Faculty of Computer Science

Master of Science in Computer Science

**[M1]** *Introduction to Automata Theory, Languages,
and Computation (3rd edition).* J.E. Hopcroft, R. Motwani,
J.D. Ullman. Addison Wesley, 2007.

**[M2]** *Lecture Notes for
Theory of Computing*. Diego Calvanese. 2009. Available as
scanned pages in pdf.

**[M3]**
*Exercises on Theory of
Computing*. Available as scanned pages in pdf.

**Topics**[M2: Part 1]- course presentation
- basic definitions about sets

**What you should know after the lecture**- the basic definitions regarding functions, relations, and their properties

**Topics**[M2: Part 1]- basic definitions about relations and functions
- cardinality of a set, countable and uncountable sets, Cantor's theorem
- basic definitions about languages

**What you should know after the lecture**- the definition of cardinality of a set
- the difference between countable and uncountable sets
- Cantor's diagonalization argument
- the formal meaning of alphabet, string, language

**Topics**[M2: Part 2]- the Turing Machine
- instantaneous description of a Turing Machine
- recursive enumerable and recursive languages

**What you should know after the lecture**- how a Turing Machine is formally defined
- design Turing Machines that recognize some simple languages

**Topics**[M2: Part 2]- examples of Turing Machines
- programming techniques for Turing Machines
- storage in the state
- multiple tracks
- subroutines and procedure calls

**What you should know after the lecture**- how one can program a TM easier by imposing structure on states and tape symbols
- how one can implement a procedure call with a TM

**Review of**basic proof techniques [M2: Part 0]- deductive proofs
- proving equivalences of sets
- proof by contradiction
- proof by induction

**Topics**[M2: Part 2]- multi-tape Turing Machines
- running time of a Turing Machine
- nondeterministic Turing Machines

**What you should know after the lecture**- how a multi-tape TM can be simulated by a single-tape TM
- how a nondeterministic TM can be simulated by a multi-tape TM (and hence also by a single-tape TM)
- the cost of simulating a nondeterministic TM by a deterministic TM

**Topics**[M2: Part 3]- classes of languages/problems
- recursive/decidable languages
- recursively enumerable (R.E.) languages
- non-R.E. languages

- Church-Turing Thesis

- classes of languages/problems
**What you should know after the lecture**- how languages/problems can be classified
- the Church-Turing Thesis and its implications

**Exercises on**deterministic and nondeterministic Turing Machines [M3: Exercise 02]

**Topics**[M2: Part 3]- closure properties of recursive and R.E. languages
- encoding Turing Machines as binary strings/integers
- enumerating binary strings/Turing Machines

**What you should know after the lecture**- how to prove closure properties of recursive and R.E. languages
- how to encode a Turing Machine as a binary string

**Topics**[M2: Part 3]- showing languages to be non-recursive/non-R.E.
- a non-R.E. language: the diagonalization languages
- a non-recursive language: the universal language
- Universal Turing Machines
- the notion of reduction between problems/languages

**What you should know after the lecture**- how to prove that the diagonalization language is non-R.E.
- how to prove that the universal language is non-recursive
- what a reduction is

**Exercises on**multitrack, multitape, and non-deterministic Turing Machines. Exercises on reductions between problems. [M3: Exercise 03]

**Topics**[M2: Part 3, M2: Part 4]- Rice's theorem
- Primitive recursive functions

**What you should know after the lecture**- how to prove Rice's theorem
- the definition of primitive recursive functions
- how to construct some simple primitive recursive functions

**Topics**[M2: Part 4]- examples of primitive recursive functions
- showing computability of primitive recursive functions
- bounded operators and bounded minimization

**What you should know after the lecture**- how to prove that every primitive recursive function is Turing computable
- how to define primitive recursive functions using bounded minimizations

**Exercises on**Turing Machines computing functions [M3: Exercise 04]

**Topics**[M2: Part 4]- Gödel numbering
- course-of-values recursion
- total computable functions that are not primitive recursive

**What you should know after the lecture**- how to encode and decode a sequence of numbers by means of a single number
- how to define functions by means of course-of-values recursion, and how to show that they are primitive recursive
- how to prove the existence of computable functions that are not primitive recursive

**Topics**[M2: Part 4]- mu-recursive functions
- arithmetization of Turing Machines

**What you should know after the lecture**- the definition of primitive recursive functions
- how to define a (primitive) recursive function that computes the trace of a Turing Machine computation

**Exercises on**primitive recursive functions [M3: Exercise 6]

**Topics**[M2: Part 4, M2: Part 5]- arithmetization of Turing Machines
- tractable and intractable problems
- the classes P and NP
- a problem in NP: SAT

**What you should know after the lecture**- how to define a mu-recursive function that simulates the computation of a Turing Machine computation
- how the classes P and NP are defined
- how to show a problem to be in NP

**Topics**[M2: Part 5]- SAT and CSAT
- poly-time reductions
- NP-hardness and NP-completeness

**What you should know after the lecture**- how to polynomially reduce one problem to another
- sketch the proof of Cook's theorem
- how to show a problem to be NP-hard

**Exercises on**the topics of the midterm exam [M3: Exercise 7]

**Topics**[M2: Part 5]- Cook's theorem

**What you should know after the lecture**- how to prove Cook's theorem

**Exercises on**the reduction from 3SAT to CSAT [M3: Exercise 8]

**Topics**- Turing Machines
- recursive and recursive enumerable languages
- recursive functions

**Topics**[M2: Part 5, M2: Part 6]- coNP-complete problems
- oracle Turing Machines

**What you should know after the lecture**- what an oracle TM is
- how complexity classes based on oracle TMs are defined

**Topics**[M2: Part 6]- the polynomial hierarchy and PSPACE
- quantified boolean formulae
- space and time bounds for Turing Machines

**What you should know after the lecture**- how the polynomial hierarchy is defined
- how the problem of QBF is defined
- relationship between the space bound and the time bound for a TM

**Topics**[M2: Part 6]- relationship between PSPACE and NPSPACE (Savitch's theorem)
- evaluation of a QBF

**What you should know after the lecture**- how to prove Savitch's theorem
- how to evaluate a QBF

**Topics**[M2: Part 6]- evaluation of a QBF in polynomial space
- PSPACE-hardness of QBF

**What you should know after the lecture**- how to evaluate a QBF in polynomial space
- how to prove PSPACE-hardness of QBF

**Exercises on**reductions to show NP-hardness [M3: Exercise 9]