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Theoretical Computer Science

• Course name: Theoretical Computer Science
• Course number: BS-18

Course Characteristics

Key concepts of the class

• Automata Theory
• Formal Grammars
• Computability

What is the purpose of this course?

A good software developer ignorant of how the mechanics of a compiler works is not better than a good pilot when it comes to fix the engine and he will definitively not be able to provide more than average solutions to the problems he is employed to solve. Like automotive engineering teach us, races can only be won by the right synergy of a good driving style and mechanics. Most importantly, limits of computation cannot be ignored in the same way we precisely know how accelerations, forces and frictions prevent us from racing at an unlimited speed. This course will investigate the prerequisites to understand compilers functioning. Although the act of compilation appears deceptively simple to most of the modern developers, great minds and results are behind the major achievements that made this possible. All starts with the Epimenides paradox (about 600 BC), which emphasizes a problem of self-reference in logic and brings us to the short time window between WWI and WW2 when, in 1936, Alan Turing proved that a general procedure to identify algorithm termination simply does not exist. Another major milestone has been reached by Noam Chomsky in 1956 with his description of a hierarchy of grammars. In this long historical timeframe we can put most of the bricks with which we build modern compilers. The course will be an historical tour through the lives of some of the greatest minds who ever lived on this planet.

Course objectives based on Bloom’s taxonomy

- What should a student remember at the end of the course?

By the end of the course, the students should be able to

• Define a formal language
• List different computational models
• Define computational models such as Finite State Automata and Pushdown Automata
• List different types of Formal Grammars
• Define computability and related concepts
• List applications for automata theory and formal grammar

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to

• Describe the basic mathematical machinery behind automata theory and how can be applied to programming languages compilers
• Explain Strengths and weaknesses of specific computational model
• Explain Finite State Automata and Pushdown Automata
• Abstract systems using the given models

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to

• Formally modelling a system
• Reasoning about verification of program properties
• Specifying a system as Finite State Automata, Pushdown Automata or Turing Machine
• Coding in programming languages an emulator of Finite State Automata
• Using proof techniques by diagonalization

Course evaluation

Course grade breakdown

type	points
Labs/seminar classes	40
Interim performance assessment	30
Exams	30

Grades range

Course grading range

grade	low	high
A	80	100
B	65	79
C	50	64
D	0	49

Resources and reference material

• Handouts supplied by the instructor
• \bibentry{houl79}
• \bibentry{Martin94}
• \bibentry{Hromkovic09}

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Section 1

Section title

Automata Theory

Topics covered in this section

• Languages
• Finite State Automata
• Pushdown Automata
• Nondeterminism
• Turing Machines

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	1
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. What is a Finite State Automaton?
2. What is a Pushdown Automaton?
3. What is a Turing Machine?
4. What is a nondeterministic automaton?
5. Given a specific language define a corresponding Finite State Automaton
6. Given a specific language define a corresponding Pushdown Automaton
7. State the difference between Finite State Automata and Pushdown Automata
8. Computing the intersection, union, complement of two automata
9. What is Pumping Lemma? Example of applications.
10. Operations on Automata

Typical questions for seminar classes (labs) within this section

1. Check if a given language is recognized by a specific Finite State Automaton
2. Prove with Pumping Lemma that a language is regular
3. Check if a given language is recognized by a specific Pushdown Automaton
4. Check if a given language is recognized by a specific Turing Machine
5. Define a correct automaton given a specific language

Test questions for final assessment in this section

1. Check if a given language is recognized by a specific Finite State Automaton
2. Prove with Pumping Lemma that a language is regular.
3. Check if a given language is recognized by a specific Pushdown Automaton
4. Check if a given language is recognized by a specific Turing Machine
5. Define a correct automaton given a specific language

Section 2

Section title

Formal Grammars

Topics covered in this section

• Chomsky Hierarchy
• Regular Expressions
• Relationships with Automata

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	0
Midterm evaluation	1
Testing (written or computer based)	0
Reports	1
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. What is Chomsky Hierarchy?
2. What is a Regular language?
3. What is a Context-free language?
4. What is a Context-sensitive language?
5. What is a regular expression?

Typical questions for seminar classes (labs) within this section

1. Given a specific Finite State Automaton define a grammar for the corresponding language
2. Given a specific Pushdown Automaton define a grammar for the corresponding language
3. Given a regular expression design the corresponding Finite state Automaton
4. Given a Finite state Automaton define the regular expression for the corresponding language

Test questions for final assessment in this section

1. What is Chomsky Hierarchy?
2. What is a Regular language?
3. Given a regular expression design the corresponding Finite state Automaton
4. Given a specific Pushdown Automaton define a grammar for the corresponding language

Section 3

Section title

Computability

Topics covered in this section

• Undecidability
• Halting Problem
• Rice Theorem

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	0
Midterm evaluation	1
Testing (written or computer based)	0
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. What is Halting Problem?
2. What is Rice theorem?
3. What is an undecidable problem
4. What is a Turing Machine?
5. What is Goedelization?

Typical questions for seminar classes (labs) within this section

1. Given a specific computational problem showing that it is undecidable (via reduction to a known problem, for example)
2. Show a proof via diagonalization of Halting Problem
3. Show a proof via diagonalization of Rice Theorem
4. Give an example of an undecidable problem
5. Show that a Turing Machine with multiple tapes is equivalent to a Turing Machine with a single tape

Test questions for final assessment in this section

1. What is a Turing Machine?
2. What is Halting Problem?
3. What is Rice theorem?
4. Show a proof via diagonalization of Halting Problem
5. Show a proof via diagonalization of Rice Theorem
6. Show that a Turing Machine with multiple tapes is equivalent to a Turing Machine with a single tape