1. CSC-470: ARTIFICIAL INTELLIGENCE
Lecturer
Muhammad Tariq Siddique
Lecture 2: Problem Solving by Search

2. Outline Problem Solving Agent Problem Examples Uninformed search
Problem formulation
Search strategies: depth-first, breadth-first
Informed search
Search strategies: best-first, A*
Heuristic functions
A* (pronounced "A star" ) is a path finding and graph traversal.

3. Problem Solving Agent
Goal formulation, based on the current situation and performance measure
Problem formulation is the process of deciding what actions and states to consider, given a goal.
an agent with several options can decide what to do by first examining different possible sequences of actions that lead to states of known value, and then choosing the best sequence.

4. Formulate, Search, Execute
Problem Formulation is the process of deciding what actions and states to consider, given a goal.
The process of looking into sequences/steps to accomplish a goal is known as search
A search algorithm takes a problem as input and return a solution.
After finding a solution, the resultant action is known as execution phase.
formulate
search
execute

5. A Simple Problem-Solving Agent
function SIMPLE-PROBLEM-SOLVING-AGENT( percept) returns an action persistent: seq, an action sequence, initially empty state, some description of the current world state goal, a goal, initially null problem, a problem formulation state  UPDATE-STATE(state, percept) if seq is empty then goal  FORMULATE-GOAL(state) problem  FORMULATE-PROBLEM(state, goal) seq SEARCH(problem) If seq = failure then return a null action action  FIRST(seq) seq  REST(seq) return action

6. Problem Solving By Search
Represent the problem as STATES and OPERATORS that transform one state into another state.
A solution to the problem is an OPERATOR SEQUENCE that transforms the INITIAL STATE into a GOAL STATE.
Finding the sequence requires SEARCHING the STATE SPACE by GENERATING the paths connecting the two.

7. Search By Generating States
Initial State 3 2
100 1 4 5 6
Goal State
1  2
1  6
2  3
2  5
6  5
3  5
5  4
Operations

8. Basic Concepts
State: finite representation of the world at a given time.
Operator(action formulation): a function that transforms a state into another (also called rule, transition, successor function, production, action)
Initial state: world state at the beginning.
Goal state: desired world state (can be several)
Goal test: test to determine if the goal has been reached.

9. Basic Concepts (Cont..)
Reachable goal: a state for which there exists a sequence of operators to reach it.
State space: set of all reachable states from initial state (possibly infinite).
Cost function: a function that assigns a cost to each operation.
Performance:
cost of the final operator sequence (path cost)
cost of finding the sequence

10. Problem Solving Agents (Example)
Goal Formulation
Based on the current situation and agent’s performance measure
E.g. Going from Los Angeles to Las Vegas
Problem Formulation
What actions and states to consider, given a goal
Not too detail, not to broad
E.g. Driving from Los Angeles to Las Vegas
If unknown environment, take random action
Observable environment  know current state (location)
Discrete environment  at any given state, only finitely many actions to choose from
Deterministic environment  each action has exactly one outcome
Solution
Action sequence

12. Problem-Solving Agent (Example)
function SIMPL-PROBLEM-SOLVING-AGENT( percept) returns an action persistent: seq, an action sequence, initially empty state, some description of the current world state goal, a goal, initially null problem, a problem formulation state  UPDATE-STATE(state, percept) if seq is empty then goal  FORMULATE-GOAL(state) problem  FORMULAE-PROBLEM(state, goal) seq SEARCH(problem) If seq = failure then return a null action action  FIRST(seq) seq  REST(seq) return action
// What is the current state?
// From LA to Vegas (given current state)
// e.g., Gas usage
// If fails to reach goal, update

13. Problems (Example) Initial state : state that agent starts in
In(Los Angeles)
Actions : applicable functions to the agents
{Go (Long Beach), Go(Santa Clarita), Go(San Bernardino)}
Transition model: what each action does (result)
RESULT(In(Los Angeles), Go (Long Beach)) = In(Long Beach)
State space = initial state + actions + transition model
Represent in a graph form
Path = sequence of states connected by a sequence of actions
Goal test: check whether a given state is a goal state
Path cost, step cost
Solution: an action sequence that leads from the initial state to a goal state.
Optimal solution: the lowest path cost among all solutions

