The language L of strings over alphabet Σ = {0, 1} which (in binary) represent even numbers is a regular language. Show this by giving both a. a regular expression and b. by drawing a finite state automaton (FSA) that recognizes strings from this language.

For any string w 6=/ E, the principal suffix of w is the string resulting by removing the first symbol from w. We will denote this string by ps(w). For any language L, define ps(L) := {ps(w) : w ∈ L ∧ w 6=/ E}. Show that if L is regular, then ps(L) is regular. (The underlying alphabet is arbitrary.) Show by giving rules to transform a regex for L into a regex for ps(L).

Consider the following finite-state automaton A over E = {a, b, c}: A = (E, {qo, 91; 92, 93 } , 8, 90, {92}) %3D where d = {(q0, a, q1), (q1, b, q0), (q1, b, q2), (q2, a, 91), (92, b, q2), (q2, a, q3), (q3, c, q3), (q3, a, q2)}. Draw the automaton A as a transition graph.

A useless state in a machine is one that is never encountered on any input string. Consider the problem of determining whether a pushdown automaton has any useless states.  Formulate this problem as a language, and show that it’s decidable.