**Class time:** TR 3-4:20 pm

**Instructor:** Vladik
Kreinovich, email vladik@utep.edu, office CCSB 3.0404,

office phone (915) 747-6951.

- The instructor's office hours are Tuesdays and Thursdays 1:30-3 pm, or by appointment.
- Preferable way of contact is by email.
- If you want to contact the instructor during the scheduled office hours, there is no need to schedule an appointment.
- If you cannot contact the
instructor during the instructor's scheduled office hours, please
schedule an appointment in the following way:
- use the instructor's appointments page http://www.cs.utep.edu/vladik/appointments.html to find the time when the instructor is not busy (i.e., when he has no other appointments), and
- send him an email, to vladik@utep.edu, indicating the day and time that you would like to meet.

**Prerequisite:** CS 3350 (Automata)

**Main Objectives:**

- to teach the foundations of computing, and
- to enable students to prove (not only to program).

**Contents**

*Turing's snakelike machine:*not very fancy, but it can compute anything (you just wait and wait and wait, ...). Finally: something purely theoretical (and not real machines):*recursive functions*. Church's bold statement: if anyone can compute anything on any machine, I can compute it on a Turing snake! Universal Turing machine. Can anyone really beat Church? We'll discuss the attempts (Gandi, Kreisel, etc) if time allows.- You are accustomed to the fact that everything is computable, and if your program does not work, that means a bad grade. Finally! Only in this course! Computational problems that cannot be solved! (and so you get a bad grade, if your program solves them - just kidding).
- If a program requires a billion years to finish its computations, only a crazy theoretician can call it an algorithm. So, to sound more reasonable, we will talk about computational complexity, realistic (polynomial-time) computations, P and NP, NC, limitations on space and on the number of processors, etc. "P=NP?" as a challenge to mankind. Will science ever stop? Again, we will find here lots of undecidability results. And maybe, as a project, you will be able to prove that some problem that you were planning to solve is undecidable.
- What to do if your problem turned out to be undecidable? For sure: not to give up. It can be still decidable in some sense: for almost all cases, by a Monte-Carlo algorithm that gives an answer with probability close to 1, etc. Few results and lots of open problems.
- Turing machine was invented in the 30s, P=NP problem appeared
when many of you guys were too young to count. What is
the modern state of the Theory of Computation? We'll try to cover
briefly:
- parallelism,
- algebraic computations,
- computations with real data (including the idea of interval mathematics),
- fuzzy algorithms,
- neural computing,
- chemical computing,
- quantum computations, etc.

**Learning Outcomes**

1. **Knowledge and Comprehension**

a. Describe the practical need for theory of computing: to know which computational problems are solvable and which are not, and which problems can be, in principle, solved within given computation time, to avoid wasting resources on trying to solve problems in such a general context that they become unsolvable

b. Describe different models of computing, including recursive functions, different versions of Turing machine, etc.

c. Describe how computability in a programming language is related to formal models of computing, on the example of primitive recursive and recursive functions

d. Understand Church-Turing thesis and understand the status of this thesis - that it is, in effect, a statement about the physical world

e. Define decidable and recursively enumerable (r.e., semi-decidable) sets

f. Understand the notion of an oracle and of computing relative to an oracle

g. Define classes P, NP, the notions of polynomial time reduction, NP-hardness, and NP-completeness; understand the motivations behind these definitions: P means feasible, NP stands for a problem; understand the difference between the formal notion of polynomial time and the practical notion of feasibility

i. Understand the difference between a proof and a sequence of reasonable arguments which does not constitute a proof

j. Know several NP-hard problems

k. Define complexity classes from the absolute and relative polynomial hierarchy; give examples of problems

l. Understand the main idea of parallelization

m. Define the class NC of parallelizable P problems, know the relation between NC and P, and an example of a P-complete problem

n. Understand the difference between the formal computation time of a parallel program and the actual time which takes communication time into account

o. Define Kolmogorov complexity, understand motivations behind this definition

p. Be aware of the main ideas behind different physical schemes of computing beyond traditional Turing machines, such as quantum computing

