2022/10/05 by Gaëtan Douéneau-Tabot, Douéneau-Tabot, Gaëtan
Computer Science · #Algorithms and Data Compression #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2210.02426
openalex publication_date 2022/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Nguyên 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.