COMP3702/7702 Artificial Intelligence Week2: Search (Russell & Norvig ch. 3)" Hanna Kurniawati"

Transcription

1 COMP3702/7702 Artificial Intelligence Week2: Search (Russell & Norvig ch. 3)" Hanna Kurniawati"

2 Last week" What is Artificial Intelligence?" Some history" Agent defined" The agent design problem" Search: A way to solve the problem" 2

3 How to build an agent?" Formulating the agent design problem:" Defining the components involved." Defining the problem." Solving the problem" Computing what the agent should do" Basically our topic most of the semester" 3

4 The Agent Design Problem The Components" The first step to formulating an agent design problem is to set the following components:" Action space (A)" Percept space (O)" State space (S)" World dynamics (T: SXA S)" Percept function (Z: S O)" Utility function (U: S real number)"

5 Formulating the agent design problem: Defining The Problem" The agent design problem: Find a mapping from sequences of percepts to action P* A that maximizes the utility function. " Given the sequences of percepts it has seen so far, what should the agent do next, so that the utility function can be maximized."

6 In the first part of this class," Assumptions on class environment:" Fully observable vs. partially observable." Does the agent know the state of the world exactly?" Deterministic vs. non-deterministic." Does an action map one state into a single other state?" Static vs. dynamic." Can the world change while the agent is thinking?" Discrete vs. continuous." Are the actions & percepts discrete?"

7 Last week" What is Artificial Intelligence?" Some history" Agent defined" The agent design problem" Search: A way to solve the problem Today" 7

8 Today" What is search?" Formulating a problem as a search problem" State graph representation" General structure of search algorithms" Performance measure of search algorithms" Blind search" Informed search" 8

9 What is search?" A general framework to solve problems by exploring alternatives." Alternatives come from knowledge about the problem." Example: " Twisted nail puzzle." 8-puzzle." 9 7" 2" 4" 5" 6" 8" 3" 1" Initial state" 1" 2" 3" 4" 5" 6" 7" 8" Goal state"

10 What is search?" 7" 2" 4" 5" 6" 8" 3" 1" Knowledge: world dynamics" 7" 2" 4" 7" 2" 4" 7" 4" 7" 2" 4" 5" 6" 5" 6" 5" 2" 6" 5" 3" 6" 8" 3" 1" 8" 3" 1" 8" 3" 1" 8" 1" 10

11 Today" What is search?" Formulating a problem as a search problem" State graph representation" General structure of search algorithms" Performance measure of search algorithms" Blind search" Informed search" 12

12 Formulating a problem as a search problem" Similar to formulation of rational agent:" Action space." Percept space." State space." World dynamics." Percept function" Utility: Cost function." Initial & goal state." Formulation of rational agent." Find a sequence of actions to move the agent from being in the initial state to being in the goal state, (such that the cost of moving is minimized)." 13

13 In this part of the class," Assumptions on class environment:" Fully observable vs. partially observable." Does the agent know the state of the world exactly?" Deterministic vs. non-deterministic." Does an action map one state into a single other state?" Static vs. dynamic." Can the world change while the agent is thinking?" Discrete vs. continuous." Are the actions & percepts discrete?" 14

14 Formulating a problem as a search problem" Similar to formulation of rational agent:" Action space." Percept space." State space." World dynamics." Percept function" Utility: Cost function." Initial & goal state." Formulation of rational agent." Due to fully observable assumption." 15

15 The 8-puzzle example" Action space (A): {up, down, left, right} for the empty cell." State space (S): All possible permutations." World dynamics (SXA S)" Utility: Each move of the blank tile cost -1 (to reach the goal with smallest #steps)." Initial & goal state: Given." 7" 2" 4" 5" 6" 8" 3" 1" Initial state" 1" 2" 3" 4" 5" 6" 7" 8" Goal state" 16

16 Introduction to search" What is search?" Formulating a problem as a search problem" State graph representation" General structure of search algorithms" Performance measure of search algorithms" Blind search" Informed search" 17

17 Recall Formulating a problem as a search problem" Similar to formulation of rational agent:" Action space." Percept space." State space." World dynamics." Percept function" Utility: Cost function." Initial & goal state." Formulation of rational agent." Find a sequence of actions to move the agent from being in the initial state to being in the goal state, (such that the cost of moving is minimized)." 18

