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What do we have so far? Basic biology of the nervous system Motivations Senses Learning Perception Memory Thinking and mental representations
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What do we have so far? All of these topics give a basic sense of
the structure and operation of our mind
What kinds of tasks does our mind engage in? Language Problem Solving Decision Making Others
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Problem Solving: Definition
A problem exists when you want to get from “here” (a knowledge
state) to “there” (another knowledge state) and the path is
not immediately obvious.
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What are problems? Everyday experiences
How to get to the airport? How to study for a quiz, complete a paper,
and finish a lab before recitation? Domain specific problems
Physics or math problems Puzzles/games
Crossword, anagrams, chess
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A Partial Problem Typology
Well-defined vs. ill-defined problems: Problems where the goal or solution is recognizable--where there is a right answer (ex. a math or physics problem) vs. problems where there is no "right" answer but a range of more or less acceptable answers.
Knowledge rich vs. knowledge lean problems: problems whose solution depends on specialized knowledge.
Insight vs. non-insight problems--those solved "all of a sudden" vs. those solved more incrementally--in a step by step fashion.
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Contents of Memory Does the contents of memory influence
how easy a problem is? Knowledge rich problems
Require domain knowledge to answer, physics problems
Knowledge lean problems Can use a general problem solving method to
solve, don’t need a lot of domain knowledge
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Some Problem Examples
Tower of Hanoi Weighing problem Traveling salesman (100 cities = 100! or 10200
or each electron, 109 operations per sec. would take 1011 years!!) but
100,000 cities within 1% in 2 days via heuristic breakup (reduce search!)
Missionaries & Cannibals Flashlight: 1, 2, 5, 10 min. walkers to cross
bridge 21 link gold necklace/21 day stay Subway Problem Vases (or 3-door)
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Early findings Zeigarnik effect, 1927
Participants were given a set of problems to solve
On some problems, they were interrupted before they could finish the problem
Participants were given a surprise recall test They remembered many more of the
interrupted problems than the uninterrupted ones
Moss et al. (2007) recent RAT results: open goals
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Early Findings Luchins water jug experiment, 1942
Participants were given a series of water jug problems
Example: You have three jugs, A holds 21 quarts, B holds 127, C holds 3. Your job is to obtain exactly 100 quarts from a well
Solution is B – A – 2C Participants solved a series of these
problems all having the same solution
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Early Findings Luchins water jug experiment, 1942
New problem: Given 23, 49, and 3 quart jugs. Goal is to get 20 quarts.
Given 28, 76, and 3 quart jugs, obtain 25 quarts Some failed to solve, others took a very long
time Mental set
People who solved series of problems using one method tended to over apply that method to new similar appearing problems
Even when other methods were easier or where the learned method no longer could solve the problem
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Early Findings Duncker’s candle problem, 1945
Problem: Find a way to fix a candle to the wall and light it without wax dripping on the floor.
Given: Candle, matches, and a bow of thumbtacks
Solution: Empty the box, tack it to the wall, place candle on box
Have to think of the box as something other than a container
People found the problem easier to solve if the box was empty with the tacks given separately
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Early Findings Duncker’s candle problem
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Maier’s two-string problem 1930
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Early Findings Functional Fixedness
Inability to realize that something familiar for a particular use may also be used for new functions
But is this really a bad thing? We learn and generalize from our experience in
order to be more efficient in most cases Is it really a good idea to sit around trying to
figure out how many potential uses a pair of pliers has?
How often do mental sets and functional fixedness save time and computation?
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General Problem Characteristics What characteristics do all problems
share? Start with an initial situation Want to end up in some kind of goal
situation There are ways to transform the current
situation into the goal situation Can we have a general theory of
problem solving?
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General Theory of Problem Solving
Newell & Simon proposed a general theory in 1972 in their book Human Problem Solving
They studied a number of problem solving tasks Proving logic theroems Chess Cryptarithmetic
DONALD D=5+ GERALD ROBERT
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General Theory of Problem Solving
Verbal Protocols Record people as they think aloud during a
problem solving task Computational simulation
Write computer programs that simulate how people are doing the task
Yields detailed theories of task performance that make specific predictions
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General Theory of Problem Solving
Problem spaces Initial state Goal state(s) Operators that transform one state into
another
Initial
Goal
Goal
Initial
Initial
………………….o1
o2
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An Example Tower of Hanoi
Given a puzzle with three pegs and three discs
Discs start on Peg 1 as shown below, and your goal is to move them all to peg 3
You can only move one at a time You can never place a larger disc on a
smaller disc
1 2 3
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An Example Tower of Hanoi problem space
Initial condition: three discs on peg 1 Goal: three discs on peg 3 Operators: Move a disc following the
problem constraints
1 2 3
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Tower of Hanoi
Taken from Zang & Norman, 1994
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Another example Missionaries and cannibals problem
Six travelers must cross a river in one boat Only two people can fit in the boat at a time Three of them are missionaries and three
are cannibals The number of cannibals on either shore of
the river can not exceed the number of missionaries
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Problem Space
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Operators How do we choose which operators to
apply given the current state of the problem? Algorithm
Series of steps that guarantee an answer within a certain amount of time
Heuristic General rule of thumb that usually leads to a
solution
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Algorithm Examples Columnar algorithm for addition
Add the ones column Carry if necessary Add the next column, etc.
People don’t have a simple algorithm for solving most problems
4 6 2+ 2 34 8 5
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Heuristics Hill climbing
Just use the operator which moves you closer to the goal no matter what
What about problems where you have to first move away from the goal in order to get to it (detour problems)?
