# 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:

- Alphabet is a finite set of symbols \(\Sigma\).
- String over an alphabet is a finite sequence of symbols from this alphabet \(\Sigma\)
- Notation for \(n\geq 0\) \(\Sigma^{n}\) is the set of all sequence of \(n\) symbol from \(\Sigma\).
- \(\Sigma^{*}=\cup_{n\geq 0}\Sigma^{n}\)
- A character is a symbol
- Language \(L\) over the alphabet \(\Sigma\) is a set of strings over \(\Sigma\) (\(L\subseteq\Sigma^{*}\))

### Main Observation

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

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

- \(\Sigma=\{0,1,...,9\}\)
- \(L_{prime}=\{2,3,5,...\}\)

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

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

- \(\Sigma=\{v_1, v_2, ...\}\), \(L_{conn}=\{v_1v_2, v_1v_3, ...\}\)
**Not Acceptable, \(\Sigma\) is infinite**

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

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

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