18 State graph representation " Graph: (V, E)" Each vertex ( v V ) represents a state" Initial & goal state: Initial & goal vertices." Edges (E) " Represents world dynamics (T: SXA S)." vv' E a A Each edge labeled by an action represents the mapping T(s, a) = sʼ, where s is the state represented by v and sʼ is the state represented by vʼ." "The edge is often labeled with cost to move from the state s to sʼ too" 19

19 State graph representation " The solution is a path from initial to goal vertices in the state graph." Cost: the sum of the cost associated with each edge in the path." Optimal solution: Path with least cost." 20

20 The 8-puzzle example" ; -1" Right ; -1" Left ; -1" Down ; -1" Top ; -1"

21 State graph representation" A way to represent the problem concretely in programs. Also a way of thinking about the problem. " We may or may not explicitly represent the state graph." In problems with continuous or very large state space, state graph is often used as a compact representation of the state space." 22

22 Today" What is search?" Formulating a problem as a search problem" State graph representation" General structure of search algorithms" Performance measure of search algorithms" Blind search" Informed search" 24

23 General structure of search algorithm" Put initial vertex in a container of states to be expanded." Loop" Select a vertex v from the container." If v is the goal vertex, then return." Expand v (i.e., put the results of successor(v) to the container )." successor(v), a function that:" Takes a vertex v as input." Output the set of immediate next vertices that can be visited from v (i.e., the end-point of out-edges from v)." 25

24 Search tree: An abstract representation of the container " State graph" Search tree" If states can be revisited, the search tree maybe infinite, even though the state graph (& state space) is finite." 26

25 Fringe nodes" Nodes in the search tree that have not been expanded yet." Fringe nodes" 27

26 General structure of search algorithm with search tree as container " Put the initial vertex as root of the search tree." Loop" Select a node t from the search tree." If t corresponds to the goal vertex, then return." Expand t" Suppose t corresponds to vertex v of the state graph." Put the results of successor(v) as children of t in the search tree." In general, only select a node from the fringe nodes." Various search methods differ in how they select a vertex from the container / a node in the search tree. " 28

27 Today" What is search?" Formulating a problem as a search problem" State graph representation" General structure of search algorithms" Performance measure of search algorithms" Blind search" Informed search" 29

28 Performance measure of search algorithms" Completeness" Complete: The algorithm will find the solution whenever one exists." What happened when no solution exists?" Optimality" Optimal: Return a minimum cost path whenever one exists." Complexity" Amount of time & memory needed to solve the problem." Use big-o notation. " 30

29 Computational complexity" Indicates how fast the problem difficulty grows as the input size increases" Here, weʼll only use big-o notation" O(g(n)) = {f(n) there exist positive constants c and n 0 such that 0 f(n) c.g(n) for all n n 0 } " If the number of operation is f(n) where n is the size of input" We say f(n) = O(g(n)) to indicate that f(n) is a member of O(g(n))" 31

30 Computational complexity example" For (i = 1 ; i < length(a) ; i ++) {" " valuetoinsert = A[i]" holepos = i" while (holepos > 0 and valuetoinsert < A[holePos-1]) {" " " A[holePos] = A[holePos-1]" holepos = holepos-1" }" A[holePos] = valuetoinsert " "}" F(n) = c 1.( n-1) + c 2.(n-1)" "n: length(a)" Complexity: O(n 2 )" For more info, please look at the resource section of the class website." 32

31 Today" What is search?" Formulating a problem as a search problem" State graph representation" General structure of search algorithms" Performance measure of search algorithms" Blind search" Informed search" 33

32 Problem example: Navigation app" Given a map, how do I move from A to B?" I! G! State graph" Weʼll use this example throughout to explain the various search algorithms." 35

33 Blind search algorithms" Blind search: Do not use additional information to estimate the cost form the current node to the goal" Breadth first search (BFS)" Depth first search (DFS)" Iterative deepening DFS "" Uniform cost search" 36

34 Breadth first search (BFS)" Cost: #steps (ignore cost on the edges)." Select a fringe node in the same level of the search tree, before selecting fringe nodes at the next level. " I! G! 37

