3 research outputs found
One Quantifier Alternation in First-Order Logic with Modular Predicates
Adding modular predicates yields a generalization of first-order logic FO
over words. The expressive power of FO[<,MOD] with order comparison and
predicates for has been investigated by Barrington,
Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated
by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that
definability in the two-variable fragment FO2[<,MOD] is decidable. In this
paper we continue this line of work.
We give an effective algebraic characterization of the word languages in
Sigma2[<,MOD]. The fragment Sigma2 consists of first-order formulas in prenex
normal form with two blocks of quantifiers starting with an existential block.
In addition we show that Delta2[<,MOD], the largest subclass of Sigma2[<,MOD]
which is closed under negation, has the same expressive power as two-variable
logic FO2[<,MOD]. This generalizes the result FO2[<] = Delta2[<] of Therien and
Wilke to modular predicates. As a byproduct, we obtain another decidable
characterization of FO2[<,MOD]
The half-levels of the FO2 alternation hierarchy
© 2016, Springer Science+Business Media New York. The alternation hierarchy in two-variable first-order logic FO 2 [ < ] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. We consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment Σm2 of FO 2 is defined by disallowing universal quantifiers and having at most m−1 nested negations. The Boolean closure of Σm2 yields the m th level of the FO 2 -alternation hierarchy. We give an effective characterization of Σm2, i.e., for every integer m one can decide whether a given regular language is definable in Σm2. Among other techniques, the proof relies on an extension of block products to ordered monoids