120,531 research outputs found
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
Synthetic Completeness for a Terminating Seligman-Style Tableau System
Hybrid logic extends modal logic with nominals that name worlds. Seligman-style tableau systems for hybrid logic divide branches into blocks named by nominals to achieve a local proof style. We present a Seligman-style tableau system with a formalization in the proof assistant Isabelle/HOL. Our system refines an existing system to simplify formalization and we claim termination from this relationship. Existing completeness proofs that account for termination are either analytic or based on translation, but synthetic proofs have been shown to generalize to richer logics and languages. Our main result is the first synthetic completeness proof for a terminating hybrid logic tableau system. It is also the first formalized completeness proof for any hybrid logic proof system
Completeness in hybrid type theory
We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret @i in propositional and first-order hybrid logic. This means: interpret @iαa, where αa is an expression of any type a, as an expression of type a that rigidly returns the value that αa receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual in hybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logic over Henkin’s logic.submittedVersionFil: Areces, Carlos Eduardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Blackburn, Patrick. University of Roskilde. Centre for Culture and Identity. Department of Philosophy and Science Studies; Dinamarca.Fil: Huertas, Antonia. Universitat Oberta de Catalunya; España.Fil: Manzano, María. Universidad de Salamanca; España.Ciencias de la Computació
Completeness in Equational Hybrid Propositional Type Theory
Equational Hybrid Propositional Type Theory (EHPTT) is a combination of
propositional type theory, equational logic and hybrid modal logic. The structures used to
interpret the language contain a hierarchy of propositional types, an algebra (a nonempty
set with functions) and a Kripke frame.
The main result in this paper is the proof of completeness of a calculus specifically
defined for this logic. The completeness proof is based on the three proofs Henkin published
last century: (i) Completeness in type theory (ii) The completeness of the first-order
functional calculus and (iii) Completeness in propositional type theory. More precisely,
from (i) and (ii) we take the idea of building the model described by the maximal consistent
set; in our case the maximal consistent set has to be named, ♦- saturated and extensionally
algebraic-saturated due to the hybrid and equational nature of EHPTT. From (iii), we use
the result that any element in the hierarchy has a name. The challenge was to deal with
all the heterogeneous components in an integrated system.publishe
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
Hybrid Rules with Well-Founded Semantics
A general framework is proposed for integration of rules and external first
order theories. It is based on the well-founded semantics of normal logic
programs and inspired by ideas of Constraint Logic Programming (CLP) and
constructive negation for logic programs. Hybrid rules are normal clauses
extended with constraints in the bodies; constraints are certain formulae in
the language of the external theory. A hybrid program is a pair of a set of
hybrid rules and an external theory. Instances of the framework are obtained by
specifying the class of external theories, and the class of constraints. An
example instance is integration of (non-disjunctive) Datalog with ontologies
formalized as description logics.
The paper defines a declarative semantics of hybrid programs and a
goal-driven formal operational semantics. The latter can be seen as a
generalization of SLS-resolution. It provides a basis for hybrid
implementations combining Prolog with constraint solvers. Soundness of the
operational semantics is proven. Sufficient conditions for decidability of the
declarative semantics, and for completeness of the operational semantics are
given
A hilbert-style axiomatisation for equational hybrid logic
This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and first-order hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics
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