recap Describing a state. En're state space vs. incremental development. Elimina'on of children. the solu'on path. Genera'on of children.

Transcription

1 Heuris'c Searches

2 recap Describing a state. En're state space vs. incremental development. Elimina'on of children. the solu'on path. Genera'on of children.

3 Heuris'c Search Heuris'cs help us to reduce the size of the search space. An evalua'on func'on is applied to each goal to assess how promising it is in leading to the goal. Heuris'c searches incorporate the use of domain- specific knowledge in the process of choosing which node to visit next in the search process.

4 Heuris'c Search Search methods that include the use of domain knowledge in the form of heuris'cs are described as weak search methods. The knowledge used is weak in that it may help but does not always help to find a solu'on. Examples of heuris'c searches : best first search, A* algorithm, hill- climbing..

5 Heuris'c Search Heuris'c searches incorporate the use of domain- specific knowledge in the process of choosing which node to visit next in the search process. Search methods that include the use of domain knowledge in the form of heuris'cs are described as weak search methods. The knowledge used is weak in that it usually helps but does not always help to find a solu'on..

6 Calcula'ng Heuris'cs Heuris'cs are rules of thumb that may find a solu'on but are not guaranteed to. Heuris'c func'ons have also been defined as evalua'on func'ons that es'mate the cost from a node to the goal node. The incorpora'on of domain knowledge into the search process by means of heuris'cs is meant to speed up the search process. Heuris'c func'ons are not guaranteed to be completely accurate..

7 Calcula'ng Heuris'cs Heuris'c values are greater than and equal to zero for all nodes. Heuris'c values are seen as an approximate cost of finding a solu'on. A heuris'c value of zero indicates that the state is a goal state. A heuris'c that never overes'mates the cost to the goal is referred to as an admissible heuris'c. Not all heuris'cs are necessarily admissible..

8 Calcula'ng Heuris'cs A heuris'c value of infinity indicates that the state is a deadend and is not going to lead anywhere. A good heuris'c must not take long to compute. Heuris'cs are omen defined on a simplified or relaxed version of the problem, e.g. the number of 'les that are out of place..

9 Calcula'ng Heuris'cs A heuris'c func'on h1 is beoer than some heuris'c func'on h2 if fewer nodes are expanded during the search when h1 is used than when h2 is used. Experience has shown that it is difficult to devise heuris'c func'ons. Furthermore, heuris'cs are fallible and are by no means perfect..

10 Example: 8- Puzzle Problem Initial State Goal State

11 Heuris'cs for the 8- Puzzle Problem Number of 'les out of place - count the number of 'les out of place in each state compared to the goal. Sum the distance that the 'les are out of place. Tile reversals - mul'ple the number of 'le reversals by 2..

12 Examples 8- Puzzle Problem State Tiles out of place Sum of distances out of place 2 x the number of direct tile reversals

13 Best- First Search The best- first search is a general search where the minimum cost nodes are expanded first. The best- first search is not guaranteed to find the shortest solu'on path. The best- first search aoempts to minimize the cost of finding a solu'on. Is a combina'on of the depth first- search and breadth- first search with heuris'cs..

14 Best- First Search A Goal States: H, L B3 C2 D2 E3 F2 G4 H1 I99 J99 K99 L3.

15 Best first search exercise A5 to B4 and C4 B4 to D6 and E5 C4 to F4 and G5 D6 to I7 and J8 I7 to K7 and L8 Start state : A Goal state : E.

16 Hill- Climbing Hill- climbing is similar to the best first search. While the best first search considers states globally, hill- climbing considers only local states. The hill- climbing algorithm generates a par'al tree/graph.

17 Hill- Climbing A Goal States: H, L B3 C2 D2 E3 F2 G4 H1 I99 J99 K99 L3

18 A to B3 and C2 B3 to D2 and E3 C2 to F2 and G4 D2 to H1 and I99 F2 to J99 G4 to K99 and L3 Start state: A Goal state: H, L Hill climbing exercise

19 Greedy Hill- Climbing Evaluate the ini'al state. Select a new operator. Evaluate the new state If it is closer to the goal state than the current state make it the current state. If it is no beoer ignore If the current state is the goal state or no new operators are available, quit. Otherwise repeat steps 2 to 4.

20 Example 1: Greedy Hill- Climbing without Backtracking A Goal States: H, L B4 C4 D2 E5 F2 G4 H1 I99 J99 K99 L3

21 Example 2: Greedy Hill- Climbing with Backtracking A Goal States: H, L B3 C2 D2 E3 F2 G4 H1 I99 J99 K99 L3

22 Example 3: Greedy Hill- Climbing without Backtracking A Goal States: H, L B3 C2 D2 E3 F2 G4 H1 I99 J99 K99 L3

23 A Algorithm The A algorithm is essen'ally the best first search implemented with the following func'on: f(n) = g(n) + h(n) where g(n) - measures the length of the path from any state n to the start state h(n) - is the heuris'c measure from the state n to the goal state

24 8- Puzzle Example f(a)=4+0=4 a) g(n)=0 f(b)=5+1=6 b) c) d) f(c)=3+1= f(d)=5+1=6 g(n)=1 e) f) g) f(e)=3+2=5 f(f)=3+2=5 f(g)=4+2=4 g(n)=2 h(n)=no. of tiles out of place

25 Admissible Algorithms Search algorithms that are guaranteed to find the shortest path are called admissible algorithms. The breadth first search is an example of an admissible algorithm, just use h(n)=0 for all n. The evalua'on func'on we have considered with the best first algorithm is f(n) = g(n) + h(n), where g(n) is the distance from the start node to some node n, e.g. the depth at which the state n is found. if the heuris'c measure h(n) <= h*(n), the op'mal distance from n to a goal, then the A algorithm is op'mal and denoted A*

26 Admissible Heuris'c and the 8- Puzzle Problem The heuris'c that we have developed for the 8- puzzle problem are bounded above by the number of moves required to move to the goal posi'on. The number of 'les out of place and the sum of the distance from each correct 'le posi'on is less than the number of required moves to move to the goal state. Thus, the best first search applied to the 8- puzzle using these heuris'cs is in fact an A* algorithm. (see paper from website)