1. Tutorial 1 Uninformed Search
CS Introduction to AI
Tutorial 1
Uninformed Search

2. Problem Representation
Initial state (InitState)
Successor function (Succ)
Goal predicate (Goal-p)
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

3. Problem Example - Binary Tree
The problem is to find the number 12 in a binary tree
A state is represented by a number
Initial State: 1
Successor Function: Succ(x) = {2*x, 2*x + 1}
Goal predicate: Goal-p(x) == true iff x == 12 1 2 3 4 5 6 7
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

4. General Search Routine
SearchFunction (States, Problem, Combiner) {
If States == {}
return ‘Search Failed’
CurrState <- Get and Remove first state in States
if (Problem.Goal-p(CurrState) == true)
return CurrState //or any other solution form
else
successors <- Problem.Succ(CurrState)
States <- Combiner(successors,States)
SearchFunction(States, Problem, Combiner) }
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

5. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Combiner examples
Append-Combiner(states1, states2) {
// First nodes in states1, then nodes in states2
Result <- states1
For s in states2
Result <- Add s to the end of Result }
Prepend-Combiner(states1, states2) {
// First nodes in states2, then nodes in states1
Result <- states2
For s in states1
Result <- Add s to the end of Result }
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

6. General Search Routine
First call to the search function:
Problem <- Initialize problem with appropriate data
Combiner <- Choose state combination method that will
characterize the search
States <- Problem.InitState
Solution = SearchFunction(States, Problem, Combiner)
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

7. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
DFS
States <- Problem.InitState
Solution = SearchFunction(States, Problem,
Append-Combiner) 1 2 3 4 5 8 9 16 17
(For this example, solution will not be found) 32 33 …
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

8. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Depth Limited DFS
LimDepth-Append-Combiner(states1, states2) {
Result <- {}
For s in states1
if s.depth <= DepthLimit
Result <- Add s to the end of Result
For s in states2 }
DepthLimit is a property of LimDepth-Append-Combiner
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

9. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Depth Limited DFS
LimDepth-Append-Combiner.DepthLimit <- 5
States <- Problem.InitState
Solution = SearchFunction(States, Problem,
LimDepth-Append-Combiner) 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 18 19 20 21 22 23 X X X X X X X X
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

10. Iterative deepening DFS
Iterative-Deepening-DFS(Problem, MaxDepth) {
for i from 1 to MaxDepth
LimDepth-Append-Combiner.DepthLimit <- i
States <- Problem.InitState
Solution = SearchFunction(States, Problem,
LimDepth-Append-Combiner)
If Solution != {}
return Solution }
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

11. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
BFS
States <- Problem.InitState
Solution = SearchFunction(States, Problem,
Prepend-Combiner) 1 2 3 4 5 6 7 8 9 10 11 12 13
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

12. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Discussion: DFS Vs. BFS
DFS BFS
Completeness  
Optimality of Solution  
Time complexity O(bd) O(bd)
Space complexity O(bd) O(bd)
Search in graphs?  
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

13. Example: 9-number puzzle
The numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 must be put in the depicted square, in such a way that the sums of the numbers in each row, column, and diagonal are equal.
Solution: 9 5 1 8 3 4 6 7 2
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

14. Example: 9-number puzzle
Problem representation
Initial state: Empty board
Successors: Put one of the missing numbers in an empty place
Goal-p: Board is full, sums are equal, as required 5 6 1 2 …
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

15. Example: 9-number puzzle
Which search algorithm should we use?
Note that the solution length is exactly 9
DFS with depth limit = 9 will find the solution and will use little memory (linear in solution depth)
BFS will find the solution at depth 9, however will require a lot of memory (will eventually hold all the tree till depth 9 in memory)
Iterative deepening DFS is unnecessary here, since we know the solution depth in advance. The solution will not be found at shallower depth.
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

16. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Example: LightzOut
5x5 board
Each cell can be “On” or “Off”
Click on a cell inverts the states of its 4 neighbors
The goal is to turn off the lights
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

17. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Example: LightzOut
Click on the
central cell
Goal:
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

18. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Example: LightzOut
Problem representation:
Initial state: The board to solve
Successors: Boards that are the result of a single click on the current board
Goal-p: Empty board
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

19. LightzOut – solution example. . . . . .
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

20. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Example: LightzOut
Which search algorithm should we use?
Given a board, we cannot predict solution length!
Note that if we are in state1, and click on the same cell twice, we return back to state1
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

21. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Example: LightzOut
DFS with no restrictions might never find a solution for the problem – can get stuck in a loop
Depth limited DFS can be used if we know the problem difficulty in advance
Iterative deepening DFS is a good solution, especially when we do not know how difficult the problem is
BFS?
The branching factor is O(N2)
BFS may be suitable for very simple problems
For difficult problems, the memory amounts required for BFS are immense
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich

22. Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich
Multiple Solutions
Do not stop at the first solution
MultGoal-SearchFunction (States, Problem, Combiner) {
Solutions <- {}
If States == {}
return ‘Search Failed’
CurrState <- Get and remove first state in States
if (Problem.Goal-p(CurrState) == true)
Solutions <- Solutions + CurrState
successors <- Problem.Succ(CurrState)
States <-Combiner.combine(successors,States)
SearchFunction(States, Problem, Combiner) }
25-Dec-18
Intro. to AI – Tutorial 1 – By Saher Esmeir & Nela Gurevich