Pebble minimization: the last theorems

TL;DR

This paper investigates the problem of minimizing recursion height in pebble transducers, providing effective algorithms for restricted models and highlighting the complexity introduced when multiple marks are visible.

Contribution

It introduces algorithms for minimizing recursion height in blind and last pebble transducers, and demonstrates the limitations when multiple marks are visible.

Findings

Effective algorithms for blind and last pebble transducer minimization.

Linear, quadratic, and cubic output size functions correspond to recursion heights 1, 2, and 3.

The key property fails when machines can see more than one mark.

Abstract

Pebble transducers are nested two-way transducers which can drop marks (named "pebbles") on their input word. Such machines can compute functions whose output size is polynomial in the size of their input. They can be seen as simple recursive programs whose recursion height is bounded. A natural problem is, given a pebble transducer, to compute an equivalent pebble transducer with minimal recursion height. This problem is open since the introduction of the model. In this paper, we study two restrictions of pebble transducers, that cannot see the marks ("blind pebble transducers" introduced by Nguy\^en et al.), or that can only see the last mark dropped ("last pebble transducers" introduced by Engelfriet et al.). For both models, we provide an effective algorithm for minimizing the recursion height. The key property used in both cases is that a function whose output size is linear (resp. quadratic, cubic, etc.) can always be computed by a machine whose recursion height is 1 (resp. 2, 3, etc.). We finally show that this key property fails as soon as we consider machines that can see more than one mark.