decomposition spaces

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Decomposition spaces Spring 2007, Juris Vīksn

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Decomposition spaces. Spring 2007, Juris V īksna. Sample problem - Towers of Hanoi. [Adapted from R.Shinghal]. Sample problem - Towers of Hanoi. [Adapted from J.Pearl]. Sample problem - Symbolic integration. [Adapted from R.Shinghal]. Sample problem - Symbolic integration. - PowerPoint PPT Presentation

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Page 1: Decomposition spaces

Decomposition spaces

Spring 2007, Juris Vīksna

Page 2: Decomposition spaces

Sample problem - Towers of Hanoi

[Adapted from R.Shinghal]

Page 3: Decomposition spaces

Sample problem - Towers of Hanoi

[Adapted from J.Pearl]

Page 4: Decomposition spaces

Sample problem - Symbolic integration

[Adapted from R.Shinghal]

Page 5: Decomposition spaces

Sample problem - Symbolic integration

[Adapted from R.Shinghal]

Page 6: Decomposition spaces

Sample problem - Block world

[Adapted from R.Shinghal]

Page 7: Decomposition spaces

Sample problem - Block world

[Adapted from R.Shinghal]

Page 8: Decomposition spaces

Sample problem - Block world

[Adapted from R.Shinghal]

Page 9: Decomposition spaces

Sample problem - Block world

[Adapted from R.Shinghal]

Page 10: Decomposition spaces

Sample problem - Coin weighting

[Adapted from J.Pearl]

Page 11: Decomposition spaces

Sample problem - Coin weighting

[Adapted from J.Pearl]

Page 12: Decomposition spaces

Decomposition spaces

[Adapted from R.Shinghal]

Page 13: Decomposition spaces

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

Page 14: Decomposition spaces

Decomposition spaces

<S,C,I,E,U,W> - decomposition space

The problem

• find a solution tree• find a solution tree with minimal weight

Page 15: Decomposition spaces

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

Page 16: Decomposition spaces

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

Page 17: Decomposition spaces

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

Page 18: Decomposition spaces

AND/OR graphs

[Adapted from J.Pearl]

Page 19: Decomposition spaces

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)

Page 20: Decomposition spaces

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

Page 21: Decomposition spaces

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

Page 22: Decomposition spaces

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

Page 23: Decomposition spaces

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)

Page 24: Decomposition spaces

AO* algorithm

[Adapted from J.Pearl]

Page 25: Decomposition spaces

Complete search

Definition

An AO* algorithm is said to be complete if it terminates with a solution when one exists.

Page 26: Decomposition spaces

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.

Page 27: Decomposition spaces

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) .

Page 28: Decomposition spaces

Completeness of AO*

Theorem

AO* algorithm is complete on locally finite state spaces.

Page 29: Decomposition spaces

Admissibility of AO*

Definition

A heuristic function h is said to be admissible if

0 h(n) h*(n) for all nS.

Page 30: Decomposition spaces

Admissibility of AO*

Theorem

AO* which uses admissible heuristic function is admissibleon locally finite state spaces.

Page 31: Decomposition spaces

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)

Page 32: Decomposition spaces

Admissibility of AO*

Theorem

AO* which uses admissible heuristic function is admissibleon locally finite state spaces.

Page 33: Decomposition spaces

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.

Page 34: Decomposition spaces

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.

Page 35: Decomposition spaces

Monotonicity and admissibility

Theorem

Every monotone heuristic is also admissible.