# Lectures A.Y. 2009/2010

## 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

• Topics [M2: Part 1]
• course presentation
• 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
• 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 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
• 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]