A Local Criterion for Polynomial-Time Stratified Computations

Abstract. This work is a consequence of studying the (un)relatedness of the principles that allow to implicitly characterize the polynomial time functions (PTIME) under two perspectives. One perspective is predicative recursion, where we take Safe Recursion on Notation as representative. The other perspective is structural proof theory, whose representative can be Light Affine Logic (LAL). A way to make the two perspectives closer is to devise polynomial sound generalizations of LAL whose set of interesting proofs-as-programs is larger than the set LAL itself supplies. Such generalizations can be found in MS. MS is a Multimodal Stratified framework that contains subsystems among which we can find, at least, LAL. Every subsystem is essentially determined by two sets. The first one is countable and finite, and supplies the modalities to form modal formulæ. The second set contains building rules to generate proof nets. We call MS multimodal because the set of modalities we can use in the types for the proof nets of a subsystem is arbitrary. MS is also stratified. This means that every box, associated to some modality, in a proof net of a subsystem can never be opened. So, inside MS, we preserve stratification which, we recall, is the main structural proof theoretic principle that makes LAL a polynomial time sound deductive system. MS is expressive enough to contain LAL and Elementary Affine Logic (EAL), which is PTIME-unsound. We supply a set of syntactic constraints on the rules that identifies the PTIME-maximal subsystems of MS, i.e. the PTIMEsound subsystems that contain the largest possible number of rules. It follows a syntactic condition that discriminates among PTIME-sound and PTIME-unsound subsystems of MS: a subsystem is PTIME-sound if its rules are among the rules of some PTIME-maximal subsystem. All our proofs widely use the techniques Context Semantics supplies, and in particular the geometrical configuration that we call dangerous spindle: a subsystem is polytime if and only if its rules cannot build dangerous spindles.