BazeleMatematicaaleCalculabilitatiiMM13_2305

COURSE PROGRAMME

* OB – Obligatory / OP – Optionally / F – Facultative

3. Total Hours (estimated per semester and activities)

3.1 Number of hours per week 4 3.2 course 2 3.3. seminary/laboratory 2

3.4 Total number of hours 56 3.5 course 28 3.6. seminary/laboratory 28

Distribution hours Individual study using textbooks, course notes, bibliography items, etc. 120

Supplimentary study (library, on-line platforms, etc.) 10

Individual study for seminary/laboratory, homeworks, projects, etc. 20

Tutoring

Examination 4

Other activities...................................

3.7 Total hours of individual activity 180

3.8 Total hours per semester 210

3.9 Credit points 7

1. Information about the programme

1.1 University University “Alexandru Ioan Cuza” of Iaşi

1.2 Faculty Faculty of Mathematics

1.3 Department Mathematics

1.4 Domain Mathematics

1.5 Cycle Master

1.6 Programme / Qualification Mathematics applied in finance and informatics

2. Information about the course

2.1 Course Name Mathematical Foundations of calculability

2.2 Course taught by Conf. dr. Dănuţ Rusu

2.3 Seminary / laboratory taught by Conf. dr. Dănuţ Rusu

2.4 Year 2.5 Semester 2.6 Type of evaluation 2.7 Course type OP

4. Pre-requisites

4.1 Curriculum Natural numbers, Real numbers, Cardinal numbers, Metric spaces, General topology

4.2 Competencies Working with the basics of algebra

5. Conditions (if necessary)

5.1 Course amphitheater / classroom

5.2 Seminary / Laboratory classroom

6. Specific competencies acquired

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C1 Manipulating notions, methods and mathematical models, specific techniques and technologies in scientific calculus and applications in economy and informatics: 2 credits C2 Data processing, analysis and interpretation using mathematical, statistical and informatics tools

C3 Being able to develop, test and validate algorithms; implementation in high level programing

languages

C4 Being able to construct and apply mathematical models for analyzing and simulating some

phenomena and processes: 2 credits

C5 Being able to develop, analyze and test computer systems and specific programming languages; being able to use them for solving problems in applied mathematics: 1 credit C6 Being able to analyze and interpret some economic processes and phenomena

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CT1 Having a responsible attitude towards scientific research and teaching, being able to fully develop the personal potential in the professional career, respecting the principles of a rigorous and efficient work in order to fulfill complex tasks, respecting the ethical norms and principles in the professional activity: 1 credit CT2 Being able to work efficiently in a team and to coordinate and efficiently lead a team or an inter-disciplinary group CT3 Being able to make a selection of information resources and to use them efficiently, in Romanian or other language of international circulation, in order to develop the professional activity and adapt it to the demands of a dynamical society: 1 credit

7. Course objectives

7.1

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tive Students will become familiar with the concepts of computability and computable function, since

Turing computable functions and recursive functions and ending with the definition and study these concepts in the abstract of the effectively given domains. Special emphasis will be placed on concrete situations: calculability on real line, calculability on complete metric spaces and calculability on Banach spaces.

7.2

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If successfull at the final examination, students will be able to: Describe the basic objects of recursive function theory and the domain theory Demonstrate their basic properties Explain the meaning and applications of the main results Use some basic techniques and concepts and problem solving exercises Calculate primitive recursive functions, real numbers based on various models of construction, etc.

8. Contents

8.1 Course Teaching methods Remarks (number oh hours, references)

1.

Computable functions (Turing machine, Turing computable functions) Recursive functions (primitive recursive functions, recursive functions, partial recursive functions, Church's thesis)

Exposure, conversation, demonstration, problematizing

2 hours [3,4]

2. Recursively enumerable sets (recursive sets, recursively enumerable sets, encodings of the plan)


2 hours [3,4]

3.

Notions and preliminary results (dcpo, ccpo, complete lattice, equivalent conditions, closure systems, the set of formal balls associated with a metric space)


2 hours [1]

4.

Scott-continuous functions (definition, examples, the Scott-continuous functions, isomorphism of dcpo)


2 hours [1]

5.

Fixed point theorems (Tarski's theorem, Markowsky’s theorem, fixed point theorem for Scott-continuous functions, applications)


2 hours [1]

6.

Approximation order (definition, properties, base in a dcpo, the lowest base, property of interpolation, continuous domains, algebraic domains)


2 hours [1]

7.

Continuous domains (characterization theorems, abstract bases, continuous domain of the ideals over an abstract basis, isomorphism theorems, role of the basis in building of domains and Scott-continuous functions)


2 hours [1]

8.

Scott topology (elements of general topology, specialized preorder, Alexandrov topology, Scott-open sets and Scott-closed sets, Scott topology, Scott continuity as a topological continuity, Scott topology on continuous domains)


2 hours [1,2,5]

9. Cartesian product of domains Exposure, conversation, demonstration, problematizing

2 hours [1]

10. Lattice domains (L-domains, bc-domains, continuous lattices, Scott domains)


2 hours [1]

11.

Effectively given domains (computable elements, computable sequences, computable functions between effectively given domains)


2 hours [1,2]

12.

Computability on the real line (the effective domain of intervals, computable numbers, effective convergence, computable real functions)


2 hours [1,2,6]

13.

