433 research outputs found
A new coalgebraic semantics for positive modal logic
Positive Modal Logic is the restriction of the modal local consequence relation defined by the class of all Kripke models to the propositional negation-free modal language. The class of positive modal algebras is the one canonically associated with PML according to the theory of Abstract Algebraic Logic. Celani and Jansana established a Priestley-style duality the category of positive modal algebras and the category of K+-spaces. In this paper, we establish a categorical equivalence between the category K+ of K+-spaces and the category Coalg(V) of coalgebras of a suitable endofunctor V on the category of Priestley spaces
Named Models in Coalgebraic Hybrid Logic
Hybrid logic extends modal logic with support for reasoning about individual
states, designated by so-called nominals. We study hybrid logic in the broad
context of coalgebraic semantics, where Kripke frames are replaced with
coalgebras for a given functor, thus covering a wide range of reasoning
principles including, e.g., probabilistic, graded, default, or coalitional
operators. Specifically, we establish generic criteria for a given coalgebraic
hybrid logic to admit named canonical models, with ensuing completeness proofs
for pure extensions on the one hand, and for an extended hybrid language with
local binding on the other. We instantiate our framework with a number of
examples. Notably, we prove completeness of graded hybrid logic with local
binding
Completeness of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
The Positivication of Coalgebraic Logics
We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing a endofunctor T\u27: Pos->Pos from an endofunctor T: Set->Set, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor L\u27: DL->DL from a syntax-building functor L: BA->BA, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case of several modal logics. We show how the semantics of a boolean coalgebraic logic can be canonically lifted to define a semantics for its positive fragment, and that weak completeness transfers from the boolean case to the positive case
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
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
Modal logics are coalgebraic
Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors' firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility
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