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Informed search CSM6120 Introduction to Intelligent Systems

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Quick review Problem definition Initial state, goal state, state space, actions, goal function, path cost function Factors Branching factor, depth of shallowest solution, maximum depth of any path in search state, complexity, etc. Uninformed techniques BFS, DFS, Depth-limited, UCS, IDS

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Informed search What well look at: Heuristics Hill-climbing Best-first search Greedy search A* search

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Heuristics A heuristic is a rule or principle used to guide a search It provides a way of giving additional knowledge of the problem to the search algorithm Must provide a reasonably reliable estimate of how far a state is from a goal, or the cost of reaching the goal via that state A heuristic evaluation function is a way of calculating or estimating such distances/cost

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Heuristics and algorithms A correct algorithm will find you the best solution given good data and enough time It is precisely specified A heuristic gives you a workable solution in a reasonable time It gives a guided or directed solution

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Evaluation function There are an infinite number of possible heuristics Criteria is that it returns an assessment of the point in the search If an evaluation function is accurate, it will lead directly to the goal More realistically, this usually ends up as seemingly-best- search Traditionally, the lowest value after evaluation is chosen as we usually want the lowest cost or nearest

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Heuristic evaluation functions Estimate of expected utility value from a current position E.g. value for pieces left in chess Way of judging the value of a position Humans have to do this as we do not evaluate all possible alternatives These heuristics usually come from years of human experience Performance of a game playing program is very dependent on the quality of the function

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Heuristics?

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Heuristic evaluation functions Must agree with the ordering a utility function would give at terminal states (leaf nodes) Computation must not take long For non-terminal states, the evaluation function must strongly correlate with the actual chance of winning The values can be learned using machine learning techniques

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Heuristics for the 8-puzzle Number of tiles out of place (h 1 ) Manhattan distance (h 2 ) Sum of the distance of each tile from its goal position Tiles can only move up or down city blocks

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The 8-puzzle Using a heuristic evaluation function: h 2 (n) = sum of the distance each tile is from its goal position

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Goal state h 1 =1 h 2 =1 h 1 =5 h 2 = =7 Current state

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Search algorithms Hill climbing Best-first search Greedy best-first search A*

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Iterative improvement Consider all states laid out on the surface of a landscape The height at any point corresponds to the result of the evaluation function

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Iterative improvement Paths typically not retained - very little memory needed Move around the landscape trying to find the lowest valleys - optimal solutions (or the highest peaks if trying to maximise) Useful for hard, practical problems where the state description itself holds all the information needed for a solution Find reasonable solutions in a large or infinite state space

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Hill-climbing (greedy local) Start with current-state = initial-state Until current-state = goal-state OR there is no change in current-state do: a) Get the children of current-state and apply evaluation function to each child b) If one of the children has a better score, then set current- state to the child with the best score Loop that moves in the direction of decreasing (increasing) value Terminates when a dip (or peak) is reached If more than one best direction, the algorithm can choose at random

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Hill-climbing (gradient ascent)

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Hill-climbing drawbacks Local minima (maxima) Local, rather than global minima (maxima) Plateau Area of state space where the evaluation function is essentially flat The search will conduct a random walk Ridges Causes problems when states along the ridge are not directly connected - the only choice at each point on the ridge requires uphill (downhill) movement

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Best-first search Like hill climbing, but eventually tries all paths as it uses list of nodes yet to be explored Start with priority-queue = initial-state While priority-queue not empty do: a) Remove best node from the priority-queue b) If it is the goal node, return success. Otherwise find its successors c) Apply evaluation function to successors and add to priority-queue

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Best-first example

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Best-first search Different best-first strategies have different evaluation functions Some use heuristics only, others also use cost functions: f(n) = g(n) + h(n) For Greedy and A*, our heuristic is: Heuristic function h(n) = estimated cost of the cheapest path from node n to a goal node For now, we will introduce the constraint that if n is a goal node, then h(n) = 0

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Greedy best-first search Greedy BFS tries to expand the node that is closest to the goal assuming it will lead to a solution quickly f(n) = h(n) aka greedy search Differs from hill-climbing – allows backtracking Implementation Expand the most desirable node into the frontier queue Sort the queue in decreasing order

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Route planning: heuristic??

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Route planning - GBFS

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Greedy best-first search

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Route planning

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Greedy best-first search

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This happens to be the same search path that hill-climbing would produce, as theres no backtracking involved (a solution is found by expanding the first choice node only, each time).

