Memoryful geometry of interaction

Abstract

Girard's Geometry of Interaction (GoI) is interaction based semantics of linear logic proofs and, via suitable translations, of functional programs in general. Its mathematical cleanness identifies essential structures in computation; moreover its use as a compilation technique from programs to state machines-"GoI implementation," so to speak-has been worked out by Mackie, Ghica and others. In this paper, we develop Abramsky's idea of resumption based GoI systematically into a generic framework that accounts for computational effects (nondeterminism, probability, exception, global states, interactive I/O, etc.). The framework is categorical: Plotkin & Power's algebraic operations provide an interface to computational effects; the framework is built on the categorical axiomatization of GoI by Abramsky, Haghverdi and Scott; and, by use of the coalgebraic formalization of component calculus, we describe explicit construction of state machines as interpretations of functional programs. The resulting interpretation is shown to be sound with respect to equations between algebraic operations, as well as to Moggi's equations for the computational lambda calculus. We illustrate the construction by concrete examples.

Cyclic proofs, system t, and the power of contraction

“…For the affine case (Corollary 6.14), Dal Lago's system H (∅) [Dal Lago 2009] is a variant of Gödel's system T which characterises primitive recursive functions and which is really close to our affine version of T. Unfortunately, we need additive pairs in order to translate affine C into affine T. Those are not available in H (∅), and it is not clear how to extend Dal Lago's proof to deal with such operations: his proof is complex and relies on a semantics based on geometry of interaction, whose extension to additives is notoriously difficult [Abramsky et al 2002;Baillot and Pedicini 2001;Girard 1995;Hoshino et al 2014]. We instead prove Corollary 6.14 by using another proof of weak normalisation for C, which works only on the image of our translation from affine T to C. This argument can be formalised into another subsystem of second order arithmetic, RCA 0 , which is known to define only the primitive recursive functions [Avigad 1996].…”

Syntax and Semantics of Quantitative Type Theory

“…Abramsky has drawn attention to the relationship between this model and concurrency models [1]. Hoshino et al [17] have used the "memoryful" aspect of this situation to further allow for effectful computations. An exciting avenue of future research will be to examine the implications of this model for an effectful QTT. )…”

“…This appears to be related to the problem of interpretating the additive connectives of Linear Logic in their axiomatic approach. Nevertheless, the connection with Geometry of Interaction style models is intriguing, and may help develop the theory of QTT into areas such as reversible computation, as described by Abramsky [2], hardware synthesis, as described by Ghica [12], and effectful computation, as described by Hoshino et al [17].…”

Efficient Implementation of Evaluation Strategies via Token-Guided Graph Rewriting

“…The GoI-style token passing itself has been adapted to implement the call-by-value evaluation strategy. Apart from the abstract machine with jumps [11] already mentioned, known adaptations [24,16] commonly use the CPS transformation [23], with the focus on correctness. However this method naively leads to an abstract machine with inefficient overhead cost [16].…”