To show the organization of the course that includes this module, follow this link:  Course organization

Learning outcomes

In this courses, the relevant notions in complexity theory are introduced. Main themes are the relationship between complexity classes and Np-completeness theory. The scope of the courses is to give formal instruments useful in the analysis of the difficulty of computational problems.

Program

This is a condensed version of the program. The detailed program (with useful notes for the students) is available in the pdf file "Diario delle Lezioni"

1)Introduction
Computational models, computational resources, tractable problems and feasible algorithms.



2) Computational models and time complexity classes
Deterministic Turing Machine with 1 and k strings
Class TIME(f(n)).
Relationship between k-MdT e 1-MdT (theorem).
Random Access Machine. 

Simulation theorems TM-RAM. 

Thesis of sequential calculus.
Linear speed-up theorem and consequences.
The class P.

Examples of problems in P.


Non deterministic Turing machine(NTM).

Class NTIME(f(n)).

Relationship between NTM and TM.

The class NP.

Examples of problems in NP.
Characterization of problem in NP with polynomial verifier.
The class EXP.



4)Space complexity
.
Input-output TM.
Classes SPACE(f(n)) and NSPACE(f(n)).

Compression Theorem
Classes L e NL.

Examples of problems in L and NL.

Relationship between space and time onf I/O TM

5)Relationship between complexity classes
Proper function.
The reachability method.
Theorems: inclusions between time and space classes. Universal TM.
Lemmata for Time Hierarchy Theorem.

Time Hierarchy Theorem. Corollary P ⊂ EXP.

Space Hierarchy Theorem. Corollary L ⊂ PSPACE.
Gap's Theorem.

Savitch's Theorem with Corollary. Corollary PSPACE=NPSPACE.



6)Reduction and completeness
Reduction and logarithmic reduction.

Examples of reduction: HAMILTON PATH ≤log SAT, PATH ≤log CIRCUIT VALUE, CIRCUIT SAT ≤log SAT.

Examples of reduction by generalization.
Property of reduction: reflexivity and transitivity.
 C-Completeness for a language.

Closure of a class C with respect to reduction.
L, NL, P, NP, PSPACE and EXP are closed w.r.t ≤log.

Table method. Computational table
CIRCUIT VALUE is P-complete.

Cook's theorem.

Examples of NP-complete problems.

7)Some notions on the complement of non deterministic classes
coC
NP and coNP

Bibliography

Examination Methods

Teaching materials

•   Diario delle Lezioni (aggiornato all'ultima lezione)   (pdf, it, 91 KB, 19/05/11)