Functional interpretation and inductive definitions

Extending G\"odel's \emph{Dialectica} interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.Comment: minor corrections and change

Toward an Algebraic Theory of Systems

We propose the concept of a system algebra with a parallel composition operation and an interface connection operation, and formalize composition-order invariance, which postulates that the order of composing and connecting systems is irrelevant, a generalized form of associativity. Composition-order invariance explicitly captures a common property that is implicit in any context where one can draw a figure (hiding the drawing order) of several connected systems, which appears in many scientific contexts. This abstract algebra captures settings where one is interested in the behavior of a composed system in an environment and wants to abstract away anything internal not relevant for the behavior. This may include physical systems, electronic circuits, or interacting distributed systems. One specific such setting, of special interest in computer science, are functional system algebras, which capture, in the most general sense, any type of system that takes inputs and produces outputs depending on the inputs, and where the output of a system can be the input to another system. The behavior of such a system is uniquely determined by the function mapping inputs to outputs. We consider several instantiations of this very general concept. In particular, we show that Kahn networks form a functional system algebra and prove their composition-order invariance. Moreover, we define a functional system algebra of causal systems, characterized by the property that inputs can only influence future outputs, where an abstract partial order relation captures the notion of "later". This system algebra is also shown to be composition-order invariant and appropriate instantiations thereof allow to model and analyze systems that depend on time

A Semantic Approach to Illative Combinatory Logic

This work introduces the theory of illative combinatory algebras, which is closely related to systems of illative combinatory logic. We thus provide a semantic interpretation for a formal framework in which both logic and computation may be expressed in a unified manner. Systems of illative combinatory logic consist of combinatory logic extended with constants and rules of inference intended to capture logical notions. Our theory does not correspond strictly to any traditional system, but draws inspiration from many. It differs from them in that it couples the notion of truth with the notion of equality between terms, which enables the use of logical formulas in conditional expressions. We give a consistency proof for first-order illative combinatory algebras. A complete embedding of classical predicate logic into our theory is also provided. The translation is very direct and natural

Transfinite Step-Indexing: Decoupling Concrete and Logical Steps

International audienceStep-indexing has proven to be a powerful technique for defining logical relations for languages with advanced type systems and models of expressive program logics. In both cases, the model is stratified using natural numbers to solve a recursive equation that has no naive solutions. As a result of this stratification, current models require that each unfolding of the recursive equation – each logical step – must coincide with a concrete reduction step. This tight coupling is problematic for applications where the number of logical steps cannot be statically bounded. In this paper we demonstrate that this tight coupling between logical and concrete steps is artificial and show how to loosen it using transfinite step-indexing. We present a logical relation that supports an arbitrary but finite number of logical steps for each concrete step