decomposition spaces

Decomposition spaces

Spring 2007, Juris Vīksna

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

[Adapted from R.Shinghal]

Sample problem - Block world

[Adapted from R.Shinghal]

Sample problem - Block world

[Adapted from R.Shinghal]

Sample problem - Block world

[Adapted from R.Shinghal]

Sample problem - Block world

[Adapted from R.Shinghal]

Sample problem - Coin weighting

[Adapted from J.Pearl]

Sample problem - Coin weighting

[Adapted from J.Pearl]

Decomposition spaces

[Adapted from R.Shinghal]

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

Decomposition spaces

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

The problem

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

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

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

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

AND/OR graphs

[Adapted from J.Pearl]

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)

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

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

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

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)

AO* algorithm

[Adapted from J.Pearl]

Complete search

Definition

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

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.

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) .

Completeness of AO*

Theorem

AO* algorithm is complete on locally finite state spaces.

Admissibility of AO*

Definition

A heuristic function h is said to be admissible if

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

Admissibility of AO*

Theorem

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

Admissibility of AO*

Theorem

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

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.

Monotonicity and admissibility

Theorem

Every monotone heuristic is also admissible.