comp3 — Computability and Computational Complexity
Academic year 2024-2025,
CS Master degree at the University of Trento, Italy.

Basic information

Important news

[2024-10-14] An updated version of the lecture notes, with changes to the solution of Exercise 36 as discussed in today's lesson, is available in the materials section.

[2024-10-11] The dates of the winter session are out, see the exam section for details.

Schedule

Next edition: Fall 2024, four hours per week.

  • Mondays: 10:30—12:30, room A107.
  • Tuesdays: 13:30—15:30, room A103.

Instructor

Mauro Brunato < mauro.brunato@unitn.it>

Please email me if you want to set up a meeting.

In order to receive urgent updates (e.g., about moved or cancelled lectures), please subscribe to the course's Moodle page.

Course syllabus

Objectives

The course aims at providing the theoretical bases for understanding two fundamental properties of computational problems.

  1. Given a problem, at first we will be concerned about its computability:
    • Is there a method to solve the problem?
    • Can we describe this method in an unequivocal way?
    • Is there a universal "language" for such task?
    • Are there any intrinsically unsolvable problems?
  2. When a problem is found out to be solvable, we can wonder how efficiently it can be done.
    • What are the meaningful criteria to express how hard a problem is?
    • Is there a way to classify problems with respect to their hardness?
    • Does this classification tell us anything about the limits of our capability to solve problems?

Entrance requirements

Students should have a basic understanding of the following topics:

  • Mathematical logic: logical connectives, quantifiers, basic propositional calculus; being able to follow the proof of simple theorems.
  • Mathematics: polynomials, logarithms, exponentials; positional representation of integers in different bases.
  • Programming, algorithms and data structures: knowledge of a specific programming language is not required, but being able to describe algorithms and data structures is essential.

Program outline

For the actual program, see below.

  • Computability
    • Turing machines.
    • Decidable languages and computable functions.
    • The Halting Problem.
    • Other undecidable languages and uncomputable functions.
    • Rice's Theorem.
  • Computational Complexity
    • Problem instance size, measures of complexity, classification of problems.
    • Time complexity classes: P, NP, EXP. NP-completeness. Cook-Levin Theorem. Study of NP-complete problems.
    • Space complexity classes: LOGSPACE, NLOGSPACE, PSPACE, NPSPACE. Savitch's theorem.
    • Polynomial hierarchy and canonical complete problems.
    • Advanced topics, depending on available time (randomized classes, counting classes, interactive proofs, approximations).

Expected learning outcomes

At the end of the course, students will be able to:

  • Describe problems and problem instances in a formally precise way.
  • Describe the solution of simple problems in terms of a universal computational model (the Turing Machine).
  • Prove that a specific problem (Turing's halting problem) is not computable, and recognize the importance of this result.
  • Apply the formal notion of reduction in order to extend properties of known problems to new ones.
  • Identify quantitative criteria to describe the complexity of an algorithm with respect to instance size.
  • Describe the main time and space complexity classes, and their inclusion relationships.
  • Define what it means for a problem to be complete for a class.
  • Prove that a specific problem (SATISFIABILITY) is complete in the NP class (Cook-Levin Theorem).
  • Discuss the P vs. NP problem and its main implications on various branches of computer science.
  • Analyze how extensions of the basic computing model (randomness, quantum computing) can benefit the solution of some problem classes.

Teaching and learning methods and activities

The course will mainly consist of lectures and discussions with students.

As soon as specific milestones are reached, exercises and papers will be proposed for self-assessment and further discussion.

Since the beginning of the course, the previous year's material will be made available on this website (in the materials section below):

  • Lecture notes
  • Example code
  • Solved exercises
  • Exam sheets from previous years
  • Relevant papers and webpages.
The material will be updated during the semester.

Textbooks

The following books may be useful, but none is mandatory. References to online material will be provided in the course program section.

Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. ISBN 9780521424264 [Draft of the book]

Christos H. Papadimitriou. Computational complexity. Addison Wesley (Pearson College Div.), 1994. ISBN 9780020153085

John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages and Computation. Pearson new International Edition, 2013. ISBN: 9781292039053

Test and assessment criteria

The exam is written, pen and paper, in English, and is composed by three open-question exercises requiring to expose theoretical topics and analyze problems and algorithms. One of the exercises is about computability, the other two are mainly about the computational complexity topic. The three exercises have an equal weight in determining the final mark.

