Family Of Calculi Research Articles

Type soundness for dependent object types (DOT)

Scala’s type system unifies aspects of ML modules, object- oriented, and functional programming. The Dependent Object Types (DOT) family of calculi has been proposed as a new theoretic foundation for Scala and similar expressive languages. Unfortunately, type soundness has only been established for restricted subsets of DOT. In fact, it has been shown that important Scala features such as type refinement or a subtyping relation with lattice structure break at least one key metatheoretic property such as environment narrowing or invertible subtyping transitivity, which are usually required for a type soundness proof. The main contribution of this paper is to demonstrate how, perhaps surprisingly, even though these properties are lost in their full generality, a rich DOT calculus that includes recursive type refinement and a subtyping lattice with intersection types can still be proved sound. The key insight is that subtyping transitivity only needs to be invertible in code paths executed at runtime, with contexts consisting entirely of valid runtime objects, whereas inconsistent subtyping contexts can be permitted for code that is never executed.

On the Relevance of Conceptual Spaces for Spatial and Temporal Reasoning

This paper argues for the use of conceptual spaces in qualitative spatial and temporal reasoning. Conceptual spaces provide a natural framework for enriching the purely combinatorial and symbolic calculi such as Allen's calculus with a geometrical or topological structure. Recent work has shown that such enrichments are useful, since they provide simpler ways of characterizing tractable subclasses of the calculi. Hence part of the computational properties can be related to geometrical properties. The paper discusses a family of calculi (including Allen's calculus) for which substantial results have been obtained using this approach. It then examines other instances of calculi where the relationship has not yet been fully investigated and lists the main open questions which arise in this context.

Modal Pure Type Systems

We present a framework for intensional reasoning in typed λ-calculus. In this family of calculi, called Modal Pure Type Systems (MPTSs), a ’’propositions-as-types‘‘-interpretation can be given for normal modal logics. MPTSs are an extension of the Pure Type Systems (PTSs) of Barendregt (1992). We show that they retain the desirable meta-theoretical properties of PTSs, and briefly discuss applications in the area of knowledge representation.

The programmatic semantics of binary predicator calculi.

One may take the view that combinatory logic is concerned with some families of calculi that share well studied morphological and transformational features. These calculi allow for a variety of interpretatio ns. There has been some interest among those concerned with the theory of programming, in these calculi. The intuitive programming interpretations of some of these calculi generally view them as abstract theories of rules that govern problem solving by idealized computing facilities (see Goodman [2]). In this paper,* we wish to present a formal programmatic semantics for a selection of combinatory calculi that is based on first-order model theoretic considerations. There may be extensions of the methods here proposed to higher order and type theoretic considerations (see Curry, Hindley, and Seldin [1], and Sanchis [4, 5]). For the present, we shall limit ourselves to the first-order case. In the following sections of the paper, we first review the morphology and transformational syntax of the calculi, we then construct a first-order referential semantics and conclude with a section that develops the programmatic semantics. The capstone of the paper is a semantic completeness theorem that exhibits how the semantic notion of programmatic equivalence is related to the syntactical concepts of reducibility and extensional equivalence.