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

Theory of Computing

Lectures A.A. 2004/2005

Prof. Diego Calvanese

Teaching material

[M1] Introduction to Automata Theory, Languages, and Computation (2nd edition). J.E. Hopcroft, R. Motwani, J.D. Ullman. Addison Wesley, 2003.

[M2] Lecture Notes for Theory of Computing. Diego Calvanese. 2004. Available as scanned pages in pdf.

[M3] Exercises on Theory of Computing. Andrea Calì. 2004. Available as scanned pages in pdf.

Lectures

Summary

Week	Topics	Monday 10:30-12:30 (lecture)	Wednesday 10:30-12:30 (lecture)	Wednesday 16:00-18:00 (exercise)
#1 Oct. 6	Course introduction	--	Course introduction, basic notions about languages (Lec 1,2)	--
#2 Oct. 11	Finite automata	Deterministic FA (Lec 3,4)	Nondeterministic FA (Lec 5,6)	DFAs and NFAs (Exer 1)
#3 Oct. 18	Regular expressions	Epsilon-NFAs (Lec 7,8)	Regular expressions, from RE to FA (Lec 9,10)	Epsilon-NFAs and RE (Exer 2)
#4 Oct. 25	Regular languages	From DFA to RE, closure properties for RL (Lec 11,12)	Pumping lemma (Lec 13,14)	Regular languages (Exer 3)
#5 Nov. 1	Properties of regular languages	--	Decision properties of regular languages (Lec 15,16)	Regular languages (Exer 4)
#6 Nov. 8	Formal grammars	Minimization of automata (Lec 17,18)	Formal grammars (Lec 19,20)	Formal grammars (Exer 5)
#7 Nov. 15	Context free grammars, pushdown automata	Context free grammars (Lec 21,22)	Pushdown automata (Lec 23,24)	Regular languages, finite automata (Exer 6)
#8 Nov. 22	Midterm exam	Topics for midterm exam (Exer 7)	Midterm exam (ME)	--
#9 Nov. 29	Context free languages	Simplifying CFGs (Lec 25,26)	Chomsky normal form, pumping lemma for CFLs (Lec 27,28)	CFGs and PDAs (Exer 8)
#10 Dec. 6	Properties of context free languages	Closure and decision properties of CFLs (Lec 29,30)	--	--
#11 Dec. 13	Turing Machines	Undecidable problems (Lec 31,32)	The Turing Machine (Lec 33,34)	Closure properties of CFLs and TMs (Exer 9)
#12 Dec. 20	Extensions of Turing Machines	Programming techniques for TMs, multi-tape TMs (Lec 35,36)	Nondeterministic TMs (Lec 37,38)	Nondeterministic TMs and extensions of TMs (Exer 10)
#13 Jan. 10	Decidability and undecidability	Closure properties of R.E. languages (Lec 39,40)	Topic lecture 2 (Lec 41,42)	Recursive and R.E. languages (Exer 11)
#14 Jan. 17	Computational complexity	Rice's theorem, (in)tractable problems (Lec 43,44)	NP-completeness (Lec 45,46)	NP-complete problems (Exer 12)
#15 Jan. 24	Complexity classes	Complexity classes above NP (Lec 47,48)	--	Exam topics (Exer 13)

Details

Week 1: Course introduction

Lectures 1,2 - 6/10/2004

Topics [M2: Part 1]
- course presentation
- basic definitions about languages
- finite state machines
What you should know after the lecture
- the formal meaning of alphabet, string, language
- how a finite-state machine works (informally)

Week 2: Finite automata

Lectures 3,4 - 11/10/2004

Topics [M2: Part 2]
- deterministic finite automata (DFA)
- extended transition function for a DFA
- nondeterministic finite automata (NFA)
What you should know after the lecture
- the formal definitions of a DFA and of an NFA, and of the language they accept
- construct a DFA and an NFA for a given language

Lectures 5,6 - 13/10/2004

Topics [M2: Part 2]
- relationship between DFAs and NFAs (subset construction)
What you should know after the lecture
- how to prove that DFAs and NFAs accept the same class of languages
- how to contstruct a DFA from a given NFA

