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Problem Solving and Reasoning

Contents

1. Problem Spaces
1.1 What are problem spaces?
1.2 Initial vs. goal states
1.3 Set of operations
1.4 Well-defined vs. ill-defined
2. Processes of Problem Solving
2.1.a Algorithms
2.1.b Heuristics
2.2 Think-aloud protocols
3. Deductive and Inductive Reasoning
3.1 Deductive reasoning
3.1.a Belief-bias effect
3.1.b Syllogisms
3.2 Inductive reasoning
3.2.a Mental sets
4. Sources


1. Problem Spaces

1.1 A Problem space is the general area containing the problem that is comprised of three parts: An initial state , a goal state, and a set of operations.
1.2 An initial state is incomplete information or unsatisfactory conditions to start with AKA "the problem", while the goal state is the information or state wished to be attained, or the desired fix, solution, outcome. Usually, to get from an initial state to a goal state (problem to solution), you must follow a set of operations.
1.3 A set of operations is simply the steps you take to solving a problem.
1.4 Much of the initial difficulty in solving a problem will arise if any of the elements in the problem space are not well defined. A well-defined problem is simply one in which the goal state, and the operations are all clearly specified, while an ill-defined problem has relatively vague elements (such as a poorly defined initial and goal state and/or an unclear set of operations). Well...Obviously, well-defined problems are most likely the easiest to solve.

2. Processes of Problem Solving

People rely on heuristics or “rules of thumb” and algorithms when they are making judgments because they in a sense apply a certain familiarity to a circumstance when faced with solving a problem. Heuristics are a critical aspect to making judgments/decisions because it takes prior knowledge and experience from past problems and attempts to apply the same strategies to solve a current problem, where as algorithms apply a series of steps to be repeated under certain circumstances to help in completing a problem more efficiently. Decision making is affected by both heuristics and algorithms in that they are applied to solve problems with greater ease.

2.1.a Algorithms

An algorithm is an effective method for solving problems expressed as a finite sequence of steps. Flowcharts are often used to represent algorithms graphically:
external image flowchart.jpg

Rubik's Cubeexternal image cube.jpg

The Rubik's Cube was invented by Erno Rubik, a Hungarian Professor of Architecture and Design. Within only a year of its launch in 1980, it became the fastest-selling puzzle the world has ever seen (with over 250 million cubes sold to date). Most cubes can be solved in only 17 moves with the aid of a computer, and theoretically there is no cube that requires more than 20 twists to solve. The cube can be solved with the application of several steps, each step involving a sequence of twists or algorithms, to move a particular square. For example:


2.1.b Heuristics

Main article: Heuristics
Heuristics serve as "rules of thumb" in that they are experience-based techniques for solving problems. Heuristics are a critical aspect of judgment and decision making

and can sometimes result in cognition-biased phenomena.
Types of heuristics include: availability, representativeness, and anchoring.

Availability Heuristic -
Availability heuristics are in use when one bases their judgment on information that is readily available in memory, for instance a person’s example bias with extensive coverage of unusual events, such as homicides and airline accidents rather than less extreme common illnesses or automobile accidents, can be fueled by watching daily media coverage. When asked to rate the probabilities of various causes of death,

a person using availability heuristics are more likely to rate “newsworthy” events as more likely because they are more readily available in memory.

Representativeness Heuristic -
Representative heuristics are in play when one assumes that if something has the characteristics considered typical of members of a category; it is, in fact, a member of that category. For example, if a person was to see an unshaven man in tattered clothing walking down the street in a large city, said person would probably assume that the man was homeless due to the fact that the person has some of the same characteristics of a homeless person.

Anchoring Heuristic -
An anchoring heuristic is another bias where one judges the probable value of some event or outcome relative to their initial impression. For example, A person waiting in line at an amusement park is told that there will be a forty minute wait. If the wait is shorter, they'll feel as if they made good time. This feeling is brought about by the fact that the person believed the wait to be longer than it was.

2.2 Think-aloud Protocols

A think-aloud protocol is a form of measuring cognitive processes in which the test subject explains their thought process out loud while performing tasks. Since cognition is extremely difficult to measure objectively, a think-aloud protocol is the most common method of measurement, and it's often considered to be the most accurate.

3. Deductive and Inductive Reasoning

Reasoning is the act of using a logical premise to come to a conclusion. In general, the idea of reasoning is split between deductive and inductive reasoning.

3.1 Deductive Reasoning

Main article: Deductive Reasoning
Deductive reasoning is the use of logic to state that a conclusion necessarily follows a set of premises. If X always causes Y and Y necessarily causes Z, then X must cause Z. This type of reasoning must be both valid and sound. Validity is determined by whether or not the premises are necessarily false if the conclusion is false. If Z does not cause X, then X must not always cause Y and/or Y must not always cause Z. A test with false premises can still be valid. Soundness is whether or not the initial premises are true. For example:
  1. All rides are roller coasters
  2. All Ferris Wheels are rides
    Sherlock Holmes uses deductive reasoning in order to solve his mysteries
    Sherlock Holmes uses deductive reasoning in order to solve his mysteries
  3. All Ferris Wheels are roller coasters
The reasoning is perfectly valid, but the initial premise that all rides are roller coasters is false, making the logic unsound.

3.1.a Belief-bias effect

The belief-bias effect is a bias that is caused by a person's own beliefs or ideas influencing
the strength of a logical argument.

3.1.b Syllogisms

Syllogisms are are logical argument in which two prepositions lead to a conclusion. The prepositions usually take the form of major and minor, with the minor supporting the major.

Major premise: All fish live in water
Minor premise: All salmon are fish
Conclusion: All salmon live in water




3.2 Inductive Reasoning

Inductive Reasoning contrasts deductive reasoning in that it doesn't deal in logical certainties, and is simply based on odds. Unlike deductive reasoning, it is not based on the premise leading necessarily to the conclusion. It is similar to an educated guess.
All of the construction workers that I've seen wear hardhats.
All construction workers must wear hardhats.

3.2.a Mental set

A mental set is a preexisting state of mind, habit, or attitude that can enhance the quality and speed of perceiving and problem solving under some conditions, while inhibit or distort the quality of one's mental activities at times when old ways of thinking and acting are non-productive in new situations.

Sources:


Gerrig, R.J., & Zimbardo, P.G. (2005). Psychology and life. Boston, Ma: Pearson.

Special thanks to Antoine Dodson, our biggest hero from the projects.