Summary

Content of this Course: Heuristics

Heuristics

Delete Relaxation Relaxed Tasks Relaxed Task Graphs

Relaxation Heuristics Abstraction

Landmarks Potential Heuristics

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Relaxed Task Graphs

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Relaxed Task Graphs

Let Π⁺ be a relaxed planning task.

Therelaxed task graph of Π⁺, in symbolsRTG(Π⁺), is an AND/OR graph that encodes

which state variablescan become true in an applicable operator sequence for Π⁺, which operators of Π⁺ can be included in an applicable operator sequence for Π⁺, if the goalof Π⁺ can be reached,

andhow these things can be achieved.

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Running Example

As a running example, consider the relaxed planning task hV,I,{o₁,o₂,o₃,o₄}, γiwith

V ={a,b,c,d,e,f,g,h}

I ={a7→T,b7→T,c 7→F,d 7→T, e 7→F,f 7→F,g 7→F,h7→F}

o1 =hc ∨(a∧b),c ∧((c∧d)Be),1i o₂ =h>,f,2i

o₃ =hf,g,1i o₄ =hf,h,1i

γ =e∧(g∧h)

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Construction

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Components of Relaxed Task Graphs

A relaxed task graph has four kinds of components:

Variable node represent the state variables.

The initial noderepresent the initial state.

Operator subgraphs represent the preconditions and effects of operators.

The goal subgraphrepresents the goal.

The idea is to construct the graph in such a way that all nodes representingreachable aspects of the task areforced true.

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Variable Nodes

Let Π⁺=hV,I,O⁺, γi be a relaxed planning task.

For eachv ∈V,RTG(Π⁺) contains an OR node nv. These nodes are called variable nodes.

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Variable Nodes: Example

a b c d e f g h

I o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Initial Node

RTG(Π⁺) contains an AND node n_I. This node is called the initial node.

For all v ∈V with I(v) =T,RTG(Π⁺) has an arc fromn_v to n_I. These arcs are called initial state arcs.

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Initial Node and Initial State Arcs: Example

a b c d e f g h

I o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Initial Node and Initial State Arcs: Example

I ={a7→T,b 7→T,c 7→F,d 7→T,e 7→F,f 7→F,g 7→F,h7→F}

o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Operator Subgraphs

For each operatoro⁺∈O⁺,RTG(Π⁺) contains anoperator subgraph with the following parts:

for each formulaϕthat occurs as a subformula of the precondition or of some effect condition of o⁺, a formula noden_ϕ (details follow)

for each conditional effect (χBv) that occurs

in the effect of o⁺, an effect noden^χ_o+ (details follow);

unconditional effects are treated as (>Bv)

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Formula Nodes

Formula nodesn_ϕ are defined as follows:

Ifϕ=v for some state variablev,n_ϕ is the variable noden_v (so no new node is introduced).

Ifϕ=>,n_ϕ is an AND node without outgoing arcs.

Ifϕ=⊥,n_ϕ is an OR node without outgoing arcs.

Ifϕ= (ϕ₁∧ϕ₂),n_ϕ is an AND node with outgoing arcs to n_ϕ₁ andn_ϕ₂. Ifϕ= (ϕ₁∨ϕ₂),n_ϕ is an OR node with outgoing arcs to n_ϕ₁ andn_ϕ₂.

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Effect Nodes

Effect nodesn^χ_o+ are defined as follows:

n^χ_o+ is an AND node

It has an outgoing arc to the formula nodes n_pre(o⁺₎ (precondition arcs) andnχ (effect condition arcs).

Exception: if χ=>, there is no effect condition arc.

(This makes our pictures cleaner.)

For every conditional effect (χBv) in the operator, there is an arc from variable noden_v to n^χ_o+ (effect arcs).

Note: identically named nodes are identical,

so if the same effect condition occurs multiple times in the same operator, this only inducesone node.

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Operator Subgraphs: Example

o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Operator Subgraphs: Example

o₁ =hc ∨(a∧b),c ∧((c∧d)Be),1i

a b c d e f g h

I o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Operator Subgraphs: Example

o₂ =h>,f,2i

o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Operator Subgraphs: Example

o₃=hf,g,1i

a b c d e f g h

I o₁,> o₁,c∧d

o2,>

o₃,>

o₄,>

γ

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Operator Subgraphs: Example

o₄ =hf,h,1i

o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Goal Subgraph

RTG(Π⁺) contains a goal subgraph, consisting of formula nodes for the goalγ and its subformulas, constructed in the same way as formula nodes for preconditions and effect conditions.

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Goal Subgraph and Final Relaxed Task Graph: Example

o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Goal Subgraph and Final Relaxed Task Graph: Example

γ =e∧(g∧h)

a b c d e f g h

I o₁,> o₁,c∧d

o2,>

o₃,> o₄,>

γ

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Reachability Analysis

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How Can We Use Relaxed Task Graphs?

We are now done with the definition of relaxed task graphs.

Now we want to usethem to derive information about planning tasks.

In the following chapter, we will use them

to compute heuristics for delete-relaxed planning tasks.

Here, we start with something simpler: reachability analysis.

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Forced True Nodes and Reachability

Theorem (Forced True Nodes vs. Reachability) LetΠ⁺=hV,I,O⁺, γi be a relaxed planning task, and let N_T be the forced true nodes of RTG(Π⁺).

For allformulas over state variablesϕ that occur in the definition ofΠ⁺:

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Forced True Nodes and Reachability: Consequences

Corollary

LetΠ⁺=hV,I,O⁺, γi be a relaxed planning task, and let NT be the forced true nodes of RTG(Π⁺). Then:

A state variablev ∈V is true in at least one reachable state iff n_v ∈N_T.

An operator o⁺∈O⁺ is part of at least one applicable operator sequence iff n_pre(o⁺₎∈N_T. The relaxed task is solvable iff n_γ ∈N_T.

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Reachability Analysis: Example

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o₁,>

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

a

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T o₁,c∧d

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Reachability Analysis: Example with Different Initial State

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Reachability Analysis: Example with Different Initial State

a

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Reachability Analysis: Example with Different Initial State

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Remarks

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Relaxed Task Graphs in the Literature

Some remarks on the planning literature:

Usually, only the STRIPScase is studied.

definitions simpler: only variable nodes andoperator nodes, no formula nodes or effect nodes

Usually, so-called relaxed planning graphs (RPGs) are studied instead of RTGs.

These are temporally unrolled versions of RTGs,

i.e., they have multiple layers (“time steps”) and are acyclic.

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Summary

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Summary

Relaxed task graphs(RTGs) represent (most of) the

information of a relaxed planning task as an AND/OR graph.

They consist of:

variable nodes an initial node

operator subgraphsincludingformula nodesandeffect nodes agoal subgraphincludingformula nodes