Monadic Second-order Quantifier Research Articles

On the Expressive Power of 2-Stack Visibly Pushdown Automata

Visibly pushdown automata are input-driven pushdown automata that recognize some non-regular context-free languages while preserving the nice closure and decidability properties of finite automata. Visibly pushdown automata with multiple stacks have been considered recently by La Torre, Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express properties beyond the class of context-free languages. At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure. These results require a restricted version of visibly pushdown automata with multiple stacks whose behavior can be split up into a fixed number of phases. In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form. We show that they are expressively equivalent to the existential fragment of monadic second-order logic. Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt words with multiple nestings. Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation. Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown automata running on infinite (nested) words. Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic.

Partially Ordered Connectives and Monadic Monotone Strict NP

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57---104, 1998) set out to search for fragments $${\mathcal{L}}$$ of $$\Sigma_1^1$$ satisfying the dichotomy property: every problem definable in $${\mathcal{L}}$$ is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form $$\exists \vec{X}\forall \vec{x} \phi$$ , where $$\phi$$ is a quantifier-free formula in a relational vocabulary; and the latter is the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1---16, 1986), Sandu and Vaananen (Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 38, 361---372 1992), and MMSNP employing a generalization C of D introduced in the present paper.

On the expressive power of monadic least fixed point logic

Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper, we take a closer look at the expressive power of MLFP. Our results are: (1) MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy. (2) On strings with built-in addition, MLFP can describe at least all languages that belong to the linear time complexity class DLIN. (3) Settling the question whether addition-invariant MLFP = ? addition-invariant MSO on finite strings or, equivalently, settling the question whether MLFP = ? MSO on finite strings with addition would solve open problems in complexity theory: “ = ” would imply that PH = PTIME whereas “ ≠ ” would imply that DLIN ≠ LINH . Apart from this we give a self-contained proof of the previously known result that MLFP is strictly less expressive than MSO on the class of finite graphs.

The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

The monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict. The proof is based on automata theoretic ideas and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. Considering grids where the width is a function of the height, we prove that the difference between the levels k+1 and k of the monadic hierarchy is witnessed by a set of grids where this function is (k+1)-fold exponential. We then transfer the hierarchy result to the class of directed (or undirected) graphs, using an encoding technique called strong reduction. It is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first-order closure of existential monadic second-order logic. We also verify that these graph properties even belong to the complexity class NLOG, which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy.

Dot-depth, monadic quantifier alternation, and first-order closure over grids and pictures

This paper presents results from two different areas. The first area is monadic second-order logic (MSO) over finite structures, in particular over the so-called grids. These are structures whose elements can be arranged as a matrix and which have two binary relations corresponding to vertical and horizontal successors. For this logic, we study the expressive power of the alternation of existential and universal monadic second-order quantifiers, i.e., set quantifiers. In Matz et al. (Information and Computation, LICS’ 97, 1999, to appear) it had been shown that these alternations cannot be limited to a fixed number without loss of expressiveness. In this paper, we strengthen this result in several ways. Firstly, we show that there are MSO formulas that have a very restricted form of k+1 set quantifiers but are not equivalent to a formula with k quantifiers. Secondly, we show that if we fix the number of such alternations, the expressive power of formulas that start with a block of universal quantifiers differs from the power of those that start with an existential one—this was previously known only for coloured grids. Thirdly, we investigate how an additional prefix of first-order (i.e., element) quantifiers influences the expressive power of MSO formulas. The second area that this paper is concerned with is two-dimensional formal language theory. We study how the alternation of (first- and monadic second-order) quantifications, on the one hand, relates to the dot-depth measure of two-dimensional (i.e., picture) languages, on the other hand. That measure is the two-dimensional version of the classical notion of dot-depth for (one-dimensional) starfree word languages. We show that the hierarchy induced by this dot-depth cuts through the hierarchy given by monadic second-order quantifications. In particular, beyond each level of the monadic second-order alternation hierarchy, there is a starfree picture language.

Classifying regular events in symbolic logic

Regular events have been classified by star-height, star-free events by the “dot-depth hierarchy,” and events of dot-depth one by the so-called β-hierarchy. It is shown here that the latter two hierarchies have appealing characterizations in symbolic logic: Referring to a first-order language L in which star-free events are described, an event is shown to be of dot-depth n iff it is defined by a Boolean combination of L-sentences in prenex normal form with a Σ n -prefix. The β-hierarchy is characterized by the quantifier-rank of L-sentences with Σ 1-prefix. A similar characterization of star-height would require the inclusion of weak monadic second-order quantifiers; it is shown that the natural classification of events depending on these quantifiers will not yield on infinite hierarchy.