The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by $\omega$-terms (this property is also known as $\omega$-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable

The FO^2 alternation hierarchy is decidable

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Th\'erien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and Vollmer

The word problem for omega-terms over the Trotter-Weil hierarchy [extended abstract]

© Springer International Publishing Switzerland 2016. Over finitewords, there is a tight connection between the quantifier alternation hierarchy inside two-variable first-order logic FO 2 and a hierarchy of finite monoids: theTrotter-Weil Hierarchy. The variousways of climbing up this hierarchy include Mal’cev products, deterministic and codeterministic concatenation as well as identities of ω-terms.We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable; this means, for every variety V of the hierarchy and every identity u = v of ω-terms, one can decide whether all monoids in V satisfy u = v. More precisely, for every fixed variety V, our approach yields nondeterministic logarithmic space (NL) and deterministic polynomial time algorithms, which are more efficient than straightforward translations of the NL-algorithms. From a language perspective, the word problem for ω- terms is the following: for every language variety V in theTrotter-Weil Hierarchy and every language varietyWgivenbyan identity of ω-terms, one can decide whether V ⊆ W. This includes the case where V is some level of the FO 2 quantifier alternation hierarchy. As an application of our results, we show that the separation problems for the so-called corners of the Trotter- Weil Hierarchy are decidable

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 $x<y$ and predicates for $x \equiv i \mod n$ 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]

Block products and nesting negations in FO2

The alternation hierarchy in two-variable first-order logic FO 2 [∈ < ∈] over words was recently shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. In this paper we consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment of FO 2 is defined by disallowing universal quantifiers and having at most m∈-∈1 nested negations. One can view as the formulas in FO 2 which have at most m blocks of quantifiers on every path of their parse tree, and the first block is existential. Thus, the m th level of the FO 2 -alternation hierarchy is the Boolean closure of. We give an effective characterization of, i.e., for every integer m one can decide whether a given regular language is definable by a two-variable first-order formula with negation nesting depth at most m. More precisely, for every m we give ω-terms U m and V m such that an FO 2 -definable language is in if and only if its ordered syntactic monoid satisfies the identity U m ∈V m. Among other techniques, the proof relies on an extension of block products to ordered monoids. © 2014 Springer International Publishing Switzerland

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

26. Theorietag Automaten und Formale Sprachen 23. Jahrestagung Logik in der Informatik: Tagungsband

Der Theorietag ist die Jahrestagung der Fachgruppe Automaten und Formale Sprachen der Gesellschaft für Informatik und fand erstmals 1991 in Magdeburg statt. Seit dem Jahr 1996 wird der Theorietag von einem eintägigen Workshop mit eingeladenen Vorträgen begleitet. Die Jahrestagung der Fachgruppe Logik in der Informatik der Gesellschaft für Informatik fand erstmals 1993 in Leipzig statt. Im Laufe beider Jahrestagungen finden auch die jährliche Fachgruppensitzungen statt. In diesem Jahr wird der Theorietag der Fachgruppe Automaten und Formale Sprachen erstmalig zusammen mit der Jahrestagung der Fachgruppe Logik in der Informatik abgehalten. Organisiert wurde die gemeinsame Veranstaltung von der Arbeitsgruppe Zuverlässige Systeme des Instituts für Informatik an der Christian-Albrechts-Universität Kiel vom 4. bis 7. Oktober im Tagungshotel Tannenfelde bei Neumünster. Während des Tre↵ens wird ein Workshop für alle Interessierten statt finden. In Tannenfelde werden • Christoph Löding (Aachen) • Tomás Masopust (Dresden) • Henning Schnoor (Kiel) • Nicole Schweikardt (Berlin) • Georg Zetzsche (Paris) eingeladene Vorträge zu ihrer aktuellen Arbeit halten. Darüber hinaus werden 26 Vorträge von Teilnehmern und Teilnehmerinnen gehalten, 17 auf dem Theorietag Automaten und formale Sprachen und neun auf der Jahrestagung Logik in der Informatik. Der vorliegende Band enthält Kurzfassungen aller Beiträge. Wir danken der Gesellschaft für Informatik, der Christian-Albrechts-Universität zu Kiel und dem Tagungshotel Tannenfelde für die Unterstützung dieses Theorietags. Ein besonderer Dank geht an das Organisationsteam: Maike Bradler, Philipp Sieweck, Joel Day. Kiel, Oktober 2016 Florin Manea, Dirk Nowotka und Thomas Wilk