Abstract:In previous papers (Kosoy 2026a,b), we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of “zipline programs” (a variant of straight-line programs we defined in Kosoy 2026b), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.
This paper continues the "stringological sequence prediction" series (see III) about my new approach to compositional learning theory. It describes a tradeoff between efficiency and expressivity, but whether this tradeoff is necessary or just a limitation of my techniques remains an interesting open problem.
Abstract: In previous papers (Kosoy 2026a,b), we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of “zipline programs” (a variant of straight-line programs we defined in Kosoy 2026b), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.
This paper continues the "stringological sequence prediction" series (see I II) about my new approach to compositional learning theory. It describes a tradeoff between efficiency and expressivity, but whether this tradeoff is necessary or just a limitation of my techniques remains an interesting open problem.