Fragment Of Monadic Second-order Logic Research Articles

Communicating finite-state machines, first-order logic, and star-free propositional dynamic logic

Message sequence charts (MSCs) naturally arise as executions of communicating finite-state machines (CFMs), in which finite-state processes exchange messages through unbounded FIFO channels. We study the first-order logic of MSCs, featuring Lamport's happened-before relation. To this end, we introduce a star-free version of propositional dynamic logic (PDL) with loop and converse. Our main results state that (i) every first-order sentence can be transformed into an equivalent star-free PDL sentence (and conversely), and (ii) every star-free PDL sentence can be translated into an equivalent CFM. This answers an open question and settles the exact relation between CFMs and fragments of monadic second-order logic. As a byproduct, we show that first-order logic over MSCs has the three-variable property.

Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

We consider a specific class of tree structures that can represent basic structures in linguistics and computer science such as XML documents, parse trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We present axiomatizations of the monadic second-order logic (MSO), monadic transitive closure logic (FO(TC1)) and monadic least fixed-point logic (FO(LFP1)) theories of this class of structures. These logics can express important properties such as reachability. Using model-theoretic techniques, we show by a uniform argument that these axiomatizations are complete, i.e., each formula that is valid on all finite trees is provable using our axioms. As a backdrop to our positive results, on arbitrary structures, the logics that we study are known to be non-recursively axiomatizable.

Temporal logics over linear time domains are in PSPACE

We investigate the complexity of the satisfiability problem of temporal logics with a finite set of modalities definable in the existential fragment of monadic second-order logic. We show that the problem is in pspace over the class of all linear orders. The same techniques show that the problem is in pspace over many interesting classes of linear orders.

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.

The monadic theory of finite representations of infinite words

In this paper, we consider existential monadic second-order logic (monadic Σ 1 ) on finite unary graphs, that is finite graphs with functional edge relation. These can be seen as finite encodings of ultimately periodic words. We show that, on these graphs, the bisimulation-invariant fragment of monadic Σ 1 equals the bisimulation-invariant fragment of monadic second-order logic itself, and that MSO-definable bisimulation closed classes of graphs coincide with classes of graphs definable by means of (an extension of) finite state ω-word automata. This result can be seen as a translation, onto finite representations of infinite words, of Büchi's automata-theoretic characterization of S1S. In terms of descriptive complexity, this result contrasts with the situation on arbitrary unary structures where bisimulation invariant monadic Σ 1 properties only define languages that are closed with respect to the prefix topology.

Fragments of Monadic Second-Order Logics Over Word Structures

In this paper, we explore the expressive power of fragments of monadic second-order logic enhanced with some generalized quantifiers of comparison of cardinality over finite word structures.The full monadic second-order fragment of the logics that we study correspond to the famous linear hierarchy, see [Y. Hachaïchi, A descriptive complexity approach to the linear hierarchy, Theoretical Computer Science 304 (2003) 421–429], and their existential fragments characterize some sequential recognizers. We prove that the first-order closure of the existential fragments of these logics is strictly beyond the existential fragments.

Logical questions concerning the μ-calculus: Interpolation, Lyndon and Łoś-Tarski

The (modal) μ-calculus ([14]) is a very powerful extension of modal logic with least and greatest fixed point operators. It is of great interest to computer science for expressing properties of processes such as termination (every run is finite) and fairness (on every infinite run, no action is repeated infinitely often to the exclusion of all others).The power of the μ-calculus is also evident from a more theoretical perspective. The μ-calculus is a fragment of monadic second-order logic (MSO) containing only formulae that are invariant for bisimulation, in the sense that they cannot distinguish between bisimilar states. Janin and Walukiewicz prove the converse: any property which is invariant for bisimulation and MSO-expressible is already expressible in the μ-calculus ([13]). Yet the μ-calculus enjoys many desirable properties which MSO lacks, like a complete sequent-calculus ([29]), an exponential-time decision procedure, and the finite model property ([25]). Switching from MSO to its bisimulation-invariant fragment gives us these desirable properties.In this paper we take a classical logician's view of the μ-calculus. As far as we are concerned a new logic should not be allowed into the community of logics without at least considering the standard questions that any logic is bothered with. In this paper we perform this rite of passage for the μ-calculus. The questions we will be concerned with are the following.