Started working on CS Theory Review Post.

blogContent/posts/programming/CS -Theory-Exam-2-Review.md

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+This a very high level review post that I am making for myself and other people taking CS Theory.
+If you want to lean about the theory behind the content in this blog post I recommed looking else where.
+This post will cover how to solve typical problems relating to topics covered by my second CS Theory exam.
+
+## Myhill-Nerode Theorem
+
+### Definition
+L is regular if and only if it has a finite index. The index is the maximum number of elements thar are pairwise distibguishable.
+Two strings are said to be pairwise distinguishable if you can append something to both of the strings and it makes one string
+accepted by the language and the other string non-accepting. 
+The size of an index set X equals the number of equivalence classes it has. Each element in the language is accepted by only
+one equivalence class.
+
+### Problem Approach
+
+Prove that language L is regular.
+
+1) Define a set X which is infinite in size - this doesn;t necesarrily need to be in the language.
+
+2) Make a general argument that show that each element in X is pairwise distinguishable.
+Pick any two elements x, y in X and show that if you append z to them one is accepted by the language and 
+the other is not in the language.
+
+### Example
+
+Prove the following language is non-regular:
+
+$$
+L={ww^r | w \in {0,1}^*}
+$$
+
+answer:
+
+1)
+
+$$
+X = {(01)^i | i \geq 0}
+$$
+
+Pick any 2 elements of X and show pairwise distinguishable
+
+$$
+x = (01)^i, y = (01)^j  | i \neq j
+$$
+
+suppose we pick
+$$
+z = (10)^i\\
+xz \in L\\
+yz \notin L
+$$
+
+
+## DFA minimization algorithm
+
+Types of Problems:
+- Prove DFA is minimal
+- Minimize the DFA
+
+The argument for DFA minimization comes from the Myhill-Nerode theorem. Given
+a DFA, if you can form a set of strings which represent each state and they are all 
+pairwise distinguishable, then the DFA is minimal with that many states.
+
+
+### Prove DFA is minimal
+
+For these types of problems you simply construct a table and show that each state is pairwise distinguishable.
+To show pairwise distinguishability you have to show that there exists a string where if appened to one element 
+makes it accepted by the language but pushes the other string out of the language.
+
+ex: Prove the following DFA is minimal.
+
+![DFA Example](media/CSTHEORY/DFAMinimalProof.png)
+
+$$
+X = {\epsilon, b, bb, ba}
+$$
+
+![DFA Example](media/CSTHEORY/Table.png)
+
+
+### Minimize the DFA
+
+
+
+## Pumping lemma for regular languages
+
+## Context-free grammars, closure properties for CFLs
+
+## Parse trees, ambiguity
+
+## Chomsky Normal Form
+
+## Pushdown automata
+
+## Construction to convert CFG to a PDA
+