# The Dynamic Geometry of Interaction Machine: A Call-by-need Graph Rewriter

## TL;DR

This paper introduces the Dynamic GoI Machine (DGoIM), a graph rewriting abstract machine that efficiently implements call-by-need evaluation for the lambda calculus by balancing token passing and graph rewriting.

## Contribution

It presents a novel token-passing abstract machine that rewrites graphs to implement call-by-need evaluation with certified time efficiency.

## Key findings

- DGoIM can implement call-by-need evaluation.
- The machine balances space and time costs effectively.
- DGoIM is classified as efficient per Accattoli's taxonomy.

## Abstract

Girard's Geometry of Interaction (GoI), a semantics designed for linear logic proofs, has been also successfully applied to programming language semantics. One way is to use abstract machines that pass a token on a fixed graph along a path indicated by the GoI. These token-passing abstract machines are space efficient, because they handle duplicated computation by repeating the same moves of a token on the fixed graph. Although they can be adapted to obtain sound models with regard to the equational theories of various evaluation strategies for the lambda calculus, it can be at the expense of significant time costs. In this paper we show a token-passing abstract machine that can implement evaluation strategies for the lambda calculus, with certified time efficiency. Our abstract machine, called the Dynamic GoI Machine (DGoIM), rewrites the graph to avoid replicating computation, using the token to find the redexes. The flexibility of interleaving token transitions and graph rewriting allows the DGoIM to balance the trade-off of space and time costs. This paper shows that the DGoIM can implement call-by-need evaluation for the lambda calculus by using a strategy of interleaving token passing with as much graph rewriting as possible. Our quantitative analysis confirms that the DGoIM with this strategy of interleaving the two kinds of possible operations on graphs can be classified as "efficient" following Accattoli's taxonomy of abstract machines.