Decision Making Under Uncertainty Lec #2: Planning with Knowledge
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Transcript of Decision Making Under Uncertainty Lec #2: Planning with Knowledge
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Decision Making Under UncertaintyLec #2: Planning with Knowledge
UIUC CS 598: Section EA
Professor: Eyal AmirSpring Semester 2005
Based on slides by José Luis Ambite, and…Paolo Traverso’s (http://sra.itc.it/people/traverso/) tutorial:
http://prometeo.ing.unibs.it/sschool/slides/traverso/traverso-slides.ps.gz, Some slides from http://www-2.cs.cmu.edu/~mmv/planning/handouts/BDDplanning.pdf by Rune Jensen http://
www.itu.dk/people/rmj
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State Space: Blocks World
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State-Transition Systems:Planning Domains
A planning domain D is a 4-tuple <F, S, A, R>:• F is a finite set of Fluents• S 2F is a finite set of states• A is a finite set of actions• R S x A x S is a transition relation
Action a A is executable in s S if s’ R(s, a, s’)
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State-Transition Systems: (Deterministic) Planning Domain Example:
• F = {loaded, locked}• S = {(loaded locked), (loaded locked), (loaded locked), (loaded locked)}• A = {lock, unlock, load, unload}• R = { [(loaded locked) unlock (loaded locked)], [(loaded locked) lock (loaded locked)], [(loaded locked) load (loaded locked)], [(loaded locked) unload (loaded locked)], [(loaded locked) lock (loaded locked)], [(loaded locked) unlock (loaded locked)] }
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State-Transition Systems:Planning Problem
A planning problem P for a planning Domain D=<F S A R> is a 3-tuple <D, I, G>:
• I S is the set of initial states
• G S is the set of goal statesI G
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State-Transition Systems: Plan
• A plan for a planning problem P=<I, G, D> in a planning domain D = <F, S, A, R> is a set of state-action pairs: – {(s, a) : s S, a A, a executable in s}– at least one (s, a) with s I
• Goal achieving plan (informally):– for each state-action pairs (s, a), either a leads from s
to the goal, R(s, a) G, or a leads from s to a state s’ such that (s’, a’) and a’ leads from s’ to the goal R(s’, a’) G, and so on.
= {(2, load), (3, lock)}
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Planning Algorithm (Regression)
I
Backward image(regress set of states)
Remove Visited
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Planning Algorithm (Regression)
IG
Backward image(regress set of states)
Remove Visited
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Planning via Symbolic Model Checking
• Problem: Realistic planning domains often have large state spaces
• Idea: exploit the work on symbolic model checking based on Ordered Binary Decision Diagrams (OBDD’s)
• OBDD’s:– Canonical form for propositional formulas– Efficient!
• Polynomial boolean operations: O(1 2) = O(|1| |2|)
• Constant time equality
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The Model Checking Problem
Determine whether a formula is true in a model1. A domain of interest is described by a
semantic model2. A desired property of the domain is described
by a logical formula3. Check if the domain satisfy the desired
property by checking whether the formula is true in the model
Motivation: Formal verification of dynamic systems
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OBDD example:
Variable orderingIs important
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Planning via Model CheckingSymbolic Representation
• Action represented by assigning true to the corresponding variable
• Transition t = <s a s’> encoded as(t) = (s) ^ (a) ^ (s’)
• Transition relation T encoded as disjunction of all the transitions
(T)= VtT (t)
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Planning Algorithm (regression)
IG
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Planning Algorithm (regression)
• OneStepPlan(S) in the regression algorithm is the backward image of the set of states S.
• Can computed as the QBF formula:
x’ (States(x’) R(x, a, x’))
• Quantified Boolean Formula (QBF): x (x y) = (0 y) (1 y)
x (x y) = (0 y) (1 y)
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Planning as model checking
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Planning as Model Checking
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Homework
1. Read readings for next time: on website