A Decidable Fragment of First Order Modal Logic: Two Variable Term Modal Logic

First order modal logic (𝖥𝖮𝖬𝖫) is built by extending First Order Logic (𝖥𝖮) with modal operators. A typical formula is of the form \(\forall x \exists y \Box P(x,y)\). Not only is 𝖥𝖮𝖬𝖫 undecidable, even simple fragments like that of restriction to unary predicate symbols, guarded fragment and two variable fragment, which are all decidable for 𝖥𝖮 become undecidable for 𝖥𝖮𝖬𝖫. In this paper we study Term Modal logic (𝖳𝖬𝖫) which allows modal operators to be indexed by terms. A typical formula is of the form \(\forall x \exists y~\Box _x P(x,y)\). There is a close correspondence between 𝖳𝖬𝖫 and 𝖥𝖮𝖬𝖫 and we explore this relationship in detail in the paper.

In contrast to 𝖥𝖮𝖬𝖫, we show that the two variable fragment (without constants, equality) of 𝖳𝖬𝖫 is decidable. Further, we prove that adding a single constant makes the two variable fragment of 𝖳𝖬𝖫 undecidable. On the other hand, when equality is added to the logic, it loses the finite model property.