Automata on infinite words are used for specification and verification of nonterminating programs. Different types of automata induce different levels of expressive power, of succinctness, and of complexity. Alternating automata have both existential and universal branching modes and are particularly suitable for specification of programs. In a weak alternating automaton, the state space is partitioned into partially ordered sets, and the automaton can proceed from a certain set only to smaller sets. Reasoning about weak alternating automata is easier than reasoning about alternating automata with no restricted structure. Known translations of alternating automata to weak alternating automata involve determinization, and therefore involve a double-exponential blow-up. In this paper we describe a quadratic translation, which circumvents the need for determinization, of Büchi and co-Büchi alternating automata to weak alternating automata. Beyond the independent interest of such a translation, it gives rise to a simple complementation algorithm for nondeterministic Büchi automata. 1

We present an algorithm to solve XPath decision problems under regular tree type constraints and show its use to statically type-check XPath queries. To this end, we prove the decidability of a logic with converse for finite ordered trees whose time complexity is a simple exponential of the size of a formula. The logic corresponds to the alternation free modal µ-calculus without greatest fixpoint, restricted to finite trees, and where formulas are cycle-free. Our proof method is based on two auxiliary results. First, XML regular tree types and XPath expressions have a linear translation to cycle-free formulas. Second, the least and greatest fixpoints are equivalent for finite trees, hence the logic is closed under negation. Building on these results, we describe a practical, effective system for solving the satisfiability of a formula. The system has been experimented with some decision problems such as XPath emptiness, containment, overlap, and coverage, with or without type constraints. The benefit of the approach is that our system can be effectively used in static analyzers for programming languages

XPath is the standard language for navigating XML documents and returning a set of matching nodes. We present a sound and complete decision procedure for containment of XPath queries, as well as other related XPath decision problems such as satisfiability, equivalence, overlap, and coverage. The considered XPath fragment covers most of the language features used in practice. Specifically, we propose a unifying logic for XML, namely, the alternation-free modal μ-calculus with converse. We show how to translate major XML concepts such as XPath and regular XML types (including DTDs) into this logic. Based on these embeddings, we show how XPath decision problems, in the presence or absence of XML types, can be solved using a decision procedure for μ-calculus satisfiability. We provide a complexity analysis of our system together with practical experiments to illustrate the efficiency of the approach for realistic scenarios.

The -calculus is an expressive specification language in which modal logic is extended with fixpoint operators, subsuming many dynamic, temporal, and description logics. Formulas of -calculus are classified according to their alternation depth, which is the maximal length of a chain of nested alternating least and greatest fixpoint operators. Alternation depth is the major factor in the complexity of -calculus model-checking algorithms. A refined classification of -calculus formulas distinguishes between formulas in which the outermost fixpoint operator in the nested chain is a least fixpoint operator ( i formulas, where i is the alternation depth) and formulas where it is a greatest fixpoint operator ( i formulas). The alternation-free -calculus (AFMC) consists of -calculus formulas with no alternation between least and greatest fixpoint operators. Thus, AFMC is a natural closure of 1 [ 1 , which is contained in both 2 and 2 . In this work we show that 2 \ 2 AFMC. In other words, if we can express a property both as a least fixpoint nested inside a greatest fixpoint and as a greatest fixpoint nested inside a least fixpoint, then we can express also with no alternation between greatest and least fixpoints. Our result refers to -calculus over arbitrary Kripke structures. A similar result, for directed -calculus formulas interpreted over trees with a fixed finite branching degree, follows from results by Arnold and Niwinski. Their proofs there cannot be easily extended to Kripke structures, and our extension involves symmetric nondeterministic Buchi tree automata, and new constructions for them.

The -calculus is an expressive specification language in which modal logic is extended with fixpoint operators, subsuming many dynamic, temporal, and description logics. Formulas of -calculus are classified according to their alternation depth, which is the maximal length of a chain of nested alternating least and greatest fixpoint operators. Alternation depth is the major factor in the complexity of -calculus model-checking algorithms. A refined classification of -calculus formulas distinguishes between formulas in which the outermost fixpoint operator in the nested chain is a least fixpoint operator ( i formulas, where i is the alternation depth) and formulas where it is a greatest fixpoint operator ( i formulas). The alternation-free -calculus (AFMC) consists of -calculus formulas with no alternation between least and greatest fixpoint operators. Thus, AFMC is a natural closure of 1 [ 1 , which is contained in both 2 and 2 . In this work we show that 2 \ 2 AFMC. In other words, if we can express a property both as a least fixpoint nested inside a greatest fixpoint and as a greatest fixpoint nested inside a least fixpoint, then we can express also with no alternation between greatest and least fixpoints. Our result refers to -calculus over arbitrary Kripke structures. A similar result, for directed -calculus formulas interpreted over trees with a fixed finite branching degree, follows from results by Arnold and Niwinski. Their proofs there cannot be easily extended to Kripke structures, and our extension involves symmetric nondeterministic B uchi tree automata, and new constructions for them.

Abstract: We present an algorithm to solve XPath decision problems under regular tree type constraints and show its use to statically type-check XPath queries. To this end, we prove the decidability of a logic with converse for finite ordered trees whose time complexity is a simple exponential of the size of a formula. The logic corresponds to the alternation free modal µ-calculus without greatest fixpoint, restricted to finite trees, and where formulas are cycle-free. Our proof method is based on two auxiliary results. First, XML regular tree types and XPath expressions have a linear translation to cycle-free formulas. Second, the least and greatest fixpoints are equivalent for finite trees, hence the logic is closed under negation. Building on these results, we describe a practical, effective system for solving the satisfiability of a formula. The system has been experimented with some decision problems such as XPath emptiness, containment, overlap, and coverage, with or without type constraints. The benefit of the approach is that our system can be effectively used in static analyzers for programming languages