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+++ b/Formal Lang.md	
@@ -59,4 +59,99 @@ Symbol: Member of alphabet
 
 String: Sequence of symbols
 
-Language: Set of strings
\ No newline at end of file
+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
+$\{\varnothing\} \neq \{\epsilon\}$
diff --git a/Formal Midterm Review.md b/Formal Midterm Review.md
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+# Pumping Lemma
+
+Show $L = \{0^n1^n | n=03 \text{ is not regular}\}$.
+Assume $L$ is regular.
+
+Pick a string and some pumping length, $P$.
+
+$s=0^p1^p$
+
+Now show that for any way to divide the string into $xyz$, pumping $s$ results in a string outside of the language.
+
+$xy^2z$ contains symbols out of order, and more 0s than 1s.
+
+# 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
+
+# Chomsky Normal Form
+
+Let $G$ be a Context Free Grammar (CFG). $G$ is in Chomsky Normal Form provided that all rules are of the form
+1) $A \rightarrow AA$
+2) $A \rightarrow u$
+
+The start state cannot appear on the right hand side. Cannot $A \rightarrow \epsilon$ unless start state. 
+
+# NFA that's the union of two languages
+
+
+
+# Misc.
+
+$\text{REG} \in \text{CFL}$
+but
+$\text{REG} \neq \text{CFL}$
+
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