The Fluted Fragment Revisited

Abstract

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. ACM Subject ClassificationTheory of computation → Complexity theory and logic; Theory of computation → Finite Model Theory I. Pratt-Hartmann and L. Tendera XX:3 respect of FO 2 , which has the finite model property and whose satisfiability problem, as just mentioned, is NExpTime-complete. The finite model property is lost when one transitive relation or two equivalence relations are allowed. For equivalence, everything is known: the (finite) satisfiability problem for FO 2 in the presence of a single equivalence relation remains NExpTime-complete, but this increases to 2-NExpTime-complete in the presence of two equivalence relations [7,8], and becomes undecidable with three. For transitivity, we have an incomplete picture: the finite satisfiability problem for FO 2 in the presence with a single transitive relation in decidable in 3-NExpTime [14], while the decidability of the satisfiability problem remains open (cf. [24]); the corresponding problems with two transitive relations are both undecidable [9].Adding equivalence relations to the fluted fragment poses no new problems. Existing results on of FO 2 with two equivalence relations can be used to show that the satisfiability and finite satisfiability problems for FL (not just FL 2 ) with two equivalence relations are decidable. Furthermore, the proof that the corresponding problems for FO 2 in the presence of three equivalence relations are undecidable can easily be seen to apply also to FL 2 . On the other hand, the situation with transitivity is much less clear. In particular, it is not known to the present authors whether the description logic SHI, the extension of SH where also role inverses can be used (a feature not expressible in FL), enjoys the finite model property. Some indication that flutedness interacts in interesting ways with transitivity is provided by known complexity results on various extensions of guarded two-variable fragment with transitive relations. The guarded fragment, denoted GF, is that fragment of first-order logic in which all quantification is of either of the forms ∀v(α → ψ) or ∃v(α ∧ ψ), where α is an atomic formula (a so-called guard) featuring all free variables of ψ. The guarded two-variable fragment, denoted GF 2 , is the intersection of GF and FO 2 . It is straightforward to show that the addition of two transitive relations to GF 2 yields a logic whose satisfiability problem is undecidable. However, as long as the distinguished transitive relations appear only in guards, we can extend the whole of GF with any number of transitive relations, yielding the so-called guarded fragment with transitive guards, whose satisfiability problem is in 2-ExpTime [23]. Intr...

Ordered fragments of first-order logic

“…It was proved in [13] that the satisfiability problem for FL is Tower-complete. The natural follow-up question is then to study what fragments of FL have more "feasible" complexity.…”

“…Informally speaking, a fragment of first-order logic is called ordered, if the syntax of the fragment restricts permutations of variables (with respect to some ordering of the variables) and the order in which the variables are to be quantified. To give an example of a fragment that belongs to this family of logics, we will mention the so-called fluted logic which has received some attention quite recently [13,14,17]. Roughly speaking, in fluted logic the order in which the variables are being quantified should be the same as the order in which these variables occur in atomic formulas.…”

“…The first logic that we consider in this article is the so-called ordered logic, which is a fragment of the fluted logic (for a formal definition see section 3). 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.…”

The Fluted Fragment with Transitive Relations

“…Then the satisfiability problem for F L m is ⌊m/2⌋-NExpTime-hard for all m ≥ 2 and in (m − 2)-NExpTime for all m ≥ 3 [18]. It follows that the satisfiability problem for F L is Tower-complete, in the framework of [24].…”

“…We refer to the formulas κ i as control formulas; observe in this regard that the binary predicates T and = count as atomic formulas of F L [m] for all m ≥ 2. The following lemma is slightly modified from [18,Lemma 4.1], where it was proved for the sub-fragment without equality. The proof, however, is virtually identical, and we may simply state: Lemma 4.…”