On Light Logics, Uniform Encodings and Polynomial Time Ugo Dal Lago * dallago@cs.unibo.it Patrick Baillot † pb@lipn.univ-paris13.fr June 7, 2004 Abstract We investigate on the extensional expressive power of Light Affine Logic, analysing the class of uniformly representable functions for various fragments of the logic. We give evidence on the incompleteness (for polynomial time) of the propositional fragment. Following previous work, we show that second order leads to polytime unsoundness. We then introduce simple constraints on second order quantification and least fixpoints, proving the obtained fragment to be polytime sound and complete. 1 Introduction Characterizing the class of functions a logic can represent helps in understanding the computa- tional expressive power of the logic. If the system under consideration enjoys a Curry-Howard correspondence, the analysis can be even more important — the class of representable functions becomes the class of functions the underlying programming language can compute. These inves- tigations become a crucial issue in the context of light logics, which have been defined precisely for capturing relevant function classes, namely complexity classes. Light Linear Logic (LLL, [6]) has been proposed by Girard as a variant of Linear Logic (LL, [5]) characterizing the class FP of deterministic polynomial time functions. It has been later simplified by Asperti into Light Affine Logic (LAL, [1]). The limited computational power of LLL/LAL is obtained by considering a weaker modality ! for resource re-use than in plain Linear Logic. LAL has been the subject of many investigations from syntactical, semantical and programming language perspectives [10, 11, 12, 14]. Another line of research in that direction is Lafont’s Soft Linear Logic (SLL, [7]) which is another variant of LL for polynomial time. Still, one can observe that these characterizations of FP via the Curry-Howard correspondence (in LLL/LAL or SLL) only hold provided data are encoded by bounded-depth proofs (the notion of box is linked to the modalities). Recently, Mairson and Neergaard [9] proved that dropping the bounded box-depth assumption makes LAL complete for doubly exponential time. In their setting, data are represented by proofs having unbounded box-depth and different conclusions. Alternative notions of encodings have also been considered in [8] for various subsystems of LL. An important point is that the encodings in [6, 2] make an extensive use of second-order quantification, which allows programming with polymorphism in the style of System F. This is an elegant and general approach, but second-order quantification comes with difficulties of its own, which are not related to LAL itself. For instance it makes the issues of provability decision problems, type-inference or semantics far more delicate. One can wonder how much of the power of second-order is really needed in LAL to get polynomial time expressivity. * Dipartimento di Scienze dell’Informazione, Universit` a di Bologna, Italy. † LIPN-CNRS, Universit´ e Paris-Nord, France. 1