2. **Application and Analysis**

a. Trace the computation of a primitive recursive or recursive function on a numerical example

b. Translate a formal description of a primitive recursive or recursive function into a program

c. Trace a Turing machine on a given input

d. Prove that satisfiability is NP-hard, and that one more problem is NP-hard

e. Know how to parallelize standard simple parallelizable algorithms (e.g., search, or dot product of two vectors)

f. Use the definition of Kolmogorov complexity K(x) to provide reasonable upper bounds for K(x) of a given string x

3. **Synthesis and Evaluation**

a. Prove, from the definition, that a given function (e.g., a given polynomial or propositional function) is primitive recursive or recursive

b. Design a Turing machine that computes a given function

c. Synthesize Turing machines that compute two functions into a Turing machine for computing their composition

d. Prove that not every problem is computable - e.g., that the halting problem is undecidable

e. Apply diagonalization to prove results similar to what we had in class

f. Prove that given simple sets are decidable and/or r.e.

g. Prove that the union, intersection, and complement of decidable sets are decidable

h. Prove that the union and intersection of r.e. sets is r.e.; prove that the complement to a r.e. set is not always r.e.

i. Prove that not every r.e. set is decidable and that not every set is r.e.

j. Prove, for a given software requirement, it is algorithmically impossible to check whether a given program satisfies this requirement

k. For problems similar to ones considered in class, prove their NP-hardness

l. Be able to parallelize simple algorithms

m. Be able to understand and present a research paper in Theory of Computing - with a minor help from a professor

**Main Source:** Michael Sipser, Introduction to the Theory of
Computation, PWS Publishing Co., 2nd or later edition

**Projects:** An important part of the class is a project.
There are three possible types of projects:

- An ideal class project is if you do something related to theory which is useful for your future thesis or dissertation. Please check with your advisor about it, maybe he or she wants you to read and report on some theory-related paper, maybe you need to do some theory-related research, whatever your advisor recommends will be a very good project for this class, just let the class instructor know what exactly you plan to do.
- If you have not yet selected an advisor, but you already know what research area you want to work in, come talks to the class instructor, we will try to find some appropriate topic -- and if you have any proposals already, great.
- If you do not have a research topic or you have a one but your advisor cannot find anything theory-related that will be helpful for your future thesis or dissertation, come talk to the instructor too.
- Maybe you like theory and want to start doing a project in theory, then come and talk to the instructor, we will try to find something that will be of interest to you.

- reviewing and reporting on a related paper, or
- doing some independent research (not research as in high school, but research as in graduate school, i.e., trying to come up with something new), or
- programming something theory-related.

**Assignments:** Reading and homework assignments will be
announced on the class website. You should expect to spend at least
10 hours/week outside of class on reading and homework.

**Homework Assignments:** Each topic means home assignments.
Howeworks will be due by the start of the next class:

- on Thursday if this homework was assigned on Tuesday, and
- on Tuesday the following week if this homework was assigned on Thursday.

One week after the homework was assigned, I will post correct solutions. I will be glad to answer questions if needed.

If you have a legitimate reason to be late, let me know, you can then submit it until the homeworks are posted. If you were simply late, you can still submit until the homeworks are posted, but then points will be taken off points for submitting late.

Since I will be posting correct solutions to homeworks, it does not make any sense to accept very late assignments: once an assignment is posted, it make no sense for you to copy it in your own handwriting, this does not indicate any understanding. So, please try to submit your assignments on time.

Things happen. If there is an emergency situation and you cannot submit it on time, let me know, you will then not be penalized -- and I will come up with a similar but different assignment that you can submit to me when you become available again.

Homework must be done individually. While you may discuss the problem in general terms with other people, your answers and your code should be written and tested by you alone. If you need help, consult the instructor.

**Exams:** There will be three tests: on February 24, March 24,
and April 28, and the final exam on May 12, 4-6:45 pm.

Similar to homeworks, I will post solution, send you the grades, and answer questions if something is not clear.

As usual, if you are unable to attend the test, let me know, I will organize a different version of the text at a time convenient for you.

**Grades:**
Each topic means home assignments (mainly on the sheets of paper,
but some on the real computer). Some of them may be graded.
Maximum number of points:

- first test: 10
- second test: 10
- third test: 15
- home assignments: 10
- final exam: 35
- project: 20

A good project can help but it cannot completely cover possible deficiencies of knowledge as shown on the test and on the homeworks. In general, up to 80 points come from tests and home assignments. So:

- to get an A, you must gain, on all the tests and home assignments, at least 90% of the possible amount of points (i.e., at least 72), and also at least 90 points overall;
- to get a B, you must gain, on all the tests and home assignments, at least 80% of the possible amount of points (i.e., at least 64), and also at least 80 points overall;
- to get a C, you must gain, on all the tests and home assignments, at least 70% of the possible amount of points (i.e., at least 56), and also at least 70 points overall.

