Free University of Bolzano/Bozen
Faculty of Computer Science
Master of Science in Computer Science

Theory of Computing

Lectures A.Y. 2009/2010

Prof. Diego Calvanese

Teaching material

[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.

Lectures

Lectures 1,2 - 7/10/2009

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

Lectures 3,4 - 8/10/2009

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

Lectures 5,6 - 14/10/2009

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

Lectures 7,8 - 15/10/2009

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

Exercise 1,2 - 16/10/2009

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

Lectures 9,10 - 21/10/2009

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

Lectures 11,12 - 22/10/2009

Topics [M2: Part 3]
- classes of languages/problems
  - recursive/decidable languages
  - recursively enumerable (R.E.) languages
  - non-R.E. languages
- Church-Turing Thesis
What you should know after the lecture
- how languages/problems can be classified
- the Church-Turing Thesis and its implications

Exercise 3,4 - 23/10/2009

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

Lectures 13,14 - 28/10/2009

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

Lectures 15,16 - 29/10/2009

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

Exercise 5,6 - 30/10/2009

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

Lectures 17,18 - 4/11/2009

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

Lectures 19,20 - 5/11/2009

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

Exercise 7,8 - 6/11/2009

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

Lectures 21,22 - 11/11/2009

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

Lectures 23,24 - 12/11/2009

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

Exercise 9,10 - 13/11/2009

Exercises on primitive recursive functions [M3: Exercise 6]

Lectures 25,26 - 25/11/2009

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

Lectures 27,28 - 26/11/2009

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

Exercise 11,12 - 26/11/2009

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

Lectures 29,30 - 27/11/2009

Topics [M2: Part 5]
- Cook's theorem
What you should know after the lecture
- how to prove Cook's theorem

Exercise 13,14 - 27/11/2009

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

Midterm exam - 2/12/2009

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

Lectures 31,32 - 3/12/2009

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

Lectures 33,34 - 4/12/2009

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

Lectures 35,36 - 9/12/2009

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

Lectures 37,38 - 10/12/2009

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

Exercise 17,18 - 11/12/2009