Searching for Solu4ons. Searching for Solu4ons. Example: Traveling Romania. Example: Vacuum World 9/8/09

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1 Searching for Solu4ons Searching for Solu4ons CISC481/681, Lecture #3 Ben Characterize a task or problem as a search for something In the agent view, a search for a sequence of ac4ons that will result in a predefined goal Many (perhaps most) computa4onal problems can be formulated as a search Copyright Ben Cartere@e 1 Copyright Ben Cartere@e 2 Example: Vacuum World Example: Traveling Romania Search for the sequence of ac4ons that will result in all rooms being clean Copyright Ben Cartere@e 3 Copyright Ben Cartere@e 4 1

2 Example: Eight Queens Puzzle Place eight queens on a chessboard such that no queen can a@ack any other More Examples Games and puzzles Each move leads to a configura4on of pieces Search over sequences of moves to find one that wins the game or solves the puzzle Scheduling and planning Math Roots of polynomials, integra4on and differen4a4on, theorem proving, Natural language processing Robo4cs Machine learning Models, features, values of model parameters Copyright Ben Cartere@e 5 Copyright Ben Cartere@e 7 Simple Search Agents Formula4ng a Search Problem Iden4fy four things: State space (a set of possible states) and ini4al state Ac4ons and successor func4on For each state, a mapping of ac4ons to states they lead to Goal test Have I arrived at the state I want to be in? Path cost Cost of traveling from star4ng state to current state Should be an addi4ve func4on A solu+on to the problem is a sequence of ac4ons from start state to goal state Copyright Ben Cartere@e 8 Copyright Ben Cartere@e 9 2

3 State Space A state is an abstract representa4on of the most important proper4es of the world In chess, a state might describe the loca4ons of all the pieces on the board Abstract because all that ma@ers is which square each piece is in, not the absolute posi4on State spaces are discrete: states are individually dis4nct and iden4fiable The state space may be infinite, though Successor Func4on For each state, a mapping indica4ng which state each possible ac4on will lead to e4 Copyright Ben Cartere@e 10 Copyright Ben Cartere@e 11 Successor Func4on From the agent program s perspec4ve, the successor func4on is a black box The agent program has no idea how it works Somehow choosing the ac4on drive to Sibiu gets me to Sibiu; I don t know how Path Cost The path cost is the total cost from the start state to a given state A path cost func4on should be addi+ve Each ac4on adds to the total cost All ac4ons have posi4ve cost Thus we may define path cost as the sum of the costs of the ac4ons taken Copyright Ben Cartere@e 12 Copyright Ben Cartere@e 13 3

4 Path Cost The arc cost is the cost of taking a par4cular ac4on Chess: arc cost = 1 Traveling Romania: arc cost = mileage It is useful to assume cost is always non zero To prevent the number of ac4ons from growing too large If the arc cost of chess were 0, two players could get stuck in an infinite loop of moving their kings one space back and forth Goal Test A search problem should have a well defined goal That goal should be represented by at least one state A state represen4ng the solu4on to the problem The goal test func4on determines whether the current state is the goal state Chess: goal is checkmate; goal func4on determines whether opponent is in check, and if so, whether opponent can move king out of check Traveling Romania: goal is Bucharest; goal func4on checks whether we re there There may be many equally viable goal states Copyright Ben Cartere@e 14 Copyright Ben Cartere@e 15 Example: Traveling Romania Example: Vacuum World State space: Ini4al state: Ac4ons: Goal test: Path cost: Solu4on: Op4mal solu4on: Copyright Ben Cartere@e 16 States: Ac4ons: Path cost: Goal test: Copyright Ben Cartere@e 17 4

5 States: Ini4al state: Ac4ons: Goal: Path cost: Example: Eight Queens Tree Search Search problems comprising states, successors, and arc costs are naturally represented as a tree Choosing the state space is a search problem (think about it) Copyright Ben Cartere@e 18 Copyright Ben Cartere@e 20 Tree Nodes Each node encodes informa4on about the search: State, parent node, ac4on, path cost g(x), depth State = Fagaras; parent = Sibiu; ac4on = drive to Fagaras; path cost = 239; depth = 3 Copyright Ben Cartere@e 21 Search Strategy Expanding a node x uses x s successor func4on to create a list of nodes containing states that can be reached from x expand(fagaras) = { (Sibiu, cost=338), (Bucharest, cost=450) } Expansion nodes added to the fringe: the list of nodes that have not yet been expanded A search strategy describes the order in which nodes are expanded Copyright Ben Cartere@e 22 5

