Minimal DFA for testing divisibility

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 present...

Minimal consistent DFA revisited

Minimal DFA for Symmetric Difference NFA

Recently, a characterization of the class of nondeterministic finite automata (NFAs) for which determinization results in a minimal deterministic finite automaton (DFA), was given in [2]. We present a similar result for the case of symmetric difference NFAs. Also, we show that determinization of any minimal symmetric difference NFA produces a minimal DFA.

State Complexity of Testing Divisibility

Under some mild assumptions, we study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m ≥ 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of mN in the Fibonacci system is exactly 2m.

non-divisibility for abelian groups

Throughout all groups are abelian. We say a group G is n-divisible if nG = G. If G has no non-zero n-divisible subgroups for all n>1 then we say that G is absolutely non-divisible. In the study of class C consisting   all absolutely non-divisible groups such as G, we come across the sub groups T_p(G) = the sum of all p-divisible subgroups and rad_p(G) = the intersection of all p^nG. The proper...