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Temporal logics for concurrent recursive programs: Satisfiability and model checking
- In MFCS’11, volume 6907 of LNCS
, 2011
"... Abstract. We develop a general framework for the design of temporal logics for concurrent recursive programs. A program execution is modeled as a partial order with multiple nesting relations. To specify properties of executions, we consider any temporal logic whose modalities aredefinable in monadi ..."
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Abstract. We develop a general framework for the design of temporal logics for concurrent recursive programs. A program execution is modeled as a partial order with multiple nesting relations. To specify properties of executions, we consider any temporal logic whose modalities aredefinable in monadic second-order logic and that, in addition, allows PDL-like path expressions. This captures, in a unifying framework, a wide range of logics defined for ranked and unranked trees, nested words, and Mazurkiewicz traces that have been studied separately. We show that satisfiability and model checking are decidable in EXPTIME and 2EXPTIME, depending on the precise path modalities. 1
Combining temporal logics for querying XML documents
- In International Conference on Database Theory
, 2006
"... Abstract. Close relationships between XML navigation and temporal logics have been discovered recently, in particular between logics LTL and CTL ⋆ and XPath navigation, and between the µ-calculus and navigation based on regular expressions. This opened up the possibility of bringing model-checking t ..."
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Abstract. Close relationships between XML navigation and temporal logics have been discovered recently, in particular between logics LTL and CTL ⋆ and XPath navigation, and between the µ-calculus and navigation based on regular expressions. This opened up the possibility of bringing model-checking techniques into the field of XML, as documents are naturally represented as labeled transition systems. Most known results of this kind, however, are limited to Boolean or unary queries, which are not always sufficient for complex querying tasks. Here we present a technique for combining temporal logics to capture nary XML queries expressible in two yardstick languages: FO and MSO. We show that by adding simple terms to the language, and combining a temporal logic for words together with a temporal logic for unary tree queries, one obtains logics that select arbitrary tuples of elements, and can thus be used as building blocks in complex query languages. We present general results on the expressiveness of such temporal logics, study their model-checking properties, and relate them to some common XML querying tasks. 1
Wreath Products of Forest Algebras, with Applications to Tree Logics
"... Abstract—We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations ..."
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Abstract—We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics. I.
On the Satisfiability of Two-Variable Logic over Data Words
"... Data trees and data words have been studied extensively in connection with XML reasoning. These are trees or words that, in addition to labels from a finite alphabet, carry labels from an infinite alphabet (data). While in general logics such as MSO or FO are undecidable for such extensions, decida ..."
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Data trees and data words have been studied extensively in connection with XML reasoning. These are trees or words that, in addition to labels from a finite alphabet, carry labels from an infinite alphabet (data). While in general logics such as MSO or FO are undecidable for such extensions, decidablity results for their fragments have been obtained recently, most notably for the two-variable fragments of FO and existential MSO. The proofs, however, are very long and nontrivial, and some of them come with no complexity guarantees. Here we give a much simplified proof of the decidability of two-variable logics for data words with the successor and data-equality predicates. In addition, the new proof provides several new fragments of lower complexity. The proof mixes database-inspired constraints with encodings in Presburger arithmetic.
Unranked tree automata with sibling equalities and disequalities
, 2006
"... We propose an extension of the tree automata with constraints between direct subtrees (Bogaert and Tison, 1992) to unranked trees. Our approach uses MSO-formulas to capture the possibility of comparing unboundedly many direct subtrees. Our main result is that the nonemptiness problem for the deter ..."
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We propose an extension of the tree automata with constraints between direct subtrees (Bogaert and Tison, 1992) to unranked trees. Our approach uses MSO-formulas to capture the possibility of comparing unboundedly many direct subtrees. Our main result is that the nonemptiness problem for the deterministic automata, as in the ranked setting, is decidable. It turns out that the main difficulty is indeed the absence of the rank, as it gives a certain bound on the number of distinct subtrees needed in order to satisfy an equality or disequality constraint. We overcome this difficulty by finding such a bound via a brute-force method.
Node Selection Query Languages for Trees
"... The study of node-selection query languages for (finite) trees has been a major topic in the recent research on query languages for Web documents. On one hand, there has been an extensive study of XPath and its various extensions. On the other hand, query languages based on classical logics, such as ..."
