14 Jan 2004CS Blind Search1 Solving problems by searching Chapter 3

14 Jan 2004CS Blind Search2 Outline Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms

14 Jan 2004CS Blind Search3 Problem-solving agents

14 Jan 2004CS Blind Search4 Example: Romania On holiday in Romania; currently in Arad. Flight leaves tomorrow to Bucharest Formulate goal: goal: be in Bucharest Formulate problem: states: various cities actions: drive between cities Find solution: sequence of cities, {Arad, Sibiu, Fagaras, Bucharest}

14 Jan 2004CS Blind Search5 Example: Romania

14 Jan 2004CS Blind Search6 Problem types Deterministic, fully observable  single-state problem Agent knows exactly which state it will be in; solution is a sequence Non-observable  sensorless problem (conformant توافقي problem) Agent may have no idea where it is; solution is a sequence Nondeterministic and/or partially observable  contingency طارئ/احتمال problem percepts provide new information about current state often interleave {search, execution} Unknown state space  exploration problem

14 Jan 2004CS Blind Search7 Example: vacuum world Single-state, start in #5. Solution?

14 Jan 2004CS Blind Search8 Example: vacuum world Single-state, start in #5. Solution? [Right, Suck] Sensorless, start in {1,2,3,4,5,6,7,8} Solution?

14 Jan 2004CS Blind Search9 Example: vacuum world Single-state, start in #5. Solution? [Right, Suck] Sensorless, start in {1,2,3,4,5,6,7,8} Solution? [Right,Suck,Left,Suck] e.g., Right goes to {2,4,6,8} Left goes to {1,3,5,7}

14 Jan 2004CS Blind Search10 Example: vacuum world Contingency Nondeterministic: Suck may dirty a clean carpet Partially observable: location, dirt at current location. Percept: [L, Clean], i.e., start in #5 or #7 Solution? [Right, if dirt then Suck]

14 Jan 2004CS Blind Search11 Single-state problem formulation A problem is defined by four items: 1. initial state e.g., "at Arad" 2. actions or successor function S(x) = set of action–state pairs e.g., S(Arad) = {, … } 3. goal test, can be explicit, e.g., x = "at Bucharest" implicit, e.g., Checkmate(x) 4. path cost (additive) e.g., sum of distances, number of actions executed, etc. c(x,a,y) is the step cost, assumed to be ≥ 0 A solution is a sequence of actions leading from the initial state to a goal state

14 Jan 2004CS Blind Search12 Selecting a state space Real world is complex state space must be abstracted for problem solving (Abstract) state = set of real states (Abstract) action = complex combination of real actions e.g., "Arad  Zerind" represents a complex set of possible routes, detours, rest stops, etc. (Abstract) solution = set of real paths that are solutions in the real world

14 Jan 2004CS Blind Search13 Vacuum world state space graph states? actions? goal test? path cost?

14 Jan 2004CS Blind Search14 Vacuum world state space graph states? integer dirt and robot location actions? Left, Right, Suck goal test? no dirt at all locations path cost? 1 per action

14 Jan 2004CS Blind Search15 Example: The 8-puzzle states? actions? goal test? path cost?

14 Jan 2004CS Blind Search16 Example: The 8-puzzle states? locations of tiles actions? move blank left, right, up, down goal test? = goal state (given) path cost? 1 per move

14 Jan 2004CS Blind Search17 Example: robotic assembly states?: real-valued coordinates of robot joint angles parts of the object to be assembled actions?: continuous motions of robot joints goal test?: complete assembly path cost?: time to execute

14 Jan 2004CS Blind Search18 Tree search algorithms Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (~expanding states)

14 Jan 2004CS Blind Search19 Tree search example

14 Jan 2004CS Blind Search20 Tree search example

14 Jan 2004CS Blind Search21 Tree search example

14 Jan 2004CS Blind Search22 Implementation: general tree search

14 Jan 2004CS Blind Search23 Implementation: states vs. nodes A state is a (representation of) a physical configuration. A node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depth. The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.

14 Jan 2004CS Blind Search24 Search strategies A search strategy is defined by picking the order of node expansion Strategies are evaluated along the following dimensions: completeness: does it always find a solution if one exists? time complexity: number of nodes generated during the search space complexity: maximum number of nodes stored in memory optimality: does it always find a least-cost solution? Time and space complexity are measured in terms of b: branching factor or the maximum number of successors of any node in the search tree. d: depth of the least-cost solution m: maximum depth of any path in the state space.

CS Blind Search25 Uninformed search strategies Uninformed search strategies use only the information available in the problem definition. Also known as Blind Search. Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search

CS Blind Search26 Breadth-first search The root node is expanded first Then all successors of the root node are expanded next, then their successors, and so on.

