263,109 research outputs found
Bisimulation and expressivity for conditional belief, degrees of belief, and safe belief
Plausibility models are Kripke models that agents use to reason about
knowledge and belief, both of themselves and of each other. Such models are
used to interpret the notions of conditional belief, degrees of belief, and
safe belief. The logic of conditional belief contains that modality and also
the knowledge modality, and similarly for the logic of degrees of belief and
the logic of safe belief. With respect to these logics, plausibility models may
contain too much information. A proper notion of bisimulation is required that
characterises them. We define that notion of bisimulation and prove the
required characterisations: on the class of image-finite and preimage-finite
models (with respect to the plausibility relation), two pointed Kripke models
are modally equivalent in either of the three logics, if and only if they are
bisimilar. As a result, the information content of such a model can be
similarly expressed in the logic of conditional belief, or the logic of degrees
of belief, or that of safe belief. This, we found a surprising result. Still,
that does not mean that the logics are equally expressive: the logics of
conditional and degrees of belief are incomparable, the logics of degrees of
belief and safe belief are incomparable, while the logic of safe belief is more
expressive than the logic of conditional belief. In view of the result on
bisimulation characterisation, this is an equally surprising result. We hope
our insights may contribute to the growing community of formal epistemology and
on the relation between qualitative and quantitative modelling
Linear Temporal Logic for Hybrid Dynamical Systems: Characterizations and Sufficient Conditions
This paper introduces operators, semantics, characterizations, and
solution-independent conditions to guarantee temporal logic specifications for
hybrid dynamical systems. Hybrid dynamical systems are given in terms of
differential inclusions -- capturing the continuous dynamics -- and difference
inclusions -- capturing the discrete dynamics or events -- with constraints.
State trajectories (or solutions) to such systems are parameterized by a hybrid
notion of time. For such broad class of solutions, the operators and semantics
needed to reason about temporal logic are introduced. Characterizations of
temporal logic formulas in terms of dynamical properties of hybrid systems are
presented -- in particular, forward invariance and finite time attractivity.
These characterizations are exploited to formulate sufficient conditions
assuring the satisfaction of temporal logic formulas -- when possible, these
conditions do not involve solution information. Combining the results for
formulas with a single operator, ways to certify more complex formulas are
pointed out, in particular, via a decomposition using a finite state automaton.
Academic examples illustrate the results throughout the paper.Comment: 35 pages. The technical report accompanying "Linear Temporal Logic
for Hybrid Dynamical Systems: Characterizations and Sufficient Conditions"
submitted to Nonlinear Analysis: Hybrid Systems, 201
Valuations in Nilpotent Minimum Logic
The Euler characteristic can be defined as a special kind of valuation on
finite distributive lattices. This work begins with some brief consideration on
the role of the Euler characteristic on NM algebras, the algebraic counterpart
of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified
version of the Euler characteristic we call idempotent Euler characteristic. We
show that the new valuation encodes information about the formul{\ae} in NM
propositional logic
Inquisitive bisimulation
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 in the
context of relational structures with two sorts, one for worlds and one for
information states. We characterise inquisitive modal logic, as well as its
multi-agent epistemic S5-like variant, as the bisimulation invariant fragment
of first-order logic over various natural classes of two-sorted structures.
These results crucially require non-classical methods in studying bisimulation
and first-order expressiveness over non-elementary classes of structures,
irrespective of whether we aim for characterisations in the sense of classical
or of finite model theory
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
A Spatial-Epistemic Logic for Reasoning about Security Protocols
Reasoning about security properties involves reasoning about where the
information of a system is located, and how it evolves over time. While most
security analysis techniques need to cope with some notions of information
locality and knowledge propagation, usually they do not provide a general
language for expressing arbitrary properties involving local knowledge and
knowledge transfer. Building on this observation, we introduce a framework for
security protocol analysis based on dynamic spatial logic specifications. Our
computational model is a variant of existing pi-calculi, while specifications
are expressed in a dynamic spatial logic extended with an epistemic operator.
We present the syntax and semantics of the model and logic, and discuss the
expressiveness of the approach, showing it complete for passive attackers. We
also prove that generic Dolev-Yao attackers may be mechanically determined for
any deterministic finite protocol, and discuss how this result may be used to
reason about security properties of open systems. We also present a
model-checking algorithm for our logic, which has been implemented as an
extension to the SLMC system.Comment: In Proceedings SecCo 2010, arXiv:1102.516
Information as Distinctions: New Foundations for Information Theory
The logical basis for information theory is the newly developed logic of
partitions that is dual to the usual Boolean logic of subsets. The key concept
is a "distinction" of a partition, an ordered pair of elements in distinct
blocks of the partition. The logical concept of entropy based on partition
logic is the normalized counting measure of the set of distinctions of a
partition on a finite set--just as the usual logical notion of probability
based on the Boolean logic of subsets is the normalized counting measure of the
subsets (events). Thus logical entropy is a measure on the set of ordered
pairs, and all the compound notions of entropy (join entropy, conditional
entropy, and mutual information) arise in the usual way from the measure (e.g.,
the inclusion-exclusion principle)--just like the corresponding notions of
probability. The usual Shannon entropy of a partition is developed by replacing
the normalized count of distinctions (dits) by the average number of binary
partitions (bits) necessary to make all the distinctions of the partition
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