Lecture 01

Overview

"To understand the fundamental capabilities and limitations of computers."

To determine which questions we can answer.

Pythagorean Triples (\(a^2+b^2=c^2\)) Proposition

There are infinite number of integer triples (\(a,b,c\)) such that \(a^2+b^2=c^2\).

Proof: If (\(a,b,c\)) is a pythagorean triple, so is (\(ka,kb,kc\)) for any integer \(k\).

Fermat's Last Theorem

For any \(n>2\), \(a^n+b^n=c^n\) has no integer solutions (conjecture).

Hilbert's Tenth Problem

Solve FLT and all its natural variants. (Polynomial equation of finite degree with a finite number of variables). Solution required: algorithm.

Davis, Putnam, Robinson, Matiyasevich Theorem

Hilbert's Tenth Problem cannot be solved.

Language

Definition:

Main Observation

Any decision problem (Y/N question) can be described as a language.

E.g. is \(k\) prime?

Is \(k\in\Sigma^*\) in \(L_{prime}\)?

E.g. is the graph \(G\) connected?

Fix: \(\Sigma=\{v,0,1,2,...,9\}\)

Another \(\Sigma=\{0,1\}\), \(L_{conn}=\{S\in\Sigma^*: S\) is matrix representation of a connected graph\(\}\)

E.g. Addition! => Additive Identities

\(\Sigma=\{0,1,...,9,+,=\}\)

\(L_{add}=\{0+0=0,...\}\)