A binary modal logic for the intersection types of lambda-calculus

Intersection types discipline allows to define a wide variety of models for the type free lambda-calculus, but the Curry–Howard isomorphism breaks down for this kind of type systems. In this paper we show that the correspondence between types and suitable logical formulas can still be recovered appealing to the fact that there is a strict connection between the semantics for lambda-calculus induced by the intersection types and a Kripke-style semantics for modal and relevant logics. Indeed, we present a modal logic hinted by the analysis of the sub-typing relation for intersection types, and we show that the deduction relation for such a modal system is a conservative extension of the relation of sub-typing. Then, we define a Kripke-style semantics for the formulas of such a system, present suitable sequential calculi, prove a completeness theorem and give a syntactical proof of the cut elimination property. Finally, we define a decision procedure for theorem-hood and we show that it yields the finite model property and cut-redundancy.     

Judgmental forecasting with interactive forecasting support systems

With an increasing use of DSS/EIS, managers are often required to process information coming from a variety of sources in making a final decision. However, we have little understanding of the efficiency with which people select and use the multiple pieces of information. This issue was examined under various conditions using a DSS in a forecasting task where multiple items of information were displayed on request in an interactive manner. Results indicate that overall people underacquired information. Moreover, people often selected less-reliable information. This sub-optimal behaviour did not diminish over time (it became worse). But an aggregation DSS was helpful at the task. This suggests that people seemed to have a problem in aggregating multiple pieces of information. It was also found that the independent preparation of an initial forecast improved forecast accuracy significantly. Perhaps, forecasters may prepare the initial forecast independently and use decision aids for the subsequent tasks of the forecasting process.     

Infinite intersection types

A type theory with infinitary intersection and union types for an extension of the λ-calculus is introduced. Types are viewed as upper closed subsets of a Scott domain and intersection and union type constructors are interpreted as the set-theoretic intersection and union, respectively, even when they are not finite. The assignment of types to λ-terms extends naturally the basic type assignment system. We prove soundness and completeness using a generalization of Abramsky’s finitary logic of domains. Finally, we apply the framework to applicative transition systems, obtaining a sound a complete infinitary intersection type assignment system for the lazy λ-calculus.     

Feasible functionals and intersection of ramified types

We show that the basic feasible functions of Cook and Urquhart's BFF [8,9] are precisely the functionals definable in a natural system of ramified recurrence that uses type intersection (for tier-variants of a common type). This further confirms the stability of BFF as a notion of computational feasibility in higher type. It also suggests the potential usefulness of type-intersection restricted to sort-variants of a common type.     

Weighted modal transition systems

Specification theories as a tool in model-driven development processes of component-based software systems have recently attracted a considerable attention. Current specification theories are however qualitative in nature, and therefore fragile in the sense that the inevitable approximation of systems by models, combined with the fundamental unpredictability of hardware platforms, makes it difficult to transfer conclusions about the behavior, based on models, to the actual system. Hence this approach is arguably unsuited for modern software systems. We propose here the first specification theory which allows to capture quantitative aspects during the refinement and implementation process, thus leveraging the problems of the qualitative setting. Our proposed quantitative specification framework uses weighted modal transition systems as a formal model of specifications. These are labeled transition systems with the additional feature that they can model optional behavior which may or may not be implemented by the system. Satisfaction and refinement is lifted from the well-known qualitative to our quantitative setting, by introducing a notion of distances between weighted modal transition systems. We show that quantitative versions of parallel composition as well as quotient (the dual to parallel composition) inherit the properties from the Boolean setting.     

Compositional characterisations of -terms using intersection types

We show how to characterise compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term.Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature.The completeness parts of the characterisations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.     

General quantitative specification theories with modal transition systems

This paper proposes a new theory of quantitative specifications. It generalizes the notions of step-wise refinement and compositional design operations from the Boolean to an arbitrary quantitative setting. Using a great number of examples, it is shown that this general approach permits to unify many interesting quantitative approaches to system design.     

Reducibility: A ubiquitous method in lambda calculus with intersection types

A general reducibility method is developed for proving reduction properties of lambda terms typeable in intersection type systems with and without the universal type Ω. Sufficient conditions for its application are derived. This method leads to uniform proofs of confluence, standardization, and weak head normalization of terms typeable in the system with the type Ω. The method extends Tait's reducibility method for the proof of strong normalization of the simply typed lambda calculus, Krivine's extension of the same method for the strong normalization of intersection type system without Ω, and Statman-Mitchell's logical relation method for the proof of confluence of βη-reduction on the simply typed lambda terms. As a consequence, the confluence and the standardization of all (untyped) lambda terms is obtained.     

Polynomial modal control for sampled data systems with delay

The problem of modal sampled-data control for continuous-time linear time-invariant plant with delay is considered. The characteristic matrix of the system is constructed. An algorithm is given for generating the set of causal discrete-time controllers that place eigenvalues of the characteristic matrix at specified points of the complex plane.     

Comparing disjunctive modal transition systems with an one-selecting variant

An expressive class of abstractions for labeled transition systems is that of disjunctive modal transition systems (DMTS), featuring may- and must transitions as well as disjunctive hypertransitions (OR). In order to describe exclusive choice adequately, we develop a variant of DMTSs called 1-selecting modal transition systems (OMTS) that, roughly speaking, interprets hypertransitions exclusively (XOR). These abstract models, DMTSs and OMTSs, are compared with respect to their expressive power. By giving transformations or showing their non-existence, we show that the two setting can express the same sets of labeled transition systems, but 1-selecting modal transition systems have a richer refinement preorder.     

