Fluted Formulas and the Limits of Decidability

In the predicate calculus, variables provide a flexible indexing service that selects the actual arguments to a predicate letter from among possible arguments that precede the predicate letter (in the parse of the formula). In the process of selection, the possible arguments can be permuted, repeated (used more than once), and skipped. If this service is withheld, so that arguments must be the immediately preceding ones, taken in the order in which they occur, the formula is said to be fluted. Quine showed that if a fluted formula contains only homogeneous conjunction (conjoins only subformulas of equal arity), then the satisfiability of the formula is decidable. It remained an open question whether the satisfiability of a fluted formula without this restriction is decidable. This paper answers that question

Inexpressiveness of First-Order Fragments

It is well-known that first-order logic is semi-decidable. Therefore, first-order logic is less than ideal for computational purposes (computer science, knowledge engineering). Certain fragments of first-order logic are of interest because they are decidable. But decidability is gained at the cost of expressiveness. The objective of this paper is to investigate inexpressiveness of fragments that have received much attention

The Fluted Fragment with Transitivity

We study the satisfiability problem for the fluted fragment extended with transitive relations. We show that the logic enjoys the finite model property when only one transitive relation is available. On the other hand we show that the satisfiability problem is undecidable already for the two-variable fragment of the logic in the presence of three transitive relations

Fluted Logic with Counting

The fluted fragment is a fragment of first-order logic in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that the fluted fragment possesses the finite model property. In this paper, we extend the fluted fragment by the addition of counting quantifiers. We show that the resulting logic retains the finite model property, and that the satisfiability problem for its (m+1)-variable sub-fragment is in m-NExpTime for all positive m. We also consider the satisfiability and finite satisfiability problems for the extension of any of these fragments in which the fluting requirement applies only to sub-formulas having at least three free variables

Craig Interpolation for Decidable First-Order Fragments

We show that the guarded-negation fragment (GNFO) is, in a precise sense, the smallest extension of the guarded fragment (GFO) with Craig interpolation. In contrast, we show that the smallest extension of the two-variable fragment (FO2), and of the forward fragment (FF) with Craig interpolation, is full first-order logic. Similarly, we also show that all extensions of FO2 and of the fluted fragment (FL) with Craig interpolation are undecidable.Comment: Submitted for FoSSaCS 2024. arXiv admin note: substantial text overlap with arXiv:2304.0808

Quine’s Fluted Fragment is Non-elementary

We study the fluted fragment, a decidable fragment of first-order logic with an unbounded number of variables, originally identified by W.V. Quine. We show that the satisfiability problem for this fragment has non-elementary complexity, thus refuting an earlier published claim by W.C. Purdy that it is in NExpTime. More precisely, we consider, for all m greater than 1, the intersection of the fluted fragment and the m-variable fragment of first-order logic. We show that this sub-fragment forces (m/2)-tuply exponentially large models, and that its satisfiability problem is (m/2)-NExpTime-hard. We round off by using a corrected version of Purdy\u27s construction to show that the m-variable fluted fragment has the m-tuply exponential model property, and that its satisfiability problem is in m-NExpTime