Artificial Intelligence

Transcription

1 Artificial Intelligence Exercises & Solutions Chapters -4: Search methods. Search Tree. Draw the complete search tree (starting from S and ending at G) of the graph below. The numbers beside the nodes represent the estimated distances from the goal state. Show how the search procedure proceeds in the tree by using: 8 5 depth-first breadth-first A D hill-climbing 0 beam-search (w=) S best-first G 9 0 methods. B C. Search traces. Consider the graph shown in the figure to the right. We can search it with a variety of different algorithms, resulting in different search trees. Each of the trees (labelled G though G7, next page) was generated by searching this graph, but with a different algorithm. Assume that children of a node are visited in alphabetical order when no other order is specified by the search. Each tree shows all the nodes that have been seen during search. Numbers next to nodes indicate the relevant score used by the algorithm for those nodes. For each tree, indicate whether it was generated with Depth first search Breadth first search niform cost search A* search Best-first (greedy) search In all cases a strict expanded list was used. Furthermore, if you choose an algorithm that uses a heuristic function, say whether we used H: heuristic = {h(a) =, h(b) = 6, h(c) = 4, h(d) = } H: heuristic = {h(a) =, h(b) =, h(c) = 0, h(d) = } Also, for all algorithms, say whether the result was an optimal path (measured by sum of link costs), and if not, why not.

2 . Search problem. Consider the -puzzle problem, which is a simpler version of the 8-puzzle where the board is x and there are three tiles, numbered,, and, and one blank. There are four operators, which move the blank up, down, left, and right. The start and goal states are given below. Show how the path to the goal can be found using: a) breath first searh b) depth first search c) A* search with the heuristic being the sum of number of moves and the number of misplaced tiles. Start Goal Assume that there is no possibility to remember states that have been visited earlier. Also, use the given operators in the given order unless the search method defines otherwise. Label each visited node with a number indicating the order in which they are visited. If a search method doesn t find a solution, explain why this happened.

3 . Search Tree solution. The complete search tree can be generated by finding all possible routes that start from the node S and lead to a node that has not yet been visited on the route. The tree below has been constructed by always choosing the rightmost (from the nodes point of view) unvisited route. Note that the branches could be in different order if routes were selected in different order. The nodes are numbered (-8) to ease the analysis of the search methods. In depth first search method the tree is searched in the depth direction. The search always branches to the leftmost unvisited node. In case a leaf node is found or there are no more unvisited nodes, the search returns to the preceding level. In this case, the tree is searched in the order indicated by the node numbers and is illustrated with the dotted line. If one solution for reaching G was enough, the search would end at the node five. If the whole tree is searched, the best route to G can be selected at the end.

4 Breadth-first search method searches the tree in the breadth direction. In this case the nodes are searched in the following order:,, 0,, 4, 7,, 5, 5, 6, 8,,, 6, 8, 9, 4 and 7. Again, if one solution was enough, the search would end at node 5. Hill-climbing is a search algorithm that proceeds like depth-first search but so that the search order of the nodes is determined by the estimated distance from a node to the goal node (depth first would choose the leftmost). When starting from the node S, the search sees the estimated distances from the nodes (9) and 0 (8) and chooses the node 0 that is estimated to be closer to the goal. From node 0 the nodes (9) and 5 (5) become visible and the search proceeds to node 5. From node 5 the goal node becomes visible and the search has found a solution. One solution is usually enough for Hill-climbing since it usually finds good solutions. However, the image illustrates the search of the whole tree to better describe the search.

5 Beam-search proceeds much like breath first, however, at each level only w number of nodes are checked. First, nodes and 0 are opened. At this point five nodes can be seen ( (0), 4 (5), 7 (8), (9) and 5 (5)). Since only two nodes can be opened, nodes 4 and 5 are selected. After those nodes are opened four more nodes become visible and as the goal node becomes visible (actually twice) the search is completed. The nodes were visited in the following order:,, 0, 4, 5 and 5. Best-first search always chooses the best node, independent of where in the tree it is located. At first, the nodes (9) and 0 (8) can be seen and the node 0 is opened. Now, the nodes (9), (9) and 5 (5) can be seen and the node 5 is opened. At this point the nodes nodes (9), (9), 6 (9) and 8 (0) can be seen and as 8 is the goal node, the search is finished. If one would want to find an alternative route to the goal node, the search would be continued from where the nodes (9), (9), 6 (9) were seen. Now all the nodes have the same distance. If the leftmost node of same valued nodes was opened first the search would continue through nodes, 4 and 5. If on the other hand the node that was seen last of the same valued nodes was to be opened first, the search would continue through nodes 6,, and 4.

6 . Search traces solution. G:. Algorithm: Breadth First Search. Heuristic (if any): None. Did it find least-cost path? If not, why? No. Breadth first search is only guaranteed to find a path with the shortest number of links; it does not consider link cost at all. G:. Algorithm: Best First Search. Heuristic (if any): H. Did it find least-cost path? If not, why? No. Best first search is not guaranteed to find an optimal path. It takes the first path to goal it finds. G:. Algorithm: A*. Heuristic (if any): H. Did it find least-cost path? If not, why? No. A* is only guaranteed to find an optimal path when the heuristic is admissible (or consistent with a strict expanded list). H is neither: the heuristic value for C is not an underestimate of the optimal cost to goal. G4:. Algorithm: Best First Search. Heuristic (if any): H. Did it find least-cost path? If not, why? Yes. Though best first search is not guaranteed to find an optimal path, in this case it did G5:. Algorithm: Depth First Search. Heuristic (if any): None. Did it find least-cost path? If not, why? No. Depth first search is an any-path search; it does not consider link cost at all. G6:. Algorithm: A*. Heuristic (if any): H. Did it find least-cost path? If not, why? Yes. A* is guaranteed to find an optimal path when the heuristic is admissible (or consistent with a strict expanded list). H is admissible but not consistent, since the link from D to C decreases the heuristic cost by, which is greater than the link cost of. Still, the optimal path was found. G7:. Algorithm: niform Cost Search. Heuristic (if any): None. Did it find least-cost path? If not, why? Yes. niform Cost is guaranteed to find a shortest path.

7 . Search problem. a) Breadth first search will start from the root node, then expands all the successors of the root node, and then all their successors and so on. Breadth first search stops when first solution is found. L D L L D R b) Depth-first search expands the deepest node in the search tree. Notice that there was no mechanism to remember states that have been visited earlier. Depth first will not find a solution as it will start oscillating between movements and D D 4 etc..

8 c) A* is a search method that opens the node with smallest cost function value. The search begins from the start state.. When the start state is opened we see two states f=+= f=0+= f=0+= L f=+=4. The leftmost state has a lower cost so that one is choosen and opened. We now see states with costs 5, and 4 f=+= f=0+= L f=+=4 D L f=+=5 f=+=. The state with lowest cost is opened and we see two more nodes including the goal node. We notice that the goal node has the lowest cost so we choose that and can finish the search. If some other node with a lower cost function value was still visible, A* search would choose that instead of the higher cost goal node. This is because A* tries to find the path with lowest cost. D f=+=5 f=+0= D f=0+= L f=+= L f=+= R f=+=5 f=+=4