A semantical introduction to differential linear logic
نویسنده
چکیده
Extending Linear Logic (LL) with differential constructs has been considered by Girard at a very early stage of the design of this system. This option appears at various places in the conclusion of [Gir86], entitled Two years of linear logic: selection from the garbage collector. In Section V.2 The quantitative attempt of that conclusion, the idea of a syntactic Taylor expansion is explicitly mentioned as a syntactic counterpart of the quantitative semantics of the λ-calculus [Gir88]. However it is contemplated there as a reduction process rather than as a transformation on terms. In Section V.5 The exponentials, the idea of reducing λ-calculus substitution to a more elementary linear operation explicitly viewed as differentiation is presented as one of the basic intuitions behind the exponential of LL. The connection of this idea with Krivine’s Machine [Kri85, Kri07] and its linear head reduction mechanism [DR99] is explicitly mentioned. In this mechanism, first considered by De Bruijn and called mini-reduction in [DB87], it is only the head occurrence of a variable which is substituted during reduction. This restriction is very meaningful in LL: the head occurrence is the only occurrence of a variable in a term which is linear. LL is based on the distinction of particular proofs among all proofs, that are linear wrt. their hypotheses. The word linear has here two deeply related meanings.
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