CMPS 260 Spring 2024
Preview of April 17 exam

Generally speaking, you should expect the exam to include problems similar to those that appeared in the homework assignments.

More specifically, you should be prepared to solve problems of the following types:

• Given an informal, but precise, description of a regular language, devise either a finite automaton (possibly restricted to being deterministic) that accepts the language or a regular expression that denotes the language.
• Given a (possibly nondeterministic) finite automaton, or a regular expression, provide a precise (even if not necessarily formal) description of the language that it accepts/denotes.
• Given a finite automaton M, devise a regular expression r such that they represent the same language (i.e., L(M) = L(r)). Or vice versa. (You will not be expected to carry out the standard NFA-to-regular expression algorithm, however, which usually produces a very complex regular expression. Instead, use your intuition.)
• Given a nondeterministic finite automaton (NFA), convert it to an equivalent deterministic finite automaton (DFA) using the "subset construction" algorithm.
• Given the description of some language operator, justify the claim that regular languages are closed under that operator. Of course, you are familiar with set-theoretic operators such as union/intersection/complement and the language operator concatenation. Another example is MAX(L), which was the basis of Problem 4 of HW #2.

  For a unary operator (such as MAX), typically this is done by showing that an FA that accepts L can be (algorithmically) modified to become an FA that accepts op(L). For a binary operator, typically this is done by showing that an FA accepting L₁ op L₂ can be built using FA's that accept L₁ and L₂.

  For some operators, a construction based upon regular expressions (or right-linear grammars) could work better.

• Given a DFA M, identify the equivalence classes of the state-indistinguishability relation and, based upon those, construct the equivalent minimal DFA M'.
• Given an informal, but precise, description of a context-free language (CFL), devise a context-free grammar (CFG) that generates that language.
• Given a CFG G, provide a precise (even if not necessarily formal) description of the language that it generates (L(G)).
• Given an ambiguous CFG, show that it is ambiguous.
• Given a Chomsky Normal Form CFG G and a (short) string x, apply the CYK algorithm to determine whether x ∈ L(G).