Problem Solving

Outline Well vs. ill-defined problems Heuristics for problem solving –Hill climbing –Means-Ends analysis –Working Backwards representation of problems –Fixedness –Analogical Reasoning In ordinary and scientific reasoning –role of expertise

Well defined vs. ill defined Problems Well defined: Examples: –geometry proofs, –logical puzzles a clearly specified goal (clear criterion on whether the goal has been achieved ) Necessary information is spelled out in the statement of the problem Ill defined Examples: –finding a perfect mate, –writing a great novel not obvious when a goal has been reached, Not obvious which is the relevant information One strategy to solve ill-defined problems is to add constraints (e.g. operationally define the goal),

General Problem-Solving “Problem-solving as search” Each problem has: –an initial state – a goal state: –a set of operators (actions that change the current state into a new state) –a path constraint –a problem space: set of all possible paths

A sample well-defined problem: The Tower of Hanoi Goal: move the tower from the left peg to the rightmost peg, Restrictions: - never placing a larger disk on top of a smaller one - only move one disk at a time.

Problem space: the set of all states that can be achieved during the course of solving a problem.

Heuristics for problem solving Hill climbing strategy: For any particular state, carry out the operation that moves you closest to the final goal state. (often not a good strategy) Means-end analysis: 1. Break down the current difference between initial state and goal into subgoals with sub-differences. 2. Choose the most important difference, then 3. find an operator that will reduce this. Working backwards: 1. Start at the goal state and 2. work backwards via means-end analysis,

Working backwards Heuristic: Example

One (painful) way to solve the water lilies problem Initial number of water lilies = 1 double the initial value 90 times Record each of these values Find the value that is 1/2 of the 90th day value. Working backwards: - value doubling every day is equivalent to say that the value is halved each preceding day - the field was full Day 90th - the field was half full on day 89th

Representations of the Problem Some problems are more easily understood and solved if they are represented in concrete terms (e.g. a mental image), others are more easily solved in abstract terms. Finding the right representation of a problem can be crucial for finding the solution.

Time of day Sunrise3:30Sunset bottom top A visual representation of the monk problem makes it obvious that the monk MUST have occupied the same spot at the same time during the two trips... Position descentascent

Starting in the square marked by the circle, draw a line through all the squares without picking up your pencil, without passing through a square more than once, without diagonal lines and without leaving the checkerboard. Possible or Impossible?

Functional Fixedness: A Problem of Representation People fixate on one potential function of an object (box = container) Fail to consider other functions (box = holder) If box is displayed empty, the second function is highlighted, better performance.

Use these three bottles to pour the perfect amount into the glass (1) (2) (3) (4) fill bottle B, pour into bottle A, then pour into bottle C twice…5 oz Rigidity in use of the same strategy

Analogical reasoning Analogy is a common and powerful form of reasoning. –In ordinary reasoning (‘love is a journey’, ‘war on drugs’) –In scientific reasoning (attentional spotlight, storehouse memory) –In problem solving Analogy is a mapping of knowledge from one domain to another. ‘Base’ domain --> ‘target’ domain (journey -> love) What is being mapped? – Elements of each map (e.g, nucleus of the atom -> sun; electrons -> planets) –Attributes of the elements –Relations among elements: rotation (planet, sun) ; rotation (electron, nucleus) The structural relations are much more important than the surface attributes knowledge from the base domain is then applied to understand the target domain and to generate inferences about it

Analogical reasoning is a 4-step process 1. Access the base. 2. Align base and target (Match Attributes & Relations) 3. Evaluate the match. 4. Make inferences about the target

Analogical Reasoning in problem solving Literal. Collapsing stars spin faster as their size shrinks. This occurs because of a principle called “conservation of angular momentum.” Metaphorical (analogical). Collapsing stars spin faster as their size shrinks. Stars are thus like ice skaters, who pirouette faster as they pull in their arms. Both stars and skaters operate by a principle called “conservation of angular momentum.”

Analogical Reasoning in problem solving: The radiation problem (alone) Very hard to come up with solution Would an analogous problem (of easier solution) help? (Duncker, 1945)

A problem with an analogous solution: Did subjects realize the connection? A general and his troops approached a fortress accessible by many heavily mined roads. If the general’s troops took only one road to the fortress, the entire column of soldiers would be killed, and the attack foiled. However, smaller groups could pass safely over the weight- sensitive mines. The general’s solution was to divide his soldiers into many small platoons and approach the fortress from different directions.

Analogical Reasoning in problem solving Read Attack problem (‘Base’ domain) Next, read Radiation problem (‘Target’ domain) Would the base problem help? –Half the subjects received a hint: “ The solution to the attack problem might be helpful as you work on the radiation problem.” –The other half received no hint Results: people could see the analogy if they were directed to do so, but noticing of this relation spontaneous was rare Gick & Holyoak (1980)

Gick and Holyoak (1983) highlighted the underlying concept of “convergence” by presenting two analogous stories (the additional story involved the cooperation of many small hoses to put out a blaze) subjects tried to solve the tumor problem. Subjects were much more likely to spot the analogy in this situation. Presumably, the repetition of the theme drew subjects’ attention to that aspect of the stories. Why do people sometimes fail to use analogy? - Emphasis on superficial similarities rather than relational similarities - Clustering of problems based on such superficial features

Expertise in Problem Solving Experts tend to notice the crucial aspects of the situation, rather than focusing on superficial features. Task: categorize simple physics problems. Subjects: novices vs. Ph.D. physics students Results: Novices grouped problems based on surface features (having an inclined plane, using a spring), Experts sorted according to the physical principles relevant to the problems. As a result, experts are better able to notice and make use of analogies when a common conceptual structure characterizes a set of problems. Chi, Feltovich and Glaser

Analogical reasoning in science ATTENTION AS SPOTLIGHT Examples "The beam of a spotlight (1) moves from one location to another, (2) moves in analogue fashion..., and (3) is characterized by a specific size." (Umiltà, 1988) “The spotlight... cannot select one or two (or more) objects that fall within the beam, or select different properties of a single object" (Logan, 1995, p. 106). MEMORY AS A STOREHOUSE

ATTENTION AS SPOTLIGHT Inferential structure source domain An agent moves her spotlight, which sheds light on part of the field. When the spotlight sheds light on the target object, the object becomes visible to the agent. target domain. Homunculus controls attention system, which expresses attention over some brain areas. When the attentional system expresses attention on a representation the representation becomes conscious.(can be seen by the homunculus)

Storehouse memory metaphor “information is held in a short-term store with very limited span. From this store it may be passed selectively to be stored for long periods" (Broadbent, 1958) Entailments: Memory is a mental space, where Items (discrete units of information) are stored. There are several stages: - input, - storage,- retrieval Topic of study: How much the subject forgets Formal aspects of memory process Measure: Quantification of memory (i.e., items) Type of questions asked: (Controlled and Generalizable) internal architecture of the store, transfer of units from among departments information loss.

Memory as Perception of the Past “the act of remembering involves the re-perception of internal representations that are created from experiences in the world” (Payne et al., p. 59) Entailments No static snapshots of the past Memories can be imperfect Memory is a reconstructive process Memories are shaped by beliefs and desires Topic of study: What the subject remembers Content of the memories Errors and distortions Measure: Accuracy of memory Type of questions asked: (Ecologically valid) Autobiographical memory; Eyewitness testimony; Memory for faces