64 research outputs found
Relating Structure and Power: Comonadic Semantics for Computational Resources
Combinatorial games are widely used in finite model theory, constraint
satisfaction, modal logic and concurrency theory to characterize logical
equivalences between structures. In particular, Ehrenfeucht-Fraisse games,
pebble games, and bisimulation games play a central role. We show how each of
these types of games can be described in terms of an indexed family of comonads
on the category of relational structures and homomorphisms. The index k is a
resource parameter which bounds the degree of access to the underlying
structure. The coKleisli categories for these comonads can be used to give
syntax-free characterizations of a wide range of important logical
equivalences. Moreover, the coalgebras for these indexed comonads can be used
to characterize key combinatorial parameters: tree-depth for the
Ehrenfeucht-Fraisse comonad, tree-width for the pebbling comonad, and
synchronization-tree depth for the modal unfolding comonad. These results pave
the way for systematic connections between two major branches of the field of
logic in computer science which hitherto have been almost disjoint: categorical
semantics, and finite and algorithmic model theory.Comment: To appear in Proceedings of Computer Science Logic 201
Relating structure and power: Comonadic semantics for computational resources
Combinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht–Fraïssé games, pebble games and bisimulation games play a central role. We show how each of these types of games can be described in terms of an indexed family of comonads on the category of relational structures and homomorphisms. The index k is a resource parameter that bounds the degree of access to the underlying structure. The coKleisli categories for these comonads can be used to give syntax-free characterizations of a wide range of important logical equivalences. Moreover, the coalgebras for these indexed comonads can be used to characterize key combinatorial parameters: tree depth for the Ehrenfeucht–Fraïssé comonad, tree width for the pebbling comonad and synchronization tree depth for the modal unfolding comonad. These results pave the way for systematic connections between two major branches of the field of logic in computer science, which hitherto have been almost disjoint: categorical semantics and finite and algorithmic model theory
Coalgebraic Weak Bisimulation from Recursive Equations over Monads
Strong bisimulation for labelled transition systems is one of the most
fundamental equivalences in process algebra, and has been generalised to
numerous classes of systems that exhibit richer transition behaviour. Nearly
all of the ensuing notions are instances of the more general notion of
coalgebraic bisimulation. Weak bisimulation, however, has so far been much less
amenable to a coalgebraic treatment. Here we attempt to close this gap by
giving a coalgebraic treatment of (parametrized) weak equivalences, including
weak bisimulation. Our analysis requires that the functor defining the
transition type of the system is based on a suitable order-enriched monad,
which allows us to capture weak equivalences by least fixpoints of recursive
equations. Our notion is in agreement with existing notions of weak
bisimulations for labelled transition systems, probabilistic and weighted
systems, and simple Segala systems.Comment: final versio
Relationship-based access control: its expression and enforcement through hybrid logic
Access control policy is typically de ned in terms of attributes, but in many applications it is more natural to de- ne permissions in terms of relationships that resources, systems, and contexts may enjoy. The paradigm of relationshipbased access control has been proposed to address this issue, and modal logic has been used as a technical foundation. We argue here that hybrid logic { a natural and wellestablished extension of modal logic { addresses limitations in the ability of modal logic to express certain relationships. Also, hybrid logic has advantages in the ability to e ciently compute policy decisions relative to a relationship graph. We identify a fragment of hybrid logic to be used for expressing relationship-based access-control policies, show that this fragment supports important policy idioms, and study its expressiveness. We also capture the previously studied notion of relational policies in a static type system. Finally, we point out that use of our hybrid logic removes an exponential penalty in existing attempts of specifying complex relationships such as \at least three friends"
Structure and Power: an emerging landscape
In this paper, we give an overview of some recent work on applying tools from
category theory in finite model theory, descriptive complexity, constraint
satisfaction, and combinatorics. The motivations for this work come from
Computer Science, but there may also be something of interest for model
theorists and other logicians.
The basic setting involves studying the category of relational structures via
a resource-indexed family of adjunctions with some process category - which
unfolds relational structures into treelike forms, allowing natural resource
parameters to be assigned to these unfoldings. One basic instance of this
scheme allows us to recover, in a purely structural, syntax-free way: the
Ehrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic;
the equivalences on structures induced by (i) the quantifier rank fragments,
(ii) the restriction of this fragment to the existential positive part, and
(iii) the extension with counting quantifiers; and the combinatorial parameter
of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the
k-pebble game, the finite-variable fragments, the corresponding equivalences,
and the combinatorial parameter of treewidth. Other instances cover modal,
guarded and hybrid fragments, generalized quantifiers, and a wide range of
combinatorial parameters. This whole scheme has been axiomatized in a very
general setting, of arboreal categories and arboreal covers.
Beyond this basic level, a landscape is beginning to emerge, in which
structural features of the resource categories, adjunctions and comonads are
reflected in degrees of logical and computational tractability of the
corresponding languages. Examples include semantic characterisation and
preservation theorems, and Lovasz-type results on counting homomorphisms.Comment: To appear in special issue for Trakhtenbrot centenary of Fundamenta
Informaticae vol. 186 no 1-
Neighbourhood Semantics for Graded Modal Logic
We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation
Isomorphism Checking for Symmetry Reduction
In this paper, we show how isomorphism checking can be used as an effective technique for symmetry reduction. Reduced state spaces are equivalent to the original ones under a strong notion of bisimilarity which preserves the multiplicity of outgoing transitions, and therefore also preserves stochastic temporal logics. We have implemented this in a setting where states are arbitrary graphs. Since no efficiently computable canonical representation is known for arbitrary graphs modulo isomorphism, we define an isomorphism-predicting hash function on the basis of an existing partition refinement algorithm. As an example, we report a factorial state space reduction on a model of an ad-hoc network connectivity protocol
Dynamic Graded Epistemic Logic
International audienceGraded epistemic logic is a logic for reasoning about uncertainties. Graded epistemic logic is interpreted on graded models. These models are generalizations of Kripke models. We obtain completeness of some graded epistemic logics. We further develop dynamic extensions of graded epistemic logics, along the framework of dynamic epistemic logic. We give an extension with public announcements, i.e., public events, and an extension with graded event models, a generalization also including nonpublic events. We present complete axiomatizations for both logics
- …