Self-Replicating Expressions in the Lambda Calculus

Larkin, J. and Stocks, P.

    The study of self-replicating structures in Computer Science has been taking place for more than half a century, motivated by the desire to understand the fundamental principles and algorithms involved in self-replication. The bulk of the literature explores self-replicating forms in Cellular Automata. Though trivially self-replicating programs have been written for dozens of languages, very little work exists that explores self-replicating forms in programming languages. This paper reports initial investigations into self- replicating expressions in the Lambda Calculus, the basis for functional programming languages. Mimicking results from the work on Cellular Automata, self- replicating Lambda Calculus expressions that also allow the application of an arbitrary program to arbitrary data are presented. Standard normal order reduction, however, will not reduce the sub-expression representing the program application. Two approaches of dealing with this, hybrid reduction and parallel reduction, are discussed, and have been im- plemented in an interpreter.
Cite as: Larkin, J. and Stocks, P. (2004). Self-Replicating Expressions in the Lambda Calculus. In Proc. Twenty-Seventh Australasian Computer Science Conference (ACSC2004), Dunedin, New Zealand. CRPIT, 26. Estivill-Castro, V., Ed. ACS. 167-173.
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