Problem Solving
Problem Solving definition: you have a
goal, and you are currently not at the goal.
Want to own a dog—don’t have a dog.
Need 3 -letter word for European blackbird—don’t know it
Want to attend graduate school—not currently in graduate school
Want to be outside—currently are inside
Problem types
Well defined problem: initial state, goal state, and
methods available to you are understood.
Example: solving an anagram.
Vince Goti
Initial state: vince goti
Goal state: any word
Methods: rearranging
Problem types
Ill defined problem: initial state, goal state, and/or
methods are poorly understood.
Example: writing a term paper
Initial state: no paper
Goal state: a paper that looks like ?
Methods: ???????????
Approaches to problem solving
• Behaviorist
• Gestalt
• Information processing
Behaviorist
Behaviorist view
Problem solving seen almost as a
matter of chance early in training. Due
to random variation, you produce
slightly better versions; these are
reinforced, so you’re more likely to do
them again.
Gestalt view
In perception, emphasized that the
relationship among parts is crucial to
what we see
Sometimes more than one organization of
parts is possible; the perception can reverse.
http://dogfeathers.com/java/necker.html
Gestalt view
Same emphasis in problem solving: Gestalt
psychologists emphasized that the way you
see the relationship of the parts of a
problem dictates the difficulty of finding a
solution:
Sometimes you reorganize the parts, the
whole looks different and the solution is
easy.
Wolfgang Köhler
Two trains are 25 miles apart and are
traveling towards one another. One is
going 35 mph and the other is going 15
mph. A fly begins on the second train
and starts to fly towards the first,
traveling 20 mph. How far does the fly
travel before the trains collide and it is
smooshed?
Gestalt view
Some problems that initially seem hard
suddenly become trivial when the
representation is changed.
You have an “Aha!” experience.
Gestalt view says little about non-Aha
problems, or about how you get the
change in representation.
Information processing view
A particular configuration of the problem is
called a state.
An operator moves you from one state to
another.
Initial
New state New state Goal
knowledge
Example
How to get Find current Know my
to Albany? location current
location
Locate target
Know my Find shortest route
Route found
between my location
location and
and target
Albany’s
Note that it’s the operators that move you through
the problem space. Problem solving is all about
choosing operators
This observation (that operators move you
through the space) makes our job clearer: the
question is how do you select operators?
Three general cases we can consider:
1) Very familiar problem--algorithm
2) you have very little relevant knowledge
3) you have some relevant knowledge
Algorithm
An algorithm for problem-solving is a
formula—it is a fixed series of steps that you
apply. If you apply them correctly, you will
arrive at a solution, possibly the desired
solution.
4 cups fresh or frozen tart cherries
1 to 1 1/2 cups granulated sugar
4 tablespoons cornstarch
1/8 tablespoon almond extract (optional)
Your favorite pie crust or pie dough recipe for 2 crust pie
1 1/2 tablespoons butter, to dot
1 tablespoon granulated sugar, to sprinkle
Place cherries in medium saucepan and place over heat. Cover. After the cherries lose
considerable juice, which may take a few minutes, remove from heat. In a small bowl, mix
the sugar and cornstarch together. Pour this mixture into the hot cherries and mix well. Add
the almond extract, if desired, and mix. Return the mixture to the stove and cook over low
heat until thickened, stirring frequently. Remove from the heat and let cool. If the filling is
too thick, add a little water, too thin, add a little more cornstarch.
Preheat the oven to 375 degrees F.
Use your favorite pie dough recipe. Prepare your crust. Divide in half. Roll out each piece
large enough to fit into an 8 to 9-inch pan. Pour cooled cherry mixture into the crust. Dot
with butter. Moisten edge of bottom crust. Place top crust on and flute the edge of the pie.
Make a slit in the middle of the crust for steam to escape. Sprinkle with sugar.
Bake for about 50 minutes. Remove from the oven and place on a rack to cool.
Algorithms are widely applicable,
and, because they are stored in
memory, save you the trouble of
having to think.
• What to do when you want a pizza.
• What to do when you need to drive to an
unfamiliar location.
• Dealing with social situations?
Meal check
yes
Wait quietly
I paid
last no
Grab check
yes time?
Get check—don’t
Friend?
make a scene
yes
no
Person to Rough split
impress? yes
no
Acquaintance
no
Split to penny
Algorithm
Algorithms are great, but they are not
applicable to all problems.
You have to have some experience
Must be a good definition of the starting
point, the goal state, and methods
available—i.e., a well-defined problem.
No algorithm for writing a novel.
Little knowledge
What will you do when you have very little
relevant knowledge?
You have to fall back on operators that are very
general-purpose, and will be more or less
applicable to any situation.
Heuristics
We do have heuristics we can use when we
are in an unfamiliar problem solving situation.
• Brute force search
• Hill climbing
• Working backwards
• Means-ends analysis
Brute force search
2 down: “friend”
A
L
You could sequentially go through the letters
“a” “b” “c” until you found a suitable solution
Brute force--problem
As the state space gets larger, you get
combinatorial explosion
Hill climbing
Hill climbing means looking ahead and
selecting an operator that you judge will bring
you closer to your goal; do not select an
operator that will move you away from your
goal.