Exercise 1 - 13/10/2004

Exercises on finite automata [M3: Exercise 1]
- [M1]: Exercises 2.2.2, 2.2.5, 2.2.8, 2.3.1

Week 3: Regular expressions

Lectures 7,8 - 18/10/2004

Topics [M2: Part 2]
- finite automata with epsilon-transitions
- eliminating epsilon-transitions
- regular expressions
What you should know after the lecture
- how to prove that epsilon-NFAs and NFAs accept the same class of languages
- how to construct an NFA from a given epsilon-NFA
- what the language generated by a regular expression is

Lectures 9,10 - 20/10/2004

Topics [M2: Part 3]
- algebraic laws for regular expressions
- converting a regular expression to a finite automaton
What you should know after the lecture
- what algebraic laws hold for regular expressions
- how to prove that each language generated by a R.E. is accepted by some finite automaton
- how to construct a finite automaton from a given R.E.

Exercise 2 - 20/10/2004

Exercises on finite automata and regular expressions [M3: Exercise 2]

Week 4: Regular languages

Lectures 11,12 - 25/10/2004

Topics [M2: Part 3, Part 4]
- converting a finite automaton to a regular expression
- closure properties of regular expressions
What you should know after the lecture
- how to prove that finite automata accept exactly the regular languages
- how to construct a R.E. from a given automaton
- that regular languages are closed under regular and boolean operations

Lectures 13,14 - 27/10/2004

Topics [M2: Part 4]
- pumping lemma for regular languages
- proving languages not to be regular
What you should know after the lecture
- how to prove that a languages is not regular
- what properties can be decided about regular languages

Exercise 3 - 27/10/2004

Exercises on regular languages [M3: Exercise 3]

Week 5: Properties of regular languages

Lectures 15,16 - 03/11/2004

Topics [M2: Part 4]
- decision problems for regular languages
- minimization of finite automata
What you should know after the lecture
- what properties about regular languages can be decided, and what is the computational cost

Exercise 4 - 03/11/2004

Exercises on regular languages [M3: Exercise 4]

Week 6: Formal grammars

Lectures 17,18 - 8/11/2004

Topics [M2: Part 4]
- minimization of finite automata
- Chomsky formal grammars
What you should know after the lecture
- how to minimize a finite automaton
- what a Chomsky formal grammar is

Lectures 19,20 - 10/11/2004

Topics [M2: Part 4]
- classification of formal grammars
What you should know after the lecture
- what the language defined by a formal grammar is
- how formal grammars are classified

Exercise 5 - 10/11/2004

Exercises on formal grammars [M3: Exercise 5]

Week 7: Context free grammars, pushdown automata

Lectures 21,22 - 15/11/2004

Topics [M2: Part 5]
- contex-free grammars
- parse-trees for contex-free grammars
- ambiguous grammars
- pushdown automata (PDAs)
What you should know after the lecture
- how to construct the parse-tree corresponding to a derivation
- recognize when a context-free grammar is ambiguous
- how a pushdown automaton is defined

Lectures 23,24 - 17/11/2004

Topics [M2: Part 6]
- istantaneous description for a pushdown automaton
- PDA acceptance by final state and by empty stack
- equivalence of PDAs that accept by final state and by empty stack
What you should know after the lecture
- how to define a PDA for a given context free language
- how to convert a PDA that accepts by final state into an equivalent one that accepts by empty stack, and vice-versa

Exercise 6 - 17/11/2004

Exercises on regular languages and finite automata [M3: Exercise 6]

Week 8: Midterm exam

Exercise 7 - 22/11/2004

Exercises on regular languages, automata, and grammars (preparation for midterm exam) [M3: Exercise 7]

Midterm exam - 24/11/2004

Topics
- regular languages
- automata
- formal grammars

Week 9: Context free languages

Lectures 25,26 - 29/11/2004

Topics [M2: Part 6, Part 7]
- relationship between PDAs and CFGs
- simplifying context free grammars (eliminating useless symbols, eliminating epsilon-transitions)
What you should know after the lecture
- how to convert a CFG to a PDA that accepts by empty stack
- how to eliminate useless symbols from a CFG
- how to eliminate epsilon-productions from a CFG

