1/21/20141 Solving problems by searching Course Teacher: Moona Kanwal.

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Solving problems by searching

Course Teacher: Moona Kanwal

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Outline

Well-defined ProblemsWell-defined Problems Solutions (as a sequence of actions).Solutions (as a sequence of actions). ExamplesExamples Search TreesSearch Trees Uninformed Search AlgorithmsUninformed Search Algorithms

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Problem-Solving

Goal formulationGoal formulation (final state): limits objectives

Problem formulationProblem formulation (set of states): actions+states

Search Search (solution): sequence of actions

ExecutionExecution (follows states in solution)

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Problem types

Deterministic, fully observableDeterministic, fully observable single-state problem: initial state known, results of action known

Non-observableNon-observable sensor less problem (conformant problem): initial state not known, results of action known

Nondeterministic and/or partially observableNondeterministic and/or partially observable contingency problem: unknown states on some results of action

Unknown state spaceUnknown state space exploration problem: unknown states, unknown actions

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Well Defined Problems

A problem is defined by four itemsA problem is defined by four items:

1.1. initial stateinitial state2.2. actions or successor functionactions or successor function S(x) = set of

action–state pairs

3.3. goal test or predicate goal test or predicate γ : S → {0, 1} is 1 iff goal state

Explicit: concrete goal implicit : abstract description of goal

path costpath cost (additive):g, assigns a numeric cost to a given path.

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Well Defined Problems

State-Space State-Space An Initial State, s0 ∈ S. A Set of Operators (or Actions), {Ai|i = 1, 2, . . }

Formally, the state space is the set of states Formally, the state space is the set of states that can be reached by any sequence of that can be reached by any sequence of actions applied to the initial state, e.g.,actions applied to the initial state, e.g.,

S = {sS = {s00, . . . ,A, . . . ,Ai1i1ss00, . . . ,A, . . . ,Ai1i1AiAi22ss00, . . . ,A, . . . ,Ai1i1AAi2i2 ・・・・・・ AAikikss00, . . . }, . . . }

A path is a particular sequence of actions A path is a particular sequence of actions applied to the initial state.applied to the initial state.

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Solutions

A solution is a sequence of actions A solution is a sequence of actions leading from the initial state to a leading from the initial state to a goal stategoal state

A solution is a path that satisfies A solution is a path that satisfies the goal test.the goal test.

An optimal solution is a solution An optimal solution is a solution with minimal cost.with minimal cost.

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Example: Romania

On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest Formulate goal:

be in Bucharest Formulate problem:

states: various cities actions: drive between cities

Find solution: sequence of cities, e.g., Arad, Sibiu, Fagaras,

Bucharest

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Example: Romania

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Example: The 8-puzzle

states? actions? goal test? path cost?

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Example: The 8-puzzle

states? locations of tileslocations of tiles actions? move blank left, right, up, downmove blank left, right, up, down goal test? = goal state (given)goal state (given) path cost?1 per move1 per move

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Example: vacuum world

Single-state, Single-state, start in #5start in #5.

Solution?

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Example: vacuum world

Single-state, Single-state, start in #5start in #5.

Solution? [Right, Suck][Right, Suck]

SensorlessSensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution?

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Example: vacuum world

SensorlessSensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution? [Right,Suck,Left,Suck]

ContingencyContingency 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?

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Example: vacuum world

SensorlessSensorless, start in {1,2,3,4,5,6,7,8} e.g., Right goes to {2,4,6,8} Solution? [Right,Suck,Left,Suck]

ContingencyContingency 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]

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Vacuum world state space graph

states? actions? goal test? path cost?