Computability on metric spaces (effective metric spaces, computable functions between effective metric spaces)


2 hours [1,2,6]

14. Computability on Banach spaces (effective Banach spaces, computable linear operators)


2 hours [1,2,6]

Bibliography Main references:

[1] S.Abramsky, A.Jung, Domain Theory, Clarendon Press, Oxford, http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf. [2] G.Gierz, K.H.Hofmann, K.Keimel, J.D.Lawson, M.Mislove, D.S.Scott, Continuous Lattices and Domains, Cambridge University Press 2003. [3] P.Odifreddi, Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, Studies in Logic and Foundations of Mathematics, vol. 125, Elsevier 1992. Other references:

[4] H.Rogers, Theory of Recursive Functions and Effective Computability, Mc-Graw Hill, New York, 1967. [5] O. Costinescu, Elemente de topologie generală, Editura tehnică, 1969. [6] A.Edalat, P.Sünderhauf, A Domain-theoretic Approach to Real Number Computation, Theoretical Computer Science, 210, (1998).

8.2 Seminary / Laboratory Teaching methods Remarks (number oh hours, references)

1. Examples of primitive recursive functions Study of Ackermann's function

Exercise conversation 2 hours [3,4]

2.

Examples of recursive and recursively enumerable sets Operations with recursive and recursively enumerable sets Study of Cantor's pair function


3. Examples of dcpo, ccpo, etc. Organizing a complete metric space as a dcpo

Exercise conversation 2 hours [1,2,8]

4. Examples of Scott-continuous functions Scott-continuous function associated with a contraction


5.

Applications of Tarski's theorem Applications of Markowsky’s theorem Applications of fixed point theorem for Scott-continuous functions

Exercise conversation 2 hours [1]

6.

Approximation order for various concrete dcpo Approximation order in the dcpo of formal balls associated with a complete metric space


7.

Examples of algebraic domains, continuous non-algebraic domains, etc The continuous non-algebraic domain of formal balls associated with a complete metric space. The form of this domain for a Banach space and the set of real numbers.


8.

Comparison between Scott topology and Alexandrov topology associated with the order relation Various characterizations of the case of equality. Concrete examples Scott topology of continuous domain of formal balls in a complete metric space

Exercise conversation 2 hours [1,2,5,8]

9.

Study invariant properties relative to the cartesian product Characterization of Scott continuity of the functions with two variables.


10.

Examples of L-domains which are not bc-domains Examples of bc-domains which are not continuous lattices Construction of new continuous domains using the cartesian product


11.

Examples of effectively given domains An algorithm for the construction of a enumeration of the computable elements in a effectively given domain


12.

Study effectively given domain of intervals Applying the general theory on this particular example


13. Domain of formal balls as an effectively given domain Operations with computable elements


14. Examples of effective Banach spaces and computable linear operators


References

[1] S.Abramsky, A.Jung, Domain Theory, Clarendon Press, Oxford, http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf. [2] G.Gierz, K.H.Hofmann, K.Keimel, J.D.Lawson, M.Mislove, D.S.Scott, Continuous Lattices and Domains, Cambridge University Press 2003. [3] P.Odifreddi, Classical Recursion Theory. The Theory of Functions and Sets of Natural Numbers, Studies in Logic and Foundations of Mathematics, vol. 125, Elsevier 1992. [4] H.Rogers, Theory of Recursive Functions and Effective Computability, Mc-Graw Hill, New York, 1967. [5] O. Costinescu, Elemente de topologie generală, Editura tehnică, 1969. [6] A.Edalat, P.Sünderhauf, A Domain-theoretic Approach to Real Number Computation, Theoretical Computer Science, 210, (1998). [7] A.Edalat, P.Sünderhauf, Computable Banach spaces via domain theory, Theoretical Computer Science, 219, (1999). [8] A.Precupanu, L.Florescu, G.Blendea, M.Cuciureanu, Spaţii metrice. Probleme, Univ. “A.I.Cuza” Iaşi, 1990.

9. Coordination of the contents with the expectations of the community representatives, professional associations and relevant employers in the corresponding domain

The current epoch is an epoch of computation. The computation is found in all electronic devices that surround us and improve our lives. Nowadays more and more electrical devices are controlled by microprocessor and default behavior performs computations in their activity. The first is of course the computers, which by virtue of communication and mobility have taken a variety of forms and features, and pc became laptop, tablet, smartphone, navigation system, etc. We continue with domestic robots, smart TVs , and so on, until the encapsulated systems, equipped with dedicated microprocessor that performs specific functions. But computational activity is not conducted only at the functional level but is found in all forms of design.So, without be visible, the computation surrounds us in a large variety of forms. During the course and the seminar the students will familiarize with the mathematical theories of calculability, i.e. recursive function theory and the domain theory.

10. Assessment and examination

Date Course coordinator

Conf. dr. Dănuţ Rusu Seminary coordinator Conf. dr. Dănuţ Rusu

Aproval date in the department Head of the departament

Prof. Răzvan Liţcanu, Ph. D.

Activity 10.1 Criteria 10.2 Modes 10.3 Weight in the final grade (%)

10.4 Course

Knowledge and use of the fundamental concepts and results, applying of the theoretical results

written exam 50

10.5 Seminary/ Laboratory

Identify of the methods for solving of exercises and problems, acquiring of computational skills, the ability to understand and present a text in discipline issues

written exam making and presenting a report

50

10.6 Minimal requirements - Knowledge of fundamental concepts, understanding the main results - Solving of problems and exercises with low-difficulty - Realization and exposure of a low-difficulty report

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BazeleMatematicaaleCalculabilitatiiMM13_2305