21,851 research outputs found
Refinement Modal Logic
In this paper we present {\em refinement modal logic}. A refinement is like a
bisimulation, except that from the three relational requirements only `atoms'
and `back' need to be satisfied. Our logic contains a new operator 'all' in
addition to the standard modalities 'box' for each agent. The operator 'all'
acts as a quantifier over the set of all refinements of a given model. As a
variation on a bisimulation quantifier, this refinement operator or refinement
quantifier 'all' can be seen as quantifying over a variable not occurring in
the formula bound by it. The logic combines the simplicity of multi-agent modal
logic with some powers of monadic second-order quantification. We present a
sound and complete axiomatization of multi-agent refinement modal logic. We
also present an extension of the logic to the modal mu-calculus, and an
axiomatization for the single-agent version of this logic. Examples and
applications are also discussed: to software verification and design (the set
of agents can also be seen as a set of actions), and to dynamic epistemic
logic. We further give detailed results on the complexity of satisfiability,
and on succinctness
A logic for n-dimensional hierarchical refinement
Hierarchical transition systems provide a popular mathematical structure to
represent state-based software applications in which different layers of
abstraction are represented by inter-related state machines. The decomposition
of high level states into inner sub-states, and of their transitions into inner
sub-transitions is common refinement procedure adopted in a number of
specification formalisms.
This paper introduces a hybrid modal logic for k-layered transition systems,
its first-order standard translation, a notion of bisimulation, and a modal
invariance result. Layered and hierarchical notions of refinement are also
discussed in this setting.Comment: In Proceedings Refine'15, arXiv:1606.0134
Logic of Negation-Complete Interactive Proofs (Formal Theory of Epistemic Deciders)
We produce a decidable classical normal modal logic of internalised
negation-complete and thus disjunctive non-monotonic interactive proofs (LDiiP)
from an existing logical counterpart of non-monotonic or instant interactive
proofs (LiiP). LDiiP internalises agent-centric proof theories that are
negation-complete (maximal) and consistent (and hence strictly weaker than, for
example, Peano Arithmetic) and enjoy the disjunction property (like
Intuitionistic Logic). In other words, internalised proof theories are
ultrafilters and all internalised proof goals are definite in the sense of
being either provable or disprovable to an agent by means of disjunctive
internalised proofs (thus also called epistemic deciders). Still, LDiiP itself
is classical (monotonic, non-constructive), negation-incomplete, and does not
have the disjunction property. The price to pay for the negation completeness
of our interactive proofs is their non-monotonicity and non-communality (for
singleton agent communities only). As a normal modal logic, LDiiP enjoys a
standard Kripke-semantics, which we justify by invoking the Axiom of Choice on
LiiP's and then construct in terms of a concrete oracle-computable function.
LDiiP's agent-centric internalised notion of proof can also be viewed as a
negation-complete disjunctive explicit refinement of standard KD45-belief, and
yields a disjunctive but negation-incomplete explicit refinement of
S4-provability.Comment: Expanded Introduction. Added Footnote 4. Corrected Corollary 3 and 4.
Continuation of arXiv:1208.184
Graphical representation of covariant-contravariant modal formulae
Covariant-contravariant simulation is a combination of standard (covariant)
simulation, its contravariant counterpart and bisimulation. We have previously
studied its logical characterization by means of the covariant-contravariant
modal logic. Moreover, we have investigated the relationships between this
model and that of modal transition systems, where two kinds of transitions (the
so-called may and must transitions) were combined in order to obtain a simple
framework to express a notion of refinement over state-transition models. In a
classic paper, Boudol and Larsen established a precise connection between the
graphical approach, by means of modal transition systems, and the logical
approach, based on Hennessy-Milner logic without negation, to system
specification. They obtained a (graphical) representation theorem proving that
a formula can be represented by a term if, and only if, it is consistent and
prime. We show in this paper that the formulae from the covariant-contravariant
modal logic that admit a "graphical" representation by means of processes,
modulo the covariant-contravariant simulation preorder, are also the consistent
and prime ones. In order to obtain the desired graphical representation result,
we first restrict ourselves to the case of covariant-contravariant systems
without bivariant actions. Bivariant actions can be incorporated later by means
of an encoding that splits each bivariant action into its covariant and its
contravariant parts.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407
The Glory of the Past and Geometrical Concurrency
This paper contributes to the general understanding of the geometrical model
of concurrency that was named higher dimensional automata (HDAs) by Pratt. In
particular we investigate modal logics for such models and their expressive
power in terms of the bisimulation that can be captured. The geometric model of
concurrency is interesting from two main reasons: its generality and
expressiveness, and the natural way in which autoconcurrency and action
