Solving problems by searching Course Teacher Moona Kanwal

Solving problems by searching Course Teacher: Moona Kanwal 12/13/2021 1

Outline n n n Well-defined Problems Solutions (as a sequence of actions). Examples Search Trees Uninformed Search Algorithms 12/13/2021 2

Problem-Solving w Goal formulation (final state): limits objectives w Problem formulation (set of states): actions+states w Search (solution): sequence of actions w Execution (follows states in solution) 12/13/2021 3

Problem types n n Deterministic, fully observable single-state problem: initial state known, results of action known Non-observable sensor less problem (conformant problem): initial state not known, results of action known Nondeterministic and/or partially observable contingency problem: unknown states on some results of action Unknown state space exploration problem: unknown states, unknown actions 12/13/2021 4

Well Defined Problems A problem is defined by four items: 2. initial state actions or successor function S(x) = set of 3. goal test or predicate γ : S → {0, 1} is 1 iff goal 1. action–state pairs state n n n Explicit: concrete goal implicit : abstract description of goal path cost (additive): g, assigns a numeric cost to a given path. 12/13/2021 5

Well Defined Problems n State-Space n n n An Initial State, s 0 ∈ S. A Set of Operators (or Actions), {Ai|i = 1, 2, . . } Formally, the state space is the set of states that can be reached by any sequence of actions applied to the initial state, e. g. , S = {s 0, . . . , Ai 1 Ai 2 s 0, . . . , Ai 1 Ai 2 ・・・Aiks 0, . . . } A path is a particular sequence of actions applied to the initial state. 12/13/2021 6

Solutions A solution is a sequence of actions leading from the initial state to a goal state n A solution is a path that satisfies the goal test. n An optimal solution is a solution with minimal cost. n 12/13/2021 7

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Example: Romania n n n On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest Formulate goal: n n Formulate problem: n n n be in Bucharest states: various cities actions: drive between cities Find solution: n sequence of cities, e. g. , Arad, Sibiu, Fagaras, Bucharest 12/13/2021 9

Example: Romania 12/13/2021 10

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Example: The 8 -puzzle n n states? actions? goal test? path cost? 12/13/2021 12

Example: The 8 -puzzle n n states? locations of tiles actions? move blank left, right, up, down goal test? = goal state (given) path cost? 1 per move 12/13/2021 13

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Example: vacuum world n Single-state, n start in #5. Solution? n 12/13/2021 15

Example: vacuum world n Single-state, n start in #5. Solution? [Right, Suck] n Sensorless, start in n {1, 2, 3, 4, 5, 6, 7, 8} e. g. , Right goes to {2, 4, 6, 8} Solution? 12/13/2021 16

Example: vacuum world n Sensorless, start in {1, 2, 3, 4, 5, 6, 7, 8} e. g. , Right goes to {2, 4, 6, 8} Solution? [Right, Suck, Left, Suck] n Contingency n n n 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? 12/13/2021 17

Example: vacuum world n Sensorless, start in {1, 2, 3, 4, 5, 6, 7, 8} e. g. , Right goes to {2, 4, 6, 8} Solution? [Right, Suck, Left, Suck] n Contingency n n n 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] 12/13/2021 18

Vacuum world state space graph n n states? actions? goal test? path cost? 12/13/2021 19

Vacuum world state space graph n n states? integer dirt and robot location actions? Left, Right, Suck goal test? no dirt at all locations path cost? 1 per action 12/13/2021 20

Example: robotic assembly n n 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 12/13/2021 21

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Tree search algorithms n Basic idea: n offline, simulated exploration of state space by generating successors of alreadyexplored states (a. k. a. expanding states) 12/13/2021 25

Tree search example 12/13/2021 26

Tree search example 12/13/2021 27

Tree search example 12/13/2021 28

Implementation: general tree search 12/13/2021 29

Implementation: states vs. nodes n n n 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 Success or Fn of the problem to create the corresponding states. 12/13/2021 30

