Expressive Power of Monadic Second-Order Logic and Modal μ-Calculus

Automata walking on graphs are a mathematical formalisation of autonomous mobile agents with limited memory operating in discrete environments. This paper establishes a framework in which to model and automatically verify that autonomous mobile agents correctly perform their tasks. The framework consists of a logical language tailored for expressing agent tasks, and an algorithm solving the parameterized verification problem, where the graphs are treated as the parameter. We reduce the parameterised verification problem to classic questions in automata theory and monadic second order logic, i.e., universality and validity problems.We illustrate the framework by instantiating it to a popular model of robot from the distributed computing literature.This work clarifies the border between classes of mobile-agent systems that have decidable parameterised verification problem, and those that do not.

ABSTRACT:(Labelled) transition systems are relatively common in theoretical computer science, chiefly as vehicles for operational semantics. The first part of this paper constructs a hierarchy of canonical transition systems and associated maps, aiming to give a strongly extensional theory of transition systems, where any two points with equivalent behaviors are identified. The cornerstone of the development is a notion of convergence in arbitrary transition systems, generalizing the idea of finite (n‐step) approximations to a given point. In particular, our canonical transition systems are also uniform spaces.The resulting hierarchy has very rich combinatorial (and topological) structure, and a lot of the first part of the paper is devoted to its study. We also discuss fixed points in this framework.In the second part of the paper, we show how to obtain a model of set theory with Aczel's Anti‐Foundation Axiom (AFA) from canonical transition systems constructed earlier. We study further the structure of the model thus obtained, and also give a few more abstract results, concerning consistency and independence in the presence of AFA.

Bidimensionality Theory is a meta-algorithmic theory whose main ingredients are the Grid Exclusion Theorem of the Graph Minors series of Robertson and Seymour and the celebrated Courcelle’s Theorem. The grid exclusion theorem states that if a graph excludes a bidimensional grid as a minor (a graph H is a minor of a graph G if H can be obtained by some subgraph of G by con- tracting edges) then its structure topologically resembles the structure of a tree (in technical terms, it has small treewidth). Intuitively, this result says that the absence of the bidimensional structure of a grid implies that a graph has the “mono-dimensional” structure of a tree. On the other side, Courcelle’s theorem states that if a problem on graphs is expressible in Monadic Second Order Logic (MSOL), then it is possible to solve it in linear time when the treewidth of their input graphs is fixed. Intuitively, this theorem expresses the fact that the mono- dimensional structure of a tree (i.e., small treewidth) makes it possible to treat a graph as the input string of a finite-state tree automaton, where the finiteness of its states is guaranteed by the MSOL-expressibility of the problem. Combin- ing these two results together, we derive that the absence of the bidimensional structure of a grid, enables the applicability of the “divide-and-conquer” tech- nique for problems of certain descriptive complexity. It appears that for many graph theoretic problems the existence of a grid-minor (or other bidimensional strucures) on the input graph provides a certificate for an immediate negative (or positive) answer and, for the remaining instances, a dynamic programming approach on graphs of bounded treewidth may give an answer to the problem. This phenomenon reveals fruitful interleave between graph structure and logic in graph algorithms. Bidimensionality Theory aims at systematizing this idea and extending its applicability in diverse paradigms of algorithm design.

We study query rewriting using views (QRV) for XML. Our queries and views are regular tree languages (RTLs) represented by tree automata over marked alphabets, where the markers serve as “node selectors”. We formally define query rewriting using views for RTLs and give an automata-based algorithm to compute the maximally contained rewriting. The formalism we use is equal in power with Monadic Second Order (MSO) logic, and our algorithm for computing QRV is the first to target this expressive class. Furthermore we prove a tight lower bound, thus showing that our algorithm is optimal. Another strength of our automata-based approach is that we are able to cast computing QRV into executing a sequence of intuitive operations on automata, thus rendering our approach practical as it can be easily implemented utilizing off-the-shelf automata toolboxes. Finally, we generalize our framework to account for more complex queries in the spirit of the FOR clause in XQuery. For this generalization as well, we give an optimal algorithm for computing the maximally contained rewriting of queries using views.KeywordsXMLView-Based RewritingTree Automata

We prove that every monadic second-order property of the unfolding<br />of a transition system is a monadic second-order property of the<br />system itself. We prove a similar result for certain graph coverings.

Completeness for μ-calculi: A coalgebraic approach

Fragments of Monadic Second-Order Logics Over Word Structures

17 Automata-theoretic techniques for temporal reasoning

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

We propose an algebraic semantics for the temporal logic CTL* and simplify it for its sublogics CTL and LTL. We abstractly represent state and path formulas over transition systems in Boolean left quantales. These are complete lattices with a multiplication that preserves arbitrary joins in its left argument and is isotone in its right argument. Over these quantales, the semantics of CTL* formulas can be encoded via finite and infinite iteration operators; the CTL and LTL operators can be related to domain operators. This yields interesting new connections between representations as known from the modal μ-calculus and Kleene/ω-algebra.

We consider a general class of timed automata parameterized by a set of “input-determined” operators, in a continuous time setting. We show that for any such set of operators, we have a monadic second order logic characterization of the class of timed languages accepted by the corresponding class of automata. Further, we consider natural timed temporal logics based on these operators, and show that they are expressively equivalent to the first-order fragment of the corresponding MSO logics. As a corollary of these general results we obtain an expressive completeness result for the continuous version of MTL.

This paper presents the investigation and comparison of TLC model checking method (TLA Checker) properties. There are two different approaches to method usage which are considered. The first one consists of a transition system states attendance by breadth-first search (BFS), and the second one by depth-first search (DFS). The Kripke structure has been chosen as a transition system model. A case study has been conducted, where composite web service usage scenario has been considered. Obtained experimental results are aimed at increasing the effectiveness of TLA+ specifications automated verification.

A possible approach to studying behavioural equivalences in labelled transition systems is that of characterizing them in terms of homomorphic transformations. This characterization permits relying on algebraic techniques for proving systems properties and reduces equivalence checking of two systems to studying the relationships among the elements of their structures. Different algebraic characterizations of bisimulation-based equivalences in terms of particular transition systems homomorphisms have been proposed in the literature. Here we show, by an example, that trace-based equivalences are not locally characterizable and thus that the above results cannot be extended to these equivalences. However, similar results can be obtained if we confine ourselves to restricted classes of transition systems. Here, the algebraic characterizations of three well known decorated-trace equivalences (ready trace, ready and failure equivalence) for tree-like structures are presented.

In this paper, the synthesis of bounded Petri nets from deterministic modal transition systems is shown to be undecidable. The proof is built from three components. First, it is shown that the problem of synthesising bounded Petri nets satisfying a given formula of the conjunctive nu-calculus (a suitable fragment of the mu-calculus) is undecidable. Then, an equivalence between deterministic modal transition systems and a language-based formalism called modal specifications is developed. Finally, the claim follows from a known equivalence between the conjunctive nu-calculus and modal specifications.

Many temporal and modal logic languages can be regarded as subsets of first order logic, i.e. the semantics of a temporal logic formula is given as a first order condition on points of the underlying models (Kripke structures). Often the set of possible models is restricted to models which are trees. A temporal logic language is (first order) expressively complete, if for every first order condition for a node of a tree there exists an equivalent temporal formula which expresses the same condition. In this paper expressive completeness of the temporal logic language with the set of operators U (until), S (since), and X k (k-next) is proved, and the result is extended to various other tree-like structures.