A Modified GoI Interpretation for a Linear Functional Programming Language and Its Adequacy

Abstract

Abstract. Geometry of Interaction (GoI) introduced by Girard provides a semantics for linear logic and its cut elimination. Several extensions of GoI to programming languages have been proposed, but it is not discussed to what extent they capture behaviour of programs as far as the author knows. In this paper, we study GoI interpretation of a linear functional programming language (LFP). We observe that we can not extend the standard GoI interpretation to an adequate interpretation of LFP, and we propose a new adequate GoI interpretation of LFP by modifying the standard GoI interpretation. We derive the modified interpretation from a realizability model of LFP. We also relate the interpretation of recursion to cyclic computation (the trace operator in the category of sets and partial maps) in the realizability model.

Measurements in Proof Nets as Higher-Order Quantum Circuits

“…Another is to accommodate recursion. We expect to be able to adapt the techniques developed in [14] and [12].…”

Computation by interaction for space-bounded functional programming

“…The idea of translating programs to circuits by means of interpretation in an interactive model is not new. In the literature one can find a number of higher-order functional languages, such as [32,23], that are translated to circuits using this approach. However, without further restrictions one cannot prove that circuits evaluate in sublinear space.…”

“…Even in languages without recursion, e.g. [23], one may encounter messages of exponential size during circuit evaluation. What is new in this paper is that we show how to refine language and model in order to obtain an expressive language for sublinear space programming.…”

Seamless Distributed Computing from the Geometry of Interaction

“…We give an interpretation of terms in our language similar to the GoI interpretations by Hoshino [9] and Mackie [10], where terms are coded into linear logic proof nets. The only difference is that the interfaces of our nets are determined by type instead of being homogeneous.…”