35 Breadth first search (BFS)" Use queue to keep fringe nodes." Queue: Abstract data structure where the most recent data is retrieved last (FIFO: First In First Out)." Set the initial vertex I as root of the search tree." Push I to the queue." Loop" t = front of the queue. " Remove t from the queue." If t is the goal vertex, then return." Otherwise," Put the results of successor(t) as children of t in the search tree." Push the results of successor(t) at the back of the queue." 38

36 Breadth first search (BFS) Properties & Analysis" b: branching factor" d: depth of shallowest goal node. " Complete?" Complete, if b is finite." Generate optimal solution?" Yes (in #steps)." Complexity" Time: O(b d ) #nodes visited" "1 + b + b b d = (b d+1-1)/(b-1)." Space: O(b d ) #nodes to remember" 39

37 Just to get some intuition " d" # Nodes" Time" Memory" 2" 110".11 msec" 107 Kbytes" 4" 11,110" 11 msec " 10.6 Mbyte" 6" ~10 6 " 1 sec" 1 Gbytes" 8" ~10 8 " ~2min" 103 Gbytes" 10" ~10 10 " ~2.8 hours" 10 Tbyte" 12" ~10 12 " ~11.6 days" 1 Pbytes" 14" ~10 14 " ~3.2 years" 99 Pbytes" Assumptions: b = 10; 1million nodes/sec; 1 Kbytes/node" 40

38 Just to get some intuition " d" # Nodes" Time" Memory" 2" 110".11 msec" 107 Kbytes" Well 4" just 11,110" wait for better 11 msec hardware " 10.6 Mbyte" After all, computer speed becomes double every 18 months " 6" ~10 6 " 1 sec" 1 Gbytes" Yeah right " The 8" waiting ~10 8 time " for ~2min" the hardware 103 maybe Gbytes" more than the 10" waiting ~10time " to solve ~2.8 hours" the problem " 10 Tbyte" 12" ~10 12 " ~11.6 days" 1 Pbytes" 14" ~10 14 " ~3.2 years" 99 Pbytes" Assumptions: b = 10; 1million nodes/sec; 1 Kbytes/node" 41

39 Bidirectional Strategy" 2 fringe queues: FRINGE1 and FRINGE2" Initial" s Goal" Time and space complexity is O(b d/2 ) << O(b d ) " 42

40 Depth first search (DFS)" Cost: #steps (ignore cost on the edges)." Expand a fringe node most recently inserted to the tree. " I! G! 43

41 Depth first search (DFS)" Use stack to keep fringe nodes." Stack: Abstract data structure where the most recent data is retrieved first (LIFO: Last In First Out)." Set the initial vertex I as root of the search tree." Push I to the stack." Loop" t = top of the stack. " Remove t from the stack." If t is the goal vertex, then return." Otherwise," Put the results of successor(t) as children of t in the search tree." Push the results of successor(t) to the stack." 44

42 Depth first search (DFS) Properties & Analysis" b: branching factor, m: maximum depth" d: depth of shallowest goal node. " Complete?" Complete, if m is finite and no loop in the state graph / avoid revisiting states." Generate optimal solution?" No." Complexity" Time: O(b m ) 1 + b + b b m = (b m+1-1)/(b-1). " Space: Can be implemented using O(bm) or O(m) using backtracking DFS. But be careful of revisiting vertices (states). " 45

43 Avoid revisiting states" Donʼt add node to the tree if a node representing the same state is already in the tree." Problem: Paths leading to the same node may have different cost." Donʼt add node to the tree if a node representing the same state has been expanded in the search tree." Valid for any search method." 46

44 Bidirectional & DFS" Will it work well?" 49

45 BFS, DFS, & Iterative Deepening DFS" BFS: " Finds minimum step path, but requires exponential space." DFS: " Efficient in space, but no path length guarantee." Iterative deepening: " Multiple DFS with increasing depth-cutoff until the goal is found." For k = 1, 2, do" Perform DFS with depth cutoff k." "Only generates nodes with depth k. " 50

46 Iterative deepening DFS" Cost: #steps (ignore cost on the edges)." Example" I! G! 51

47 Iterative Deepening DFS Properties & Analysis" b: branching factor, m: maximum depth" d: depth of shallowest goal node. " Complete?" Yes. If b is finite." Generate optimal solution?" Yes (in terms of #steps)." Complexity" Time: O(b d ) " "db + (d-1)b (1)b d " Space: O(bd) " 52