  In CSL'99 Roversi pointed out that the Turing machine encoding of Girard's
seminal paper "Light Linear Logic" has a flaw. Moreover he presented a working
version of the encoding in Light Affine Logic, but not in Light Linear Logic.
In this paper we present a working version of the encoding in Light Linear
Logic. The idea of the encoding is based on a remark of Girard's tutorial paper
on Linear Logic. The encoding is also an example which shows usefulness of
additive connectives. Moreover we also consider a nondeterministic extension of
Light Linear Logic. We show that the extended system is NP-complete in the same
meaning as P-completeness of Light Linear Logic.