14. Example Problems Real-world problems Toy problems Route finding
Touring and travelling salesperson
VLSI layout
Robot navigation
Assembly sequencing
Toy problems
The Maze
The 8 and 15-puzzle
The 8-queen/n- queens
Cryptarithmetic
Robotic Assembly
The vacuum world
Missionaries and cannibals
The River Problem
Water Jugs

15. The Maze A maze can be represented as a state space
Each state represents “where you are” in the maze
The start state represents your starting position
The goal state represents the exit from the maze
Operators (for a rectangular maze) are: move north, move south, move east, and move west
Each operator takes you to a new state (maze location)
Operators may not always apply, because of walls in the maze
Initial
state
Goal state

16. The 8 and 15-puzzle The 8-puzzle is a small single board player game:
Tiles are numbered 1 through 8 and one blank space on a 3 x 3 board.
A 15-puzzle, using a 4 x 4 board, is commonly sold as a child's puzzle.

17. The 8-Puzzle
Given an initial configuration of 8 numbered tiles on a 3 x 3 board, move the tiles in such a way so as to produce a desired goal configuration of the tiles.

18. The 8-Puzzle (Cont)
States: Integer location of tiles (ignore intermediate positions)
Operators: Move blank left, right, up, down
Goal Test: goal state (given)
Path Cost: 1 per move
Constraints: Can only move blank in a direction if it stays in puzzle

19. The 8-Puzzle (Cont)
States: a state description specifies the location of each of the eight tiles in one of the nine squares.
Operators: blank moves left, right, up,, or down.
Goal test: state matches the goal configuration shown in the right.
Path: each step costs 1, so that the path cost is just the length of the path.

20. The 8-Puzzle (Cont) 5 4 6 8 2 3 1 7 Goal: 5 4 6 8 2 3 1 7 5 4 6 8 2 3
Goal: 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7 5 4 6 8 2 3 1 7

21. The 15-puzzle
Start state:
The start state is some (almost) random configuration of the tiles
The goal state is as shown
Operators are
Move empty space up
Move empty space down
Move empty space right
Move empty space left
Operators apply if not against edge
Goal state:

22. The 8-queens problem
The objective is to place eight queens on a chessboard such that no queen threatens another.
A queen threatens another queen in the same row, column or in diagonal
In this example,
the two queens
on the corners
are the only
queens threatening
each other.

23. The 8-queens problem (cont)
The states and operators for this problem could be:
States: any arrangement of 0 to 8 queens on the board.
Operators: add a queen to any square.
This is a bad choice because there are 64^8 possible sequences to investigate.
A better formulation would be to choose a smarter operator that
Operators: place a queen in the left-most empty column such that it is not attacked by any other queen.

24. The 8-queens problem (cont)

25. Robotic Assembly
states: real-valued coordinates of robot joint angles parts of the object to be assembled operators: continuous motions of robot joints goal test: complete assembly with no robot included!
path cost: time to execute

26. Vacuum World
states: integer dirt and robot location (ignore dirt amount) operators: Left, Right, Suck goal test: no dirt
path cost: 1 per operator

27. Cryptarithmetic Problem
States: a cryptarithmetic puzzle with some letters replaced by digits
Operators: replace all occurrences of a letter with a digit not already appearing in the puzzle.
Goal test: puzzle contains only digits, and represents a correct sum.
Path cost: zero. All solutions equally valid.