Hill climbing works great for for digging a hole
Hill climbing good for getting to the top of a hill.
When would hill climbing not work?
I’m not on the highest peak--how will I get there?
If this dog wanted to sniff you, what would it do
and what would it need to do?
Hill climbing won’t work when you need to move
away from the goal for a little while in order to
eventually reach the goal.
Working backwards
As the name suggests, working backwards
means imagining being at the goal of the
problem space and seeing if you can figure out
a way to get to the initial state, or closer to it.
http://www.transience.com.au/pearl3.html
Means ends analysis
1. Compare the current state to the goal state. If there is no
difference between them, the problem is solved.
2. If there is a difference between the current state and the goal
state, set as a goal to solve that difference. If there is more than
one difference, set as a goal to solve the largest difference.
3. Select an operator that will solve the difference identified in
step 2.
4. If the operator can be applied, apply it. If it cannot, set as a new
goal to reach a state that would allow the application of the
operator.
5. Return to step 1 with the new goal set in step 4.
Means ends analysis allows the setting of
subgoals. That’s what’s missing from the
other heuristics. Sometimes you must set a
subgoal, which means temporarily moving
away from the goal. It’s a combination of
moving forwards & backwards.
Example
Suppose I want to make Etouffee
Step 1: Do I have Etouffee? No.
Step 2: Goal: get Etouffee. Differences = knowledge, ingredients. Subgoal,
get knowledge.
Step 3: What has Etouffee knowledge? A cookbook.
Step 4: Do I have a cookbook? No.
Step 5: Set new subgoal: get cookbook, return step 1
Step 1: Do I have a cookbook? No.
Step 2: Goal: get cookbook. Difference = distance to cookbook.
Step 3: What reduces distance? A car
Step 4: Can I use car? No. friend has keys.
Step 5: Set new subgoal: get car keys.
What would my solution be if I used hill-climbing?
Newell, Shaw & Simon
The General Problem Solver was a computer
program designed to solve problems using
means ends analysis, and it was very effective.
More important, it not only solved problems,
the authors argued that it solved problems in
ways similar to the ways humans solved
problems.
Those four heuristics were for
situations where you have little or
no relevant knowledge.
What happens if you have some
relevant knowledge, but aren’t
familiar with this particular
problem?
You would think that some knowledge is
better than none. The truth is that
sometimes this knowledge helps and
sometimes it doesn’t.
Knowledge can help. . .
Knowledge can help when it allows you to
recognize patterns based on past experience
Experts recognize patterns,
allowing them to limit the
search space (e.g, they
immediately perceive that
the Queen is in trouble)
Knowledge can help. . .
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Prior knowledge can also impede
Prior knowledge can impede
Functional fixedness
People fixate on the typical function
of an object and fail to realize it can
be used in an atypical way to solve a
problem
Functional fixedness depends on
prior knowledge—knowing the
typical function.
Pliers cut things, don’t serve as weight
Cutting a link allows joining of the
attached length of chain
Other prior knowledge
You can be fixated not only on the function of
an object, but also on the procedures used to
solve a problem--try to solve a problem as you
have before, even if the procedures are no
longer appropriate.
Luchins’s water jars
Problem number Jug A Jug B Jug C Required Amount
1 18 43 10 5
2 21 127 33 40
3 14 163 25 99
7 23 49 3 20
8 15 39 3 18
9 28 76 3 25
Formula B -A- 2C works, but not for 9, and 7 and 8 have easier
solutions
You should bear in mind that being an expert
almost always helps you. The problem arises
where it looks like knowledge you have will
be applicable, but it actually leads you down
the wrong solution path.
What makes an expert?
Surprising upshot: it doesn’t seem to be that
they are better at choosing operators. Rather,
they seem to be better at restricting the search
space. They can do that by recognizing
familiar patterns.
Chess experts may have as many as
50,000 board positions in memory.
When a chess master plays many people
simultaneously (at an exhibition) his play
is not much worse than when he’s at a
tournament, even though he has much
less time per move. This indicates that
the amount of time spent considering
moves is not so important.
This has led to the more general hypothesis
that experts are not so different than novices
in terms of how they select operators,but are
are different in terms of how they think of the
search space.
Example
0.8
This is amount of
Novice information about cases
0.6
Trainee remembered by doctors with
Expert different levels of expertise.
Experts remember LESS
Memory
0.4
after a delay, but the
0.2 information they remember
is all the stuff necessary to
0 diagnosis; experts remember
Immediate One week
fewer unimportant details.
Retention Interval
How do you get to be an expert?
Practice like crazy for 10 years
This is an estimate of the
number of hours of
practice each day among
aspiring performers
identified as potential
soloists, aspiring
performers identified as
competent, and aspiring
music teachers,
Broad summary
• If you have no clue about a problem, you
apply a heuristic
• If some elements of a problem seem familiar,
it may help, but if the part that is familiar
leads you to a solution that is ineffective, it
will hurt.
• Experts probably don’t solve problems
differently than you, they just know more
than you do.