88,606 research outputs found
Regular Cost Functions, Part I: Logic and Algebra over Words
The theory of regular cost functions is a quantitative extension to the
classical notion of regularity. A cost function associates to each input a
non-negative integer value (or infinity), as opposed to languages which only
associate to each input the two values "inside" and "outside". This theory is a
continuation of the works on distance automata and similar models. These models
of automata have been successfully used for solving the star-height problem,
the finite power property, the finite substitution problem, the relative
inclusion star-height problem and the boundedness problem for monadic-second
order logic over words. Our notion of regularity can be -- as in the classical
theory of regular languages -- equivalently defined in terms of automata,
expressions, algebraic recognisability, and by a variant of the monadic
second-order logic. These equivalences are strict extensions of the
corresponding classical results. The present paper introduces the cost monadic
logic, the quantitative extension to the notion of monadic second-order logic
we use, and show that some problems of existence of bounds are decidable for
this logic. This is achieved by introducing the corresponding algebraic
formalism: stabilisation monoids.Comment: 47 page
Bisimulation in Inquisitive Modal Logic
Inquisitive modal logic, InqML, is a generalisation of standard Kripke-style
modal logic. In its epistemic incarnation, it extends standard epistemic logic
to capture not just the information that agents have, but also the questions
that they are interested in. Technically, InqML fits within the family of
logics based on team semantics. From a model-theoretic perspective, it takes us
a step in the direction of monadic second-order logic, as inquisitive modal
operators involve quantification over sets of worlds. We introduce and
investigate the natural notion of bisimulation equivalence in the setting of
InqML. We compare the expressiveness of InqML and first-order logic, and
characterise inquisitive modal logic as the bisimulation invariant fragments of
first-order logic over various classes of two-sorted relational structures.
These results crucially require non-classical methods in studying bisimulations
and first-order expressiveness over non-elementary classes.Comment: In Proceedings TARK 2017, arXiv:1707.0825
Computing Distances between Probabilistic Automata
We present relaxed notions of simulation and bisimulation on Probabilistic
Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve
the usual notions of bisimulation and simulation on PAs. We give logical
characterisations of these notions by choosing suitable logics which differ
from the elementary ones, L with negation and L without negation, by the modal
operator. Using flow networks, we show how to compute the relations in PTIME.
This allows the definition of an efficiently computable non-discounted distance
between the states of a PA. A natural modification of this distance is
introduced, to obtain a discounted distance, which weakens the influence of
long term transitions. We compare our notions of distance to others previously
defined and illustrate our approach on various examples. We also show that our
distance is not expansive with respect to process algebra operators. Although L
without negation is a suitable logic to characterise epsilon-(bi)simulation on
deterministic PAs, it is not for general PAs; interestingly, we prove that it
does characterise weaker notions, called a priori epsilon-(bi)simulation, which
we prove to be NP-difficult to decide.Comment: In Proceedings QAPL 2011, arXiv:1107.074
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
A Characterization Theorem for a Modal Description Logic
Modal description logics feature modalities that capture dependence of
knowledge on parameters such as time, place, or the information state of
agents. E.g., the logic S5-ALC combines the standard description logic ALC with
an S5-modality that can be understood as an epistemic operator or as
representing (undirected) change. This logic embeds into a corresponding modal
first-order logic S5-FOL. We prove a modal characterization theorem for this
embedding, in analogy to results by van Benthem and Rosen relating ALC to
standard first-order logic: We show that S5-ALC with only local roles is, both
over finite and over unrestricted models, precisely the bisimulation invariant
fragment of S5-FOL, thus giving an exact description of the expressive power of
S5-ALC with only local roles
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