Automatic and Polynomial-Time Algebraic Structures

Abstract

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

Foundations of Online Structure Theory

“…This solves a problem left open in the brief conference survey [51]. The proof is too technical to be discussed in this survey, but [5] contains an extended informal discussion of the proof. Suprisingly, the same diagonalisation technique with insignificant adjustments allows to prove: Theorem 2.4 ( [5]).…”

“…The proof is too technical to be discussed in this survey, but [5] contains an extended informal discussion of the proof. Suprisingly, the same diagonalisation technique with insignificant adjustments allows to prove: Theorem 2.4 ( [5]). The index set {e : M e has an automatic presentation} is Σ 1 1 -complete.…”

“…The index set {e : M e has an automatic presentation} is Σ 1 1 -complete. Theorem 2.5 ( [5]). The index set {e : M e has a polynomial-time presentation} is Σ 1 1 -complete.…”

Punctual Categoricity and Universality

“…In fact, as discussed in [2,22], very often eliminating unbounded search is the crucial step in turning a general Turing computable algebraic procedure into, say, a polynomial time or a polylogspace one; see, for example, [5,6,7,16]. A nontrivial illustration of this phenomenon is the recent solution [3] to a problem of Khouissainov and Nerode on the characterization of automatic structures ( [24], Question 4.9). The key step in the proof in [3] is a simpler argument for primitive recursive structures; with some extra work it is then pushed to the extremely narrow class of automatic structures.…”

“…A nontrivial illustration of this phenomenon is the recent solution [3] to a problem of Khouissainov and Nerode on the characterization of automatic structures ( [24], Question 4.9). The key step in the proof in [3] is a simpler argument for primitive recursive structures; with some extra work it is then pushed to the extremely narrow class of automatic structures.…”