Turing machines and register machines have been used for decades in theoretical computer science as abstract models of computation. Also the $\lambda$-calculus has played a central role in this domain as it allows to focus on the notion of functional computation, based on the substitution mechanism, while abstracting away from implementation details. The present article starts from the observation that the equivalence between these formalisms is based on the Church-Turing Thesis rather than an actual encoding of $\lambda$-terms into Turing (or register) machines. The reason is that these machines are not well-suited for modelling $\lambda$-calculus programs. We study a class of abstract machines that we call "addressing machine" since they are only able to manipulate memory addresses of other machines. The operations performed by these machines are very elementary: load an address in a register, apply a machine to another one via their addresses, and call the address of another machine. We endow addressing machines with an operational semantics based on leftmost reduction and study their behaviour. The set of addresses of these machines can be easily turned into a combinatory algebra. In order to obtain a model of the full untyped $\lambda$-calculus, we need to introduce a rule that bares similarities with the $\omega$-rule and the rule $\zeta_\beta$ from combinatory logic.

• 03B38 - Type theory
• 18M05 - Monoidal categories, symmetric monoidal categories
• 68N15 - Theory of programming languages
• 68N19 - Other programming paradigms (object-oriented, sequential, concurrent, automatic, etc.)
• 68Q12 - Quantum algorithms and complexity in the theory of computing
• 81P68 - Quantum computation