A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new i...

The Abiteboul and Beeri algebra for complex objects can express a query whose meaning is transitive closure, but the algorithm naturally associated to this query needs exponential space. We show that any other query in the algebra which expresses transitive closure needs exponential space, under a \call by value" evaluation strategy. This proves that in general the powerset is an intractable op...

We investigate experimentally, alternative approaches to the distributed parallel computation of a class of problems related to the generic transitive closure problem and the algebraic path problem. Our main result is the comparison of two parallel algorithms for transitive closure, { a straightforward coarse-grained parallel implementation of the Warshall algorithm named Block-Processing (whic...

We study extensions of first-order logic over the reals with different types of transitive-closure operators as query languages for constraint databases that can be described by Boolean combinations of polynomial inequalities. We are in particular interested in deciding the termination of the evaluation of queries expressible in these transitive-closure logics. It turns out that termination is ...

We survey results on transitive-closure spanners and their applications. Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in different areas of computer science, and properties of these spanners h...

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments o...

In this paper, we investigate an extension of the description logic SHIQ––a knowledge representation formalism used for the Semantic Web––with transitive closure of roles occurring not only in concept inclusion axioms but also in role inclusion axioms. We start by proving that adding transitive closure of roles to SHIQ without restriction on role hierarchies may lead to undecidability. An analy...

First-order logic with k-ary deterministic transitive closure has the same power as two-way k-head deterministic automata that use a finite set of nested pebbles. This result is valid for strings, ranked trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toru...

Let 2N be the class of families of problems solvable by families of two-way nondeterministic finite automata of polynomial size. We characterize 2N in terms of families of formulas of transitive-closure logic. These formulas apply the transitive-closure operator on a quantifier-free disjunctive normal form of first-order logic with successor and constants, where (i) apart from two special varia...

In dynamic epistemic logic and other elds, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that three fragments of the ...