# Chapter 0

dont do np hard problems

# Define

## Computability Theory

1930s - 1950s

What can be computed?

## Automata Theory

1930s - 1950s

How does making changes o the underlying model effect computational power?

## Complexity Theory

1960s - Present

What is computable in practice?

P vs. NP

# Graph Theory?

## Define

### Subgraph

Let $G = \{V_G, E_G\}$ and $H = \{V_H, E_H\}$

$G$ is a subgraph of $H$ if
1. $V_A = V_H$
2. $E_G \subset E_H + \forall(v_1, v_2)$

### Path
Hits vertices in a graph

Simple Graph: only hits vertices once

Cycle: There's some sort of loop in the graph

Simple Cycle: Only repeated node is the first and last one

Tree

Directed Graph: a graph that is made up of a set of vertices connected by directed edges

Strongly Connected: A directed graph is strongly connected if a directed path connects every two nodes.

# Strings & Languages

Alphabet: Non-empty finite set

Symbol: Member of alphabet

String: Sequence of symbols

Language: Set of strings

# Lecture Jan 24

idk he was talking about proofs in stuff I wasn't paying attention

# Lecture Jan 27

## 1.1 Finite Automata

One of the simplest models of computation is the finite state machine also called finite automata

### Example

Smoke Alarm

| Sensors      | Values               |
| ------------ | -------------------- |
| Smoke Sensor | Smoke, No smoke      |
| Button       | Not pressed, pressed |

#### State Diagram

![[fsm-diagram.png]]

#### State Transition Table

|          | No Smoke, unpressed | No smoke, pressed | Smoke, unpressed | Smoke, pressed |
| -------- | ------------------- | ----------------- | ---------------- | -------------- |
| No Alarm | No Alarm            | No Alarm          | Alarm            | No Alarm       |
| Alarm    | No Alarm            | No Alarm          | Alarm            | No Alarm       |


### Formal Definition of a Finite State Machine

A finite automaton is a 5 tuple (Q, $\Sigma$, $\delta$, q, F) where
1) Q is a finite set called the states
2) $\Sigma$ is a finite set called the _alphabet_
3) $\delta = Q \times \Sigma \rightarrow Q$ is the transition function
4) $q \in Q$ is the start state
5) $F \subset Q$ is the set of accept state

# Lecture Jan 29

__Define__ A language is called a _regular language_ if some FSM recognizes it.

# Lecture Feb 5

The transition function is taking in the current state ($Q$), a symbol ($\Sigma$), and outputs a new state ($Q$).

DFA - Determinant Finite Automata

$\delta((q_{01}, q_2), b) = (\delta(q_{01}, b), \delta(q_2, b))$

# Lecture Feb 7th

![[Pasted image 20250207085048.png]]

^ NFA that accepts any string that has a zero three places from the end

## Formal Definition of an NFA

An NFA is a 5 tuple ($Q$, $\Sigma$, $\delta$, $q_o$, F) where

1) $Q$ is a finite set of states
2) $\Sigma$ is the alphabet
3) $\delta : Q \times \Sigma_\epsilon \rightarrow P(Q)$
4) $q_0 \in Q$


# Lecture Feb 10th

When reading a particular symbol $a$ in a state $R_1$ our DFA simulator will apply the following the transition $f^n$

$\delta^1(R,a) = {r \in R} | \delta(r,a)$

## NFA to DFA example

![[NFA-to-DFA.png]]

Top is an NFA, lower is DFA version of it

# Lecture Feb 14th

$A^\star = \{x | x \in \{0, \text{None}\}\}$
$B^\star = \{x | x \in \{1, \text{None}\}\}$

Is $01 \in \{A^\star \cup B^\star \}$? No.
$\{A^\star \cup B^\star \} = \{\{0\}^\star \cup \{1\}^\star \}$

Is $01 \in (A \cup B)^\star$? Yes.
$= (0 \cup 1)^\star$
$A \cup B = \{0, 1\}$

Empty set != empty string
$\{\varnothing\} \neq \{\epsilon\}$