Remarks on recognizable subsets and local rank [article]

Christopher D. C. Hawthorne
2019 arXiv   pre-print
Given a monoid (M,ε,· ) it is shown that a subset A⊆ 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 Th(M,ε ,· ,A), where φ (x;u) is the formula xu∈ 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
more » ... φ' (x;u,v) given by uxv∈ A, with the φ'-multiplicity now equal to the size of the syntactic monoid of A.
arXiv:1803.07234v2 fatcat:t3hjxt47qreorgoq3sviasqqdq