Unary Self-verifying Symmetric Difference Automata

Abstract

We investigate self-verifying nondeterministic finite automata, in the case of unary symmetric difference nondeterministic finite automata (SV-XNFA). We show that there is a family of languages L n≥2 which can always be represented non-trivially by unary SV-XNFA. We also consider the descriptional complexity of unary SV-XNFA, giving an upper and lower bound for state complexity.

Descriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata

“…In [6], self-verifying symmetric difference automata (SV-XNFA) were defined as a combination of the notions of symmetric difference automata and self-verifying automata, but only the unary case was examined. We now restate the definition of SV-XNFA in order to present results on larger alphabets in Section 4.…”

“…We now restate the definition of SV-XNFA in order to present results on larger alphabets in Section 4. Note, however, that the definition is slightly amended: in [6], the implicit assumption was made that no SV-XNFA state could be both an accept state and a reject state. This assumption is explored in detail for the unary case in [5], but for our current purposes it suffices to say that such a requirement removes the equivalence between XNFA and weighted automata over GF (2), which is essential for certain operations on XNFA, such as minimisation [7].…”

“…We now work towards establishing a lower bound on state complexity. First, we restate the following lemma from [6] for the unary case.…”

“…We now present a witness language for any n to show that 2 n−1 is a lower bound on the state complexity of SV-XNFA with non-unary alphabets. First, we restate the following theorem from [6].…”

“…It was shown in [4] that e Θ √ n ln n is an upper bound for the unary case, but not a tight bound, while in the non-unary case, g(n), where g(n) grows like 3 n 3 , is a tight upper bound. In [6], we extended the notion of self-verification (SV) to XNFA to obtain SV-XNFA. We showed that 2 n − 1 is not a tight upper bound for SV-XNFA in the case of a unary alphabet.…”