Computability theory

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
R1
ECTS:
6
Language:
slovenian, english
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

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (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]