Everardo Bárcenas

Global Numerical Constraints on Trees

We introduce a logical foundation to reason on tree structures with constraints on the number of node occurrences. Related formalisms are limited to express occurrence constraints on particular tree regions, as for instance the children of a given node. By contrast, the logic introduced in the present work can concisely express numerical bounds on any region, descendants or ancestors for instance. We prove that the logic is decidable in single exponential time even if the numerical constraints are in binary form. We also illustrate the usage of the logic in the description of numerical constraints on multi-directional path queries on XML documents. Furthermore, numerical restrictions on regular languages (XML schemas) can also be concisely described by the logic. This implies a characterization of decidable counting extensions of XPath queries and XML schemas. Moreover, as the logic is closed under negation, it can thus be used as an optimal reasoning framework for testing emptiness, containment and equivalence.

Expressive Reasoning on Tree Structures: Recursion, Inverse Programs, Presburger Constraints and Nominals

The Semantic Web lays its foundations on the study of graph and tree logics. One of the most expressive graph logics is the fully enriched μ-calculus, which is a modal logic equipped with least and greatest fixed-points, nominals, inverse programs and graded modalities. Although it is well-known that the fully enriched μ-calculus is undecidable, it was recently shown that this logic is decidable when its models are finite trees. In the present work, we study the fully-enriched μ-calculus for trees extended with Presburger constraints. These constraints generalize graded modalities by restricting the number of children nodes with respect to Presburger arithmetic expressions. We show that the logic is decidable in EXPTIME. This is achieved by the introduction of a satisfiability algorithm based on a Fischer-Ladner model construction that is able to handle binary encodings of Presburger constraints.

Reasoning on expressive description logics with arithmetic constraints

Description logics (DL) is a well-known knowledge representation formalism. DL have been applied as reasoning framework in diverse domains, including the Semantic Web and Context-Aware Systems. It is an open question whether or not the expressive description logic ALCQIOreg is decidable. This logic is equipped with negation, conjunction, regular roles, inverse roles, nominals and qualified number restrictions. In this paper, we show this logic is decidable when interpreted over tree models. Moreover, it is shown that μALCTIO, which is known to be undecidable and that subsumes ALCQIOreg, is in EXPTIME in the case of tree models. μALCTIO generalizes regular roles with fixed-point constructors, and qualified number restrictions with arithmetic constraints. This EXPTIME bound holds even if the arithmetic constraints are coded in binary. Furthermore, we show that knowledge base reasoning, TBoxes and ABoxes, can also be decided in EXPTIME. These results are achieved via a polynomial reduction to the satisfiability problem of the propositional modal ¡-calculus extended with Presburger arithmetic constraints interpreted over tree models.

On the Model Checking of the Graded \(\mu \)-calculus on Trees

The \(\mu \)-calculus is an expressive propositional modal logic augmented with least and greatest fixed-points, and encompasses many temporal, program, dynamic and description logics. The model checking problem for the \(\mu \)-calculus is known to be in NP \(\cap \) Co-NP. In this paper, we study the model checking problem for the \(\mu \)-calculus extended with graded modalities. These constructors allow to express numerical constraints on the occurrence of accessible nodes (worlds) satisfying a certain formula. It is known that the model checking problem for the graded \(\mu \)-calculus with finite models is in EXPTIME. In the current work, we introduce a linear-time model checking algorithm for the graded \(\mu \)-calculus when models are finite unranked trees.

Depth-first search satisfiability of the μ-calculus with converse over trees

The μ-calculus is a modal logic with least and greatest fixed-point operators, encompassing many temporal, program and description logics such as LTL, PDL, CTL and ALCQIOreg. Many decision procedures have been proposed for the μ-calculus, however few implementations have been shown useful in practice. In this paper, we propose a satisfiability algorithm for the μ-calculus with converse interpreted on finite unranked trees. In contrast with current state of the art algorithms, mostly automata-based, we propose an algorithm based on a depth-first search. We prove the algorithm to be correct (sound and complete) and optimal (EXPTIME). We also provide an implementation, which shows significant performance improvement with respect to a known breadth-first search based algorithm.

On Regular Paths with Counting and Data Tests

Regular path expressions represent the navigation core of the XPath query language for semi-structured data (XML), and it has been characterized as the First Order Logic with Two Variables (FO2). Data tests refers to (dis)equality comparisons on data tree models, which are unranked trees with two kinds of labels, propositions from a finite alphabet, and data values from a possibly infinite alphabet. Node occurrences on tree models can be constrained by counting/arithmetic constructors. In this paper, we identify an EXPTIME extension of regular paths with data tests and counting operators. This extension is characterized in terms of a closed under negation Presburger tree logic. As a consequence, the EXPTIME bound also applies for standard query reasoning (emptiness, containment and equivalence).

Reasoning about the past on temporal specifications for motion planning

Due to the nice balance between the expressive power and the computational cost of associated algorithms, propositional temporal logic (PTL) has recently been used with great success as a specification language for robot motion planning, which is the problem of finding a collision-free route from an initial to a final configuration in a given environment. Elaborated specifications involving temporal ordering such as sequencing or coverage can be succinctly expressed by PTL formulae. The generation of plans satisfying PTL formulae are then reduced to the model checking problem, which concerns the satisfaction of PTL formulae with respect to a given model (in this case the environment). However, in case there is not a plan satisfying the given specification in the current environment, one may also be interested in finding a plan satisfying such specification in some other environment. In the present work, this problem is studied in the context of PTL specifications enriched with past operators. These constructs are used to express backward navigation in the environment. Other interesting reasoning problems such as the equivalence or subsumption of specifications are also studied.