Exam on Thursday

1. Exam on Thursday • Review session • Tuesday, 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 modified this morning

2. Problem Solving

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

4. 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

5. 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

6. General Questions • Why study problem solving? • Studying a task will help us understand the system performing the task • Its something that everyone engages in daily • What can our findings about the mind tell us about problem solving? • Mental representations • Memory constraints • Perception

7. 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

8. 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

9. 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

10. 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

11. 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?

12. 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?

13. 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

14. 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

15. Initial Goal o1 …………………. Initial o2 Goal Initial General Theory of Problem Solving • Problem spaces • Initial state • Goal state(s) • Operators that transform one state into another

16. 1 2 3 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

17. 1 2 3 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

18. Tower of Hanoi Taken from Zang & Norman, 1994

19. 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

20. ProblemSpace Taken from Jeffries et al., 1977

21. 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

22. 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

23. 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

24. 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

25. 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

26. Simple Example • Difference-operator table Operators Differences

27. 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

28. 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

29. 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

30. 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

31. Art by AARON

32. 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

33. 1 2 3 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?

34. Isomorphs Taken from Kotovsky, Hayes, & Simon, 1985

35. Isomorphs Taken from Kotovsky, Hayes, & Simon, 1985

36. Results of Isomorphs Adapted from Kotovsky, Hayes, & Simon, 1985

37. 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

38. 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

39. 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

40. 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

41. 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

42. Mutilated Checkerboard • Place dominoes on the mutilated checkerboard until it is entirely covered Taken from Kaplan & Simon, 1990

43. 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

44. 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

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

46. 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?