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

Theory of Computing

Lectures A.Y. 2015/2016

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. 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 - 19/10/2015

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 - 20/10/2015

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 - 22/10/2015

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

Lectures 5,6 - 26/10/2015

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 - 28/10/2015

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
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

Lectures 9,10 - 29/10/2015

Topics [M2: Part 2]
- programming techniques for Turing Machines
  - multiple tracks
  - 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

Exercise 3,4 - 2/11/2015

Exercises on deterministic Turing Machines [M5: Exercise 02]

Lectures 11,12 - 3/11/2015

Topics [M2: Part 3]
- nondeterministic Turing Machines
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

Lectures 13,14 - 5/11/2015

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

Exercise 5,6 - 9/11/2015

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 15,16 - 10/11/2015

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 - 11/11/2015

Topics [M2: Part 3]
- 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

Lectures 19,20 - 12/11/2015

Topics [M2: Part 4, M3: Chapter 13]
- 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 7,8 - 16/11/2015

Exercises on reductions between problems. [M5: Exercise 04]

Lectures 21,22 - 17/11/2015

Topics [M2: Part 4, M3: Chapter 13]
- 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

Exercise 9,10 - 18/11/2015

Exercises on Turing Machines computing functions [M5: Exercise 05]

Lectures 23,24 - 19/11/2015

Topics [M2: Part 4, M3: Chapter 13]
- 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 - 23/11/2015

Exercises on primitive recursive functions [M5: Exercise 06]

Lectures 25,26 - 24/11/2015

Topics [M2: Part 4, M3: Chapter 13]
- 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 - 25/11/2015

Exercises on the topics of the midterm exam [M5: Exercise 07]

Lectures 27,28 - 30/11/2015

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 - 1/12/2015

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

Midterm exam - 2/12/2015

Topics
- Turing Machines
- decidability and undecidability
- recursive and recursively enumerable languages
- recursive functions

Exercise 17,18 - 14/12/2015

Exercises on the reduction from 3SAT to CSAT and on variations of SAT [M5: Exercise 08]

Lectures 31,32 - 15/12/2015

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

Lectures 33,34 - 17/12/2015

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 - 21/12/2015

Exercises on variations of satisfiability and their complexity [M5: Exercise 09]

Lectures 35,36 - 22/12/2015

Topics [M2: Part 6]
- relationship between PSPACE and NPSPACE (Savitch's theorem)
- PSPACE-completeness
What you should know after the lecture
- how to prove Savitch's theorem

Lectures 37,38 - 7/1/2016

Topics [M2: Part 6]
- evaluation of a QBF
- membership in PSPACE of QBF
What you should know after the lecture
- how to evaluate a QBF in polynomial space

Exercise 21,22 - 7/1/2016

Exercises on NP-complete vs. polynomial problems [M5: Exercise 10]

Lectures 39,40 - 11/1/2016

Topics [M2: Part 6]
- PSPACE-hardness of QBF
- the classes EXPTIME, EXPSPACE, and the exponential hierarchy
- logarithmic space
What you should know after the lecture
- how to prove PSPACE-hardness of QBF
- typical problems complete for the classes EXPTIME and EXPSPACE
- how space is measured for sublinear space computations

Lectures 41,42 - 12/1/2016

Topics [M2: Part 7]
- composition of LogSpace computations
What you should know after the lecture
- how the trivial, naive, and emulative compositions of function computations are defined

Lectures 43,44 - 14/1/2016

Topics [M2: Part 7]
- LogSpace reductions
- the complexity classes L and NL
- the relationship between N and NL with graph connectivity
- the complement of NL
What you should know after the lecture
- how LogSpace reductions are defined
- how the complexity classes L and NL are defined
- how to prove that connectivity in directed graphs in NL-complete
- what a log-space reduction is

Exercise 23,24 - 18/1/2016

Exercises on on space complexity [M5: Exercise 11]

Lectures 45,46 - 19/1/2016

Topics [M4: Part 8]
- tiling problems (defined using colored tiles and using adjacency relations)
- complexity of variants of tiling problems
- relationship between tilings and Turing machine computations
What you should know after the lecture
- how tiling problems are defined
- how one can convert tilings defined with colored tiles into tilings defined with adjacency relations
- the master reduction from Turing machine computations to tilings