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

Theory of Computing

Lectures A.Y. 2007/2008

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. 2007. Available as scanned pages in pdf.

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

Lectures

Summary

Week	Topics	Monday 10:30-12:30 (lecture)	Wednesday 8:30-10:30 (lecture)	Wednesday 16:00-18:00 (exercise)
#1 Oct. 8	Course introduction	Course introduction, basic notions about sets (Lec 1,2)	Basic notions about relations, functions, languages (Lec 3,4)	Formal proofs (Exer 1)
#2 Oct. 15	Undecidability	Undecidable problems (Lec 5,6)	The Turing Machine (Lec 7,8)	Turing Machines (Exer 2)
#3 Oct. 22	(Extended) Turing Machines	Programming techniques for TMs (Lec 9,10)	Multitape and nondeterministic TMs (Lec 11,12)	Nondeterministic TMs and extensions of TMs (Exer 3)
#4 Oct. 29	Decidability and undecidability	Church-Turing Thesis (Lec 13,14)	Closure properties of R.E. languages (Lec 15,16)	TMs computing functions (Exer 4)
#5 Nov. 19	Reductions	Showing languages to be non-recursive/non-R.E. (Lec 17,18)	Rice's theorem, primitive recursive functions (Lec 19,20)	Recursive and R.E. languages (Exer 5)
#6 Nov. 26	Recursive functions	Primitive recursive functions (Lec 21,22)	Gödel numbering, non primitive recursive functions (Lec 23,24)	Primitive recursive functions (Exer 6)
#6bis Nov. 29	Mu-recursive functions	Mu-recursive functions (Lec 25,26)	--	Preparation for midterm exam (Exer 7)
#7 Dec. 3	NP-completeness	P and NP (Lec 27,28)	NP-hardness (Lec 29,30)	Midterm exam (ME)
#8 Dec. 10	NP and coNP	Cook's theorem (Lec 31,32)	Oracle TMs (Lec 33,34)	Reduction CSAT to 3SAT (Exer 8)
#8bis Dec. 13	PSPACE	PSPACE (Lec 35,36)	--	Reductions for NP-hardness (Exer 9)
#9 Jan. 7	PSPACE and NPSPACE	Savitch's theorem (Lec 37,38)	PSPACE-completeness of QBF (Lec 39,40)	Tiling problems (Exer 10)
#10 Jan. 14	Non-uniform complexity	Circuits (Lec 41,42)	Non-uniform complexity classes (Lec 43,44)	Boolean circuits (Exer 11)
#10bis Jan. 16	Non-uniform complexity	Non-uniform TMs and circuits (Lec 45,46)	BDDs (Lec 47,48)	--

Details

Week 1: Course introduction

Lectures 1,2 - 8/10/2007

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 - 10/10/2007

Topics [M2: Part 1]
- basic definitions about sets, relations, 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

Exercise 1 - 10/10/2007

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

Week 2: Undecidability

Lectures 5,6 - 15/10/2007

Topics [M2: Part 2]
- an example for an undecidable problem: the Hello-world problem
- proving undecidability by reduction to an undecidable problem
What you should know after the lecture
- the notion of reduction
- how to prove that a problem is undecidable by reduction to another undecidable problem

Lectures 7,8 - 17/10/2007

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

Exercise 2 - 17/10/2007

Exercises on Turing Machines [M3: Exercise 02]

Week 3: Turing Machines and their extensions

Lectures 9,10 - 22/10/2007

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

Lectures 11,12 - 24/10/2007

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

Exercise 3 - 24/10/2007

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

Week 4: Decidability and undecidability

Lectures 13,14 - 29/10/2007

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

Lectures 15,16 - 31/10/2007

Topics [M2: Part 3]
- showing languages to be non-recursive/non-R.E.
- encoding Turing Machines as binary strings/integers
- enumerating binary strings/Turing Machines
What you should know after the lecture
- how to encode a Turing Machine as a binary string

Exercise 4 - 31/10/2007

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

Week 5: Reductions

Lectures 17,18 - 19/11/2007

Topics [M2: Part 3]
- 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

Lectures 19,20 - 21/1/2007

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

Exercise 5 - 21/11/2007

Exercises on recursive and R.E. languages [M3: Exercise 5]

Week 6: Recursive functions

Lectures 21,22 - 26/11/2007

Topics [M2: Part 4]
- showing computability of primitive recursive functions
- bounded operators
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
- what a reduction is

Lectures 23,24 - 28/11/2007

Topics [M2: Part 4]
- Gödel numbering
- course-of-values recursion
- 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

Exercise 6 - 28/11/2007

Exercises on primitive recursive functions [M3: Exercise 6]

Week 6 bis: Mu-recursive functions

Lectures 25,26 - 29/11/2007

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 7 - 30/11/2007

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

Week 7: NP-completeness

Lectures 27,28 - 3/12/2007

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 29,30 - 5/12/2007

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

Midterm exam - 5/12/2007

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

Week 8: NP and coNP

Lectures 31,32 - 10/12/2007

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

Lectures 33,34 - 12/12/2007

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

Exercise 8 - 12/12/2007

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

Week 8 bis: PSPACE

Lectures 35,36 - 13/12/2007

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

Exercise 9 - 14/12/2007

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

Week 9: PSPACE and NPSPACE

Lectures 37,38 - 7/1/2008

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 39,40 - 9/1/2008

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 10 - 9/1/2008

Exercises on on tiling problems [M3: Exercise 10]

Week 10: Non-uniform complexity

Lectures 41,42 - 14/1/2008

Topics [M2: Part 6, M2: Part 7]
- the classes EXPTIME, EXPSPACE, and the exponential hierarchy
- basic definitions about circuits
What you should know after the lecture
- typical problems complete for the classes EXPTIME and EXPSPACE
- how circuits are defined
- what the size and depth of a circuit are

Lectures 43,44 - 15/1/2008

Topics [M2: Part 7]
- circuit families
- the non-uniform complexity classes SIZE(f(n)) and DEPTH(f(n))
- upper bounds on SIZE and DEPTH for a language L
What you should know after the lecture
- how the complexity classes SIZE(f(n)) and DEPTH(f(n)) are defined
- how to prove that every language L has an exponential upper bound on SIZE
- how to prove that some languages L has an exponential lower bound on SIZE
- how to prove that every language L has a linear upper bound on DEPTH

Exercise 11 - 16/1/2008

Exercises on on boolean circuits [M3: Exercise 11]

Week 10 bis: Non-uniform complexity

Lectures 45,46 - 16/1/2008

Topics [M2: Part 7]
- simulation of TMs by uniform circuits
- non-uniform TMs
- simulation of circuits by non-uniform TMs
What you should know after the lecture
- how to simulate a TMs by a uniform circuit
- how a non-uniform TM is defined
- how to simulate a circuit by a non-uniform TM

Lectures 47,48 - 17/1/2008

Topics [M2: Part 7]
- Binary Decision Diagrams (BDDs)
- branching program complexity of a boolean functions
- simulation of BDDs by space-bounded non-uniform TMs
- simulation of space-bounded TMs by BDDs
What you should know after the lecture
- how BDDs are defined
- what the complexity measures for BDDs are
- how a BDD can be simulated by a space-bounded non-uniform TM
- how a space-bounded TM can be simulated by a BDD

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Last modified: Sunday, 22-Mar-2009 16:14:05 CET

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