Final slide (bsc csit) chapter 3

Introduction to State Space
Search
1AI, Subash Chandra Pakhrin

Instructional Objectives for this module
• The student should be familiar with the
following algorithms, and should be able to
code the algorithms
– Greedy search
– DFS
– BFS
– Uniform cost search
– Iterative deepening search
– Bidirectional search

Instructional Objectives for this module
• The student should understand the state space
representation, and gain familiarity with some
common problems formulated as state space
search problems.
• Given a problem description, the student should
be able to formulate it in terms of a state space
search problem.
• The student should understand how implicit state
spaces can be unfolded during search.
• Understand how states can be represented by
features.

Intelligent Agent

Goal Directed Agent
• A goal directed agent needs to achieve certain
goals.
• Many problems can be represented as a set of
states and a set of rules of how one state is
transformed to another
• The agent must choose a sequence of actions
to achieve the desired goal.

Problem Solving
• It is a systematic search through range of
possible actions in order to reach some
predefined goal or solution.
There are two kinds of problem solving
methods:
– Special-purpose method
– General-purpose method
AI, Subash Chandra Pakhrin 6

Problem Solving
• Special-purpose method:
– It is made for particular problem and often
exploits very specific features of the situation in
which the problem is embedded.
• General-purpose method:
– It is applicable for wide variety of problems.
– End analysis
• A step by step
• Difference between the current state and the final state

General steps in problem solving:
• Goal formulation
– What are successful world states
• Problem formulation
– What actions and states to consider given the goal
• Search
– Determine the possible sequence of actions that lead
to the state of known values and then choosing the
best sequence.
• Execute
– Give the solution perform the actions.

Problem Formulation:
• A problem is defined by:
– An initial state
– Successor function
– Goal Test
– Path Cost:
• Sum of cost of each path from initial state to the given state.
• A solution is a sequence of actions from initial to
goal state
• Optimal solution has the lowest path cost.

• Each state is an abstract representation of the
agent’s environment. It is an abstraction that
denoted a configuration of the agent.
• Initial state: The description of the starting
configuration of the agent.
• An action/operator takes the agent from one
state to another state. A state can have a
number of successor states.
• A plan is a sequence of actions.

• A goal is a description of a set of desirable
states of the world. Goal states are often
specified by a goal test which any goal state
must satisfy
• Path cost: path -> positive number
Usually path cost = sum of step costs

• Problem formulation: means choosing a
relevant set of states to consider, and a
feasible set of operators for moving from one
state to another.
• Search is the process of imagining sequences
of operators applied to the initial state, and
checking which sequence reaches a goal state.

Why Search ?
• To achieve goals or to maximize our utility we
need to predict what the result of our actions
in the future will be.
• There are many sequences of actions, each
with their own utility.
• We want to find, or search for, the best one.

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 or choose next city
• Find solution:
– Sequence of cities, e.g., Arad, Sibiu, Fagaras,
Bucharest

Example: Romania

Search Problem
• S: the full set of states
• S0 : the initial state
• A: S -> S set of operators
• G: the set of final states. G
is subset of S
• Search problem: Find a
sequence of actions
which transforms the
agent from the initial
state to a goal state gє G

Searching process
• Check the current state
• Execute allowable action to move to the next
state
• Check if the new state is a solution state
– If it is not, the new state becomes the current
state and the process is repeated until a solution is
found or the state space is exhausted.

State Space

State Space
• Commonly defined as a directed graph in
which each node is a state and each arc
represents the application of an operator
transforming a state to a successor state.
• A solution is a path from the initial state to a
goal state.

Vacuum World State Space
Representation :

Vacuum World State Space
Representation :
• States ?
___2___ X ___ 2___X ___2___ = 8
Dirty/clean Suck / remain ideal move left / right
• Initial States ? Any state can be initial
• Actions ? { Left, Right, Suck }
• Goal Test ? Check whether squares are clean.
• Path cost ? Number of actions to reach goal.