Answers are evaluated on the basis of the following criteria:

  • Correctness: the answers must follow a logical path, consistent with the course topics and with the question; however, given that the discipline is still young, personal opinions are welcome, as long as they are motivated.
  • Relevance: going off-topic in the hope to collect some points doesn't usually work.
  • Motivation: the answers provided by the student must enable the examiner to follow the logical path leading to their formulation; answers without explanations are not taken into account.
  • Clarity: short and precise answers, with the correct use of technical terminology and without ambiguities are preferred.
Students have free access to the exam sheets of the previous sessions on the course website.

Quick stats

In the winter and summer exam sessions of 2024, 61 registrants out of 66 have passed the exam (38 out of 42 in the first exam in January).

Course materials

This section contains course material and exercises.

Lecture notes

  • 2024-10-16 version: please see the included changelog for details about revisions (only a few typos revised since version 2024-10-14).

Lecture videos

Example code

External links

(See the lecture notes for details)

Past exams

Exercises from previous exams that are compatible with this year's program appear in the lecture notes. Here are links to the exam sheets as they were proposed during the academic years 2018-2019, 2019-2020 and 2023-2024 (the three missing years were taught by another instructor):

Exams

  • See the course syllabus above for a comprehensive description of the exam and of the assessment criteria.
  • All tests from the previous years are available in the materials section above.
  • The next exam session will take place on Winter 2025 on the following dates (still subject to change due to possible conflicts with other courses):
    Date/timeRoom
    First callThursday, January 9th, 2025 09:00a,—12:00pmB107 (conference room in Povo2)
    Second callMonday, February 17th, 2025 09:00a,—12:00pmB107 (conference room in Povo2)

Detailed program and lecture videos

This Section will be updated during the year.

2024-09-09

  • Basic definitions: alphabet, string, language. Enumerating strings.
  • Enumeration of countable sets, diagonal proofs of uncountability.

2024-09-10

  • Examples of potentially infinite computations based on some unsolved mathematical conjectures (Collatz).
  • Intuitive definitions of recursive and recursively enumerable sets. Diagonal procedure to enumerate R.E. sets.
  • Turing machines: intuition, introduction.

2024-09-16

  • Turing machines: examples. Using the states to “store” finite amounts of information (e.g., bits).
  • Multiple-tape Turing Machines.
Note: the recording was interrupted at 28:15 and resumed a few minutes later; the missing part is the transcription of the state diagram of the Turing machine on the blackboard into the simulator; the completed transcription can be seen in the upper left picture after the cut.

2024-09-17

  • Computational power of Turing machines: multiple tapes, alphabet size. Reduction to 1-tape, 2-symbol machines with quadratic time loss.

2024-09-23

  • Universal Turing machines: string encodings of TMs, simulation of TMs.
  • Proof of uncomputability by diagonalization.

2024-09-24

  • Uncomputability of the halting problem.
  • The Church-Turing thesis.
  • Recursive and recursively enumerable languages: Recursive enumerability of HALT.
Note: an intermediate 20' segment was lost because the computer entered sleep mode. Likewise, the final 30' are lost. The following is an approximate timeline of the lecture with pointers to the lecture notes for the missing parts.
  • 00'00"—19'50" from the video, first recorded segment: summary of the previous lecture; why is UC uncomputable?
  • 20' missing footage: in order to compute UC, we need to know if the computation of Ms(s) halts, so we would like to have a TM that tells us if a generic computation M(s) halts. Let us call this machine H (text before Theorem 4 in the notes, page 14).
  • 20'00"—40'00" from the video, second recorded segment: proof by contradiction that machine H, computing function HALT, cannot exist (Theorem 4 in the notes, pages 14–15).
  • 30' missing footage: The Church-Turing Thesis (section 1.2.4 in the notes, page 13); although HALT is not recursive, it is recursively enumerable (Section 1.2.6 of the notes, pages 15–16).

2024-09-30

  • Recursive and recursively enumerable languages, clarifications about the equivalence of different definitions.
  • Turing reductions (introduction).

2024-10-01

  • Co-recursively enumerable languages, relationship between R, RE and coRE.
  • Variations of the halting problem: machines on empty inputs.
  • The Busy Beaver game.

2024-10-07

  • The Busy Beaver game (continued).
  • Uncomputable busy beaver numbers: links to propositional logic via the unprovable consistency of the Zermelo-Fraenkel axiomatic set theory.

2024-10-08

  • Properties of TMs as decision functions or sets.
  • Trivial properties; semantic properties.
  • Statement and proof of Rice's Theorem

2024-10-14

  • Discussion of Exercise 36 from the lecture notes (recursive and RE sets, recursivity of TM properties).
Note: the initial 20' were not recorded, however the initial part of the exercise is quite straightforward.

2024-10-15

  • Turing completeness.
  • An example of Turing-complete problem: Post's Correspondence Problem.