# 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

• Topics [M2: Part 1]
• course presentation
• What you should know after the lecture
• the basic definitions regarding sets, 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

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

• Topics [M2: Part 2]
• 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

• 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

• 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

• 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

• 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]

• 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

• 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.

• 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

• 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

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

• 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

• 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

• 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

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

• 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

• 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

• 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

• 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

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

• 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