Abstract. We introduce a multimodal stratified framework MS that generalizes an idea hidden in the definitions of Light Linear/Affine logical systems: "More modalities means more expressiveness". MS is a set of building-rule schemes that depend on parameters. We interpret the values of the parameters as modalities. Fixing the parameters yields deductive systems as instances of MS, that we call subsystems. Every subsystem generates stratified proof nets whose normalization preserves stratification, a structural property of nodes and edges, like in Light Linear/Affine logical systems. A first result is a sufficient condition for determining when a subsystem is strongly polynomial time sound. A second one shows that the ability to choose which modalities are used and how can be rewarding. We give a family of subsystems as complex as Multiplicative Linear Logic -they are linear time and space sound -that can represent Church numerals and some common combinators on them.
We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels (LALL), derived from the logic ML 4 . LALL is an intuitionistic deductive system, with a polynomial time cut elimination strategy. The embedding allows to represent every term t of SRN as a family of nets ⌈t⌉ l l∈N in LALL. Every net ⌈t⌉ l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the "fuzzy" borders of paragraph boxes that LALL inherits from ML 4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of ⌈t⌉ l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of ⌈t⌉ l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principles SRN relies on, and the proof theoretic one, called stratification, at the base of Light Linear Logic.
Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is considered as computation. We introduce modal impredicativity as a new form of impredicativity causing the complexity of cut-elimination to be problematic from a complexity point of view. Modal impredicativity occurs when, during reduction, the conclusion of a residual of a box b interacts with a node that belongs to the proof net inside another residual of b. Technically speaking, superlazy reduction is a new notion of reduction that allows to control modal impredicativity. More specifically, superlazy reduction replicates a box only when all its copies are opened. This makes the overall cost of reducing a proof net finite and predictable. Specifically, superlazy reduction applied to any pure proof nets takes primitive recursive time. Moreover, any primitive recursive function can be computed by a pure proof net via superlazy reduction.
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