Week 2 a - Search.ppsx this is used to search things

1
By
Engr. Dr. Jawwad Ahmad
Search
ARTIFICIAL
INTELLIGENCE &
EXPERT SYSTEMS

2
Today’s Goal
 Search
 Problem Solving as Search
 Problem Types
 Components of Problem Solving
 Problem Solving Process
 Problem Formulation
 Trees, Graphs, & Representation

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Why Need Search?
 A farmer went to a market and purchased a wolf, a goat, and
a cabbage.
 On his way home, the farmer came to the bank of a river and
rented a boat.
 But crossing the river by boat, the farmer could carry only
himself and a single one of his purchases: the wolf, the goat, or
the cabbage.
 If left unattended together, the wolf would eat the goat, or the
goat would eat the cabbage.
 The farmer's challenge was to carry himself and his purchases to
the far bank of the river, leaving each purchase intact.

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Search & Search Algorithm
 Search is the process of looking for a sequence of actions that
reaches the goal.
 A search algorithm takes a problem as input and returns a
solution in the form of an action sequence.
 Once a solution is found, the actions it recommends can be
carried out.
 There are many sequences of actions, each with their own utility,
and we want to find, or search for, the best one.
Now Consider another Search Problem: Travel in Romania

5
Travel in Romania

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Travel in 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
 State Space
 Cities
 Actions
 Drive down different roads
 Transition Function
 Roads lead to different Cities
 Cost
 Distance
 Initial State
 Arad
 Goal Test/State
 Bucharest

7
Travel in Romania
733
607
450
418

Engr. Dr. Jawwad Ahmad 8
Problem Type
 Static / Dynamic
 Previous problem was static: no attention to changes
in environment
 Observable / Partially Observable / Unobservable
 Previous problem was observable: it knew its initial
state
 Deterministic / Stochastic
 Previous problem was deterministic: no new percepts
were necessary, we can predict the future perfectly
 Discrete / continuous
 Previous problem was discrete: we can enumerate all
possibilities

A Well – Defined Problem
 State Space
 Set of all states reachable from the initial state by
any sequence of actions
 Initial state, actions, and transition model implicitly
define the State Space of the problem
 Solution to the problem is a sequence of actions that
lead from the initial state to a goal state
 Quality of Solution – measured by cost function
 Optimal Solution has the lowest path cost among all
solutions
The process of removing detail from a
representation is called ABSTRACTION.
The choice of a good abstraction thus
involves removing as much detail as possible
while retaining validity and ensuring that
the abstract actions are easy to carry out.
A good problem formulation has the right
level of detail.

Components of Problem Solving
 State Space forms “Directed Network” or “Graph”
 Path is the sequence of states
 Initial State
 The state the agent starts in
 Possible Actions
 The set of actions that can be executed in any state s
 Transition Model / Results (description of each action)
 The state or result obtained by implementing action a in
state s
 Goal test determines / evaluate whether the given state is a
goal state or not
 Path Cost: Function that assigns a numeric cost to each
path.

Problem Solving Process
 The Environment
 We consider only the simplest environments:
 Episodic
 Single agent
 Fully observable
 Deterministic
 Static
 Discrete
 Known
Phases of
Problem
Solving
Process
Calculating
Cost /
Fitness of
Path
Testing the
Goal State
State Space
(with Initial
State)
Problem
Formulatio
n
Goal
Formulatio
n

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) = {<Arad  Zerind, Zerind>, … }
3. goal test,
 e.g., x = “at Bucharest, 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

Vacuum World State Space Graph
States: integer, dirt and robot locations 2 x 22
= 8 States (ignore dirt amounts etc.)
Actions: Left, Right, Suck, NoOp
Initial State: any
Goal Test: no dirt at all locations
Path Cost: 1 per action (0 for NoOp)

Goal State
The 8-Puzzle
States: integer, locations of tiles (ignore intermediate positions)
Actions: move blank left, right, up,
down (ignore unjamming etc.)
Initial State: any
Goal Test: Goal State
Path Cost: 1 per move
Start State

The 8 Queens Problem
States: any arrangement of n<=8 queens or in leftmost n columns, 1 per column, such
that no queen attacks any other
Actions: add queen to empty or leftmost
empty square such that it is not attacked by
other queens.
Initial State: no queens on the board
Goal Test: Goal State (8 queens on
the board, none attacked)
Path Cost: 1 per move
Goal State
Start State

Trees, Graphs, & Representation
 Graph
 Finite set of vertices (nodes) that are connected by edges
(arcs)
 Cycle
 An arc leading back to the original node
 Connected Graph
 Every pair of nodes is connected by an arc
 Complete Graph
 Every node is connected to every other node
 Weighted Graph
 Arc may carry a weight implying cost is associated with a
path
 Adjacency Matrix
 One of the simplest ways of representing a graph is by N by N
matrix, where N = no of nodes
 Rows identify source nodes whereas columns identify
Undirected graph – symmetric matrix
Directed graph – non symmetric matrix

Trees, Graphs, & Representation
 Tree
 Connected graph with NO cycles
 Trees usually have roots and are drawn with roots on top
 Two nodes within a tree at most have one path between them
 Since tree has only one path of access to any other node it is
impossible to have cycles.

Tree Search Example

Search Algorithm
 When solving a problem
1. Consider solution space as a number of actions that can be taken
2. Performing these actions result in a change of state of the environment
3. Taking one of the possible action, the environment may add alternatives
for new actions
4. Each action has its associate cost
 Solution to the problem is finding sequence of operators that
transition from the start to the goal state.
 A search algorithm takes a problem as input and returns a solution
in the form of an action sequence
 As in general problem solving
1. Some paths lead to dead – ends and some lead to solution
2. Possibility of multiple solutions
3. One solution may be better than the other
 Avoiding the dead ends and selecting the best solution is the result of
the chosen search strategy

Search Algorithm Strategies
 A search algorithm 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
 space complexity: maximum number of nodes in
memory
 optimality: does it always find a least-cost solution?
 Time and space complexity are measured in terms of
 b: maximum branching factor of the search tree
 d: depth of the least-cost solution
 m: maximum depth of the state space (may be ∞)

The Towers of Hanoi
 The puzzle was invented by the French mathematician
Édouard Lucas in 1883

The Towers of Hanoi

Thank you

Week 2 a - Search.ppsx this is used to search things

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Week 2 a - Search.ppsx this is used to search things