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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.
-
- 
-
- $$
- X = {\epsilon, b, bb, ba}
- $$
-
- 
-
-
- ### 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
-
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