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Problem Solving Based on Searching Module 2
- 1. MODULE 2-Problem Solving basedon Searching Introduction to Problem Solving by searching Methods-State Space search, Uninformed Search Methods – Uniform Cost Search, Breadth First Search- Depth First Search-Depth limited search, Iterative deepening depth-first, Informed Search Methods- Best First Search, A* Search
- 2. Artificial intelligence • Intelligence: “Ability to learn, understand, and think”. • The study of the capacity of machines to simulate intelligent human behavior. • AI is the study of how to make computers make things which at the moment people do better. ARTIFICIAL INTELLIGENCE 2
- 3. Formal description for defininga problem •Representation : • State Space • Initial State • Goal State • Transformation Rules ARTIFICIAL INTELLIGENCE 3
- 4. Problem solving System •Definethe problem. •Analyze the problem. •Isolate & represent the task knowledge necessary to solve the problem. •Choose the best problem- solving techniques & apply it to the problem. ARTIFICIAL INTELLIGENCE 3
- 5. Production Systems • ProductionSystem Consists of • A set of rules : (Pattern) 🡺Operation to be performed • One / more Knowledge/ databases • Control strategy –specifies order in which rules will be compared • Rule applier ARTIFICIAL INTELLIGENCE 5
- 6. Example 1: 8-Puzzleproblem ARTIFICIAL INTELLIGENCE 6 URL:http://www.cs.mun.ca/~oram/cs3754/AI6.pdf
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- 8. 8-PUZZLE • Initial StateGoal State • Apply 2 heuristic functions; • Misplaced Tile & Manhattan Distance
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- 12. Example 2: GamePlaying ARTIFICIAL INTELLIGENCE 12
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- 14. Chess game ARTIFICIAL INTELLIGENCE 14 Transformation Rules: whitepawn at square(file e, rank 2) AND square(file e, rank 3) is empty AND square(file e, rank 4) is empty Initial state: Current board position Goal state: opponent does not have a legal move and his /her king is under attack. Move pawn from square(file e, rank 2) to square(file e, rank 4)
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- 17. Problem types • Deterministic,fully observable ⇒ single state problem • Agent knows exactly which state it will be in • solution is a sequence. • E.g:Chess game • Partial knowledge of states and actions: • Non-observable ⇒ sensorless or conformant problem • Agent may have no idea where it is; • solution (if any) is a sequence. • Nondeterministic (stochastic) and/or partially observable ⇒ contingency problem E.g: Self Driving Cars • Unknown state space ⇒ exploration problem (“online”) • When states and actions of the environment are unknown. ARTIFICIAL INTELLIGENCE 17
- 18. Problem characteristics • Isthe problem decomposable into a set of independent smaller or easier sub problems ? • E.g: ∫(x^2+ 3x + sin2^x * cos2^x) dx • Can solution steps be ignored or at least undone if they prove unwise? • In real life, there are three important types of problems: • Ignorable ( theorem proving) • Recoverable ( 8-puzzle) • Irrecoverable ( Chess) • Is the problem’s universe predictable? • Is a good solution absolute or relative? ARTIFICIAL INTELLIGENCE 18
- 19. Problem characteristics.. • Isthe solution a state or path? • Is a large amount of knowledge absolutely required to solve the problem , or is knowledge important only to constrain the search? • Can a computer that is simply given the problem return the solution , or will the solution of the problem require interaction between the computer and a person? ARTIFICIAL INTELLIGENCE 19
- 20. Production System characteristics •Monotonic : application of rule never prevents later application of another rule when both are used at the same time (that could have been applied at the time first rule was selected) • Non Monotonic : this is not true • Partially commutative : If the application of a particular sequence of rules transform x🡺y, then any permutation of those rules ie allowable also transform x🡺 y • Commutative : both Monotonic & Partially commutative ARTIFICIAL INTELLIGENCE 20
- 21. Examples of Problems •“Toy”Problems : • Water jug • 8 – Queens • 8 Puzzle ARTIFICIAL INTELLIGENCE 21 • “Real” Problems : • Schedules • Traveling Salesman. • Robot navigation. • Language Analysis (Parsers, Grammars). • VLSI design. • Speech Recognition
- 22. Issues in thedesign of Search programs • Direction in which to conduct the search (forward Vs Backward reasoning) • How to select applicable rules (matching) • How to represent each node of the search process ARTIFICIAL INTELLIGENCE 22
- 23. Control /Search Strategies •Control Strategy decides which rule to apply next during the process of searching for a solution to a problem • Requirements for a good Control Strategy • It should cause motion. • It should explore the solution space in a systematic manner • Types • Uninformed • Heuristic • ARTIFICIAL INTELLIGENCE 23
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- 25. Search strategies-performance measure •A strategy is defined by picking the order of node expansion. • Problem-solving performance is measured in four ways: • Completeness; Does it always find a solution if one exists? • Optimality; Does it always find the least-cost solution? • Time Complexity; Number of nodes generated/expanded? • Space Complexity; Number of nodes stored in memory during search? ARTIFICIAL INTELLIGENCE 25
