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Chinese Ring Puzzle and its Isomorphs (Kotovsky & Simon)
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Non-conscious problem-solving
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Announcements Midterm grades Quiz tomorrow in recitation (thru today’s lec.)
Exam Thursday Oct. 30 Review session
Tuesday, Oct. 28, 7pm, DH 2210 A very brief outline of the material will be shown and you will be given time to ask questions about the material
Review notes sample exam up soon
Problem Solving
Definition of a problem 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 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 General architecture of mind
What kinds of tasks does our mind engage in? Language Problem Solving Decision Making Others
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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
A 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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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
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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 Functional Fixedness
Inability to realize that something known to have a particular use may also be used for performing 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 nail clippers 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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ProblemSpace
Taken from Jeffries et al., 1977
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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?
Fractionation and Subgoaling Break the problem into a series or hierarchy of smaller problems
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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
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Simple Example Difference-operator table
Subtract number from both sides
Add number to both sides
Multiply both sides by number
Divide both sides by number
Extra number added on one side
Extra number subtracted on both sides
Extra constant multiplier for x, neither of the first two differences
Extra constant divisor for x, neither of the first two differences
Operators
Differences
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First AI programs Newell & Simon
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
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
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
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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Model-Human Agreement
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Chinese Ring Puzzle and its Isomorphs (Kotovsky & Simon)
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Non-conscious problem-solving
Strategy acquisition can be unconscious--
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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
Expertise Hayes on ten year rule
Expertise: What’s being Learned in the Ten Years? DeGroot and Chase & Simon’s work on chunking and chess
Estimates of knowledge base size
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 What are some methods of problem solving? Why are some problems harder than others?
