1 research outputs found
Linear Depth Increase of Lambda Terms along Leftmost-Outermost Beta-Reduction
Performing steps of -reduction to a given term in the
-calculus can lead to an increase in the size of the resulting term
that is exponential in . The same is true for the possible depth increase of
terms along a -reduction sequence. We explain that the situation is
different for the leftmost-outermost strategy for -reduction: while
exponential size increase is still possible, depth increase is bounded linearly
in the number of steps. For every -term with depth , in every
step of a leftmost-outermost -reduction rewrite sequence starting from
the term depth increases by at most . Hence the depth of the -th
reduct of in such a rewrite sequence is bounded by .
We prove the lifting of this result to -term representations as
orthogonal first-order term rewriting systems, which can be obtained by the
lambda-lifting transformation. For the transfer to lambda-calculus, we rely on
correspondence statements via lambda-lifting. We argue that the
linear-depth-increase property can be a stepping stone for an alternative proof
of, and so can shed new light on, a result by Accattoli and Dal Lago (2015)
that states: leftmost-outermost -reduction rewrite sequences of length
in the lambda-calculus can be implemented on a reasonable machine with an
overhead that is polynomial in and the size of the initial term