Vacuum World

Pegs and Disks / Tower of Hanoi
• Consider the
following
problem. We
have three pegs
and 3 disks.
• Operators: one
may move the
topmost disk
on any needle
to the topmost
position to any
other needle
Goal Configuration
Initial State

Actions
• Move top most peg from start to end
• Move From start to auxiliary
• Move from start to End
• Move from auxiliary to start
• Move form end to auxiliary
• Move from start to auxiliary
• Move from end to auxiliary
And finally we have the desired configuration

Production System
• A production system (or production rule system)
is a computer program typically used to provide
some form of AI, which consists primarily of a set
of rules about behavior but it also includes the
mechanism necessary to follow those rules as the
system responds to states of the.
• Productions systems, are found useful
in automated planning, expert
systems and action selection.

Production System
• Productions consist of two parts: a sensory precondition
(or "IF" statement) and an action (or "THEN").
• If a production's precondition matches the current
state of the world, then the production is said to
be triggered. If a production's action is executed, it is said
to have fired.
• A production system also contains a database,
sometimes called working memory, which maintains data
about current state or knowledge, and a rule interpreter.
• The rule interpreter must provide a mechanism for
prioritizing productions when more than one is triggered.

Production System
• Production systems represents knowledge in
the form of condition – action pairs called
production rules:
• If the condition C is satisfied then the action A
is appropriate.

Types of Production rules
• Situation-action rule
– If it is raining then open the umbrella.
• Inference rules
– If Hari is a man then Hari is a person
• Production system is also called rule-based
system

Architecture of Production System
Short
Term Memory

Architecture of Production System
• Short Term Memory
– Contains the description of current state.
• Set of production rules
– Set of condition-action pairs
– It defines a single chunk of problem solving
knowledge.
• Interpreter
– A mechanism to examine the short term memory and
to determine which rules to fire (BFS, DFS, A* etc)

Production System:
The water Jug Problem

You are given two jugs, a 4-gallon one and a 3-
gallon one, a pump which has unlimited water
which you can use to fill the jug, and the
ground on which water may be poured.
Neither jug has any measuring markings on it.
How can you get exactly 2 gallons of water in
the 4-gallon jug?

• we will represent a state of the problem as a
ordered pair (x, y) where x represents the
amount of water in the 4-gallon jug and y
represents the amount of water in the 3-
gallon jug.
• Note 0 ≤ x ≤ 4, and 0 ≤ y ≤ 3.
• Initial state: (0,0)
• Goal Predicate – state = (2,y) where 0 ≤ y ≤ 3

Define a set of operators that will take us from one state to another:
1. Fill 4-gal jug (x, y) → (4,y) x < 4
2. Fill 3-gal jug (x, y) → (x,3) y < 3
3. Empty 4-gal jug on ground (x, y) → (0,y) x > 0
4. Empty 3-gal jug on ground (x, y) → (x,0) y > 0
5. Pour water from 3-gal jug (x, y) → (4, y - (4 - x)) to fill 4-gal jug 0 <
x + y ≥ 4 and y > 0
6. Pour water from 4-gal jug (x, y) → (x - (3-y), 3) to fill 3-gal-jug 0 <
x + y ≥ 3 and x > 0
7. Pour all of water from 3-gal jug (x, y) → (x+ y, 0) into 4-gal jug
0 < x + y ≤ 4 and y ≥ 0
8. Pour all of water from 4-gal jug (x, y) → (0, x + y) into 3-gal jug 0 <
x + y ≤ 3 and x ≥ 0

Through Graph Search, the following solution is
found :
Gals in 4-gal jug Gals in 3-gal jug Rule Applied
0 0 1. Fill 4
4 0 6. Pour 4 into 3 to fill
1 3 4. Empty 3
1 0 8. Pour all of 4 into 3
0 1 1. Fill 4
4 1 6. Pour into 3
2 3

The water Jug Problem:
Representation

8 Queens
• The N Queens Problem
is the problem of
placing eight queens on
an N×N chessboard
such that none of them
attack one another (no
two are in the same
row, column, or
diagonal). Alongside
we can see the
arrangements of an
N=8 queen solution:

