Decidable fragments of first-order logic and of first-order linear arithmetic with uninterpreted predicates

Separateness of Variables -- A Novel Perspective on Decidable First-Order Fragments

“…The proof of this result can be found in [69] (Theorem 3.1.5). It is based on techniques described by Behmann [8] in the context of second-order quantifier elimination for the monadic second-order fragment -see [70], Section 13.2, for a modern account.…”

“…It is worth noting that [68] uses a much more fine-grained measure than the number of ∀∃ alternations. Even more details can be found in [69], Section 3.2.…”

“…Due to space limitations, we will not elaborate any further on this topic. The interested reader will find more details in [68] and in [69], Sections 3.2 and 3.5.…”

“…More recently identified decidable fragments that are neither covered in the present paper nor in the mentioned texts are treated in [62,41,49,61,12]. A much more detailed but still brief overview of past and recent developments in the area is given in the beginning of Chapter 3 of [69].…”

“…Note to editors and reviewers: The present paper is (except for Theorem 8.6) a condensation of material taken from the author's PhD thesis [69], mainly Chapter 3. Moreover, results from previous conference publications are recapitulated [65,68] (with citations).…”