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Complexity theory and combinatorial optimization Class #2 – 17 th of March e we deal with decision problems, finite automata, Turing mac nk dogs, …. o P, NP, NP-completeness, …..
Complexity theory and combinatorial optimizationClass #2 – 17th of March
…. where we deal with decision problems, finite automata, Turing machines pink dogs, ….
But also P, NP, NP-completeness, …..
Introduction to computational intractability
Is my problem efficiently solved by a computer?
an automatic m
achine
an algorithm?
What’s a problem?
Decision problems: each instance is a question
Formal definition with language theory
More natural problems: “meta-language”
encoding scheme
That is the problem ….
What’s an algorithm?
• The pink dog question
Does a pink dog exist?Does a pink dog exist (outside London)?
Since the answer is yes, it can be answered
Since there are only a finite number of dogs (outside London) and since for each one a can decide whether it is pink or notit can be answered.
Will be ever exist any pink dog outside London?
To answer it one needs a formal model of dogs.
What’s an algorithm?
• Computability models
o Lambda-calculus (A. Church, 1931)o General recursive functions (K. Gödel, 1934)o Turing machines (A. Turing, 1936)
o Random-Access Machines, …
• Church thesis
Before the first computer
• The pink dog question
What’s an algorithm: the Turing machine model
• From finite states automaton ….. to Turing machines
• 1-tape (deterministic) Turing Machine (DTM)
• multi-tape Turing machines
• non-deterministic Turing machines (NDTM)
o transition function transition relationo put non-determinism at the beginning
• equivalence between all these Turing machines models
• universal Turing Machine
Example
Problems solved by Turing machines
• Is L recognized by M?
• Is L decided by L?
• The halting problem: an example of undecidable problem
M: DTM, L a language on the same input alphabet
Decision problem solved by an algorithm?(through an encoding scheme)
Complexity of Turing machines
• Complexity of DTM (halting for each instance)
• Complexity of NDTM
• Polynomial-time: considered as efficiency (Cobham-Edmond’s thesis)
• Difference between DTM and NDTM (from complexity point of view)
• From languages to problems (reasonable encoding schemes)
A notion of efficiency
P, NP and NP-completeness
• The class P
• The class NP
• Exponentially solving problems in NP
• Polynomial reductions
• NP-complete problems
Some NP-complete problems
• SAT
• Cook’s theorem (1971)
•How to prove NP-completeness after Cook?
• 3-SAT
• to be continued during the next class
Enjoy your vacation
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What is the accepted language?