N queen problem formulation 1
• Any arrangement of 0 to 8 queens on the
board
• Initial state: 0 queens on the board
• Successor function: Add a queen in any square
• Goal test: 8 queens on the board, none are
attacked

N queen problem formulation 1
• States: Any arrangement of 8 queens on the
board
• Initial state: All queens are at column 1
• Successor function: Change the position of
any one queen
attacked

N queens problem formulation 3
• States: Any arrangement of k queens in the
first k rows such that none are attacked
• Initial state: 0 queens on the board
• Successor function: Add a queen to the (k+1)th
row so that none are attacked.
attacked

Explicit vs. Implicit state space
• The state space may be explicitly represented
• Typically it is implicitly represented and
generated when required.
• The agents knows
– The initial state
– The operators
• An operator is a function which “expands” a node
– Compute the successor node(s)

Problem Definition – Example, 8 puzzle
1 2 3
4 8 -
7 6 5
1 2 3
4 5 6
7 8 -
Start State Goal State
Initial State: {(1,2,3), (4,8,0), (7,6,5)}
Successor State: {(1,2,3), (4,8,5), (7,6,-)}; move 5 up or – down
Successor State: {(1,2,3), (4,8,5), (7,-,6)}; move 6 right or – to left
Successor State: {(1,2,3), (4,-,5), (7,8,6)}; move – up or 8 down
Successor State: {(1,2,3), (4,5,-), (7,8,6)}; move 5 to left or – to right
Goal State: {(1,2,3), (4,5,-), (7,8,6)}; move – down or 6 up
PATH COST = 5

Problem Definition – Example, 8 puzzle
/ n2-1 puzzle
• States
– A description of each of the eight tiles in each
location that it can occupy.
• Operation/Action
– The blank moves left, right, up or down
• Goal Test
– The current state matches a certain goal state.
• Path Cost
– Each move of the blank costs 1

Problem Definition – Example, tic-tac-toe

Search Through a state space
• Input
– Set of states
– Operators [and costs]
– Start state
– Goal state [test]
• Output
– Path: start => a state satisfying goal test
– [May require shortest path]

Basic Search algorithm
Let fringe be a list containing the initial state
LOOP
if fringe is empty return failure
Node <- remove – first (fringe)
if Node is a goal
then return the path from initial state to Node
else generate all successors of Node,
and merge the newly generated nodes into fringe
END LOOP

Basic Search Algorithm: Key Issues
• Search tree may be unbounded
– Because of loops
– Because state space is infinite
• Return a path or a node ?
• How are merge and select done ?
– Is the graph weighted or un weighted ?
– How much is known about the quality of intermediate
states?
– Is the aim to find a minimal cost path or any path as
soon as possible ?

Search strategy
• Measuring problem solving performance:
– Completeness: Is the strategy guaranteed to find a
solution if one exists ?
– Optimality: Does the solution have low cost or the
minimal cost ?
– What is the search cost associated with the time
and memory required to find a solution ?

Search Strategies
• Blind Search
– Depth First Search
– Breadth First Search
– Iterative Deepening Search
– Iterative Broadening Search
• Informed Search
• Constraint Satisfaction
• Adversary Search

Questions for Lecture 3
• Given the initial state, goal test, successor function,
and cost function for each of the following.
Choose a formulation that is precise enough to be
implemented.
1. You have to color a planar map using only four
colors, in such a way that no two adjacent regions
have the same color.
2. In the travelling salesperson problem (TSP) there
is a map involving N cities some of which are
connected by roads. The aim is to find the shortest
tour that starts from a city, visits all the cities
exactly once and comes back to the starting city.

Questions for Lecture 3
• Missionaries & Cannibals problem: 3
missionaries & 3 cannibals are on one side of
the river. 1 boat carries 2.
Missionaries must never be outnumbered by
cannibals. Give a plan for all to cross the river.

Final slide (bsc csit) chapter 3

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