Algebraic classifications for fragments of first-order logic and beyond

Abstract

Complexity and decidability of logical systems is a major research area currently involving a huge range of different systems from fragments of first-order logic to modal logic, description logics, et cetera. Due to the sheer number of different frameworks investigated, a systematic approach could be beneficial. We introduce a research program based on an algebraic approach to systematic complexity classifications of fragments of first-order logic and beyond. Our base system GRA, or general relation algebra, is equiexpressive with FO. The system GRA resembles cylindric algebra and Codd's relational algebra, but employs a finite signature with seven different operators. We provide a comprehensive classification of the decidability and complexity of the systems obtained by limiting the allowed sets of operators. Furthermore, we also investigate algebras with sets of operators not included in the list of the seven basic ones and use such operator sets to give algebraic characterizations to some of the best known decidable first-order fragments. To lift the related studies beyond FO, we also define a notion of a generalized relation operator. These operators can be seen to slightly generalize the notion of a generalized quantifier due to Mostowski and Lindström.

Ordered fragments of first-order logic

“…Our third undecidability result is that the extension of fluted logic with swap is undecidable. Since the restriction of GRA(e, s, ¬, ∩, ∃) to vocabularies of arity at most two is sententially equivalent with two-variable logic [6], this result seems to indicate that there does not exists a natural decidable logic that extends both the two-variable logic and the fluted logic.…”

“…The purpose of this section is to present the algebraic framework introduced in [5,6,9] for classifying fragments of FO. We will be working with purely relational vocabularies with no constants and function symbols.…”

“…The following definition was introduced in [6], where it was called arity-regular relation operator.…”

“…Recently a research program was introduced in [5,6,9] for classifying fragments of first-order logic within an algebraic framework. In a nutshell, the basic idea is to identify fragments of first-order logic with finite algebraic signatures (for more details, see the next section).…”

“…The satisfiability problem for this fragment was proved to be in Pspace in [4], which should be compared with the fact that the satisfiability problem for fluted logic is Tower-complete, which was proved in [13]. We note that the problem is in fact Pspace-complete, and the proof for the Pspace-hardness will be given in section 5.…”