M850: Complexity and Computability
Study Board of Science
Teaching language: Danish or English depending on the teacher, but English if international students are enrolled
EKA: N310035102
Assessment: Second examiner: External
Grading: 7-point grading scale
Offered in: Odense
Offered in: Spring
Level: Master
STADS ID (UVA): N310035101
ECTS value: 10
Date of Approval: 29-10-2018
Duration: 1 semester
Version: Archive
Comment
The course is co-read with: DM533
The course cannot be chosen by students who: have passed DM508 or DM553
NEW course Spring 2019.
Entry requirements
Academic preconditions
Students taking the course are expected to:
- Have knowledge of basic datastructures
- Have knowledge of basic algorithms for manipulating (representations of) sets of numbers and graphs as well as implementations of such in an imperative programming language.
- Be able to use basic mathematical argumentation, including
- proof by induction, proof by contradiction and logic expressions
- Be familiar with the use of combinatorial and probabilistic techniques to develop algorithms.
Course introduction
The aim of the course is to enable the student to
- Apply formalisms of formal languages in order to formulate decision problems precisely
- Construct finite automata, regular expressions, push-down automata and context-free grammars as elements in an algorithmic solution of more complicated problems.
- Decide the complexity of new problems based on knowledge of the complexity of important examples of problems from the course.
- Judge whether a given problem may be solved by a computer or is undecidable.
- Argue that problems are NP-complete.
- Judge the possibility to develop an approximation algorithm for a given NP-hard optimization problem.
- Give lower bounds for the complexity of problem that are similar in nature to those studied in the course.
These competencies are important both when one wishes to develop new algorithms for a given problem and when one wants to judge whether a given problem may be possible to solve efficiently (possibly only approximately) by a computer.
The course builds on the knowledge acquired in the courses MM541 Combinatorial mathematics and MM544 Probability Theory.
The course forms the basis for doing a bachelor project as well as elective candidate level courses containing one or more of the following elements: complexity of algorithms, approximation algorithms and computability.
Together with courses as above this course also provides a basis for doing a masters thesis on algorithmic and complexity theoretic subjects.
In relation to the competence profile of the degree it is the explicit focus of the course to:
- Give the competence to analyze complexity of (decision) problems.
- Give knowledge about the computational power of different models of computation.
- Enable the student to construct finite automata and regular expressions for simple languages.
- Enable the student to construct push-down automata and context-free grammars for simple languages.
- Equip the students with important tools to prove that a given language cannot be recognized by a finite automation, a push-down automaton or a Turing machine.
- Enable the student to prove lower bounds for the complexity of algorithms for a given problem.
- Enable the student to develop new approximation algorithms.
- Give the student important tools for proving that a given decision problem is NP-complete or undecidable.
Expected learning outcome
The learning objectives of the course is that the student demonstrates the ability to:
- Judge the complexity of (decision) problems.
- Judge the computational power of various models of computation.
- Construct finite automata and regular expressions for simple languages.
- Construct push-down automata and context-free grammars for simple languages.
- Prove that a given language, which in nature resembles those from the course, cannot be recognized by a finite automaton, a push-down automaton or a Turing machine.
- Prove lower bounds for the complexity of algorithms for a given problem which in nature resembles those from the course.
- Design new approximation algorithms for a given problem which in nature resembles those from the course.
- Prove that a given decision problem which in nature resembles those from the course is NP-complete or undecidable.
Content
The following main topics are contained in the course:
- Finite automata and push-down automata
- Regular languages and context-free languages
- Grammars
- Turing machines
- Decidability
- Problem reductions
- Lower bounds (information theoretical and adversary arguments)
- The complexity classes P and NP
- The theory of NP-completeness
- Approximation algorithms
Literature
Examination regulations
Exam element a)
Timing
June
Tests
Oral exam
EKA
N310035102
Assessment
Second examiner: External
Grading
7-point grading scale
Identification
Student Identification Card
Language
Normally, the same as teaching language
Examination aids
To be announced during the course
ECTS value
10
Additional information
The Exam consists of:
- An assignment to be done independently, which will count towards the final grade.
- Two assignments which can be done in groups, which will count towards the final grade.
In the exam term when the assignments were done, the final grade is given on the basis of an overall assessment of the three assignments and the oral exam. The external examiner will be able to see the assignments.
The re-exam is an oral exam. External examiner, Danish 7 mark scale.
Indicative number of lessons
Teaching Method
At the faculty of science, teaching is organized after the three-phase model ie. intro, training and study phase.
Activities during the stydy phase:
- Self study of the teaching materials
- Solving weekly assignments in order to discuss these at the tutorials
- Written assignments as part of the exam
- Selfguided followup on the intro and tutorial classes
- Repetition for the exam