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Vacuum world state space graph

states? integer dirt and robot locationinteger dirt and robot location actions? Left, Right, SuckLeft, Right, Suck goal test? no dirt at all locationsno dirt at all locations path cost? 11 per actionper action

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Example: robotic assembly

states?: real-valued coordinates of robot joint real-valued coordinates of robot joint angles parts of the object to be assembledangles parts of the object to be assembled

actions?: continuous motions of robot jointscontinuous motions of robot joints goal test?: complete assemblycomplete assembly path cost?: time to executetime to execute

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Tree search algorithms

Basic ideaBasic idea: offline, simulated exploration of state offline, simulated exploration of state

space by generating successors of already-space by generating successors of already-explored states (a.k.a.expanding states)explored states (a.k.a.expanding states)

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Tree search example

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Tree search example

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Tree search example

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Implementation: general tree search

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Implementation: states vs. nodes

A state is a (representation of) a physical configurationA state is a (representation of) a physical configuration A node is a data structure constituting part of a search tree A node is a data structure constituting part of a search tree

includes state, parent node, action, path cost g(x), depthincludes state, parent node, action, path cost g(x), depth

The Expand function creates new nodes, filling in the various The Expand function creates new nodes, filling in the various fields and using the Success or Fn of the problem to create the fields and using the Success or Fn of the problem to create the corresponding states.corresponding states.

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Search strategies

Time and space complexity are measured Time and space complexity are measured in terms of in terms of b: maximum branching factor of the b: maximum branching factor of the

search treesearch tree d: depth of the least-cost solutiond: depth of the least-cost solution m: maximum depth of the state space m: maximum depth of the state space

(may be (may be ∞∞))

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Uninformed search strategies

Uninformed search strategies use only the Uninformed search strategies use only the information available in the problem information available in the problem definitiondefinition

Breadth-first searchBreadth-first search Uniform-cost searchUniform-cost search Depth-first searchDepth-first search Depth-limited searchDepth-limited search Iterative deepening searchIterative deepening search

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Breadth-first search

Expand shallowest unexpanded nodeExpand shallowest unexpanded node Implementation:

fringe is a FIFO queue, i.e., new successors go at end

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Breadth-first search

Expand shallowest unexpanded nodeExpand shallowest unexpanded node Implementation:

fringe is a FIFO queue, i.e., new successors go at end

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Breadth-first search

Expand shallowest unexpanded node Implementation:

fringe is a FIFO queue, i.e., new successors go at end

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Breadth-first search

Expand shallowest unexpanded node Implementation:

fringe is a FIFO queue, i.e., new successors go at end

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Properties of breadth-first search

Complete? Yes (if b is finite) Time? 1+b+b2+b3+… +bd + b(bd-1) =

O(bd+1) Space? O(bd+1) (keeps every node in

memory) Optimal? Yes (if cost = 1 per step)

Space is the bigger problem (more than time)

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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 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, O(bceiling(C*/ ε))

Optimal? Yes – nodes expanded in increasing order of g(n)

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search

Expand deepest unexpanded node Implementation:

fringe = LIFO queue, i.e., put successors at front

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Properties of depth-first search

Complete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path

complete in finite spaces

Time? O(bm): terrible if m is much larger than d but if solutions are dense, may be much faster

than breadth-first Space? O(bm), i.e., linear space! Optimal? No

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Depth-limited search

= depth-first search with depth limit l,i.e., nodes at depth l have no successors

Recursive implementation:

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Iterative deepening search

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Iterative deepening search

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Iterative deepening search

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Iterative deepening search

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Iterative deepening search

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Iterative deepening search

Number of nodes generated in a depth-limited search to depth d with branching factor b:

NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd

Number of nodes generated in an iterative deepening search to depth d with branching factor b:

NIDS = (d+1)b0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd

For b = 10, d = 5, NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 =

111,111 NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 =

123,456

Overhead = (123,456 - 111,111)/111,111 = 11%

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Properties of iterative deepening search

Complete? Yes Time? (d+1)b0 + d b1 + (d-1)b2 + … +

bd = O(bd) Space? O(bd) Optimal? Yes, if step cost = 1

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Summary of algorithms

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Repeated states

Failure to detect repeated states can turn a linear problem into an exponential one!

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Graph search

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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