| Summary: | We examine the relationship between the algebraic lambda-calculus, a fragment
of the differential lambda-calculus and the linear-algebraic lambda-calculus, a
candidate lambda-calculus for quantum computation. Both calculi are algebraic:
each one is equipped with an additive and a scalar-multiplicative structure,
and their set of terms is closed under linear combinations. However, the two
languages were built using different approaches: the former is a call-by-name
language whereas the latter is call-by-value; the former considers algebraic
equalities whereas the latter approaches them through rewrite rules. In this
paper, we analyse how these different approaches relate to one another. To this
end, we propose four canonical languages based on each of the possible choices:
call-by-name versus call-by-value, algebraic equality versus algebraic
rewriting. We show that the various languages simulate one another. Due to
subtle interaction between beta-reduction and algebraic rewriting, to make the
languages consistent some additional hypotheses such as confluence or
normalisation might be required. We carefully devise the required properties
for each proof, making them general enough to be valid for any sub-language
satisfying the corresponding properties.
|
|---|