2.5 KiB
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.
-
Define a set X which is infinite in size - this doesn;t necesarrily need to be in the language.
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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:
$$ X = {(01)^i | i \geq 0} $$
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.
$$ X = {\epsilon, b, bb, ba} $$
