DCP 1172 Introduction to Artificial Intelligence

1. DCP 1172Introduction to Artificial Intelligence Lecture notes for Chap. 4 [AIMA] Chang-Sheng Chen

2. Last time: Problem-Solving • Problem solving: • Goal formulation • Problem formulation (states, operators) • Search for solution • Problem formulation: • Initial state • Operators • Goal test • Path cost • Problem types: • single state: fully observable and deterministic environment • multiple state: partially observable and deterministic environment • contingency: partially observable and nondeterministic environment • exploration: unknown state-space DCP 1172, Ch.4

3. Last time: Finding a solution Function General-Search(problem, strategy) returns a solution, or failure initialize the search tree using the initial state problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal statethen return the corresponding solution else expand the node and add resulting nodes to the search tree end • Solution: is a sequence of operators that bring you from current state to the goal state • Basic idea:offline, systematic exploration of simulated state-space by generating successors of explored states (expanding) • Strategy: The search strategy is determined by the order in which the • nodes are expanded. DCP 1172, Ch.4

4. A Clean Robust Algorithm Function UniformCost-Search(problem, Queuing-Fn) returns a solution, or failure open make-queue(make-node(initial-state[problem])) closed [empty] loop do if open is empty then return failure currnode  Remove-Front(open) if Goal-Test[problem] applied to State(currnode) then return currnode children  Expand(currnode, Operators[problem]) whilechildren not empty [… see next slide …] end closed  Insert(closed, currnode) open  Sort-By-PathCost(open) end DCP 1172, Ch.4

5. A Clean Robust Algorithm [… see previous slide …] children  Expand(currnode, Operators[problem]) whilechildren not empty child  Remove-Front(children) if no node in open or closed has child’s state open  Queuing-Fn(open, child) else if there exists node in open that has child’s state if PathCost(child) < PathCost(node) open  Delete-Node(open, node) open  Queuing-Fn(open, child) else if there exists node in closed that has child’s state if PathCost(child) < PathCost(node) closed  Delete-Node(closed, node) open  Queuing-Fn(open, child) end [… see previous slide …] DCP 1172, Ch.4

6. Last time: search strategies Uninformed: Use only information available in the problem formulation • Breadth-first • Uniform-cost • Depth-first • Depth-limited • Iterative deepening Informed: Use heuristics to guide the search • Best first • A* DCP 1172, Ch.4

7. Evaluation of search strategies • Search algorithms are commonly evaluated according to the following four criteria: • Completeness: does it always find a solution if one exists? • Time complexity: how long does it take as a function of number of nodes? • Space complexity: how much memory does it require? • Optimality: does it guarantee the least-cost solution? • Time and space complexity are measured in terms of: • b – max branching factor of the search tree • d – depth of the least-cost solution • m – max depth of the search tree (may be infinity) DCP 1172, Ch.4

8. Last time: uninformed search strategies Uninformed search: Use only information available in the problem formulation • Breadth-first • Uniform-cost • Depth-first • Depth-limited • Iterative deepening DCP 1172, Ch.4

9. This time: informed search Informed search: Use heuristics to guide the search • Best first • A* • Heuristics • Hill-climbing • Simulated annealing DCP 1172, Ch.4

10. Best-first search • Idea: use an evaluation function for each node; estimate of “desirability” • expand most desirable unexpanded node. • Implementation: QueueingFn = insert successors in decreasing order of desirability • Special cases: greedy search A* search DCP 1172, Ch.4

11. Romania with step costs in km 374 253 329 DCP 1172, Ch.4

12. Greedy search • Estimation function: h(n) = estimate of cost from node n to goal (heuristic) • For example: hSLD(n) = straight-line distance from n to Bucharest • Greedy search expands first the node that appears to be closest to the goal, according to h(n). DCP 1172, Ch.4

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17. Properties of Greedy Search • Complete? No – can get stuck in loops e.g., Iasi > Neamt > Iasi > Neamt > … • Complete in finite space with repeated-state checking. • Time? O(b^m) but a good heuristic can give dramatic improvement • Space? O(b^m) – keeps all nodes in memory • Optimal? No. DCP 1172, Ch.4

18. A* search • Idea: avoid expanding paths that are already expensive • evaluation function: f(n) = g(n) + h(n) with: g(n) – cost so far to reach n h(n) – estimated cost to goal from n f(n) – estimated total cost of path through n to goal • A* search uses an admissible heuristic, that is, h(n)  h*(n) where h*(n) is thetrue cost from n. • For example: hSLD(n)never overestimates actual road distance. • Theorem: A* search is optimal DCP 1172, Ch.4

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25. 1 Optimality of A* (standard proof) Suppose some suboptimal goal G2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G1. DCP 1172, Ch.4

