ASMs and Operational Algorithmic Completeness of Lambda Calculus

TL;DR

This paper demonstrates that lambda calculus can simulate any sequential deterministic algorithm over various data types through a fixed point technique, establishing its operational completeness.

Contribution

It introduces a formal method to simulate algorithms within lambda calculus, extending to partial computable functions and finite computations with exceptions.

Findings

Lambda calculus can simulate any sequential deterministic algorithm.

Simulation requires a fixed number of lambda reductions per algorithm step.

The approach extends to partial functions and computations ending with exceptions.

Abstract

We show that lambda calculus is a computation model which can step by step simulate any sequential deterministic algorithm for any computable function over integers or words or any datatype. More formally, given an algorithm above a family of computable functions (taken as primitive tools, i.e., kind of oracle functions for the algorithm), for every constant K big enough, each computation step of the algorithm can be simulated by exactly K successive reductions in a natural extension of lambda calculus with constants for functions in the above considered family. The proof is based on a fixed point technique in lambda calculus and on Gurevich sequential Thesis which allows to identify sequential deterministic algorithms with Abstract State Machines. This extends to algorithms for partial computable functions in such a way that finite computations ending with exceptions are associated to finite reductions leading to terms with a particular very simple feature.