What do we have so far?

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

3. 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

5. 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.

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 oreach 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: Prob. Solv’ Set • 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: P.S. Set • 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. Prob. A B C Goal • 1 21 127 3 100 • 2 14 163 25 99 • 3 18 43 10 5 • 4 9 42 6 21 • 5 20 59 4 31 • 6 23 49 3 20 • 7 15 39 3 18 • 8 28 76 3 25 • 9 8 48 4 22 • 10 14 36 8 6

16. Early Findings:Functional Fixedness • 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:Funct. Fixedness • Duncker’s candle problem

18. Maier’s two-string problem 1930

19. Functional Fix’dness: Conclusion • 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 theorems • 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. 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

24. 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

25. Another ex.: Detour Problems • 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

26. Problem Space

28. 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

29. 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

30. Common Heuristics “Weak Methods” • 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

31. Problem Space: Subgoaling

32. Heuristics • Working Backwards from the goal • Works well if there are fewer branches going from the goal to the initial state • Only works if you can reverse the operators • Ex. Lily Pond Problem

33. 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

34. 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

35. Centrality of Representation • Problem space and representation • Problem difficulty and representation • The interaction of representation and processing limitations (problem isomorphs)

36. Representation: Example • Number scrabble • 1 2 3 4 5 6 7 8 9

37. 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

38. 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

39. Limits of AI? • Can AI programs be applied to ill-defined problems? • Engineering Design (ex. A-Design, Campbell, Cagan, Kotovsky) • AARON • Program created by Harold Cohen • Produces paintings using a number of heuristics and general conceptions of aesthtics

40. Art by AARON

41. 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

42. 1 2 3 Memory constraints • Work on isomorphs of the Tower of Hanoi (Kotovsky, Hayes & Simon, 1985) • 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?

43. Isomorphs Hayes & Simon, 1977, Kotovsky, Hayes, & Simon, 1985

44. Isomorphs Hayes & Simon, 1977, Kotovsky, Hayes, & Simon, 1985

45. Isomorph Difficulty (Kotovsky, Hayes & Simon, 1985)

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

47. 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

48. Computational Model (From Kushmerick & Kotovsky, 1993) • 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

49. Model-Human Agreement