1. Brute Force Search Depth-first or Breadth-first search
Do not apply any heuristics to aid in solving the problem
Therefore, these techniques are considering blind forms of search
Usually very inefficient (intractable)
Used when there is no knowledge to apply
8 queens problem, traveling salesman problem
In this chapter, we will consider heuristic forms of search and look at several examples

2. Heuristic Search Heuristic - a rule of thumb used to help guide search
Heuristic Function - function applied to a state in a search space to indicate a likelihood of success if that state is achieved
Best-first search – a variation on brute force search where a heuristic is used to guide the search
Heuristic search can be powerful but very also general
We examine several forms of heuristic search here

3. Weak Methods:
Heuristic search methods known as “weak methods” because of their generality and because they do not apply a great deal of knowledge
They all attempt to reduce the amount of search required to solve the problem, try to make the problem tractable
We will consider:
Generate-and-test
Hill Climbing and variations
Best-first Search and variations
Constraint Satisfaction
Means-End Analysis

4. Generate-and-Test
Generate a possible solution
A path through the search space, or a single state in the case of interpretation or diagnosis
Test to see if solution reaches a goal state
If so, quit, otherwise repeat
Obviously, this is not a very useful method – it relies on randomness
Was used as part of Dendral’s process for generating chemical analyses of a mass spectrogram reading – but Dendral also used constraint satisfaction and user feedback
We will concentrate on more reasonable methods that apply heuristics

5. A Heuristic Function Consider the 8-puzzle as an example problem
We want to select the best move to make next
What move is best? We might compare the current state to the goal state and see how they differ. Now look at each possible move and determine which one takes us closest to the goal state
Goal: Current: Moves:
Move 7 down
Move 6 right
Move 8 left
5 6
7 8

6. Hill Climbing
Visualize the choices as an 2-dimensional space, apply a heuristic and seek the state that takes you uphill the maximum amount
In actuality, many problems will be viewed in more than 2 dimensions
In 3-dimensions, the heuristic worth represents “height”
To solve a problem, pick a next state that moves you “uphill”
Examples:
Simple Hill Climbing
Steepest Ascent Hill Climbing
Simulated Annealing

7. Simple Hill Climbing Given initial state perform the following
Set initial state to current
Loop on the following until goal is found or no more operators available
Select an operator and apply it to create a new state
Evaluate new state
If new state is better than current state, perform operator making new state the current state
Once loop is exited, either we will have found the goal
This algorithm only tries to improve during each selection, but not find the best solution

8. Steepest Ascent Hill Climbing
Here, we attempt to improve on the previous Hill Climbing algorithm
Given initial state perform the following
Set initial state to current
Loop on the following until goal is found or a complete iteration occurs without change to current state
Generate all successor states to current state
Evaluate all successor states using heuristic
Select the successor state that yields the highest heuristic value and perform that operator
Notice that this algorithm can lead us to a state that has no better move, this is called a local maximum (other phenomenon are plateaus and ridges)

9. Examples 8 Puzzle Blocks World
Heuristic: add 1 point for every block that is resting on the thing it is supposed to be resting on and subtract 1 point for every block that is sitting on the wrong thing
This is a local heuristic, it only considers a block in isolation, not the substructure
Better heuristic: for each block that has the correct substructure, add 1 point for every block in the substructure, and for each block that has an incorrect substructure, subtract 1 point for every block in the substructure
8 Puzzle
Heuristic: add 1 point for each tile that is in its proper location and subtract 1 point for each tile in its wrong location
Better heuristic: add 1 point for each tile that is in its proper location and subtract 1 point for each move that it would take to move a tile from its improper location to its proper location

10. Simulated Annealing
A variation where some downhill moves can be made early on in the search
The idea is that early in the search, we haven’t invested much yet, so we can make some downhill moves
In the 8 puzzle, we have to be willing to “mess up” part of the solution to move other tiles into better positions
The heuristic worth of each state is multiplied by a probability and the probability becomes more stable as time goes on (see the formula on page 70)
Simulated annealing is actually applied to neural networks

11. Best-first search
One problem with hill climbing is that you are throwing out old states when you move uphill and yet some of those old states may wind up being better than a few uphill moves
Algorithm uses two sets, open nodes (which can be selected) and closed nodes (already selected)
Start with Open containing the initial state
While current <> goal and there are nodes left in Open do
Set current = best node in Open and move current to Closed
Generate current’s successors
Add successors to Open if they are not already in Open or Closed
A (5)
B (4) C (3) D (6)
G (6) H (4) E (2) F (3)
I (3) J (8)

12. Variations of Best-first
A* Algorithm – add to the heuristic the cost of getting to that node
For instance, if a solution takes 20 steps and another takes 10 steps, even though the 10-step solution may not be as good, it takes less effort to get there
Problem-Reduction – use AND/OR graphs where some steps allow choices and others require combined steps
AO* Algorithm – A* variation for AND/OR graphs
Alpha-Beta Pruning – use a threshold to remove any possible steps that look too poor to consider
Agendas – Evaluating different AND/OR paths using different heuristics, the agenda is a list of tasks that can be applied to a given state in the search space

13. Constraint Satisfaction
Many branches of a search space can be ruled out due to constraints
Constraint satisfaction is a form of best-first search where constraints are applied to eliminate branches
Consider the Cryptorithmetic problem, we can rule out several possibilities for some of the letters
After making a decision, propagate any new constraints that come into existence
Constraint Satisfaction can also be applied to planning where a certain partial plan may exceed specified constraints and so can be eliminated
SEND
+ MORE
MONEY
M = 1 
S = 8 or 9 
O = 0 or 1 
O = 0 …

14. Means-Ends Analysis Compare the current state to the goal state
Pick the operator which moves the problem most towards the goal state
Repeat until the goal state has been reached
Forward and backward chaining may be used
Subgoaling required
Generating intermediate states or steps
Means-Ends is often used in planning problems, but can be applied in other situations too such as math theorem proving
Means-ends analysis is much like top-down design in programming