We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued accessibility relation. We define two notions of bisimulation that allow us to obtain truth invariance results. We provide game semantics and, for the more interesting and complicated notion, we are able to provide characteristic formulae and prove a Hennessy-Milner type theorem. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformulated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw inspiration from this formulation, which is of independent interest to Knowledge Representation applications.
Abstract-In all-optical networks that use WDM technology it is often the case that several communication requests have to be blocked, due to bandwidth and technology limitations. Minimizing request blocking is therefore an important task calling for algorithmic techniques for efficient routing and wavelength assignment.Here we study the problem for rings under both the undirected and the directed settings, corresponding to symmetric and oneway communication respectively. The problem in graph-theoretic terms can be formulated as the MAXIMUM ROUTING AND PATH COLORING PROBLEM. We present a Chain-and-Matching technique for routing requests and coloring the corresponding paths which gives constant approximations for both the undirected and the directed cases. For the undirected problem we obtain a
Boolean grammars [A. Okhotin, Boolean grammars, Information and Computation 194 (1) (2004) 19-48] are a promising extension of context-free grammars that supports conjunction and negation in rule bodies. In this paper, we give a novel semantics for Boolean grammars which applies to all such grammars, independently of their syntax. The key idea of our proposal comes from the area of negation in logic programming, and in particular from the so-called well-founded semantics which is widely accepted in this area to be the "correct" approach to negation. We show that for every Boolean grammar there exists a distinguished (three-valued) interpretation of the non-terminal symbols, which satisfies all the rules of the grammar and at the same time is the least fixed-point of an operator associated with the grammar. Then, we demonstrate that every Boolean grammar can be transformed into an equivalent (under the new semantics) grammar in normal form. Based on this normal form, we propose an O(n 3) algorithm for parsing that applies to any such normalized Boolean grammar. In summary, the main contribution of this paper is to provide a semantics which applies to all Boolean grammars while at the same time retaining the complexity of parsing associated with this type of grammars.
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