arXiv subjects

Christopher D. C. Hawthorne

Publications and source records attributed to Christopher D. C. Hawthorne.

2 recordsLinked to original sources

arXiv2020

Automata and tame expansions of $(\mathbb{Z},+)$

The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}$ whose set of representations base $d$ is recognized by a finite automaton, a necessary condition is found for $x+y\in A$ to be a stable formula in $\operatorname{Th}(\mathbb{Z},+,A)$. Combined with a theorem of Moosa and Scanlon this gives a combinatorial characterization of the $d$-automatic $A\subseteq \mathbb{Z}$ such that $(\mathbb{Z},+,A)$ is stable. This characterization is in terms of what were called "$F$-sets" by Moosa and Scanlon and "elementary $p$-nested sets" by Derksen. Automata-theoretic methods are also used to produce some NIP expansions of $(\mathbb{Z},+)$, in particular the expansion by the monoid $(d^\mathbb{N},\times )$.

math.LO↗

arXiv2019

Remarks on recognizable subsets and local rank

Given a monoid $(M,\varepsilon,\cdot )$ it is shown that a subset $A\subseteq M$ is recognizable in the sense of automata theory if and only if the $φ$-rank of $x=x$ is zero in the first-order theory $\operatorname{Th}(M,\varepsilon ,\cdot ,A)$, where $φ(x;u)$ is the formula $xu\in A$. In the case where $M$ is a finitely generated free monoid on a finite alphabet $Σ$, this gives a model-theoretic characterization of the regular languages over $Σ$. If $A$ is a regular language over $Σ$ then the $φ$-multiplicity of $x=x$ is the state complexity of $A$. Similar results holds for $φ' (x;u,v)$ given by $uxv\in A$, with the $φ' $-multiplicity now equal to the size of the syntactic monoid of $A$.

math.LO↗