Abstract Parametric Classes and Abstract Data Types defined by Classical and Constructive Logical Methods

AbstractWe introduce a methodology to treat abstract data types (ADT), abstract parametric classes (APC) and subclasses, together with appropriate inheritance properties, by means of first order theories. The notion of a first order theory axiomatizing an ADT is based on the notion of isoinitial model and has been proposed by the authors in previous papers (Bertoni et al. (1979), Bertoni et al. (1983), Bertoni et al. (1984)). A theory formalizing an APC is seen, in this paper, as a theory T incompletely axiomatizing an ADT. Given a class C of ADT's, the class formalized by T can be seen (under suitable soundness conditions on T) as the class of the instances of T over C. An instantiation of T by an ADT I of C completes T into a T′ formalizing an ADT I′, which extends I and inherits the properties of the APC T.We use both classical and constructive methods in the following sense: on the one hand, the semantics is based on classical model theory; on the other hand, the soundness of a consistent axiomatization can be analyzed by purely syntactical methods, in terms of provability within suitable constructive systems.A theory T formalizing an APC (or an ADT) is not given by a list of axioms, but by a suitable "APC-expression", which explicitly or implicitly (but effectively ) defines the axioms of T. We have APC-expressions to define APC's, to extend already defined APC's and to instantiate APC's (into ADT's or subclasses). We allow also "recurrence APC-expressions". At the end of the paper we give some examples showing how the proposed methodology works

A Survey of Languages for Specifying Dynamics: A Knowledge Engineering Perspective

A number of formal specification languages for knowledge-based systems has been developed. Characteristics for knowledge-based systems are a complex knowledge base and an inference engine which uses this knowledge to solve a given problem. Specification languages for knowledge-based systems have to cover both aspects. They have to provide the means to specify a complex and large amount of knowledge and they have to provide the means to specify the dynamic reasoning behavior of a knowledge-based system. We focus on the second aspect. For this purpose, we survey existing approaches for specifying dynamic behavior in related areas of research. In fact, we have taken approaches for the specification of information systems (Language for Conceptual Modeling and TROLL), approaches for the specification of database updates and logic programming (Transaction Logic and Dynamic Database Logic) and the generic specification framework of abstract state machine

Towards a Rule Interchange Language for the Web

This articles discusses rule languages that are needed for a a full deployment of the SemanticWeb. First, it motivates the need for such languages. Then, it presents ten theses addressing (1) the rule and/or logic languages needed on the Web, (2) data and data processing, (3) semantics, and (4) engineering and rendering issues. Finally, it discusses two options that might be chosen in designing a Rule Interchange Format for the Web

Relational Parametricity for Computational Effects

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects, and are not accounted for by the standard theory of relational parametricity. In this paper, we develop a foundational framework for extending the notion of relational parametricity to programming languages with effects.Comment: 31 pages, appears in Logical Methods in Computer Scienc

Predicativity and parametric polymorphism of Brouwerian implication

A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten

Relational Parametricity and Control

We study the equational theory of Parigot's second-order λμ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λμ-terms. On the other hand, the unconstrained relational parametricity on the λμ-calculus turns out to be inconsistent with this CPS semantics. Following these facts, we propose to formulate the relational parametricity on the λμ-calculus in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc

QPCF: higher order languages and quantum circuits

qPCF is a paradigmatic quantum programming language that ex- tends PCF with quantum circuits and a quantum co-processor. Quantum circuits are treated as classical data that can be duplicated and manipulated in flexible ways by means of a dependent type system. The co-processor is essentially a standard QRAM device, albeit we avoid to store permanently quantum states in between two co-processor's calls. Despite its quantum features, qPCF retains the classic programming approach of PCF. We introduce qPCF syntax, typing rules, and its operational semantics. We prove fundamental properties of the system, such as Preservation and Progress Theorems. Moreover, we provide some higher-order examples of circuit encoding

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200