On simulation,subtyping and substitutability in sequential object systems

The aim of this article is to clarify the concepts of behavioural subtype and substitutability that are used to establish the safety of dynamic binding and run-time polymorphism in object-oriented programming. A new model of object behaviour is introduced and used to define the states of an object and the notion of object type. A notion of simulation between object behaviours is defined and a subtype relation between object types is derived from it. The syntax and structural operational semantics of an elementary OOPL is given, and the concepts of object and type substitutability are defined. It is shown formally that simulation is equivalent to object substitutability and that subtyping is equivalent to type substitutability.This research was supported by SERC grant GR/H16629.     

Union and intersection types to support both dynamic and static typing

Although static typing provides undeniable benefits for the development of applications, dynamically typed languages have become increasingly popular for specific scenarios. Since each approach offers different benefits, the StaDyn programming language has been designed to support both dynamic and static typing. This paper describes the minimal core of the StaDyn programming language. Its type system performs type reconstruction over both dynamic and static implicitly typed references. A new interpretation of union and intersection types allows statically gathering the type information of dynamic references, which improves runtime performance and robustness. The evaluation of the generated code has shown how our approach offers an important runtime performance benefit.     

Exploring inconsistencies between modal transition systems

It is commonplace to have multiple behaviour models that describe the same system but have been produced by different stakeholders or synthesized from different sources. Although in practice, such models frequently exhibit inconsistencies, there is a lack of tool support for analyzing them. There are two key difficulties in explaining why two behavioural models are inconsistent: (1) explanations often require branching structures rather than linear traces, or scenarios; and (2) there can be multiple sources of inconsistency and many different ways of explaining each one. In this paper, we present an approach that supports exploration of inconsistencies between modal transition systems, an extension to labelled transition systems. We show how to produce sound graphical explanations for inconsistencies, how to compactly represent all possible explanations in a composition of the models being compared, and how modelers can use this composition to explore the explanations encoded therein.     

Constructing modal control systems for plants with meromorphic transfer function

A problem of constructing an infinite-dimensional controller that ensures the given desired distribution of poles and a part of zeros of the transfer function of the closed-loop system is solved for the plant with distributed parameters. Transfer functions of the plant, controller and closed-loop system are considered in the class of meromorphic functions. The modal controller is synthesized directly based on the desired transfer function of the closed-loop system. The method of synthesizing the controller is reduced to searching an interpolation series. An example is given.     

On determinism in modal transition systems

Modal transition system (MTS) is a formalism which extends the classical notion of labelled transition systems by introducing transitions of two types: must transitions that have to be present in any implementation of the MTS and may transitions that are allowed but not required.The MTS framework has proved to be useful as a specification formalism of component-based systems as it supports compositional verification and stepwise refinement. Nevertheless, there are some limitations of the theory, namely that the naturally defined notions of modal refinement and modal composition are incomplete with respect to the semantic view based on the sets of the implementations of a given MTS specification. Recent work indicates that some of these limitations might be overcome by considering deterministic systems, which seem to be more manageable but still interesting for several application areas.In the present article, we provide a comprehensive account of the MTS framework in the deterministic setting. We study a number of problems previously considered on MTS and point out to what extend we can expect better results under the restriction of determinism.     

Rewrite rule systems for modal propositional logic

This paper explains new results relating modal propositional logic and rewrite rule systems. More precisely, we give complete term rewriting systems for the modal propositional systems known as K, Q, T, and S5. These systems are presented as extensions of Hsiang's system for classical propositional calculus. We have checked local confluence with the rewrite rule system K.B. (cf. the Knuth-Bendix algorithm) developed by the Formel project at INRIA. We prove that these systems are noetherian, and then infer their confluence from Newman's lemma. Therefore each term rewriting system provides a new automated decision procedure and defines a canonical form for the corresponding logic. We also show how to characterize the canonical forms thus obtained.     

Direct digital modal control of cascaded-vehicle systems

In this paper control laws for the direct digital control of cascaded-vehicle systems are derived on the basis of modal control theory. These laws are analogous to these previously obtained by Porter and Crossley (1970) for the continuous-time modal control of such systems.     

Type inference with non-structural subtyping

We present an O(n³) time type inference algorithm for a type system with a largest type, a smallest type , and the usual ordering between function types. The algorithm infers type annotations of least shape, and it works equally well for recursive types. For the problem of typability, our algorithm is simpler than the one of Kozen, Palsberg, and Schwartzbach for type inferencewithout . This may be surprising, especially because the system with is strictly more powerful.     

Integrating coercion with subtyping and multiple dispatch

Coercion can greatly improve the readability of programs, especially in arithmetic expressions. However, coercion interacts with other features of programming languages, particularly subtyping and overloaded functions and operators, in ways that can produce surprising behavior. We study examples of such surprising behavior in existing languages. This study informs the design of the coercion mechanism of Fortress, an object-oriented language with multiple dynamic dispatch, multiple inheritance and user-defined coercion. We describe this design and show how its restrictions on overloaded declarations prevent ambiguous calls due to coercion.