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The study of node-selection query languages for (finite) trees has been a major topic in the recent research on query languages for Web documents. On one hand, there has been an extensive study of XPath and its various extensions. On the other hand, query languages based on classical logics, such as first-order logic (FO) or monadic second-order logic (MSO), have been considered. Results in this area typically relate an Xpath-based language to a classical logic. What has yet to emerge is an XPath-related language that is expressive as MSO, and at the same time enjoys the computational properties of XPath, which are linear query evaluation and exponential query-containment test. In this paper we propose µXPath, which is the alternation-free fragment of XPath extended with fixpoint operators. Using two-way alternating automata, we show that this language does combine desired expressiveness and computational properties, placing it as an attractive candidate as the definite query language for trees.
Algorithmic meta theorems for circuit classes of constant and logarithmic depth
, 2011
"... An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic second-order (mso) definable problems are linear-time solvable on gr ..."
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An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic second-order (mso) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that mso-definable problems are (a) solvable by uniform constant-depth circuit families (AC 0 for decision problems and TC 0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC 1 for decision problems and #NC 1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC 0-completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC 1. Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata’s computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC 0, which encapsulates most of the technical difficulties surrounding earlier results connecting tree automata and NC 1.
An easy completeness proof for the modal µ-calculus on finite trees
- FOSSACS 2010, volume 6014 of LNCS
"... Abstract. We give a complete axiomatization for the modal μ-calculus on finite trees. While the completeness of our axiomatization already follows from a more powerful result by Igor Walukiewicz in [11], our proof is easier and uses very different tools, inspired from model theory. We show that our ..."
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Abstract. We give a complete axiomatization for the modal μ-calculus on finite trees. While the completeness of our axiomatization already follows from a more powerful result by Igor Walukiewicz in [11], our proof is easier and uses very different tools, inspired from model theory. We show that our approach generalizes to certain axiomatic extensions, and to the extension of the μ-calculus with graded modalities. We hope that the method might be helpful for other completeness proofs as well. The μ-calculus is an extension of modal logic with a fixpoint operator. In 1983, Dexter Kozen suggested an axiomatization and showed completeness for the aconjunctive fragment of the μ-calculus (see, e.g., [7]). It took more than ten years to prove completeness. This proof is due to Igor Walukiewicz [11] and is quite involved. It uses tableaux and the notion of disjunctive formula. We propose here a simpler proof in a particular case. More precisely, we prove the completeness of the Kozen axiomatization K μ extended with the axiom μx.□x with respect to the class of finite tree models. Finite trees are a fundamental
A direct translation from XPath to nondeterministic automata.
, 2011
"... Since navigational aspects of XPath correspond to first-order definability, it has been proposed to use the analogy with the very successful technique of translating LTL into automata, and produce efficient translations of XPath queries into automata on unranked trees. These translations can then b ..."
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Since navigational aspects of XPath correspond to first-order definability, it has been proposed to use the analogy with the very successful technique of translating LTL into automata, and produce efficient translations of XPath queries into automata on unranked trees. These translations can then be used for a variety of reasoning tasks such as XPath consistency, or optimization, under XML schema constraints. In the verification scenarios, translations into both nondeterministic and alternating automata are used. But while a direct translation from XPath into alternating automata is known, only an indirect translation into nondeterministic automata- going via intermediate logics- exists. A direct translation is desirable as most XML specifications have particularly nice translations into nondeterministic automata and it is natural to use such automata to reason about XPath and schemas. The goal of the paper is to produce such a direct translation of XPath into nondeterministic automata.
Effective characterizations of tree logics
, 2008
"... A survey of effective characterizations of tree logics. If L is a logic, then an effective characterization for L is an algorithm, which inputs a tree automaton and replies if the recognized language can be defined by a formula in L. The logics L considered include path testable languages, frontier ..."
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A survey of effective characterizations of tree logics. If L is a logic, then an effective characterization for L is an algorithm, which inputs a tree automaton and replies if the recognized language can be defined by a formula in L. The logics L considered include path testable languages, frontier testable languages, fragments of Core XPath, and fragments of monadic second-order logic.