**Special Accommodations:** If you have a disability and need
classroom accommodations, please contact the Center for
Accommodations and Support Services (CASS) at 747-5148 or by email
to cass@utep.edu, or visit their
office located in UTEP Union East, Room 106. For additional
information, please visit the CASS website at http://www.sa.utep.edu/cass.
CASS's staff are the only individuals who can validate and if need
be, authorize accommodations for students.

**Scholastic Dishonesty:** Any student who commits an act of
scholastic dishonesty is subject to discipline. Scholastic
dishonesty includes, but not limited to cheating, plagiarism,
collusion, submission for credit of any work or materials that are
attributable to another person.

*Cheating* is:

- copying from the test paper of another student;
- communicating with another student during a test to be taken individually;
- giving or seeking aid from another student during a test to be taken individually;
- possession and/or use of unauthorized materials during tests (i.e. crib notes, class notes, books, etc.);
- substituting for another person to take a test;
- falsifying research data, reports, academic work offered for credit.

- using someone's work in your assignments without the proper citations;
- submitting the same paper or assignment from a different course, without direct permission of instructors.

*Collusion* is unauthorized collaboration with another person
in preparing academic assignments.

Instructors are required to -- and will -- report academic dishonesty and any other violation of the Standards of Conduct to the Dean of Students.

NOTE: When in doubt on any of the above, please contact your instructor to check if you are following authorized procedure.

**Daily Schedule:** (tentative and subject to change)

*January 18:* topics to cover:

- what is Theory of Computation and why it is important, see lecture; see also the following file
- Church-Turing Thesis, see lecture
- how can we use Church-Turing Thesis to prove results?; see lecture.

*January 20 and 25:* topics to cover:

- primitive recursive functions as description of for-loops, see lecture; see also primitive recursive parts of of file 1 and file 2, and notes

*January 27:* topics to cover:

- many functions and constructions are primitive recursive, see lecture

*February 1:* topics to cover:

- proving that not all computable functions are primitive recursive, see lecture

*February 3:* topics to cover:

- mu-recursive functions as description of while-loops, see lecture; see mu-recursive parts of file 1 and file 2

*February 8:* topics to cover:

- Turing machines, see lecture

*February 10:* topics to cover:

- equivalence of Turing machines and mu-recursive functions, see lecture.

*February 15:* topics to cover:

- how to simulate a Turing machine; see lecture;
- how to represent a Turing machine as a finite automaton with two stacks; see paper.

*February 17:* topics to cover:

- halting problem, see lecture
- impossibility to check whether a program -- that always halts -- is correct, see lecture

*February 22:* topics to cover:

- decidable and enumerable sets, see lecture
- overview for Test 1

* March 1:* overview of Test 1 results.

*March 3:* topics to cover:

- feasible and non-feasible algorithms; intuitive idea, precise definition, and why this definition is not perfect,
- P, NP, notion of reduction, NP-hard

* March 8:* topics to cover:

* March 10:* topics to cover -- other examples of NP-complete
problems:

* March 22:* topics to cover:

- interval computations are NP-hard, see lecture
- overview for Test 2.

*March 24:* Test 2.

*March 29:* overview of Test 2 results.

* March 31:* topics to cover:

*April 5:* topics to cover:

- probabilistic algorithms; see lecture,
- greedy algorithms; see lecture,
- a natural feasible algorithm that checks satisfiability of 2-CNF formulas, see lecture

*April 7:* topics to cover:

- quantum computing, see lecture

* April 12:* topics to cover:

- polynomial hierarchy; see lecture
- unconventional computation schemes; see lecture 1, lecture 2, lecture 3, and lecture 4

* April 14:* topics to cover:

- Kolmogorov complexity; see lecture

* April 19:* project presentations

* April 21:* project presentations

*April 26:* preview for Test 3

*April 28:* Test 3.

*May 3:* overview of Test 3 results.

*May 5:* preview for final exam.