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Search strategies Time and space complexity are measured in terms of n b: maximum branching factor of the search tree n d: depth of the least-cost solution n m: maximum depth of the state space (may be ∞) 12/13/2021 32

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Uninformed search strategies use only the information available in the problem definition n Breadth-first search n Uniform-cost search n Depth-first search n Depth-limited search n Iterative deepening search 12/13/2021 34

Breadth-first search n Expand shallowest unexpanded node n Implementation: n fringe is a FIFO queue, i. e. , new successors go at end 12/13/2021 35

Breadth-first search n Expand shallowest unexpanded node n Implementation: n fringe is a FIFO queue, i. e. , new successors go at end 12/13/2021 36

Breadth-first search n n Expand shallowest unexpanded node Implementation: n fringe is a FIFO queue, i. e. , new successors go at end 12/13/2021 37

Breadth-first search n n Expand shallowest unexpanded node Implementation: n fringe is a FIFO queue, i. e. , new successors go at end 12/13/2021 38

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Properties of breadth-first search n Complete? Yes (if b is finite) Time? 1+b+b 2+b 3+… +bd + b(bd-1) = O(bd+1) Space? O(bd+1) (keeps every node in memory) Optimal? Yes (if cost = 1 per step) n Space is the bigger problem (more than time) n n n 12/13/2021 53

Uniform-cost search n n Expand least-cost unexpanded node Implementation: n fringe = queue ordered by path cost n Equivalent to breadth-first if step costs all equal Complete? Yes, if step cost ≥ ε Time? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε)) where C* is the cost of the optimal solution Space? # of nodes with g ≤ cost of optimal solution, n Optimal? Yes – nodes expanded in increasing order of g(n) n n n O(bceiling(C*/ ε)) 12/13/2021 54

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Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 79

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 80

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 81

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 82

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 83

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 84

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 85

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 86

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 87

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 88

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 89

Depth-first search n n Expand deepest unexpanded node Implementation: n fringe = LIFO queue, i. e. , put successors at front 12/13/2021 90

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Properties of depth-first search n Complete? No: fails in infinite-depth spaces, spaces with loops n Modify to avoid repeated states along path complete in finite spaces n Time? O(bm): terrible if m is much larger than d n n n but if solutions are dense, may be much faster than breadth-first Space? O(bm), i. e. , linear space! Optimal? No 12/13/2021 103

Depth-limited search = depth-first search with depth limit l, i. e. , nodes at depth l have no successors n Recursive implementation: 12/13/2021 104

Iterative deepening search 12/13/2021 105

Iterative deepening search 12/13/2021 106

Iterative deepening search 12/13/2021 107

Iterative deepening search 12/13/2021 108

Iterative deepening search 12/13/2021 109

Iterative deepening search n n Number of nodes generated in a depth-limited search to depth d with branching factor b: NDLS = b 0 + b 1 + b 2 + … + bd-2 + bd-1 + bd Number of nodes generated in an iterative deepening search to depth d with branching factor b: NIDS = (d+1)b 0 + d b^1 + (d-1)b^2 + … + 3 bd-2 +2 bd-1 + 1 bd For b = 10, d = 5, n NDLS = 1 + 100 + 1, 000 + 100, 000 = 111, 111 n NIDS = 6 + 50 + 400 + 3, 000 + 20, 000 + 100, 000 = 123, 456 Overhead = (123, 456 - 111, 111)/111, 111 = 11% 12/13/2021 110

Properties of iterative deepening search n Complete? Yes n Time? O(bd) (d+1)b 0 + d b 1 + (d-1)b 2 + … + bd = O(bd) n Space? n Optimal? Yes, if step cost = 1 12/13/2021 111

Summary of algorithms 12/13/2021 112

Repeated states n Failure to detect repeated states can turn a linear problem into an exponential one! 12/13/2021 113

Graph search 12/13/2021 114

Summary n n n 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 12/13/2021 115