4,211 research outputs found
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
PSPACE Reasoning for Graded Modal Logics
We present a PSPACE algorithm that decides satisfiability of the graded modal
logic Gr(K_R)---a natural extension of propositional modal logic K_R by
counting expressions---which plays an important role in the area of knowledge
representation. The algorithm employs a tableaux approach and is the first
known algorithm which meets the lower bound for the complexity of the problem.
Thus, we exactly fix the complexity of the problem and refute an
ExpTime-hardness conjecture. We extend the results to the logic Gr(K_(R \cap
I)), which augments Gr(K_R) with inverse relations and intersection of
accessibility relations. This establishes a kind of ``theoretical benchmark''
that all algorithmic approaches can be measured against
The Complexity of Enriched Mu-Calculi
The fully enriched μ-calculus is the extension of the propositional
μ-calculus with inverse programs, graded modalities, and nominals. While
satisfiability in several expressive fragments of the fully enriched
μ-calculus is known to be decidable and ExpTime-complete, it has recently
been proved that the full calculus is undecidable. In this paper, we study the
fragments of the fully enriched μ-calculus that are obtained by dropping at
least one of the additional constructs. We show that, in all fragments obtained
in this way, satisfiability is decidable and ExpTime-complete. Thus, we
identify a family of decidable logics that are maximal (and incomparable) in
expressive power. Our results are obtained by introducing two new automata
models, showing that their emptiness problems are ExpTime-complete, and then
reducing satisfiability in the relevant logics to these problems. The automata
models we introduce are two-way graded alternating parity automata over
infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite
forests. The former are a common generalization of two incomparable automata
models from the literature. The latter extend alternating automata in a similar
way as the fully enriched μ-calculus extends the standard μ-calculus.Comment: A preliminary version of this paper appears in the Proceedings of the
33rd International Colloquium on Automata, Languages and Programming (ICALP),
2006. This paper has been selected for a special issue in LMC
Enriched MU-Calculi Module Checking
The model checking problem for open systems has been intensively studied in
the literature, for both finite-state (module checking) and infinite-state
(pushdown module checking) systems, with respect to Ctl and Ctl*. In this
paper, we further investigate this problem with respect to the \mu-calculus
enriched with nominals and graded modalities (hybrid graded Mu-calculus), in
both the finite-state and infinite-state settings. Using an automata-theoretic
approach, we show that hybrid graded \mu-calculus module checking is solvable
in exponential time, while hybrid graded \mu-calculus pushdown module checking
is solvable in double-exponential time. These results are also tight since they
match the known lower bounds for Ctl. We also investigate the module checking
problem with respect to the hybrid graded \mu-calculus enriched with inverse
programs (Fully enriched \mu-calculus): by showing a reduction from the domino
problem, we show its undecidability. We conclude with a short overview of the
model checking problem for the Fully enriched Mu-calculus and the fragments
obtained by dropping at least one of the additional constructs
A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics
Graded modal logic is the formal language obtained from ordinary
(propositional) modal logic by endowing its modal operators with cardinality
constraints. Under the familiar possible-worlds semantics, these augmented
modal operators receive interpretations such as "It is true at no fewer than 15
accessible worlds that...", or "It is true at no more than 2 accessible worlds
that...". We investigate the complexity of satisfiability for this language
over some familiar classes of frames. This problem is more challenging than its
ordinary modal logic counterpart--especially in the case of transitive frames,
where graded modal logic lacks the tree-model property. We obtain tight
complexity bounds for the problem of determining the satisfiability of a given
graded modal logic formula over the classes of frames characterized by any
combination of reflexivity, seriality, symmetry, transitivity and the Euclidean
property.Comment: Full proofs for paper presented at the IEEE Conference on Logic in
Computer Science, 200
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