26. Optimality of A* (more useful proof) DCP 1172, Ch.4

27. f-contours How do the contours look like when h(n) =0? DCP 1172, Ch.4

28. Properties of A* • Complete? Yes, unless infinitely many nodes with f  f(G) • Time? Exponential in [(relative error in h) x (length of solution)] • Space? Keeps all nodes in memory • Optimal? Yes – cannot expand fi+1 until fi is finished DCP 1172, Ch.4

29. Proof of lemma: pathmax DCP 1172, Ch.4

30. Admissible heuristics DCP 1172, Ch.4

31. Relaxed Problem • Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem. • If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution. • If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution. DCP 1172, Ch.4

32. Recall: breadth-first search, step by step DCP 1172, Ch.4

33. Implementation of search algorithms Function General-Search(problem, Queuing-Fn) returns a solution, or failure nodes  make-queue(make-node(initial-state[problem])) loop do if nodes is empty then return failure node  Remove-Front(nodes) if Goal-Test[problem] applied to State(node) succeeds then return node nodes  Queuing-Fn(nodes, Expand(node, Operators[problem])) end Queuing-Fn(queue, elements) is a queuing function that inserts a set of elements into the queue and determines the order of node expansion. Varieties of the queuing function produce varieties of the search algorithm. DCP 1172, Ch.4

34. Recall: breath-first search, step by step DCP 1172, Ch.4

35. Breadth-first search Node queue: initialization # state depth path cost parent # 1 Arad 0 0 -- DCP 1172, Ch.4

36. Breadth-first search Node queue: add successors to queue end; empty queue from top (i.e., FIFO) # state depth path cost parent # 1 Arad 0 0 -- 2 Zerind 1 1 1 3 Sibiu 1 1 1 4 Timisoara 1 1 1 DCP 1172, Ch.4

37. Breadth-first search Node queue: add successors to queue end; empty queue from top # state depth path cost parent # 1 Arad 0 0 -- 2 Zerind 1 1 1 3 Sibiu 1 1 1 4 Timisoara 1 1 1 5 Arad 2 2 2 6 Oradea 2 2 2 (get smart: e.g., avoid repeated states like node #5) DCP 1172, Ch.4

38. Depth-first search DCP 1172, Ch.4

39. Depth-first search Node queue: initialization # state depth path cost parent # 1 Arad 0 0 -- DCP 1172, Ch.4

40. Depth-first search Node queue: add successors to queue front; empty queue from top (i.e., FILO, or stack ) # state depth path cost parent 2 Zerind 1 1 1 3 Sibiu 1 1 1 4 Timisoara 1 1 1 1 Arad 0 0 -- DCP 1172, Ch.4

41. Depth-first search Node queue: add successors to queue front; empty queue from top # state depth path cost parent # 5 Arad 2 2 2 6 Oradea 2 2 2 2 Zerind 1 1 1 3 Sibiu 1 1 1 4 Timisoara 1 1 1 1 Arad 0 0 -- DCP 1172, Ch.4

42. Last time: search strategies Uninformed: Use only information available in the problem formulation • Breadth-first • Uniform-cost • Depth-first • Depth-limited • Iterative deepening Informed: Use heuristics to guide the search • Best first: • Greedy search -- queue first nodes that maximize heuristic “desirability” based on estimated path costfrom current node to goal; • A* search – queue first nodes that minimizesum of path cost so far and estimated path cost to goal. DCP 1172, Ch.4

43. This time – Local Search • Iterative improvement • Hill climbing • Simulated annealing DCP 1172, Ch.4

44. Iterative improvement • In many optimization problems, path is irrelevant; the goal state itself is the solution. • Then, state space = space of “complete” configurations. Algorithm goal: - find optimal configuration (e.g., TSP), or, - find configuration satisfying constraints (e.g., n-queens) • In such cases, can use iterative improvement algorithms: keep a single “current” state, and try to improve it. DCP 1172, Ch.4

45. Iterative improvement example: vacuum world Simplified world: 2 locations, each may or not contain dirt, each may or not contain vacuuming agent. Goal of agent: clean up the dirt. If path does not matter, do not need to keep track of it. DCP 1172, Ch.4

46. Iterative improvement example: n-queens • Goal: Put n chess-game queens on an n x n board, with no two queens on the same row, column, or diagonal. • Here, goal state is initially unknown but is specified by constraints that it must satisfy. DCP 1172, Ch.4

47. Hill climbing (or gradient ascent/descent) • Iteratively maximize “value” of current state, by replacing it by successor state that has highest value, as long as possible. (健忘症) DCP 1172, Ch.4

48. Question: What is the difference between this problem and our problem (finding global minima)? DCP 1172, Ch.4

49. Hill climbing • Note: • minimizing a “value” function v(n) is equivalent to maximizing –v(n), thus both notions are used interchangeably. • Notion of “extremization”: find extrema (minima or maxima) of a value function. DCP 1172, Ch.4

50. Hill climbing • Problem: depending on initial state, may get stuck in local extremum. DCP 1172, Ch.4