Quantification in Frame Semantics with Binders and Nominals of Hybrid Logic

International audienceThis paper aims at integrating logical operators into frame-based semantics. Frames are semantic graphs that allow to capture lexical meaning in a fine-grained way but that do not come with a natural way to integrate logical operators such as quantifiers. The approach we propose starts from the observation that modal logic is a powerful tool for describing relational structures, hence frames. We use its hybrid logic extension in order to incorporate quantification and thereby allow for inference and reasoning. We integrate our approach to a type theoretic compositional semantics, formulated within Abstract Categorial Grammars. We also show how the key ingredients of hybrid logic, nominals and binders, can be used to model semantic coercion, such as the one induced by the begin predicate. In order to illustrate the effectiveness of the proposed syntax-semantics interface, all the examples can be run and tested with the Abstract Categorial Grammar development toolkit

Modal Hybrid Logic

This is an extended version of the lectures given during the 12-th Conference on Applications of Logic in Philosophy and in the Foundations of Mathematics in Szklarska Poręba (7–11 May 2007). It contains a survey of modal hybrid logic, one of the branches of contemporary modal logic. In the first part a variety of hybrid languages and logics is presented with a discussion of expressivity matters. The second part is devoted to thorough exposition of proof methods for hybrid logics. The main point is to show that application of hybrid logics may remarkably improve the situation in modal proof theory

For-Adverbials and Aspectual Interpretation: An LTAG Analysis Using Hybrid Logic and Frame Semantics

International audienceIn this paper, we propose to use Hybrid Logic (HL) as a means to combine frame-based lexical semantics with quantification. We integrate this into a syntax-semantics interface using LTAG (Lexicalized Tree Adjoining Grammar) and show that this architecture allows a fine-grained description of event structures by quantifying, for instance, over subevents. As a case study, we provide an analysis of for-adverbials and the aspectual interpretations they induce. The basic idea is that for-adverbials introduce a universal quantification over subevents that are characterized by the predication contributed by the verb. Depending on whether these subevents are bounded or not, the resulting overall event is then an iteration or a progression. We show that by combining the HL approach with standard techniques of underspecification and by using HL to formulate general constraints on event frames, we can account for the aspectual coercion triggered by these adverbials. Furthermore, by pairing this with syntactic building blocks in LTAG, we provide a working syntax-semantics interface for these phenomena

Behavioural and abstractor specifications revisited

In the area of algebraic specification there are two main approaches for defining observational abstraction: behavioural specifications use a notion of observational satisfaction for the axioms of a specification, whereas abstractor specifications define an abstraction from the standard semantics of a specification w.r.t. an observational equivalence relation between algebras. Earlier work by Bidoit, Hennicker, Wirsing has shown that in the case of first-order logic specifications both concepts coincide semantically under mild assumptions. Analogous results have been shown by Sannella and Hofmann for higher-order logic specifications and recently, by Hennicker and Madeira, for specifications of reactive systems using a dynamic logic with binders. In this paper, we bring these results into a common setting: we isolate a small set of characteristic principles to express the behaviour/abstractor equivalence and show that all three mentioned specification frameworks satisfy these principles and therefore their behaviour and abstractor specifications coincide semantically (under mild assumptions). As a new case we consider observational modal logic where observational satisfaction of Hennessy–Milner logic formulae is defined “up to” silent transitions and observational abstraction is defined by weak bisimulation. We show that in this case the behaviour/abstractor equivalence can only be obtained, if we restrict models to weakly deterministic labelled transition systems.publishe

A Logical Foundation for Environment Classifiers

Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with a sound type system using the notion of environment classifiers. They are special identifiers, with which code fragments and variable declarations are annotated, and their scoping mechanism is used to ensure statically that certain code fragments are closed and safely runnable. In this paper, we investigate the Curry-Howard isomorphism for environment classifiers by developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to multi-modal logic that allows quantification by transition variables---a counterpart of classifiers---which range over (possibly empty) sequences of labeled transitions between possible worlds. This interpretation will reduce the "run" construct---which has a special typing rule in {\lambda}{\alpha}---and embedding of closed code into other code fragments of different stages---which would be only realized by the cross-stage persistence operator in {\lambda}{\alpha}---to merely a special case of classifier application. {\lambda}|> enjoys not only basic properties including subject reduction, confluence, and strong normalization but also an important property as a multi-stage calculus: time-ordered normalization of full reduction. Then, we develop a big-step evaluation semantics for an ML-like language based on {\lambda}|> with its type system and prove that the evaluation of a well-typed {\lambda}|> program is properly staged. We also identify a fragment of the language, where erasure evaluation is possible. Finally, we show that the proof system augmented with a classical axiom is sound and complete with respect to a Kripke semantics of the logic