Fractionation and Subgoaling Break the problem into a series or hierarchy
of smaller problems
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Problem Space: Subgoaling
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Heuristics Working Backwards from the goal
Works well if there are fewer branchings going from the goal to the initial state
Only works if you can reverse the operators
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Heuristics Means-ends analysis
Always choose an operator that reduces the difference between your current state and the goal state
Tests for their applicability of the operator on the current problem state
Adopts subgoals if there is no move that will take you to the goal in one step
Must have a difference-operator table or its equivalent Tells you what operator(s) to use given the current
difference between the state of the problem and the goal. Might have to modify operator if none can be applied in current state
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First AI programs Newell & Simon (1956)
Logic Theorist (LT) LT completed proofs for a number of logic
theorems General Problem Solver (GPS)
GPS incorporated means-ends analysis, capable of solving a number of problems
Planning problems Cryptarithmetic Logic proofs
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Centrality of Representation
Problem space and representation Problem difficulty and representation The interaction of representation and
processing limitations (problem isomorphs)
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Representation: Example Number scrabble
1 2 3 4 5 6 7 8 9
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Limitations of GPS What about problems where there is no
explicit test for a goal state? Well-defined problems have a clearly
defined goal state Ill-defined problems don’t have a clearly
defined goal state GPS and other AI programs work only on
well-defined problems
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Examples of ill-defined problems Engineering Design Architecture Painting Sculpture How to run a business? A number of other creative or difficult
tasks that people engage in
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Limits of AI? Can AI programs be applied to ill-
defined problems? AARON
Program created by Harold Cohen Produces paintings using a number of
heuristics and general conceptions of aesthtics
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Art by AARON
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What makes problems hard? Large problem spaces are usually
harder to search than small ones Compare playing tic-tac-toe to chess
What factors from our architecture of mind play a role in determining how hard a problem is? Memory constraints Memory contents Types of mental representations we use
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Memory constraints Kotovsky, Hayes, & Simon, 1985
Work on isomorphs of the Tower of Hanoi An isomorph of a problem is one in which the
structure of the problem space is the same but the appearance of the problem is different
Remember the Tower of Hanoi?
1 2 3
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IsomorphsTaken from Kotovsky, Hayes, & Simon, 1985
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IsomorphsTaken from Kotovsky, Hayes, & Simon, 1985
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Isomorph Difficulty
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Results of IsomorphsAdapted from Kotovsky, Hayes, & Simon, 1985
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Memory constraints In the original Tower of Hanoi and in the
condition with monster models there was an external memory aid
Change problems are harder than move problems Takes more processing to assess whether a
change is valid than it does for a move Spatial proximity of the information Working with unchanging discs (stable
representation) vs. changing discs
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Computational Model
Tested understanding via a computer model that was:
Goal driven, subgoaling, limited memory capable of perfect behavior except for limited working memory
To see if we were in right “ballpark” To separate actions of various mechanisms
to see which had the most control/influence To be able to experiment with the separate
postulated mechanisms
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From Kushmerick & Kotovsky, 1993)
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Model-Human Agreement
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Chinese Ring Puzzle
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From Kotovsky & Simon, 1990
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Two-Phase Problem Solving: Exploratory & Final Path
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Non-conscious problem-solvingFrom Reber & Kotovsky, 1997
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Strategy acquisition can be unconscious--
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Expertise Hayes on ten year rule
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Expertise: What’s being Learned in the Ten Years? DeGroot and Chase & Simon’s work on
chunking and chess Estimates of knowledge base size Ericsson: idea of deliberate practice
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Practice Makes Perfect! Power law of practice: Ta = cPb + d
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Expertise Physics (Simon et al., 1980)
Physics experts approach physics problems differently than do novices
Chess (Chase & Simon, 1973) Given a mid-game chessboard, grandmasters can
reconstruct it almost perfectly after studying it for only 5 seconds
Novices can only place 3-5 pieces correctly after the same amount of study
However, if the pieces are randomly placed on the board, novices and experts perform at the same level
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Knowledge in Chess Why do experts and novices perform
differently? Experts have more knowledge and
experience But the organization of this knowledge is
crucial Experts can chunk the chess board into
meaningful units that are already in memory Novices have no such chunking mechanism Random placement of pieces eliminates this
chunking from an expert’s performance
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Mental Representations Insight problems
Insight is a seemingly sudden understanding of a problem or strategy that aids in solving the problem
Sometimes require a change in mental representation before the problem can be solved
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Mutilated Checkerboard Place dominoes
on the mutilated checkerboard until it is entirely covered
Taken from Kaplan & Simon, 1990
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Mutilated Checkerboard Subjects had
difficulty solving this problem
Average of 38 minutes
Requires parity to be part of the representation
Taken from Kaplan & Simon, 1990
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Learning in Problem Solving Can knowledge learned on one problem
be transferred to another problem? Sometimes, if people notice a similarity
between the source and target problems How do people map knowledge from a
source problem to a target problem Analogy
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Analogy Classic example (Gick & Holyoak, 1983)
Army problem Cancer problem Mapping between the two leads to a
solution for the cancer problem
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Conclusions Problem solving is an everyday activity We can use findings from problem
solving to further our understanding of the mind and its processes
We can use our knowledge of the mind’s structure and operation to understand elements of problem solving
Different types of problems and different contributions to problem difficulty