44,044 research outputs found
An algebraic generalization of Kripke structures
The Kripke semantics of classical propositional normal modal logic is made
algebraic via an embedding of Kripke structures into the larger class of
pointed stably supported quantales. This algebraic semantics subsumes the
traditional algebraic semantics based on lattices with unary operators, and it
suggests natural interpretations of modal logic, of possible interest in the
applications, in structures that arise in geometry and analysis, such as
foliated manifolds and operator algebras, via topological groupoids and inverse
semigroups. We study completeness properties of the quantale based semantics
for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization
for S5 which does not use negation or the modal necessity operator. As
additional examples we describe intuitionistic propositional modal logic, the
logic of programs PDL, and the ramified temporal logic CTL.Comment: 39 page
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Connectionist modal logic: Representing modalities in neural networks
AbstractModal logics are amongst the most successful applied logical systems. Neural networks were proved to be effective learning systems. In this paper, we propose to combine the strengths of modal logics and neural networks by introducing Connectionist Modal Logics (CML). CML belongs to the domain of neural-symbolic integration, which concerns the application of problem-specific symbolic knowledge within the neurocomputing paradigm. In CML, one may represent, reason or learn modal logics using a neural network. This is achieved by a Modalities Algorithm that translates modal logic programs into neural network ensembles. We show that the translation is sound, i.e. the network ensemble computes a fixed-point meaning of the original modal program, acting as a distributed computational model for modal logic. We also show that the fixed-point computation terminates whenever the modal program is well-behaved. Finally, we validate CML as a computational model for integrated knowledge representation and learning by applying it to a well-known testbed for distributed knowledge representation. This paves the way for a range of applications on integrated knowledge representation and learning, from practical reasoning to evolving multi-agent systems
Logical Step-Indexed Logical Relations
Appel and McAllester's "step-indexed" logical relations have proven to be a
simple and effective technique for reasoning about programs in languages with
semantically interesting types, such as general recursive types and general
reference types. However, proofs using step-indexed models typically involve
tedious, error-prone, and proof-obscuring step-index arithmetic, so it is
important to develop clean, high-level, equational proof principles that avoid
mention of step indices. In this paper, we show how to reason about binary
step-indexed logical relations in an abstract and elegant way. Specifically, we
define a logic LSLR, which is inspired by Plotkin and Abadi's logic for
parametricity, but also supports recursively defined relations by means of the
modal "later" operator from Appel, Melli\`es, Richards, and Vouillon's "very
modal model" paper. We encode in LSLR a logical relation for reasoning
relationally about programs in call-by-value System F extended with general
recursive types. Using this logical relation, we derive a set of useful rules
with which we can prove contextual equivalence and approximation results
without counting steps
An interval logic for higher-level temporal reasoning
Prior work explored temporal logics, based on classical modal logics, as a framework for specifying and reasoning about concurrent programs, distributed systems, and communications protocols, and reported on efforts using temporal reasoning primitives to express very high level abstract requirements that a program or system is to satisfy. Based on experience with those primitives, this report describes an Interval Logic that is more suitable for expressing such higher level temporal properties. The report provides a formal semantics for the Interval Logic, and several examples of its use. A description of decision procedures for the logic is also included
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