28. Cryptarithmetic Problem (cont..)
The puzzle
SEND + MORE = MONEY, after solving,
will appear like this:
S E N D
+ M O R E
M O N E Y

29. Missionaries and Cannibals
An old puzzle is the “Missionaries and cannibals” problem (in various guises)
The missionaries and cannibals wish to cross a river
They have a canoe that can hold two people
It is unsafe to have cannibals outnumber missionaries M C
Initial state M C
Goal state

30. Operations An operation takes us from one state to another
Here are five possible operations:
Canoe takes 1 missionary across river (1m)
Canoe takes 1 cannibal across river (1c)
Canoe takes 2 missionaries across river (2m)
Canoe takes 2 cannibals across river (2c)
Canoe takes 1 missionary and 1 cannibal across river (1m1c)
We don’t have to specify “west to east” or “east to west” because only one of these will be possible at any given time

31. The State Space
A state could be (CanLeft, MissLeft, BoatPos, CanRight, MissRight)
e.g. (2, 2, RIGHT, 1, 1) i.e. 2 cannibals and 2 missionaries on the left bank of the river, the boat is on the right side, together with 1 cannibal and 1 missionary.
Operators: A legal move is one which involves moving one or two people to the opposite bank, such that cannibals don't outnumber missionaries on either bank.
An initial state is: (3, 3, LEFT, 0, 0)
Possible moves are:
from (3, 3, LEFT, 0, 0) to (2, 2, RIGHT, 1, 1)
from (2, 2, RIGHT, 1, 1) to (2, 3, LEFT, 1, 0)
A goal state is: (0, 0, RIGHT, 3, 3)
An example action:
Assume the current state is: (cLeft, mLeft, boatPos, cRight, mRight)
Action: move 1 missionary and 1 cannibal from the left bank to the right bank.
Preconditions:
boatPos = LEFT
cLeft >= 1 AND mLeft >= 1
(mLeft-1 >= cLeft-1) OR mLeft = 0
(mRight+1 >= cRight+1) OR mRight = 0
New state would become: (cLeft-1, mLeft-1, RIGHT, cRight+1, mRight+1)
This action could be applied to the state: (3, 3, LEFT, 0, 0) and would give the new state: (2, 2, RIGHT, 1, 1). This action could not be applied to the state: (2, 2, RIGHT, 1, 1)

32. The State Space

33. The State Space

35. The state space etc. 3m, 2c 3m, 1c 2m, 3c 1m, 3c 2m, 2c 1m 2m 2c 1m1c
3m, 3c, canoe
1m1c 1c
3m, 2c, canoe
2m, 3c, canoe 1m

36. Missionaries and Cannibals Solution
Near side Far side 0 Initial setup: MMMCCC B - 1 Two cannibals cross over: MMMC B CC 2 One comes back: MMMCC B C 3 Two cannibals go over again: MMM B CCC 4 One comes back: MMMC B CC 5 Two missionaries cross: MC B MMCC 6 A missionary & cannibal return: MMCC B MC 7 Two missionaries cross again: CC B MMMC 8 A cannibal returns: CCC B MMM 9 Two cannibals cross: C B MMMCC 10 One returns: CC B MMMC 11 And brings over the third: - B MMMCCC

37. The River Problem Let’s consider the River Problem:
A farmer wishes to carry a wolf, a duck and corn across a river, from the south to the north shore. The farmer is the proud owner of a small rowing boat called Bounty which he feels is easily up to the job. Unfortunately the boat is only large enough to carry at most the farmer and one other item. Worse again, if left unattended the wolf will eat the duck and the duck will eat the corn.
How can the farmer safely transport the wolf, the duck and the corn to the opposite shore?
River
boat
Farmer, Wolf,
Duck and Corn

38. F=Farmer W=Wolf D=Duck C=Corn /=River
The River Problem
The River Problem:
F=Farmer W=Wolf D=Duck C=Corn /=River
How can the farmer safely transport the wolf, the duck and the corn to the opposite shore?
-/FWCD
FWCD/-

39. (F-Takes-W, F-Takes-D, F-Takes-C, F-Takes-Self [himself only])
The River Problem
Problem formulation:
State representation: location of farmer and items in both sides of river
[items in South shore / items in North shore] : (FWDC/-, FD/WC, C/FWD …)
Initial State: farmer, wolf, duck and corn in the south shore FWDC/-
Goal State: farmer, duck and corn in the north shore
-/FWDC
Operators: the farmer takes in the boat at most one item from one side to the other side
(F-Takes-W, F-Takes-D, F-Takes-C, F-Takes-Self [himself only])
Path cost: the number of crossings

