The $\aleph$ Calculus

TL;DR

The paper introduces the $\u2113$ calculus, a reversible, declarative, concurrent model of computation inspired by molecular architectures, along with examples, a programming language, and a type system.

Contribution

It presents the $\u2113$ calculus as a novel reversible computation model suitable for molecular environments, including language design, implementation, and analysis tools.

Findings

Designs for molecular implementations of arithmetic operations

Development of the alethe programming language and interpreter

Proposal of a type system for reversible computation

Abstract

Motivated by a need for a model of reversible computation appropriate for a Brownian molecular architecture, the $\aleph$ calculus is introduced. This novel model is declarative, concurrent, and term-based--encapsulating all information about the program data and state within a single structure in order to obviate the need for a von Neumann-style discrete computational 'machine', a challenge in a molecular environment. The name is inspired by the Greek for 'not forgotten', due to the emphasis on (reversibly) learning and un-learning knowledge of different variables. To demonstrate its utility for this purpose, as well as its elegance as a programming language, a number of examples are presented; two of these examples, addition/subtraction and squaring/square-rooting, are furnished with designs for abstract molecular implementations. A natural by-product of these examples and accompanying syntactic sugar is the design of a fully-fledged programming language, alethe, which is also presented along with an interpreter. Efficiently simulating $\aleph$ on a deterministic computer necessitates some static analysis of programs within the alethe interpreter in order to render the declarative programs sequential. Finally, work towards a type system appropriate for such a reversible, declarative model of computation is presented.