1,130 research outputs found
An expressive completeness theorem for coalgebraic modal mu-calculi
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. We then consider invariance under behavioral equivalence
of MSO-formulas. More specifically, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of the monadic second-order
language for a given functor. Using automatatheoretic techniques and building
on recent results by the third author, we show that in order to provide such a
characterization result it suffices to find what we call an adequate uniform
construction for the coalgebraic type functor. As direct applications of this
result we obtain a partly new proof of the Janin-Walukiewicz Theorem for the
modal mu-calculus, avoiding the use of syntactic normal forms, and bisimulation
invariance results for the bag functor (graded modal logic) and all exponential
polynomial functors (including the "game functor"). As a more involved
application, involving additional non-trivial ideas, we also derive a
characterization theorem for the monotone modal mu-calculus, with respect to a
natural monadic second-order language for monotone neighborhood models.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0721
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
Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. Similar to well-known results for monadic second-order
logic over trees, we provide a translation of this logic into a class of
automata, relative to the class of coalgebras that admit a tree-like supporting
Kripke frame. We then consider invariance under behavioral equivalence of
formulas; more in particular, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of monadic second-order
logic. Building on recent results by the third author we show that in order to
provide such a coalgebraic generalization of the Janin-Walukiewicz Theorem, it
suffices to find what we call an adequate uniform construction for the functor.
As applications of this result we obtain a partly new proof of the
Janin-Walukiewicz Theorem, and bisimulation invariance results for the bag
functor (graded modal logic) and all exponential polynomial functors.
Finally, we consider in some detail the monotone neighborhood functor, which
provides coalgebraic semantics for monotone modal logic. It turns out that
there is no adequate uniform construction for this functor, whence the
automata-theoretic approach towards bisimulation invariance does not apply
directly. This problem can be overcome if we consider global bisimulations
between neighborhood models: one of our main technical results provides a
characterization of the monotone modal mu-calculus extended with the global
modalities, as the fragment of monadic second-order logic for the monotone
neighborhood functor that is invariant for global bisimulations
Separability in the Ambient Logic
The \it{Ambient Logic} (AL) has been proposed for expressing properties of
process mobility in the calculus of Mobile Ambients (MA), and as a basis for
query languages on semistructured data. We study some basic questions
concerning the discriminating power of AL, focusing on the equivalence on
processes induced by the logic . As underlying calculi besides MA we
consider a subcalculus in which an image-finiteness condition holds and that we
prove to be Turing complete. Synchronous variants of these calculi are studied
as well. In these calculi, we provide two operational characterisations of
: a coinductive one (as a form of bisimilarity) and an inductive one
(based on structual properties of processes). After showing to be stricly
finer than barbed congruence, we establish axiomatisations of on the
subcalculus of MA (both the asynchronous and the synchronous version), enabling
us to relate to structural congruence. We also present some
(un)decidability results that are related to the above separation properties
for AL: the undecidability of on MA and its decidability on the
subcalculus.Comment: logical methods in computer science, 44 page
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
Disjunctive bases: normal forms and model theory for modal logics
We present the concept of a disjunctive basis as a generic framework for
normal forms in modal logic based on coalgebra. Disjunctive bases were defined
in previous work on completeness for modal fixpoint logics, where they played a
central role in the proof of a generic completeness theorem for coalgebraic
mu-calculi. Believing the concept has a much wider significance, here we
investigate it more thoroughly in its own right. We show that the presence of a
disjunctive basis at the "one-step" level entails a number of good properties
for a coalgebraic mu-calculus, in particular, a simulation theorem showing that
every alternating automaton can be transformed into an equivalent
nondeterministic one. Based on this, we prove a Lyndon theorem for the full
fixpoint logic, its fixpoint-free fragment and its one-step fragment, a Uniform
Interpolation result, for both the full mu-calculus and its fixpoint-free
fragment, and a Janin-Walukiewicz-style characterization theorem for the
mu-calculus under slightly stronger assumptions.
We also raise the questions, when a disjunctive basis exists, and how
disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla
formulas provide disjunctive bases for many coalgebraic modal logics, but there
are cases where disjunctive bases give useful normal forms even when nabla
formulas fail to do so, our prime example being graded modal logic. We also
show that disjunctive bases are preserved by forming sums, products and
compositions of coalgebraic modal logics, providing tools for modular
construction of modal logics admitting disjunctive bases. Finally, we consider
the problem of giving a category-theoretic formulation of disjunctive bases,
and provide a partial solution
Expressiveness of the modal mu-calculus on monotone neighborhood structures
We characterize the expressive power of the modal mu-calculus on monotone
neighborhood structures, in the style of the Janin-Walukiewicz theorem for the
standard modal mu-calculus. For this purpose we consider a monadic second-order
logic for monotone neighborhood structures. Our main result shows that the
monotone modal mu-calculus corresponds exactly to the fragment of this
second-order language that is invariant for neighborhood bisimulations
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