And or graph problem reduction using predicate logic
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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.
And or graph problem reduction using 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
- 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