Finite satisfiability for two‐variable, first‐order logic with one transitive relation is decidable

Pratt-Hartmann, Ian

doi:10.1002/malq.201700055

Cited by 7 publications

(28 citation statements)

References 15 publications

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“…Due to space limits the proof is given in Section A.2. The following lemma employs a technique from [13] to eliminate binary predicates.…”

Section: Proof a Structurementioning

confidence: 99%

“…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.…”

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confidence: 99%

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The Fluted Fragment Revisited

Pratt-Hartmann

Szwast²,

Tendera³

2019

J. symb. log.

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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...

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“…Due to space limits the proof is given in Section A.2. The following lemma employs a technique from [13] to eliminate binary predicates.…”

Section: Proof a Structurementioning

confidence: 99%

mentioning

confidence: 99%

The Fluted Fragment Revisited

Pratt-Hartmann

Szwast²,

Tendera³

2019

J. symb. log.

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“…Lemma 5.13 (Adaption of Lemma 4.2 in [Pra18]). If a class K of structures allows cloning then for all ϕ ∈ FO 2 (K) one can effectively find formulas (ϕ C ) C∈C , for some finite set C, in spread normal form such that the following conditions are equivalent:…”

Section: Decision Procedures For a Treementioning

confidence: 99%

“…A particular interest has been in deciding the satisfiability problem for such extensions. Recently a decision procedure for the finite satisfiability problem for ESO 2 with one transitive relation and for ESO 2 with one partial order have been obtained [Pra18]. Previously ESO 2 with two equivalence relations [KO12,KT09] and ESO 2 with two "forest" relations have been shown to be decidable [CW13].…”

Section: Introductionmentioning

confidence: 99%

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

Toruńczyk

Zeume

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register values in subtrees. We show that the emptiness problem for these automata is decidable. As an application, we prove decidability of the countable satisfiability problem for two-variable logic in the presence of a tree order, a linear order, and arbitrary atoms that are MSO definable from the tree order. As a consequence, the satisfiability problem for two-variable logic with arbitrary predicates, two of them interpreted by linear orders, is decidable.

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“…For equivalence, everything is known: the (finite) satisfiability problem for FO 2 in the presence of a single equivalence relation remains NExp-Time-complete, but this increases to 2-NExpTime-complete in the presence of two equivalence relations [9,10], 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 is decidable in 3-NExpTime [16], while the decidability of the satisfiability problem remains open (cf. [26]); the corresponding problems with two transitive relations are both undecidable [11].…”

Section: Introductionmentioning

confidence: 99%

The Fluted Fragment with Transitive Relations

Pratt-Hartmann¹,

Tendera²

2020

Preprint

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We study the satisfiability problem for the fluted fragment extended with transitive relations. The logic enjoys the finite model property when only one transitive relation is available and the finite model property is lost when additionally either equality or a second transitive relation is allowed. We show that the satisfiability problem for the fluted fragment with one transitive relation and equality remains decidable. 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 (or two transitive relations and equality).

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Finite satisfiability for two‐variable, first‐order logic with one transitive relation is decidable

Cited by 7 publications

References 15 publications

The Fluted Fragment Revisited

The Fluted Fragment Revisited

Register Automata with Extrema Constraints, and an Application to Two-Variable Logic

The Fluted Fragment with Transitive Relations

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