refinement are captured. Logics for this model, though, are not well
investigated, where a simple, yet adequate, modal logic over HDAs was only
recently introduced. As this modal logic, with two existential modalities,
during and after, captures only split bisimulation, which is rather low in the
spectrum of van Glabbeek and Vaandrager, the immediate question was what small
extension of this logic could capture the more fine-grained hereditary history
preserving bisimulation (hh)? In response, the work in this paper provides
several insights. One is the fact that the geometrical aspect of HDAs makes it
possible to use for capturing the hh-bisimulation, a standard modal logic that
does not employ event variables, opposed to the two logics (over less
expressive models) that we compare with. The logic that we investigate here
uses standard past modalities and extends the previously introduced logic
(called HDML) that had only forward, action-labelled, modalities. Besides, we
try to understand better the above issues by introducing a related model that
we call ST-configuration structures, which extend the configuration structures
of van Glabbeek and Plotkin. We relate this model to HDAs, and redefine and
prove the earlier results in the light of this new model. These offer a
different view on why the past modalities and geometrical concurrency capture
the hereditary history preserving bisimulation. Additional correlating insights
are also gained.Comment: 17 pages, 7 figure
Fractional-valued modal logic
This paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval [0,1] . Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to the standard Kripkean semantics (a new proof of Dugundji’s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic
Hennessy-Milner Logic with Greatest Fixed Points as a Complete Behavioural Specification Theory
There are two fundamentally different approaches to specifying and verifying
properties of systems. The logical approach makes use of specifications given
as formulae of temporal or modal logics and relies on efficient model checking
algorithms; the behavioural approach exploits various equivalence or refinement
checking methods, provided the specifications are given in the same formalism
as implementations.
In this paper we provide translations between the logical formalism of
Hennessy-Milner logic with greatest fixed points and the behavioural formalism
of disjunctive modal transition systems. We also introduce a new operation of
quotient for the above equivalent formalisms, which is adjoint to structural
composition and allows synthesis of missing specifications from partial
implementations. This is a substantial generalisation of the quotient for
deterministic modal transition systems defined in earlier papers
Labelled transition systems as a Stone space
A fully abstract and universal domain model for modal transition systems and
refinement is shown to be a maximal-points space model for the bisimulation
quotient of labelled transition systems over a finite set of events. In this
domain model we prove that this quotient is a Stone space whose compact,
zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree
of bisimilarity such that image-finite labelled transition systems are dense.
Using this compactness we show that the set of labelled transition systems that
refine a modal transition system, its ''set of implementations'', is compact
and derive a compactness theorem for Hennessy-Milner logic on such
implementation sets. These results extend to systems that also have partially
specified state propositions, unify existing denotational, operational, and
metric semantics on partial processes, render robust consistency measures for
modal transition systems, and yield an abstract interpretation of compact sets
of labelled transition systems as Scott-closed sets of modal transition
systems.Comment: Changes since v2: Metadata updat
Complexity Jumps In Multiagent Justification Logic Under Interacting Justifications
The Logic of Proofs, LP, and its successor, Justification Logic, is a
refinement of the modal logic approach to epistemology in which
proofs/justifications are taken into account. In 2000 Kuznets showed that
satisfiability for LP is in the second level of the polynomial hierarchy, a
result which has been successfully repeated for all other one-agent
justification logics whose complexity is known.
We introduce a family of multi-agent justification logics with interactions
between the agents' justifications, by extending and generalizing the two-agent
versions of the Logic of Proofs introduced by Yavorskaya in 2008. Known
concepts and tools from the single-agent justification setting are adjusted for
this multiple agent case. We present tableau rules and some preliminary
complexity results. In several cases the satisfiability problem for these
logics remains in the second level of the polynomial hierarchy, while for
others it is PSPACE or EXP-hard. Furthermore, this problem becomes PSPACE-hard
even for certain two-agent logics, while there are EXP-hard logics of three
agents
When are prime formulae characteristic?
In the setting of the modal logic that characterizes modal refinement over modal transition systems, Boudol and Larsen showed that the formulae for which model checking can be reduced to preorder checking, that is, the characteristic formulae, are exactly the consistent and prime ones. This paper presents general, sufficient conditions guaranteeing that characteristic formulae are exactly the consistent and prime ones. It is shown that the given conditions apply to the logics characterizing all the semantics in van Glabbeek's branching-time spectrum
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