Online and feasible presentability: from trees to modal algebras
Nikolay Bazhenov, Dariusz Kalociński, Michał Wrocławski
TL;DR
The paper analyzes whether every computable member of a class of structures has a punctual or $P$-TIME copy, focusing on trees, lattices, and modal algebras. It establishes that the class of relational predecessor ordered trees (r.p.o. trees) is both punctually robust and $P$-TIME robust, while poset trees are not punctually robust, and several intermediate lattice-like classes (including join semilattices, meet semilattices, lattices, complemented lattices, and non-distributive lattices) fail to be punctually robust; a corollary notes that semilattices and lattices are not punctually robust. The results contribute to the program of effective and feasible algebra and clarify the landscape of online presentations for countable structures, using diagonalization and constructive methods. A key finding is that, contrary to Boolean algebras, modal algebras are not punctually robust, and open questions remain for distributive lattices.
Abstract
We investigate whether every computable member of a given class of structures admits a fully primitive recursive (also known as punctual) or fully P-TIME copy. A class with this property is referred to as punctually robust or P-TIME robust, respectively. We present both positive and negative results for structures corresponding to well-known representations of trees, such as binary trees, ordered trees, sequential (or prefix) trees, and partially ordered (poset) trees. A corollary of one of our results on trees is that semilattices and lattices are not punctually robust. In the main result of the paper, we demonstrate that, unlike Boolean algebras, modal algebras - that is, Boolean algebras with modality - are not punctually robust. The question of whether distributive lattices are punctually robust remains open. The paper contributes to a decades-old program on effective and feasible algebra, which has recently gained momentum due to rapid developments in punctual structure theory and its connections to online presentations of structures.