Implicit complexity through linear realisability: polynomial time and probabilistic classes - Archive ouverte HAL
Preprints, Working Papers, ... Year : 2023

Implicit complexity through linear realisability: polynomial time and probabilistic classes

Thomas Seiller

Abstract

Based on work on realisability models for linear logic, the author recently proposed a new approach of implicit computational complexity. He showed how to characterise in this way a hierarchy of sub-linear space non-deterministic complexity classes by means of group actions. These classes, defined by means of two-way non-deterministic automata, range from regular languages to NLogspace (non-deterministic logarithmic space). In the present paper, we extend those results in two directions. First, we show how the same techniques can be used to characterise the sub-linear space complexity classes defined by both deterministic and probabilistic two-way multi-head automata. We thus obtain characterisations of deterministic complexity classes between regular languages and Logspace, as well as a hierarchy of probabilistic classes between stochastic languages and PLogspace (unbounded error probabilistic logarithmic space). Second, we exhibit a monoid action capturing polynomial time computation based on pushdown machines, characterising both Ptime and PPtime (unbounded error probabilistic polynomial time).
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Dates and versions

hal-02458358 , version 1 (28-01-2020)
hal-02458358 , version 2 (10-02-2023)

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Thomas Seiller. Implicit complexity through linear realisability: polynomial time and probabilistic classes. 2023. ⟨hal-02458358v2⟩
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