Math 441, Computability and Logic

Fall 1996, taught by Peter Cholak

Comment 1: This course as I taught is very similar to COMP 411
(same book) and in some sense is a really computer science
course.  To make it a math class, one has to do the proofs and
force the students to write proofs for homework and exams and we
cannot cover as much.

Comment 2: The student's background with proving theorems is very
poor. Maybe I should spend more time on background material with
them. Also an issue is their lack of critical thinking skills.  I
tried do improve both of these skills.

Comment 3: I am willing to share my notes, exams, etc.  I also
have copies of everything Anand Pillay used to teach the cousre
before me which I can share.

Book: "Elements of the theory of computation" by Lewis and
Papadimtriou.  I thought that the book was good.  Many of the
problems were challenging.  Other possible books that could be
used are an upcoming book by Kozen (this cover about the same
stuff) and the book by Cutland.  This last book would not cover
automata but focus on Turing machines. Maybe this is more
meaningful for math majors.  However the problems about Turing
machines can be very hard; hence there is not a good supply of
problems.  There is also some computer programs by Barwise and
etal. for understanding these objects.  I have not explored these
options but feel using them may move the course away from doing
proofs.

Syllabus  (most of the headings follow the book)

	1 -- Background (Chapter 1.1 -- 1.7) 
		type of binary of relations
		closures
		finite and infinite sets
		proof techniques (more time next round)

	2 -- Regular Languages (Mostly Chapter 2)
		definitions (1.9)
		deterministic finite automata (FA)
		non-deterministic finite automata
		equivalence of the two
		properties of regular languages
		equivalence of regular languages and languages
			accepted by FA
		Pumping Lemma
		Examples of non-regular languages

	3 -- Context-free Languages (Mostly Chapter 3)
		context-free languages 
		regular and context-free language
		equivalence of pushdown automata and context-free languages 
			(this is a big pain -- here was the one
			place I do not do all the proofs)
		Properties of Context-free Languages
		Deterministic pushdown automata (3.6.1 only)

	4 -- Turing Machines
		Definition and some examples (4.1 and some of 4.2)	
		Universal Turing Machines (5.7)		
		Church Thesis (5.1 not enough examples of
			different possible definitions of
			"algorithm" for them to truly believe it
		The halting set (6.1)

Timing: About 3-5 days on 1, 15 each on 2 and 3 and the rest on 4
(which I sure will prove not to be enough time).  I spent a lot
of class time reworking their homework in the hopes they would
learn to write better proofs and improve their critical thinking
skills.

Grades: One inclass midterm, one takehome midterm and a final.
Homework every week when possible.

Option: One possible option is remove the Turing material and
replacing it will some stuff from control theory which uses FA.
This stuff looks nice but someone would have to write a handout
for the students. This would take some research.