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$$\}$$

$$\Sigma=\{0,1,...,9,+,=\}$$
$$L_{add}=\{0+0=0,...\}$$