Minimal DFAs for Testing Divisibility

The following exercise is typical in introductory texts on deterministic finite automata (DFAs): “produce an automaton that recognizes the set of binary strings that, when interpreted as binary numbers, are divisible by k.” For example, exercise 1.30 in [2] asks the student to prove that the language {x |x is a binary number that is a multiple of k} is regular for each k ≥ 1; explicitly presenting an automaton is the easiest solution. The traditional (and correct) answer constructs a k-state automaton that keeps track not only of divisibility by k, but also the current residue modulo k. For example, if the input read was 1101, the machine would remember “13 mod k”. The transitions between states are simple: if the automaton’s current state is “r mod k”, and the input symbol read is “0”, it moves to state (2r) mod k; if the input symbol read is “1”, it moves to state (2r + 1) mod k. (This example also generalizes to bases other than binary. Furthermore, even if the input string is encoded in base b, the canonical DFA will still have k states. It will, however, contain b transitions from each state.) The traditional answer, unfortunately, in general fails to produce a minimal DFA. This paper addresses the considerably more difficult question of “how many states does a minimal DFA that recognizes the set of base-b numbers divisible by k have?” We denote this number by fb(k) and derive a closed-form expression; in the proof, we also describe the states of the minimal DFA in more detail. The function fb(k) may be computed by algorithmic means. The author used two implementations of the Hopcroft minimization algorithm: an original Perl program and the highly-optimized AT&T FSM Package. According to experts in the field, no prior work addresses the general case of this problem except through such computational alleys.