Decomposition spaces
description
Transcript of Decomposition spaces
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Decomposition spaces
Spring 2007, Juris Vīksna
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Sample problem - Towers of Hanoi
[Adapted from R.Shinghal]
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Sample problem - Towers of Hanoi
[Adapted from J.Pearl]
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Sample problem - Symbolic integration
[Adapted from R.Shinghal]
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Sample problem - Symbolic integration
[Adapted from R.Shinghal]
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Sample problem - Block world
[Adapted from R.Shinghal]
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Sample problem - Block world
[Adapted from R.Shinghal]
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Sample problem - Block world
[Adapted from R.Shinghal]
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Sample problem - Block world
[Adapted from R.Shinghal]
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Sample problem - Coin weighting
[Adapted from J.Pearl]
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Sample problem - Coin weighting
[Adapted from J.Pearl]
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Decomposition spaces
[Adapted from R.Shinghal]
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Decomposition spaces
<S,C,I,E,U,W> - decomposition space
S - set of problemsC= {{(x,y1),...,(x,yk)}|x,yiS} - set of connectors
IS - the initial problemES - set of elementary problemsUS - set of unsolvable problemsW: CR+ - weight function
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Decomposition spaces
<S,C,I,E,U,W> - decomposition space
The problem
• find a solution tree• find a solution tree with minimal weight
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Solution tree
Definition
T(n) is a solution tree for node n, if
• T(n)={n} and n is an elementary problem• T(n) = {T(n1),....,T(nk)}, where T(n1),...,T(nk) are solution trees for nodes n1,...,nk and there is a connector {(n,n1),...,(n,nk)} C
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Solution tree
Maximum weight
We define w(T(n)) as follows:
• w(T(n)) = 0, if T(n)={n} • w(T(n)) = max{w(T(n1)),....,w(T(nk))} + W(cT), if T(n) = {T(n1),....,T(nk)}, where cT= {(n,n1),...,(n,nk)} C
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Solution tree
Summary weight
We define w(T(n)) as follows:
• w(T(n)) = 0, if T(n)={n}
• w(T(n)) = {w(T(n1)),....,w(T(nk))} + W(cT), if T(n) = {T(n1),....,T(nk)}, where cT= {(n,n1),...,(n,nk)} C
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AND/OR graphs
[Adapted from J.Pearl]
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Heuristics
<S,C,I,E,U,W> - decomposition space
h*(x) - a minimum weight for solution tree T(x)
h(x) - heuristic estimate of h*(x)
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Potential solution tree
<S,C,I,E,U,W> - decomposition spaceA S - set of already discovered problems
T(n) is a potential solution tree for node n, if
• T(n)={n} , if nA and the children of n does not belong to A
• T(n) = {T(n1),....,T(nk)}, where T(n1),...,T(nk) are potential solution trees for nodes n1,...,nk and there is a connector {(n,n1),...,(n,nk)} C
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Potential solution tree
(Maximum) weight of potential solution tree
We define w(T(n)) as follows:
• w(T(n)) = 0, if T(n)={n} and n is shown to be in E • w(T(n)) = h(n), if T(n)={n} and n is not shown to be in E • w(T(n)) = max{w(T(n1)),....,w(T(nk))} + W(cT), if T(n) = {T(n1),....,T(nk)}, where cT= {(n,n1),...,(n,nk)} C
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Potential solution tree
(Summary) weight of potential solution tree
We define w(T(n)) as follows:
• w(T(n)) = 0, if T(n)={n} and n is shown to be in E • w(T(n)) = h(n), if T(n)={n} and n is not shown to be in E
• w(T(n)) = {w(T(n1)),....,w(T(nk))} + W(cT), if T(n) = {T(n1),....,T(nk)}, where cT= {(n,n1),...,(n,nk)} C
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Most promising solution tree
A potential solution tree T(n) is most promising, if it has the minimal weight (of all potential solution trees)
We denote the cost of the most promising solution tree by e(n)
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AO* algorithm
[Adapted from J.Pearl]
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Complete search
Definition
An AO* algorithm is said to be complete if it terminates with a solution when one exists.
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Admissible search
Definition
An AO* algorithm is admissible if it is guaranteed to return an optimal solution (solution tree with minimum possible weight) whenever a solution exists.
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Locally finite state spaces
Definition
A decomposition space <S,C,I,E,U,W> is locally finite, if • for every xS, there is only a finite number of ySsuch that (x,y)c for some c C
• there exists > 0 such that for all cC we haveW(c) .
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Completeness of AO*
Theorem
AO* algorithm is complete on locally finite state spaces.
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Admissibility of AO*
Definition
A heuristic function h is said to be admissible if
0 h(n) h*(n) for all nS.
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Admissibility of AO*
Theorem
AO* which uses admissible heuristic function is admissibleon locally finite state spaces.
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Admissibility of AO*
Lemma
If AO* uses admissible heuristic function h, then at any time before AO* terminates:
• e(n) h*(n) for nodes from Open• if n is marked as solved then e(n)=h*(n)
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Admissibility of AO*
Theorem
AO* which uses admissible heuristic function is admissibleon locally finite state spaces.
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Monotone heuristic functions
Definition
A heuristic function h is said to be monotone, if
h(n) min max{h(n1),....,h(nk)} + W(c), where the minimumis taken for all c={(n,n1),...,(n,nk)}C.
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Monotone heuristic functions
Definition
A heuristic function h is said to be monotone, if
h(n) min {h(n1),....,h(nk)} + W(c), where the minimumis taken for all c={(n,n1),...,(n,nk)}C.
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Monotonicity and admissibility
Theorem
Every monotone heuristic is also admissible.