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Greedy best-first search Complete No, GBFS can get stuck in loops (e.g. bouncing back and forth between cities) Time complexity O(b m ) but a good heuristic can have dramatic improvement Space complexity O(b m ) – keeps all the nodes in memory Optimal No! (A – S – F – B = 450, shorter journey is possible)

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Practical 2 Implement greedy best-first search for pathfinding Look at code for AStarPathFinder.java

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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 start h(n) g(n) n goal

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A* search When h(n) = h*(n) (h*(n) is actual cost to goal) Only nodes in the correct path are expanded Optimal solution is found When h(n) < h*(n) Additional nodes are expanded Optimal solution is found When h(n) > h*(n) Optimal solution can be overlooked

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Route planning - A*

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A* search

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Complete and optimal if h(n) does not overestimate the true cost of a solution through n Time complexity Exponential in [relative error of h x length of solution] The better the heuristic, the better the time Best case h is perfect, O(d) Worst case h = 0, O(b d ) same as BFS, UCS Space complexity Keeps all nodes in memory and save in case of repetition This is O(b d ) or worse A* usually runs out of space before it runs out of time

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A* exercise Node Coordinates SL Distance to K A (5,9) 8.0 B (3,8) 7.3 C (8,8) 7.6 D (5,7) 6.0 E (7,6) 5.4 F (4,5) 4.1 G (6,5) 4.1 H (3,3) 2.8 I (5,3) 2.0 J (7,2) 2.2 K (5,1) 0.0

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Solution to A* exercise

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GBFS exercise Node Coordinates Distance A (5,9) 8.0 B (3,8) 7.3 C (8,8) 7.6 D (5,7) 6.0 E (7,6) 5.4 F (4,5) 4.1 G (6,5) 4.1 H (3,3) 2.8 I (5,3) 2.0 J (7,2) 2.2 K (5,1) 0.0

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Solution

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To think about... f(n) = g(n) + h(n) What algorithm does A* emulate if we set h(n) = -g(n) - depth(n) h(n) = 0 Can you make A* behave like Breadth-First Search?

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A* search - Mario Control of Super Mario by an A* search Source code available Various videos and explanations Written in Java

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Admissible heuristics A heuristic h(n) is admissible if for every node n, h(n) h * (n), where h * (n) is the true cost to reach the goal state from n An admissible heuristic never overestimates the cost to reach the goal Example: h SLD (n) (never overestimates the actual road distance) Theorem: If h(n) is admissible, A * is optimal (for tree-search)

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Optimality of A* (proof) Suppose some suboptimal goal G 2 has been generated and is in the frontier. Let n be an unexpanded node in the frontier such that n is on a shortest path to an optimal goal G f(G 2 ) = g(G 2 )since h(G 2 ) = 0 (true for any goal state) g(G 2 ) > g(G) since G 2 is suboptimal f(G) = g(G)since h(G) = 0 f(G 2 ) > f(G)from above

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Optimality of A* (proof) f(G 2 ) > f(G) h(n) h*(n)since h is admissible g(n) + h(n) g(n) + h*(n) f(n) f(G)(f(G) = g(G) = g(n) + h*(n)) Hence f(G 2 ) > f(n), and A* will never select G 2 for expansion

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Admissible heuristic example: for the 8-puzzle h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance i.e. no of squares from desired location of each tile h 1 (S) = ?? h 2 (S) = ?? Heuristic functions

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Admissible heuristic example: for the 8-puzzle h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance i.e. no of squares from desired location of each tile h 1 (S) = 6 h 2 (S) = = 14 Heuristic functions

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Dominance/Informedness if h 2 (n) h 1 (n) for all n (both admissible) then h 2 dominates h 1 and is better for search Typical search costs: (8 puzzle, d = solution length) d = 12 IDS = 3,644,035 nodes A*(h 1 ) = 227 nodes A*(h 2 ) = 73 nodes d = 24 IDS 54,000,000,000 nodes A*(h 1 ) = 39,135 nodes A*(h 2 ) = 1,641 nodes Heuristic functions

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Admissible heuristic example: for the 8-puzzle h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance i.e. no of squares from desired location of each tile h 1 (S) = 6 h 2 (S) = = 14 But how do we come up with a heuristic? Heuristic functions

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Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem E.g. If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h 1 (n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h 2 (n) gives the shortest solution Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem Relaxed problems

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Choosing a strategy What sort of search problem is this? How big is the search space? What is known about the search space? What methods are available for this kind of search? How well do each of the methods work for each kind of problem?

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Which method? Exhaustive search for small finite spaces when it is essential that the optimal solution is found A* for medium-sized spaces if heuristic knowledge is available Random search for large evenly distributed homogeneous spaces Hill climbing for discrete spaces where a sub-optimal solution is acceptable

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Summary What is search for? How do we define/represent a problem? How do we find a solution to a problem? Are we doing this in the best way possible? What if search space is too large? Can use other approaches, e.g. GAs, ACO, PSO...

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Finally Try the A* exercise on the course website (solutions will be made available later) Next seminar on Monday at 9:30am See the algorithms in action: baur.de/cs.web.mashup.pathfinding.html baur.de/cs.web.mashup.pathfinding.html

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