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15-816 Linear Logic |
The inversion principles discussed so far leave a gaping hole in that the dereliction rule (DL) is applicable at any point, and may be applied to any unrestricted hypothesis without reducing any connective, and without being invertible. In this lecture we analyse it further, considering special instances of the dereliction rule (one for each connective and quantifier) which, in totality, replace the unrestricted derelection rule by many specialized left rules for unrestricted hypotheses. Many of these will be invertible, further reducing non-determinism.
We also consider how natural numbers and equations over natural numbers can be encoded in linear logic in such a way that a set of linear equations has a solution if and only if their representation has a proof. The image of this translation uses only !, *, 1, and -o, the so-called multiplicative-exponential fragment of linear logic, whose decidability (I believe) is still open.