1,232 research outputs found
On P-transitive graphs and applications
We introduce a new class of graphs which we call P-transitive graphs, lying
between transitive and 3-transitive graphs. First we show that the analogue of
de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we
show that the mu-calculus fixpoint hierarchy is infinite for P-transitive
graphs. Both results contrast with the case of transitive graphs. We give also
an undecidability result for an enriched mu-calculus on P-transitive graphs.
Finally, we consider a polynomial time reduction from the model checking
problem on arbitrary graphs to the model checking problem on P-transitive
graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081
On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components
For every positive integer k we consider the class SCCk of all finite graphs
whose strongly connected components have size at most k. We show that for every
k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level
Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1
and Pi1). This contrasts with the class of all graphs, where
Delta2=Comp(Sigma1,Pi1)
Succinctness in subsystems of the spatial mu-calculus
In this paper we systematically explore questions of succinctness in modal
logics employed in spatial reasoning. We show that the closure operator,
despite being less expressive, is exponentially more succinct than the
limit-point operator, and that the -calculus is exponentially more
succinct than the equally-expressive tangled limit operator. These results hold
for any class of spaces containing at least one crowded metric space or
containing all spaces based on ordinals below , with the usual
limit operator. We also show that these results continue to hold even if we
enrich the less succinct language with the universal modality
The universal tangle for spatial reasoning
The topological -calculus has gathered attention in recent years as a
powerful framework for representation of spatial knowledge. In particular,
spatial relations can be represented over finite structures in the guise of
weakly transitive wK4 frames. In this paper we show that the topological
-calculus is equivalent to a simple fragment based on a variant of the
`tangle' operator. Similar results were proven for transitive frames by Dawar
and Otto, using modal characterisation theorems for the corresponding classes
of frames. However, since these theorems are not available in our setting,
which has the upshot of providing a more explicit translation and upper bounds
on formula size.Comment: 20 page
Gauge field theories: various mathematical approaches
This paper presents relevant modern mathematical formulations for (classical)
gauge field theories, namely, ordinary differential geometry, noncommutative
geometry, and transitive Lie algebroids. They provide rigorous frameworks to
describe Yang-Mills-Higgs theories or gravitation theories, and each of them
improves the paradigm of gauge field theories. A brief comparison between them
is carried out, essentially due to the various notions of connection. However
they reveal a compelling common mathematical pattern on which the paper
concludes.Comment: 33 pages. To be published in the book: Mathematical Structures of the
Universe (Copernicus Center Press, Krak\'ow, Poland, 2014
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
The Modal μ-Calculus Hierarchy on Restricted Classes of Transition Systems
We discuss the strictness of the modal µ-calculus hierarchy over some restricted classes of transition systems. First, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for reflexive systems the same results holds for finite models. Second, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the finite model theorem for transitive transition systems is also proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment
The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity
It is known that the alternation hierarchy of least and greatest fixpoint
operators in the mu-calculus is strict. However, the strictness of the
alternation hierarchy does not necessarily carry over when considering
restricted classes of structures. A prominent instance is the class of infinite
words over which the alternation-free fragment is already as expressive as the
full mu-calculus. Our current understanding of when and why the mu-calculus
alternation hierarchy is not strict is limited. This paper makes progress in
answering these questions by showing that the alternation hierarchy of the
mu-calculus collapses to the alternation-free fragment over some classes of
structures, including infinite nested words and finite graphs with feedback
vertex sets of a bounded size. Common to these classes is that the connectivity
between the components in a structure from such a class is restricted in the
sense that the removal of certain vertices from the structure's graph
decomposes it into graphs in which all paths are of finite length. Our collapse
results are obtained in an automata-theoretic setting. They subsume,
generalize, and strengthen several prior results on the expressivity of the
mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202
The Arity Hierarchy in the Polyadic -Calculus
The polyadic mu-calculus is a modal fixpoint logic whose formulas define
relations of nodes rather than just sets in labelled transition systems. It can
express exactly the polynomial-time computable and bisimulation-invariant
queries on finite graphs. In this paper we show a hierarchy result with respect
to expressive power inside the polyadic mu-calculus: for every level of
fixpoint alternation, greater arity of relations gives rise to higher
expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282
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