The document discusses the AO* algorithm for solving problems represented as AND/OR graphs. It begins by explaining AND/OR graphs and how they can represent achieving subgoals simultaneously or independently. It then introduces the AO* algorithm, which extends A* search to AND/OR graphs by examining multiple nodes simultaneously. The algorithm is described in pseudocode and an example is provided. Finally, the document shows an example of generating a proof tree using forward and backward chaining on a set of logical statements and translating the statements into predicate logic.

1. SUBMITTED TO: SUBMITTED BY:
MR. RAVI KHATWAL KRATI KATYAL
MCA 5TH SEM
AO* ALGORITHM &
PROBLEM REDUCTION

2. Topic Covered
 AND/OR Graph
 AO* Problem Reduction Algorithm
 Forward Chaining
 Backward Chaining
 Proof Tree Generated by Forward/Backward Chaining
 References

3. AND/OR graphs
 Some problems are best represented as achieving subgoals, some of which
achieved simultaneously and independently (AND)
 Up to now, only dealt with OR options
Complete Assignment
Copy Assignment Study Topic Do Assignment

4. Searching AND/OR graphs
 A solution in an AND-OR tree is a sub tree whose leafs are included in the goal
set
 Cost function: sum of costs in AND node
f(n) = f(n1) + f(n2) + …. + f(nk)
 How can we extend A* to search AND/OR trees? The AO* algorithm.

5. AND/OR search
 We must examine several nodes simultaneously when choosing the next move
A
B C D(3)
(4)
(5)
(9)
A
B C D
38
E F G H I J
17 9 27
(5) (10) (3) (4) (15) (10)

6. AND/OR Best-First-Search
 Traverse the graph (from the initial node) following the best current path.
 Pick one of the unexpanded nodes on that path and expand it. Add its
successors to the graph and compute f for each of them
 Change the expanded node’s f value to reflect its successors. Propagate the
change up the graph.
 Reconsider the current best solution and repeat until a solution is found

7. AND/OR Best-First-Search example

8. AND/OR Best-First-Search example

9. A Longer path may be better

10. AO* algorithm
1. Let G be a graph with only starting node INIT.
2. Repeat the followings until INIT is labeled SOLVED or
h(INIT) > FUTILITY
a) Select an unexpanded node from the most promising path from
INIT (call it NODE)
b) Generate successors of NODE. If there are none, set h(NODE) =
FUTILITY (i.e., NODE is unsolvable); otherwise for each
SUCCESSOR that is not an ancestor of NODE do the following:
i. Add SUCCESSSOR to G.
ii. If SUCCESSOR is a terminal node, label it SOLVED and set
h(SUCCESSOR) = 0.
iii. If SUCCESSPR is not a terminal node, compute its h

11. AO* algorithm (Cont.)
c) Propagate the newly discovered information up the graph by doing the
following: let S be set of SOLVED nodes or nodes whose h values have
been changed and need to have values propagated back to their parents.
Initialize S to Node. Until S is empty repeat the followings:
i. Remove a node from S and call it CURRENT.
ii. Compute the cost of each of the arcs emerging from CURRENT. Assign
minimum cost of its successors as its h.
iii. Mark the best path out of CURRENT by marking the arc that had the
minimum cost in step ii
iv. Mark CURRENT as SOLVED if all of the nodes connected to it through
new labeled arc have been labeled SOLVED
v. If CURRENT has been labeled SOLVED or its cost was just changed,
propagate its new cost back up through the graph. So add all of the
ancestors of CURRENT to S.

12. An Example

13. An Example

14. An Example

15. An Example

19. Proof Tree Generated by
Forward/Backward Chaining
Consider The Following Sentences
1. It is a crime for an American to sell weapons to hostile Nations.
2. Pakistan has some missiles.
3. All of its(Paksitan’s) missiles were sold to it by Colonel West.
4. Missiles are weapons
5. Enemy of America counts as “hostile”.
6. West is American.
7. The country Pakistan is an enemy of America.

20. a) Translate these sentenses into Formula in Predicate Logic.
b) Prove that “West is criminal” using forward chaining.
c) Prove that “West is criminal” using Backward chaining.

21. Solution A:
1. ∀X, Y, Z american(X) ⋀ weapon(Y) ⋀ sells(X,Y,Z) ⋀ hostile(Z) → criminal(X)
2. ∃X owns(pakistan, X) ⋀ missile(X)
3. ∀X missile(X) ⋀ owns(pakistan, X) → sells(west, X, pakistan)
4. Missile(X) → weapon(X)
5. Enemy(X, america) → hostile(X)
6. American(west)
7. Enemy(pakistan,america)

22. Solution B:
 Own(pakistan,m1)⋀missile(m1) ⋀ american(west) ⋀ enemy(pakistan, america)
↓ Rule(3) and {m1/X}
 Own(pakistan,m1)⋀missile(m1) ⋀ sells(west,m1, pakistan) ⋀ american(west)
⋀enemy(pakistan, america)
↓ Rule(4) and {m1/X}
 Own(pakistan,m1) ⋀ missile(m1) ⋀ weapon(m1) ⋀ sells(west, m1, pakistan,) ⋀
enemy(pakistan, america) ⋀ american(west)
↓ Rule(5) and {pakistan/X}
 Own(pakistan,missile(m1) ⋀weapon(m1) ⋀ sells(west, m1, pakistan) ⋀
hostile(pakistan) ⋀ enemy(pakistan,america) ⋀ american(west)
↓ Rule(1) {west/X, Y/m1, Z/pakistan}
 criminal (west)

23. Showing Procedure Using AND/OR Graph

24. Solution C:
 Here, we start with our goal as given in figure
 Criminal(west)
 ↑ Rule(1) & {west/X}
 american(west) ⋀ weapon(Y) ⋀ sells(wet, Y, Z) ⋀ hostile(Z)
 ↑ Rule(6)
 weapon(Y) ⋀ sells(west, Y, Z) ⋀ hostile(Z)
 ↑ Rule(5) & {Z/X}
 weapon(Y) ⋀ sells(west, Y, Z) ⋀ enemy(Z, America)
 ↑ Rule(7) & {pakistan/Z}
 weapon(Y) ⋀ sells(west, Y, pakistan)
 ↑ Rule(4) & {Y/X}

25.  missile(Y) ⋀ sells(west, Y, pakistan)
 ↑ Rule(2,b) and {Y/m1}
 sells(west, Y, pakistan)
 ↑ Rule(3) {Y/X}
 missile(Y) ⋀ owns(pakistan, Y)
 ↑ Rule(2, a)and (2, b) and {Y/m1}
 Nil

26. Showing Procedure Using AND/OR Graph

27. References
 Book Reference
 Artificial Intelligence – Amit Barjatiya & Dinesh Diggiwal
 Artificial Intelligence – Russell & Norvig
 Web Reference
 www.wikipedia.org
 artificialintelligence-notes.blogspot.in