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Computability theory

2019/2020
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
R1
ECTS:
6
Language:
slovenian, english
Course director:

Prof. Dr. Andrej Bauer, Prof. Dr. Marko Petkovšek

Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Turing machines and computable functions. Universal machine. Undecidable problems and non-computable functions.
Basic theorems and notions: s-m-n and u-t-m theorems, recursion theorem, computable and computably enumerable sets and their properties, non-separable sets, Rice's theorem, Rice-Shapiro theorem.
Oracle computations, Turing reducibility and degrees.
If time permits: computable functionals, continuity of functionals, KLS theorem, computable real numbers, basic results in computable analysis.

Readings

J. E. Hopcroft, J. D. Ullman: Uvod v teorijo avtomatov, jezikov in izračunov, FER, Ljubljana, 1990.
P. Odifreddi: Classical Recursion Theory, North-Holland, 1989.

Objectives and competences

Knowledge of basic notions and results in computability theory.

Intended learning outcomes

Knowledge and understanding:
Understanding of the connections between computability notions, such as Turing machines, and basic mathematical notions, such as sets of numbers.
Application:
The subject matter provides a general theoretical foundation for computer science.
Reflection:
The influence of the notion of computability on foundations of mathematics.
Transferable skills:
Analytic and abstract thinking about the theoretical frontiers of computer science.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Andrej Bauer:
AWODEY, Steve, BAUER, Andrej. Propositions as [Types]. Journal of logic and computation, ISSN 0955-792X, 2004, vol. 14, no. 4, str. 447-471. [COBISS-SI-ID 13374809]
BAUER, Andrej. First steps in synthetic computability theory. V: Proceedings of the 21st Annual Conference on Mathematical Foundations of Programming Semantics (MFPS XXI), (Electronic notes in theoretical computer science, ISSN 1571-0661, Vol. 155). Amsterdam: Elsevier, 2006, str. 5-31. [COBISS-SI-ID 14631001]
BAUER, Andrej. A ralationship between equilogical spaces and Type Two Effectivity. Mathematical logic quarterly, ISSN 0942-5616, 2002, vol. 48, suppl. 1, str. 1-15. [COBISS-SI-ID 12033369]
Marko Petkovšek:
PETKOVŠEK, Marko. Ambiguous numbers are dense. American mathematical monthly, ISSN 0002-9890, 1990, let. 97, št. 5, str. 408-411. [COBISS-SI-ID 8040537]
PETKOVŠEK, Marko, WILF, Herbert S., ZEILBERGER, Doron. A=B. Wellesley (Massachusetts): A. K. Peters, cop. 1996. VII, 212 str. ISBN 1-56881-063-6. [COBISS-SI-ID 4085337] xii + 212 str. (ISBN 1-56881-063-6).
PETKOVŠEK, Marko. Letter graphs and well-quasi-order by induced subgraphs. Discrete Mathematics, ISSN 0012-365X. [Print ed.], 2002, vol. 244, no. 1-3, str. 375-388. [COBISS-SI-ID 11414873]