1

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

2

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

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.

4

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 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.

5

6

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

7

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)

12

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

13

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

14

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

15

16

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

17

Early Findings Duncker’s candle problem

Maier’s two-string problem 1930

18

19

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?

20

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?

21

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

22

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

23

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

24

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

25

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

26

Tower of Hanoi

Taken from Zang & Norman, 1994

27

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

Problem Space

30

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

31

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

32

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

Problem Space: Subgoaling

34

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

35

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

36

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

Centrality of Representation

Problem space and representation Problem difficulty and representation The interaction of representation and

processing limitations (problem isomorphs)

37

Representation: Example Number scrabble

1 2 3 4 5 6 7 8 9

38

39

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

40

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

41

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

42

Art by AARON

43

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

44

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

45

IsomorphsTaken from Kotovsky, Hayes, & Simon, 1985

46

IsomorphsTaken from Kotovsky, Hayes, & Simon, 1985

Isomorph Difficulty

47

48

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

49

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

50

From Kushmerick & Kotovsky, 1993)

Model-Human Agreement

51

52

Chinese Ring Puzzle

53

From Kotovsky & Simon, 1990

55

Two-Phase Problem Solving: Exploratory & Final Path

Non-conscious problem-solvingFrom Reber & Kotovsky, 1997

Strategy acquisition can be unconscious--

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 Ericsson: idea of deliberate practice

Practice Makes Perfect! Power law of practice: Ta = cPb + d

66

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

67

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

68

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

69

Mutilated Checkerboard Place dominoes

on the mutilated checkerboard until it is entirely covered

Taken from Kaplan & Simon, 1990

70

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

71

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

72

Analogy Classic example (Gick & Holyoak, 1983)

Army problem Cancer problem Mapping between the two leads to a

solution for the cancer problem

73

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