Exam on Thursday

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Chinese Ring Puzzle and its Isomorphs (Kotovsky & Simon)

Non-conscious problem-solving

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.

What do we have so far? Basic biology of the nervous system Motivations Senses Learning Perception Memory Thinking and mental representations

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

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.

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)

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

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

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

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

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?

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?

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

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

Problem spaces Initial state Goal state(s) Operators that transform one state into another

Initial

………………….o1

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

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

Tower of Hanoi

Taken from Zang & Norman, 1994

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

ProblemSpace

Taken from Jeffries et al., 1977

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

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

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

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

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

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

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)

Representation: Example Number scrabble

1 2 3 4 5 6 7 8 9

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

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

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

Art by AARON

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

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?

IsomorphsTaken from Kotovsky, Hayes, & Simon, 1985

Isomorph Difficulty

Results of IsomorphsAdapted from Kotovsky, Hayes, & Simon, 1985

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

Model-Human Agreement

Chinese Ring Puzzle and its Isomorphs (Kotovsky & Simon)

Non-conscious problem-solving

Strategy acquisition can be unconscious--

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

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

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

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

Mutilated Checkerboard Place dominoes on the mutilated checkerboard until it is entirely covered

Taken from Kaplan & Simon, 1990

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

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

Analogy Classic example (Gick & Holyoak, 1983) Army problem Cancer problem Mapping between the two leads to a solution for the cancer problem

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?

Exam on Thursday

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