M7970 Approximation in the mathematical foundations of computer science

Faculty of Science
Autumn 2005
Extent and Intensity
0/0. 2 credit(s). Recommended Type of Completion: k (colloquium). Other types of completion: zk (examination), z (credit).
Teacher(s)
Profesor Achim Jung (lecturer), prof. RNDr. Jan Paseka, CSc. (deputy)
Guaranteed by
prof. RNDr. Jan Paseka, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
PREREQUISITES: I hope that this course will be attractive to both mathematicians (interested in applications to computer science) and computer scientists (interested in the mathematical foundations of their subject). Mathematicians should have some background in topology, order theory, category theory, or logic; computer scientists should have some experience of denotational semantics. I hope to be able to fill in the missing bits for both types of participants, and are very willing to have the course move in whichever direction the audience prefers.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
Mathematics often deals with infinite objects without much consideration whether such objects have a representation in a computer. For example, real numbers in general require infinitely many digits to write down, and functions on either the reals or the natural numbers may require an infinite amount of information to specify. Over the last 35 years, computer scientists have revisited many such mathematical structures and examined the ways in which they might be represented in a computer. Central to this enterprise is the notion of "approximation" where the ideal infinite object is seen as the limit of a sequence of finite parts. The course will explore this topic from various angles, and will aim to present the results of some very recent research.
Syllabus
  • I. Domain Theory: Scott's "domains" as a technique to model recursion and iteration, and as a way to solve recursive domain equations; continuous domains as a way to deal with real numbers and probability. II. Exact Real Number Computation: Representing real numbers as streams of generalised digits; programming languages for exact real number computation; parallelism and sequentiality. III. Domain Theory in Logical Form: Stone duality as a link between semantics and program logics; Stone type dualities for domains; partial predicates and Stone dualities for continuous domains.
Assessment methods (in Czech)
The lectures will be given in a week from 27.6. - 1.7.2005.
Language of instruction
English
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
The course is taught only once.
The course is taught: in blocks.
Credit evaluation note: 2 kr. zápočet, 3 kr. kolokvium, 4 kr. zkouška.
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