Lectures 27,28 - 1/12/2004

Topics [M2: Part 7b]
- eliminating unit-productions
- Chomsky normal form for a CFG
- pumping lemma for context-free languages
What you should know after the lecture
- how to eliminate unit-productions from a CFG
- how to convert a CFG into Chomsky normal form
- how to prove that a language is not context-free by applying the pumping lemma

Exercise 8 - 1/12/2004

Exercises on context-free grammars and pushdown automata [M3: Exercise 8]

Week 10: Properties of contex-free languages

Lectures 29,30 - 6/12/2004

Topics [M2: Part 7b]
- closure properties for context-free languages
- decision properties for context-free languages
What you should know after the lecture
- under which operations contex-free languages are closed and not closed
- which decision problems related to context-free can be solved by an algorithm, and what is the computational cost of the algorithm

Week 11: Turing Machines

Lectures 31,32 - 13/12/2004

Topics [M2: Part 8]
- 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
- a technique to prove that a problem is undecidable
- how one can exploit an undecidable problem to show that other problems are also undecidable

Lectures 33,34 - 15/12/2004

Topics [M2: Part 8]
- 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
- relationship between acceptance by a TM and acceptance by a FA/PDA

Exercise 9 - 15/12/2004

Exercises on closure properties of contex-free languages and Turing Machines [M3: Exercise 9]

Week 12: Extensions of Turing Machines

Lectures 35,36 - 20/12/2004

Topics [M2: Part 8]
- programming techniques for Turing Machines
  - storage in the state
  - multiple tracks
  - subroutines and procedure calls
- multi-tape Turing Machines
What you should know after the lecture
- how one can program a TM easier by imposing structure on states and tape symbols
- how a multi-tape TM can be simulated by a single-tape TM

Lectures 37,38 - 22/12/2004

Topics [M2: Part 8]
- running time of a Turing Machine
- 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

Exercise 10 - 22/12/2004

Exercises on nondeterministic Turing Machines and extensions of Turing Machines [M3: Exercise 10]

Week 13: Decidability and undecidability

Lectures 39,40 - 10/1/2005

Topics [M2: Part 9]
- classes of languages/problems
  - recursive/decidable languages
  - recursively enumerable (R.E.) languages
  - non-R.E. languages
- closure properties of recursive and R.E. languages
What you should know after the lecture
- how languages/problems can be classified
- how to prove closure properties of recursive and R.E. languages

Lectures 41,42 - 12/1/2005

Topics [M2: Part 9]
- showing languages to be nonrecursive/non-R.E.
- encoding Turing Machines as binary strings/integers
- enumerating binary strings/Turing Machines
- a non-R.E. language: the diagonalization languages
- a non-recursive language: the universal language
What you should know after the lecture
- how to encode a Turing Machine as a binary string
- how to prove that the diagonalization language is non-R.E.
- how to prove that the universal language is non-recursive

Exercise 11 - 12/1/2005

Exercises on recursive and R.E. languages [M3: Exercise 11]

Week 14: Computational complexity

Lectures 43,44 - 17/1/2005

Topics [M2: Part 9, M2: Part 10]
- Rice's theorem
- tractable and intractable problems
- the classes P and NP
- a problem in NP: SAT
What you should know after the lecture
- how to prove Rice's theorem
- how the classes P and NP are defined
- how to show a problem to be in NP

Lectures 45,46 - 19/1/2005

Topics [M2: Part 10]
- SAT and CSAT
- poly-time reductions
- NP-hardness and NP-completeness
- Cook's theorem
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

Exercise 12 - 19/1/2005

Exercises on NP-complete problems [M3: Exercise 12]

Week 15: Complexity classes

Lectures 47,48 - 24/1/2005

Topics [M2: Part 10]
- coNP-complete problems
- oracle Turing Machines
- the polynomial hierarchy and PSPACE
- the classes EXPTIME, EXPSPACE, and the exponential hierarchy
What you should know after the lecture
- what an oracle TM is
- how the polynomial hierarchy is defined
- typical problems complete for the classes EXPTIME and EXPSPACE

Exercise 13 - 26/1/2005

Exercises on exam topics [M3: Exercise 13]

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Last modified: Tuesday, 7-Jun-2005 1:56:01 CEST

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