40. The River Problem Problem solution: (path Cost = 7)
While there are other possibilities here is one 7 step solution to the river problem F W D C
F-Takes-D
Initial State
WC/FD
Goal State
F-Takes-S
FD/WC
F-Takes-C
D/FWC
FDC/W
F-Takes-W
C/FWD
FWC/D

41. Water Jugs You’re given A spigot A 3 Gallon jug A 4 Gallon jug
The goal: Get 2 gallons of water in the 4 Gallon jug.
Actions: Filling jugs from spigot, dumping water in jugs onto ground, dumping 4 gallon into 3 gallon jug until 3 gallon jug is full. Dumping 3 gallon jug into 4 gallon jug until empty or until 4 gallon is full, etc, etc.
Think of states, actions, goals state, operators and a solution.

42. Water Jugs States: How full are the two jugs? Initial State:
State Representation 4G 3G
Constraints
0  4G  4
0  3G  3
Initial State:
4G = 0
3G=0
Goal State
4G=2

43. Operators F3: Fill the 3 Gallon jug from the tap.
E4: Empty the 4-Gallon jug on the ground
P43: Pour water from 4G jug into the 3G jug until 3G jug is full
P34: Pour water from 3G jug into the 4G jug

44. Search Space
The state space for this problem can be described as the set of ordered pairs of integers (x, y) such that x = 0, 1,2, 3 or 4 and y = 0,1,2 or 3; x represents the number of gallons of water in the 4-gallon jug and y represents the quantity of water in 3-gallon jug
Production rules for Water Jug Problem
The operators to be used to solve the problem can be described as follows:
The start state is (0,0)
The goal state is (2,n)

46. 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
parent, children, depth, path cost g(n).
States do not have parents, children, depth, or path cost!
A Node is a data structure with five components:
the state in the state space to which the node corresponds;
the node in search tree that generated this node (this is called the parent node);
the operator that was applied to generate the node;
the number of nodes on the path from the root to this node (the depth of the node);
the path cost of the path from the initial state to the node.
Datatype node
components: STATE, PARENT-NODE, OPERATOR, DEPTH, PATH-COST

47. Nodes and States
Node is a data structure to represent search tree for a particular problem
State represents a configuration of the world
Fringe or frontier
A collection of nodes that are waiting to be expanded
a set of nodes can be represented as a queue with following operations
MAKE-QUEUE(Elements) creates a queue with the given elements
EMPTY?(Queue) returns true if there are no more elements in the queue.
REMOVE-FRONT(Queue) removes the element at the front of the queue and returns it
QUEUE-FN(Element, Queue) inserts a set of elements into the queue. Different varieties of the queuing function produce different varieties of the search algorithm

48. Implementation of Search Algorithms
Nodes: state, parent-node,operator, depth, path cost
Function GENERAL-SEARCH (problem, queing-fn)
returns a solution or failure
nodes  MAKE-QUEUE (MAKE-NODE(INITIAL-STATE[problem]))
loop do
if nodes is empty, then return failure
node  Remove-Front(nodes)
if GOAL-TEST [problem] applied to STATE(node) succeeds
then return node
else nodes  QUEING-FN(nodes, EXPAND(node, operators[problem]))
end

49. Conclusions
Interesting problems can be cast as search problems, but you’ve got to set up state space
Problem formulation usually requires abstracting away real world details to define a state space that can feasibly be explored

50. Real-life example: VLSI Layout
Given schematic diagram comprising components (chips, resistors, capacitors, etc) and interconnections (wires), find optimal way to place components on a printed circuit board, under the constraint that only a small number of wire layers are available (and wires on a given layer cannot cross!)
“optimal way”??
minimize surface area
minimize number of signal layers
minimize number of vias (connections from one layer to another)
minimize length of some signal lines (e.g., clock line)
distribute heat throughout board
etc.

51. Real-life example: VLSI Layout

52. Use automated tools to place components
and route wiring.

53. Real-life example: VLSI Layout