entire characterization of regular languages: provide necessary and sufficient conditions for regular languages

Statement: a language \(L\) is regular IFF the number of equivalence classes of \(\equiv_{L}\) is finite

corollary

\(L\) is not regular IFF there are infinitely many equivalance classes of \(\equiv_{L}\), meaning there are infinitely many strings $w_1, w_2, …$ such that \(w_{i} \neq w_{j}\) and those strings are also distinguishable to \(L\) meaning, there is at least one \(z \in \Sigma^{*}\) such that exactly one of \(w_{i}z\) and \(w_{j}z\) is in \(L\).

Main goals: assign a probability of each sequence of words existing:

\begin{equation} P(W) = P(w_1, \dots, w_{n}) \end{equation}

closely related is the NLG formulation of predicting an upcoming word:

\begin{equation} P(w_5|w_1, \dots, w_{n}) \end{equation}

either of these we call a “grammar”, or “Language Model”.

Chain Rule Language Modeling

Recall probability chain rule. Now, the probability of a sequence like:

\begin{equation} P(its\ water\ is\ so\ transparent) \end{equation}

gives:

\begin{equation} P(its) \times P(water|its) \times P(is | its\ water) \dots \end{equation}

NACC is a large, longitudinal dataset for neurodegentitive disease as a project in collaboration with Dr. Alyssa Weakley at UC Davis.

Dr. Alyssa Weakley is interested in

• Early Cognitive Change
• Mild Cognitive Impairment (MCI)

“How early can we detect, using NACC, change?”

dataset construction

• Participants are given a battery of mental capacity tests, these values are tracked over time
• There are also family member questionnaire
• Neuroimaging and biomarker data

Other things tracked in the data—

The National Banking Act unified Financial Markets.

natural numbers (\(\mathbb{N}\)) are the counting numbers: 1,2,3,4….

Zero is not part of it; this produces interesting results like set of natural number under addition is not a group because there is no identity (tbh nor inverse (inverse of 1 is -1 which is not in the set.))