- 26. Search strategies- performance measure •Time and space complexity are measured in terms of problem difficulty defined by: • b - maximum branching factor of the search tree(the number of children at each node, the outdegree) • d - depth of the least-cost solution ( the depth of its deepest leaf-longest path from node to leaf) • m - maximum depth of the state space (may be ∞) ARTIFICIAL INTELLIGENCE 26
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- 28. Uninformed search strategies •Blindsearch • uses only the information available in problem definition. •Types • Breadth-first search (BFS) • Depth-first search (DFS) • Depth-limited search • Iterative deepening search. ARTIFICIAL INTELLIGENCE 28
- 29. BF-search :Breadth-First Search •At each level we expand all nodes(possible solutions) • Expand shallowest unexpanded node ARTIFICIAL INTELLIGENCE 29 A
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- 34. BF-search: Evaluation • Completeness: •YES (if b is finite) • Time complexity: • Total number of nodes generated • T (b) = 1+b2+b3+.......+ bd= O (bd) • Where the d= depth of shallowest solution and b is a node at every state • Space complexity: • Memory requirements are a bigger problem than its execution time. • O (bd) • Optimality: • Does it always find the least-cost solution? • In general YES unless actions have different cost. ARTIFICIAL INTELLIGENCE 34
- 35. Example :Water Jugproblem ARTIFICIAL INTELLIGENCE 35 .Example: Water Jug Problem • You are given two jugs, a 4-gallon one and a 3-litre one. Neither have any measuring markers on it. There is a pump that can be used to fill the jugs with water. How can you get exactly 2 gallons of water into 4- gallon jug. • Let x and y be the amounts of water in 4-gallon and 3-gallon Jugs respectively. • (x,y) refers to water available at any time in 4-gallon, 3-gallon jugs. • (x,y) 🡪 (x-d,y+dd) means drop some unknown amount d of water from 4- gallon jug and add dd onto 3-gallon jug.
- 36. Example :Water Jugproblem ARTIFICIAL INTELLIGENCE 36
- 37. Elements of Production Systems •State Space:(x,y) • Where • x=0,1,2,3,4 • y=0,1,2,3 • Initial State: (0,0) • Goal State: (2,n) • Production Rules ARTIFICIAL INTELLIGENCE 37
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- 40. BF-search: Evaluation • Completeness: •YES • Time complexity: • Total number of nodes generated • O (bd) • Space complexity: • Memory requirements are a bigger problem than its execution time. • O (bd) • Optimality: • YES ARTIFICIAL INTELLIGENCE 40
- 41. DF-search: Depth-First Search •Expands one of the nodes at the deepest level ARTIFICIAL INTELLIGENCE 41 A
- 42. DF-search, an example •Expand deepest unexpanded node ARTIFICIAL INTELLIGENCE 42 A B C S-A-B-D-E-C-G
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- 44. DF-search: evaluation • Completeness: •NO unless search space is finite. • Time complexity : • T(n)= 1+ n2+ n3 +.........+ nm=O(nm) • Where, m= maximum depth of any node and this can be much larger than d (Shallowest solution depth) • Space complexity: • Backtracking search uses even less memory. • O(bm). • Optimality :No • Same issues as completeness ARTIFICIAL INTELLIGENCE 44
- 45. Uniform-cost Search Algorithm • Uniform-costsearch is a searching algorithm used for traversing a weighted tree or graph. • This algorithm comes into play when a different cost is available for each edge. • The primary goal of the uniform-cost search is to find a path to the goal node which has the lowest cumulative cost. • A uniform-cost search algorithm is implemented by the priority queue. • It gives maximum priority to the lowest cumulative cost. • Uniform cost search is equivalent to BFS algorithm if the path cost of all edges is the same. ARTIFICIAL INTELLIGENCE 45
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- 47. Depth-Limited Search Algorithm •A depth-limited search algorithm is similar to depth-first search with a predetermined limit. Depth-limited search can solve the drawback of the infinite path in the Depth-first search. • In this algorithm, the node at the depth limit will treat as it has no successor nodes further. • Depth-limited search can be terminated with two Conditions of failure: • Standard failure value: It indicates that problem does not have any solution. • Cutoff failure value: It defines no solution for the problem within a given depth limit. ARTIFICIAL INTELLIGENCE 47
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- 49. Iterative deepening depth-firstSearch • The iterative deepening algorithm is a combination of DFS and BFS algorithms. • This search algorithm finds out the best depth limit and does it by gradually increasing the limit until a goal is found. • This algorithm performs depth-first search up to a certain "depth limit", and it keeps increasing the depth limit after each iteration until the goal node is found. • This Search algorithm combines the benefits of Breadth-first search's fast search and depth-first search's memory efficiency. • The iterative search algorithm is useful uninformed search when search space is large, and depth of goal node is unknown. ARTIFICIAL INTELLIGENCE 49
- 50. ARTIFICIAL INTELLIGENCE 50 1'st Iteration-----> A 2'ndIteration----> A, B, C 3'rd Iteration------>A, B, D, E, C, F, G 4'th Iteration------>A, B, D, H, I, E, C, F, K, G In the fourth iteration, the algorithm will find the goal node.