14 Jan 2004CS Blind Search27 Breadth-first search All nodes are expanded at a given depth in the search tree before any nodes at the next level are expanded.

14 Jan 2004CS Blind Search28 Breadth-first search Nodes that are visited first will be expanded first. Implementation: fringe is a FIFO queue, i.e., new successors go at end.

14 Jan 2004CS Blind Search29 Breadth-first search Expand shallow nodes before deeper nodes Implementation: fringe is a FIFO queue, i.e., new successors go at end

14 Jan 2004CS Blind Search30 Properties of breadth-first search Complete? Yes (if b is finite) Time? b+b 2 +b 3 +… +b d + (b d+1 -b) = O(b d+1 ) Space? b+b 2 +b 3 +… +b d +(b d+1 - b) (keeps every node in memory) Optimal? Yes (if cost = 1 per step)

Time & Memory requirements for Breadth-first search MemoryTimeNodesDepth 1 mega byte0.11 sec mega byte11 sec gega byte19 min tera byte31 hours tera byte129 days peta byte35 year Jan 2004CS Blind Search31 Space is the bigger problem (more than time) Ex: b = 10, speed = nodes/sec, space = 1000 bytes/node

14 Jan 2004CS Blind Search32 Uniform-cost search Expand least-cost unexpanded node Implementation: fringe = queue ordered by path cost Equivalent to breadth-first if step costs all equal Complete? Yes, if step cost ≥ ε (+ve constant) Time? # of nodes with g ≤ cost of optimal solution, O(b ceiling(C*/ ε) ) where C * is the cost of the optimal solution Space? # of nodes with g ≤ cost of optimal solution, O(b ceiling(C*/ ε) ) Optimal? Yes – nodes expanded in increasing order of g(n)

14 Jan 2004CS Blind Search33 Depth-first search Expand the deepest node in the fringe of search tree expanded nodes are dropped from the fringe.

14 Jan 2004CS Blind Search34 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search35 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search36 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search37 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search38 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search39 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search40 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search41 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search42 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search43 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

14 Jan 2004CS Blind Search44 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

CS Blind Search45 Properties of depth-first search Complete? No: fails in infinite-depth spaces, spaces with loops (unbounded tree) Modify to avoid repeated states along path Time? O(b m ): where m is maximum depth. if solutions are dense, much faster than breadth-first Space? O(bm)= bm+1 nodes space. Ex: memory = 118 kB (instead of 10 peta B for breadth-first, for d=12) Optimal? No (in case of multi-nodes goal)

14 Jan 2004CS Blind Search46 Depth-limited search To solve the problem of unbounded trees. = depth-first search with depth limit L i.e., nodes at depth L treated as they have no successors In case of L < d, the goal at depth d is beyond the limit, and hence fail to get the goal. In case of L > d, the solution is non-optimal.

CS Blind Search47 Iterative deepening search it combines the benefits of depth-first (low memory & breadth-first (complete) search. Total number of nodes generated: N(IDS) = D*b + (d-1)*b 2 + … * b d Ex: if b=10, d=5, N(IDS) = 123,450 N(BFS) = 1,111,100 Hence, Iterative deepening is the preferred uninformed search method when there is large search space and the depth of the solution is not known.

14 Jan 2004CS Blind Search48 Iterative deepening search L =0

14 Jan 2004CS Blind Search49 Iterative deepening search L =1

14 Jan 2004CS Blind Search50 Iterative deepening search L =2

14 Jan 2004CS Blind Search51 Iterative deepening search L =3

14 Jan 2004CS Blind Search52 Iterative deepening search Number of nodes generated in a depth-limited search N DLS to depth d with branching factor b: N DLS = b 0 + b 1 + b 2 + … + b d-2 + b d-1 + b d Number of nodes generated in an iterative deepening search N IDS to depth d with branching factor b: N IDS = (d+1)b 0 + d b^ 1 + (d-1)b^ 2 + … + 3b d-2 +2b d-1 + 1b d For b = 10, d = 5, N DLS = , , ,000 = 111,111 N IDS = , , ,000 = 123,456 Overhead = (123, ,111)/111,111 = 11%

14 Jan 2004CS Blind Search53 Properties of iterative deepening search Complete? Yes Time? (d+1)b 0 + d b 1 + (d-1)b 2 + … + b d = O(b d ) Space? O(bd) Optimal? Yes, if step cost = 1

14 Jan 2004CS Blind Search54 Summary of algorithms

14 Jan 2004CS Blind Search55 Summary Problem formulation usually requires abstracting away real- world details to define a state space that can feasibly be explored Variety of uninformed search strategies Iterative deepening search uses only linear space and not much more time than other uninformed algorithms