On Regularity-Preserving Functions

1 Some Homework Exercises In introductory automata theory, one can nd a wealth of entertaining automata-theoretic puzzles such as the classical rst halves problem: Show that if A is a regular set, then so is the set of all rst halves of strings in A: FirstHalves(A) = fx j 9y jyj = jxj and xy 2 Ag : Once students master the basic pebbling technique for solving such problems, they can move on to more challenging variants: Show that if A is a regular set, then so are the following: A n 2 = fx j 9y jyj = jxj 2 and xy 2 Ag A 2 n = fx j 9y jyj = 2 jxj and xy 2 Ag A 2 2 n = fx j 9y jyj = 2 2 jxj and xy 2 Ag : Students are often quite surprised at rst that these sets should be regular, since the presence of the nonlinear functions seems to contradict their emerging intuition about regularity. An eeective tool in all these problems is the Boolean transition matrix of an automaton for A. This is the square Boolean matrix indexed by states of the automaton with 1 in position uv ii the automaton contains a transition u a ?! v for some symbol a. The Boolean powers n give the n-step transition relations. To solve the problem above for A 2 n, for example, one only has to determine how to get from 2 n to 2 n+1 in one step. This is done by squaring 1