1 DFA-NFA Equivalence🔗

As part of the proof that NFAs also recognize Regular Languages (as presented in the lecture version of Sipser 1.40), we needed a function \mathrm{CONVERT}_\textsf{DFA-NFA} : \textsf{DFA} \rightarrow \textsf{NFA} where given some input DFA M (where we have a proof that M \operatorname{IS-A} \texttt{DFA} Machine), \mathrm{CONVERT}_\textsf{DFA-NFA} M= an NFA that recognizes the same language as M.

For your Examples Table you may use the template below, or from lecture, but correct answers must demonstrate knowledge—of how an NFA computes, how it compares and differs to an analogous DFA’s computation, and an understanding of the specific transformation function above—by being exact and precise. Merely submitting a table from a different example will not receive any credit.

As usual, proofs and definitions in submitted answers must be valid mathematics, e.g., does not have unbound variables, follows correct syntax and formal definitions, and has correct types. Correct answers should not need any extraneous information, which will make the answer incorrect.

2 "Closed" Operation Practice🔗

Every programming language comes with built-in string functions and one of the most commonly used ones is string "join".

Say we have the following function on languages \mathrm{JOI} : \textsf{Language}\times\textsf{Language}\rightarrow\textsf{Language}, which lifts the string join operation to work on languages:

\mathrm{JOI}(C,D) = \left\{c\texttt{+}d \mid c\in C, d\in D\right\}

To simplify the problem, assume that both C and D always have the same alphabet \Sigma, and that the alphabet for \mathrm{JOI}(C,D) is \Sigma \cup \{\texttt{+}\}, where \texttt{+}\notin\Sigma.

Prove that set of regular languages is closed under the \mathrm{JOI} operation by:

• Using the proof terminology of this semester’s course, give the equivalent IF-THEN statement that must be proved (see lectures for examples of this kind of problem. usually it will say "in other words ...").

  Note: submissions that prove the wrong statement will not receive credit.

  You may ask on Piazza if your statement is correct. (But note that if you start the assignment too late and the course staff is no longer available, then you’ll have to proceed without getting this approval.)

• Give a proof of this IF-THEN statement.

  The proof must be properly formatted using the appropriate proof constructors and/or "Statements and Justifications" tables when appropriate

  Since we dont know the input languages, then any Examples Tables will need an additional "Justification" column, as explained in lecture.

  Your answer must define and use in the proof a function \mathrm{JOIN}_\textsf{NFA} : \textsf{NFA}\times\textsf{NFA}\rightarrow\textsf{NFA} that combines two NFA machines into another one with the appropriate behavior.

  Example Tables should have at least 4 examples, two in the joined language and two not in the joined language.

3 Another "Closed" Operation🔗

Another common string function available in nearly every programming language is string "upcase".

• Define a function \mathrm{UP} : \textsf{Language}\rightarrow\textsf{Language}, which lifts the string upcase operation to work on languages

  To simplify the problem, assume the input language always uses the alphabet containing the 26 lowercase letters, and that the output language uses the alphaget containing the 26 uppercase letters.

• Prove that set of Regular languages is closed under the \mathrm{UP}. The proof must use NFAs.