  The computable model theory of modal logic was initiated by Suman Ganguli and
Anil Nerode in [4]. They use an effective Henkin-type construction to
effectivize various completeness theorems from classical modal logic. This
construction has the feature of only producing models whose frames can be
obtained by adding edges to a tree digraph. Consequently, this construction
cannot prove an effective version of a well-known completeness theorem which
states that every $\mathsf{S4.3.1}$-theory has a model whose accessibility
relation is a linear order of order type $\omega$. We prove an effectivization
of that theorem by means of a new construction adapted from that of Ganguli and
Nerode.
