1 research outputs found
Weak Affine Light Typing is complete with respect to Safe Recursion on Notation
Weak affine light typing (WALT) assigns light affine linear formulae as types
to a subset of lambda-terms of System F. WALT is poly-time sound: if a
lambda-term M has type in WALT, M can be evaluated with a polynomial cost in
the dimension of the derivation that gives it a type. The evaluation proceeds
under any strategy of a rewriting relation which is a mix of both call-by-name
and call-by-value beta-reductions. WALT weakens, namely generalizes, the notion
of "stratification of deductions", common to some Light Systems -- those
logical systems, derived from Linear logic, to characterize the set of
Polynomial functions -- . A weaker stratification allows to define a
compositional embedding of Safe recursion on notation (SRN) into WALT. It turns
out that the expressivity of WALT is strictly stronger than the one of the
known Light Systems. The embedding passes through the representation of a
subsystem of SRN. It is obtained by restricting the composition scheme of SRN
to one that can only use its safe variables linearly. On one side, this
suggests that SRN, in fact, can be redefined in terms of more primitive
constructs. On the other, the embedding of SRN into WALT enjoys the two
following remarkable aspects. Every datatype, required by the embedding, is
represented from scratch, showing the strong structural proof-theoretical roots
of WALT. Moreover, the embedding highlights a stratification structure of the
normal and safe arguments, normally hidden inside the world of SRN-normal/safe
variables: the less an argument is "polyomially impredicative", the deeper, in
a formal, proof-theoretical sense, it is represented inside WALT. Finally,
since WALT is SRN-complete it is also polynomial-time complete since SRN is