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We study the algorithmic properties of first-order monomodal logics of frames$\langle \mathbb{N}, \leq \rangle$, $\langle \mathbb{N}, < \rangle$, $\langle\mathbb{Q}, \leq \rangle$, $\langle \mathbb{Q}, < \rangle$, $\langle\mathbb{R}, \leq \rangle$, $\langle \mathbb{R}, < \rangle$, as well as somerelated logics, in languages with restrictions on the number of individualvariables as well as the number and arity of predicate letters. We show thatthe logics of frames based on $\mathbb{N}$ are $\Pi^1_1$-hard -- thus, notrecursively enumerable -- in languages with two individual variables, onemonadic predicate letter and one proposition letter. We also show that thelogics of frames based on $\mathbb{Q}$ and $\mathbb{R}$ are $\Sigma^0_1$-hardin languages with the same restrictions. Similar results are obtained for anumber of related logics.

Algorithmic properties of first-order modal logics of linear Kripke frames in restricted languages