- 51. Informed search strategies •Uses problem specific knowledge beyond the definition of the problem. • Types of Heuristic search algorithms • Local search algorithms • Generate and test • Hill Climbing • Global search algorithms • A* search (Best First Search) • AO* search (Problem reduction) • Constraint Satisfaction Problem(CSP) • Means-ends Analysis(MEA) ARTIFICIAL INTELLIGENCE 51
- 52. Local search algorithm: Generateand Test Steps: 1. Generate a possible solution 2. Test to see if this is a solution by comparing the chosen point / end point to the set of acceptable goal states 3. If a solution has been found, quit. Otherwise return to step 1 ARTIFICIAL INTELLIGENCE 52 • Not an Effective Technique • Use Combination- Plan _Generate _ Test • Combination = G&T+ CSP
- 53. • Example -Traveling Salesman Problem (TSP) • Traveler needs to visit n cities. • Know the distance between each pair of cities. • Want to know the shortest route that visits all the cities once. ARTIFICIAL INTELLIGENCE 53
- 54. • TSP -generation of possible solutions is done in lexicographical order of cities: • 1. A - B - C - D • 2. A - B - D - C • 3. A - C - B - D • 4. A - C - D - B • 5. A - D - C - B • 6. A - D - B - C • n=80 will take millions of years to solve exhaustively! ARTIFICIAL INTELLIGENCE 54
- 55. Hill Climbing (HC) •Variant of G&T based on Feedback • In G& T, Test Procedure 🡪 Yes (or) No • Here, “Test Function is Augmented with heuristic“ • HC often used when a good heuristic Function is available & when no useful knowledge available • Example : To reach a place in Unfamiliar city without a map ARTIFICIAL INTELLIGENCE 55
- 56. Hill Climbing • Considerall possible successors as “one step” from the current state on the landscape. • At each iteration, go to • The best successor (steepest ascent) • Any uphill move (first choice) • Any uphill move but steeper is more probable (stochastic) • All variations get stuck at local maxima • Types: ARTIFICIAL INTELLIGENCE 56 •Simple Hill climbing •Steepest –Ascent Hill climbing •Simulated Annealing
- 57. Simple Hill climbing •Algorithm Evaluate the initial state. Loop until a solution is found or there are no new operators left to be applied: - Select and apply a new operator - Evaluate the new state: goal → quit better than current state → new current state • Example: 8 puzzle problem • Here, h(n) = the number of misplaced tiles (not including the blank), the Manhattan Distance heuristic helps us quickly find a solution to the 8-puzzle. ARTIFICIAL INTELLIGENCE 57 Click to add text
- 58. • Advantages ofHill Climbing • Estimates how far away the goal is. • Is neither optimal nor complete. • Can be very fast. • ARTIFICIAL INTELLIGENCE 58
- 59. Global Search Strategies •Best-First Search – an algorithm in which a node is selected for expansion based on an evaluation function f(n) -node with the lowest evaluation function is selected • Choose the node that appears to be the best ARTIFICIAL INTELLIGENCE 59
- 60. A Quick Review •g(n) = cost from the initial state to the current state n • h(n) = estimated cost of the cheapest path from node n to a goal node • f(n) = evaluation function to select a node for expansion (usually the lowest cost node) • Example: in route planning the estimate of the cost of the cheapest path might be the straight line distance between two cities ARTIFICIAL INTELLIGENCE 60
- 61. Greedy Best-First Search •Expand the node that is closest to the goal assuming it will lead to a solution quickly - aka “Greedy Search” • f(n) = h(n) • Implementation • expand the “most desirable” node into the fringe queue • sort the queue in decreasing order of desirability • Example: • consider the straight-line distance heuristic hSLD • Expand the node that appears to be closest to the goal ARTIFICIAL INTELLIGENCE 61
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- 65. Greedy Best-First Search ARTIFICIAL INTELLIGENCE 65 Shortestpath cost is 450
- 66. A* Search • A*(A star) is the most widely known form of Best-First search • It evaluates nodes by combining g(n) and h(n) • f(n) = g(n) + h(n) • Where • 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 ARTIFICIAL INTELLIGENCE 66
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- 71. A* Search ARTIFICIAL INTELLIGENCE 71 Shortest pathcost A->S->R->P->B=418 415=317+98
- 72. A* Search • Complete •Yes • Time • Exponential • The better the heuristic, the better the time • Space • Keeps all nodes in memory and save in case of repetition • A* usually runs out of space before it runs out of time • Optimal • Yes, cannot expand fi+1 unless fi is finished ARTIFICIAL INTELLIGENCE 72
- 73. References • http://www.cs.mun.ca/~oram/cs3754/AI6.pdf • https://www.codingninjas.com/blog/2021/03/24/solving- water-jug-problem-using-bfs/ •Kevin Night and Elaine Rich, Nair B., “Artificial Intelligence (SIE)”, Mc Graw Hill-2008.