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 - decomposition space S - set of problems C= {{(x,y 1 ),...,(x,y k )}|x,y i  S}- set of connectors I  S - the initial problem E  S- set of elementary problems U  S- set of unsolvable problems W: C  R + - weight function

Decomposition spaces - 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(n 1 ),....,T(n k )}, where T(n 1 ),...,T(n k ) are solution trees for nodes n 1,...,n k and there is a connector {(n,n 1 ),...,(n,n k )}  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(n 1 )),....,w(T(n k ))} + W(c T ), if T(n) = {T(n 1 ),....,T(n k )}, where c T = {(n,n 1 ),...,(n,n k )}  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(n 1 )),....,w(T(n k ))} + W(c T ), if T(n) = {T(n 1 ),....,T(n k )}, where c T = {(n,n 1 ),...,(n,n k )}  C

AND/OR graphs [Adapted from J.Pearl]

Heuristics - decomposition space h*(x)- a minimum weight for solution tree T(x) h(x)- heuristic estimate of h*(x)

Potential solution tree - decomposition space A  S - set of already discovered problems T(n) is a potential solution tree for node n, if T(n)={n}, if n  A and the children of n does not belong to A T(n) = {T(n 1 ),....,T(n k )}, where T(n 1 ),...,T(n k ) are potential solution trees for nodes n 1,...,n k and there is a connector {(n,n 1 ),...,(n,n k )}  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(n 1 )),....,w(T(n k ))} + W(c T ), if T(n) = {T(n 1 ),....,T(n k )}, where c T = {(n,n 1 ),...,(n,n k )}  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(n 1 )),....,w(T(n k ))} + W(c T ), if T(n) = {T(n 1 ),....,T(n k )}, where c T = {(n,n 1 ),...,(n,n k )}  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 is locally finite, if for every x  S, there is only a finite number of y  S such that (x,y)  c for some c  C there exists  > 0 such that for all c  C we have W(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 n  S.

Admissibility of AO* Theorem AO* which uses admissible heuristic function is admissible on 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 admissible on locally finite state spaces.

Monotone heuristic functions Definition A heuristic function h is said to be monotone, if h(n)  min max{h(n 1 ),....,h(n k )} + W(c), where the minimum is taken for all c={(n,n 1 ),...,(n,n k )}  C.

Monotone heuristic functions Definition A heuristic function h is said to be monotone, if h(n)  min  {h(n 1 ),....,h(n k )} + W(c), where the minimum is taken for all c={(n,n 1 ),...,(n,n k )}  C.

Monotonicity and admissibility Theorem Every monotone heuristic is also admissible.