Lukasiewicz's logic as an architecture model of arithmetic

We obtain a complete formal description of various architecture models of computer realizations of arithmetic in terms of the finite-valued Lukasiewicz logic and logics enriched by it. The weak incompleteness property of these logics enables one to study the structural characteristics of operations independent of the representation of numbers, such as the forms of admissible overflows. We describe some properties of the lattice of these logics and find bases suitable for applications. We consider logical structures that arise in the realization of arithmetic operations and the representation of overflow in positional number systems. We indicate paths for the development of hardware-based realizations of arithmetic. We present a new view of the nature of Lukasiewicz's logic.