6 Tree Search Measuring Search Strategy Performance Completeness: will it find a solu4on if one exists? Op+mality: is the solu4on it finds op4mal? Time & space complexity: how many nodes expanded and how many in memory? Branching factor b: maximum number of successors of any node Depth d: depth of least cost solu4on Depth m: maximum depth of the search Copyright Ben Cartere@e 23 Copyright Ben Cartere@e 24 Big O Nota4on Review f(n) = O(g(n)) Describes rate of growth of 4me/space to compute f(n) as a func4on g of n g(n) = 1: constant growth; input size does not determine 4me/space costs g(n) = log n: logarithmic growth g(n) = n: linear growth g(n) = n k, k > 1: polynomial growth g(n) = k n, k > 1: exponen4al growth Complexity Review EXP PSPACE NP P Copyright Ben Cartere@e 25 Copyright Ben Cartere@e 26 6

7 Algorithms Uninformed or Blind Search Breadth first Uniform cost Depth first Depth limited Itera4ve deepening Their implementa4on depends mainly on how the fringe nodes are ordered for expansion Copyright Ben 27 Copyright Ben 28 Environment Proper4es Assump4ons behind uninformed search: Sta4c: states can t change Discrete: states are individually dis4nct; there is not a smooth con4nuum between states Observable: the world is visible from each state Determinis4c: state proper4es, arc costs, etc are fixed Breadth First Search Expand nodes in the order they appear Treat fringe as a queue (first in, first out) A and B expanded Fringe = C, D, E Expand C Proper4es: Complete: it will find a solu4on Op4mal: the solu4on will be op4mal Exponen4al 4me & space complexity: O(b d ) Copyright Ben Cartere@e 29 Copyright Ben Cartere@e 30 7

8 Uniform Cost Search Just like breadth first, except order nodes on the fringe by increasing path cost Expand least cost node on fringe Proper4es: Complete assuming arc cost ε > 0 Op4mal Time complexity: number of nodes with g cost of op4mal solu4on = O(b C*/ε ), C* = cost of op4mal Space complexity: number of nodes with g cost of op4mal solu4on = O(b C*/ε ) Depth First Search Expand the most recently added node Treat fringe as a stack (first in, last out) A, B, D, E, H, I, J, K expanded Fringe = C Expand C next Proper4es: Not complete: may get stuck in loops; may deepen infinitely Not op4mal: it finds the first solu4on, not the best Exponen4al 4me O(b m ) but linear space O(bm) Copyright Ben Cartere@e 31 Copyright Ben Cartere@e 32 Depth Limited Search Depth first, but with a depth limit l Do not con4nue to expand nodes at the limit Solves infinite path problem But s4ll incomplete why? Itera4ve Deepening Use depth limited search itera4vely, increasing limit by one each 4me For each value of l, every possible node at depth l is tested Like breadth first Proper4es: Complete: it will find a solu4on Op4mal: if arc cost is constant Complexity: exponen4al 4me O(b d ), linear space O(bd) No worse than depth first (d m) Copyright Ben Cartere@e 33 Copyright Ben Cartere@e 34 8

9 Algorithm Comparison Prac4cal 4me/space implica4ons when b=10: Bidirec4onal Search Start a search from the beginning and a search from the end Stop when their fringes intersect Requires a predecessor func4on Breadth first Itera+ve deepening D=2 0.11s; 1Mb 0.01s; 20Kb D=4 11s; 106Mb 1s; 40Kb D=8 31h; 1Tb 2.8h; 80Kb D=16 317,000y; 10Eb 32,000y; 1.6Mb Copyright Ben Cartere@e 35 Copyright Ben Cartere@e 36 Avoiding Repeated States Graph Search A finite state space can result in an infinite search E.g. when states can be revisited Copyright Ben Cartere@e 37 Copyright Ben Cartere@e 38 9