Computability Theory: Syllabus

Class will be lecture and discussion. There will be two short tests, each 30-45 minutes long, and a term paper instead of a final exam. Homework will be a major component of your grade; it will be assigned ahead of class and is due at the end of each Monday - as long as it is under my door when I come in Tuesday morning it is on time (or you may turn it in Monday in class). At the foot of the page is a section on the honor code.

• Mini-tests (2): 50 points each, in class
• Homework: 100 points total
• Term paper: 50 points (outline 10, first draft 15, final version 25)

Schedule:
(to be updated with topics as we go: semicolons divide each days' sections)
------- Week 1 (Mar 28-Apr 1) -------
introduction, formalizations of computability, coding sets
1.2, 3.2; middle of 2.4, 3.3, middle of 3.1, 3.5; third part of 3.1, 3.4

------- Week 2 (Apr 4-8) -------
basic tools of computability, unsolvability
loose ends of 3.4 and 3.5, 4.1, 4.2; 4.3, 4.4; 4.4, 4.5

------- Week 3 (Apr 11-15) -------
chapter 4 recap, computable and c.e. sets, noncomputable sets
5.1-2; 5.2-3; 5.4

------- Week 4 (Apr 18-22) -------
first mini-test, Friday Apr 22 (accommodations possible if you will be fasting that day)     Test Topics
Turing reduction (6.1); review; test

------- Week 5 (Apr 25-29) -------
Wednesday office hours canceled!
finite injury priority arguments, Turing degrees, relativization and the Turing jump, the arithmetic hierarchy
6.2; 6.2, 7.1; 7.2

------- Week 6 (May 2-6) -------
More on the arithmetic hierarchy, discussing term paper topics, the limit lemma     Chapter 7 supplemental notes

------- Week 7 (May 9-13) -------
Tuesday May 10 office hours canceled
topic and outline of term paper due Wednesday May 11, to be returned May 13 -- update: May 13 also okay; to be returned May 16    Topic Ideas    Sample Outline
Using the limit lemma to build a noncomputable low set, musings on priority arguments, Chp 5-8 recap, algorithmic randomness

------- Week 8 (May 16-20) -------
second mini-test, Friday May 20     Test Topics
Randomness, possibly other Chapter 9 material, review, test

------- Week 9 (May 23-27) -------
presentations begin midweek; first version of term paper due Wednesday May 25, to be returned May 27
Chapter 9 material, presentations

------- "Week" 10 (June 1) -------
presentations conclude; final version of term paper due

Disabilities: I encourage anyone with disabilities, including "invisible" disabilities, to discuss appropriate accommodations with me. All such conversations will be confidential, except that the Academic Skills Center may be consulted to verify documentation and advise me on appropriate accommodations. Please talk to me well in advance of your need!

More on homework: Please give a summary of the problem you are solving before giving your solution, and list each problem by its full chapter.section.number identity. Homework will include some short proofs; I do not expect you to know how to write proofs at the start of class, so leave plenty of space for my comments! Homework page is here.

Grading, per problem or part, will be on a scale of 0 to 5:
5: The problem and the solution are explained thoroughly, clearly, and completely. It is easy to read and understand, is correct, and contains nothing extraneous. It shows a deep understanding of the problem, includes all necessary details, and addresses any subtleties. Scores of 5 will be assigned very selectively, and indeed may not be possible on all problems.
4: The solution is correct and written up clearly and completely, with at most minor omissions, tangents, or errors.
3: Either the solution is correct but unclearly written, or it is clear but only partially correct, with some significant error.
2: The solution is only partially correct, with some significant error, and the explanation is also unclear, incomplete, or includes nonsense, irrelevancies, or egregiously misused terminology or notation. A 2 may also be given for a good explanation of a completely incorrect solution, or a strong attempt at a solution which did not get anywhere.
1: It is apparent some reasonable attempt was made to solve the problem.
0: No attempt was made or what was written is so incoherent that it fails to communicate the fact that an attempt was made.
You may interpret the numbers roughly as letter grades, where 4 is an A, 3 a B, etc. 4 is considered the maximum for letter grade purposes. Note I will make no effort to ensure the numbers actually add up to 100; your overall score will be scaled.

Term papers: Every term paper will include mathematical exposition, but the paper may be aimed at a variety of topics:

• Pure computability theory
• Computability-theoretic results about another area of math
• Mathematical theory behind a programming language
• Result of putting computer science-inspired restrictions on mathematical computability
• A computability-theoretic result in its historical context (its effect on continuing research, for example)
• A computability-theoretic result and its philosophical ramifications

The term papers will have three drafts: the first is just the topic selection and a detailed outline. There is no limit on how detailed it can be, so if you want to put a lot of work in on this step to save work later I will give you feedback at the level of detail you submit. At minimum it should be clear you know what you are looking for and have a full term paper outlined. The second draft should be nearly complete: the sort of thing you would hand in as the final draft if you were pressed for time. It may lack polish but should have no content gaps and should, in particular, include a complete bibliography. The papers should be in the vicinity of 8 pages, though the content often dictates length: if it is a full paper at six and a half pages and lengthening it would be artificial, then make it six and a half pages. If it really requires nine, and not because you have redundancy or excessively long explanations, then it should be nine. One of the purposes of the first draft being complete is to address these concerns.

The Honor Principle: For the mini-exams, keep your eyes on your own paper and don't use crib sheets. That is clear. For the homework, please feel free to discuss problems with each other, but write up your solutions individually and state with whom you collaborated on the write-up. If you use outside sources, cite them as well; it is a breach of the honor code to read another student's solutions in order to write your own. For the term paper, cite all sources. The writing must be your own. It can be difficult to find ways to set out proofs different from the way they are in your source, so try to put proofs in your own words as much as possible, and cite the source right at the proof in the paper.

Please ask me if you are uncertain whether some collaboration or source use is acceptable under the honor code!

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