https://scribbletogether.com/whiteboard/4E34104E-2EF5-4249-8C6A-B427580E8F76

Overview

• One model for all inputs
• Single-pass, fixed memory
• DFAs closed under NOT, AND, OR

Today:

• More operations on DFAs
• Non-determinism
• NFA

Concatenation

Let $f_1$ and $f_2$ take bit strings and output a bit. $f_1\, \mathrm{CAT}\,f_2$ returns $1$ if we can split up the input such that $f_1(x_1) = 1$ and $f_2(x_2) = 1$.

Let $L_1$ and $L_2$ be languages.

$L_1\cdot L_2 = \{ x_1\cdot x_2 | x_1\in L_1, x_2\in L_2\}$

Example

$f_1(x) = 1$ for all $x$.

$f_2(x) = 1$ if $x$ starts with a $1$ and has length exactly $4$.

What is the function concatenating the 2?

It returns $1$ if the 4th bit from the end is a $1$.

Is this computable by a DFA? Yes.