The $\aleph$ Calculus

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.