add state machine and reg ex lecture

lectures/index.md

@@ -3,3 +3,4 @@
 * [languages and machines](./chomsky.md)
 * [all about names](./names.md)
 * [lambda calculus](./lambda.md)
+* [state machines and regular expression](./stateMachinesAndRegularExprs.md)

lectures/stateMachinesAndRegularExprs.md

@@ -0,0 +1,35 @@
+# Finite State Machines
+A **deterministic finite automata** is sometimes called a **DFA**. The DFA named, A, is defined mathematically with a 5-tuple. 
+
+A = (Q, Σ, δ, q<sub>0</sub>, F)
+
+* Q is a finite set of states
+* Σ is a finite set of symbols (an alphabet)
+* δ is a transition function δ:QxΣ->Q
+* q<sub>0</sub> is the start state
+* F is a set of final or accepting states F ⊆ Q
+
+**Σ*** denotes the set of all finite strings that can be made using letters from Σ.
+The empty string is denoted **ε**.
+
+The δ function has a natural extension to a function δ̂:QxΣ<sup>*</sup>->Q. We call this function **delta-hat**. It is defined to be the result of iteratively applying each letter in the second argument to produce the next state. 
+
+δ̂(q, 123) = δ(δ(δ(q<sub>0</sub>, 1), 2), 3)
+
+We can define the **language of a state machine** A in the following way:
+
+L(A) = {w∈Σ* | δ̂(q<sub>0</sub>, w)∈F}
+
+In plain English, The language of A is the set of all words from the alphabet Σ that will cause the delta-hat function to return a final state when beginning at the start state.
+
+# Regular Expressions
+The language of finite state machines are the regular expressions. There are five rules the define a regular expression.
+
+* A symbol taken from the alphabet is a regular expression.
+* If E is a regular expression, then (E) is also a regular expression with the same language. L(E) = L((E)).
+* If A and B are regular expressions, then A+B is a regular expression. L(A+B) == L(A) ∪ L(B)
+* If A and B are regular expressions, then AB is a regular expression. (concatenation A followed by B) L(AB) = L(A) X L(B)
+* If A is a regular expression, then A* is a regular expression. (closure zero or more occurrences of A)
+The precedence of the last three operators are in order from lowest precedence, to highest.
+
+