Free University of Bolzano/Bozen
Faculty of Computer Science
Master of Science in Computer Science
Theory of Computing
Lectures A.Y. 2014/2015
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. 2013. Available
as scanned pages in pdf.
[M3] Languages and Machines (3rd
edition). Thomas A. Sudkamp. Addison Wesley, 2005.
Only Chapter 13.
[M4]
The Convenience of
Tilings. Peter van Emde Boas. In Complexity, Logic, and
Recursion Theory, volume 187 of Lecture Notes in Pure and
Applied Mathematics, pages 331-363, 1997.
[M5]
Exercises on Theory of
Computing. Available as scanned pages in pdf.
Lectures
Lectures 1,2 - 1/10/2014
- Topics
[M2: Part 1]
- course presentation
- basic definitions about sets
- What you should know after the lecture
- the basic definitions regarding sets, relations, and their
properties
Lectures 3,4 - 7/10/2014
- Topics
[M2: Part 1]
- basic definitions about relations and functions
- cardinality of a set, countable and uncountable sets, Cantor's
theorem
- 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
Exercise 1,2 - 7/10/2014
- Review of basic proof techniques
[M2: Part 0]
- deductive proofs
- proving equivalences of sets
- proof by contradiction
- proof by induction
Lectures 5,6 - 8/10/2014
- Topics
[M2: Part 2]
- basic definitions about languages
- the Turing Machine
- What you should know after the lecture
- the formal meaning of alphabet, string, language
- how a Turing Machine is formally defined
- design Turing Machines that recognize some simple languages
Lectures 7,8 - 14/10/2014
- Topics
[M2: Part 2]
- instantaneous description of a Turing Machine
- recursive and recursive enumerable languages
- examples of Turing Machines
- programming techniques for Turing Machines
- storage in the state
- multiple tracks
- 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 3,4 - 14/10/2014
Lectures 9,10 - 15/10/2014
- Topics
[M2: Part 2]
- programming techniques for Turing Machines
- subroutines and procedure calls
- multi-tape Turing Machines
- running time of a Turing Machine
- What you should know after the lecture
- how a multi-tape TM can be simulated by a single-tape TM
- the cost of simulating a multi-tape TM by a single-tape TM
Lectures 11,12 - 21/10/2014
- Topics
[M2: Part 3]
- nondeterministic Turing Machines
- classes of languages/problems
- recursive/decidable languages
- recursively enumerable (R.E.) languages
- non-R.E. languages
- What you should know after the lecture
- 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
- how languages/problems can be classified
Exercise 5,6 - 21/10/2014
- Exercises on non-deterministic Turing Machines and further Turing
Machines extensions. Exercises on the correspondence between function
computation and language recognition by Turing Machines.
[M5:
Exercise 03]
Lectures 13,14 - 22/10/2014
- Topics
[M2: Part 3]
- Church-Turing Thesis
- closure properties of recursive and R.E. languages
- What you should know after the lecture
- the Church-Turing Thesis and its implications
- how to prove closure properties of recursive and R.E. languages
Lectures 15,16 - 28/10/2014
- Topics
[M2: Part 3]
- encoding Turing Machines as binary strings/integers
- enumerating binary strings/Turing Machines
- 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
- What you should know after the lecture
- how to encode Turing Machines as binary strings
- how to prove that the diagonalization language is non-R.E.
- how to prove that the universal language is R.E.
Lectures 17,18 - 28/10/2014
- Topics
[M2: Part 3,
Part 4]
- the notion of reduction between problems/languages
- properties of R.E. languages
- Rice's theorem
- What you should know after the lecture
- what a reduction is
- how to prove that the universal language is non-recursive
- how to prove Rice's theorem
Exercise 7,8 - 29/10/2014
Lectures 19,20 - 4/11/2014
- Topics
[M2: Part 4]
- Primitive recursive functions
- examples of primitive recursive functions
- showing computability of primitive recursive functions
- What you should know after the lecture
- the definition of primitive recursive functions
- how to construct some simple primitive recursive functions
- how to prove that every primitive recursive function is Turing
computable
Exercise 9,10 - 4/11/2014
- Exercises on Turing Machines computing functions
[M5:
Exercise 05]
Lectures 21,22 - 5/11/2014
- Topics
[M2: Part 4]
- bounded operators and bounded minimization
- Gödel numbering
- What you should know after the lecture
- how to define primitive recursive functions using bounded
minimizations
- how to encode and decode a sequence of numbers by means of a single
number
Lectures 23,24 - 11/11/2014
- Topics
[M2: Part 4]
- course-of-values recursion
- total computable functions that are not primitive recursive
- mu-recursive functions
- Turing computability of mu-recursive functions
- What you should know after the lecture
- 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
- the definition of mu-recursive functions
- how to show that every mu-recursive function is Turing computable
Exercise 11,12 - 11/11/2014
Lectures 25,26 - 12/11/2014
- Topics
[M2: Part 4]
- arithmetization of Turing Machines
- What you should know after the lecture
- how to define a (primitive) recursive function that computes the
trace of a Turing Machine computation
- how to define a mu-recursive function that simulates the
computation of a Turing Machine computation
Exercise 13,14 - 12/11/2014
Midterm exam - 18/11/2014
- Topics
- Turing Machines
- decidability and undecidability
- recursive and recursively enumerable languages
- recursive functions
Lectures 27,28 - 18/11/2014
- Topics
[M2: Part 5]
- tractable and intractable problems
- the classes P and NP
- a problem in NP: SAT
- SAT and CSAT
- poly-time reductions
- What you should know after the lecture
- how the classes P and NP are defined
- how to show a problem to be in NP
- how to polynomially reduce one problem to another
Lectures 29,30 - 19/11/2014
- Topics
[M2: Part 5]
- NP-hardness and NP-completeness
- Cook's theorem
- What you should know after the lecture
- how to show a problem to be NP-hard
- how to prove Cook's theorem
Lectures 31,32 - 24/11/2014
- Topics
[M2: Part 5]
- NP-completeness of Vertex Cover
- coNP-complete problems
- What you should know after the lecture
- hot to carry out an NP-hardness proof
- how NP and coNP are related to each other
Exercise 17,18 - 24/11/2014
Lectures 33,34 - 25/11/2014
- Topics
[M2: Part 6]
- oracle Turing Machines
- the polynomial hierarchy
- Quantified Boolean Formulae (QBF)
- the classes PSPACE and NPSPACE.
- What you should know after the lecture
- what an oracle TM is
- how complexity classes based on oracle TMs are defined
- 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
Exercise 19,20 - 25/11/2014
- Exercises on variations of satisfiability and their complexity
[M5:
Exercise 09]
Lectures 35,36 - 26/11/2014
- Topics
[M2: Part 6]
- relationship between PSPACE and NPSPACE (Savitch's theorem)
- What you should know after the lecture
- how to prove Savitch's theorem
Course home page
Last modified:
Tuesday, 25-Nov-2014 23:54:33 CET