A fine-grained hierarchy of hard problems in the separated fragment

Voigt, Marco

doi:10.1109/lics.2017.8005094

Cited by 6 publications

(11 citation statements)

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“…For example, the classical decidable quantifier-prefix fragments [2], the guarded fragment [3] and the two-variable fragment (mentioned above) all have elementary complexity. One first-order fragment that comprises a similar hierarchy of hard problems to FL, however, is the recently discovered separated fragment [23].…”

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confidence: 99%

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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confidence: 99%

The Fluted Fragment Revisited

Pratt-Hartmann

Szwast²,

Tendera³

2019

J. symb. log.

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“…Concerning computational complexity, the hierarchy of k-NExpTime-complete subproblems of SF-satisfiability presented in [22] together with the containment of SF in GBSR leads to the observation that GBSR-satisfiability is non-elementary.…”

Section: Proofmentioning

confidence: 99%

“…On the other hand, the lower bound proof for SF-satisfiability in [22] is based on SF-formulas that encode computationally hard domino problems. One can use the same formulas to show that every set GBSR k contains an infinite subset SF k of SF formulas of degree 1 k such that there is a polynomial reduction from some k-NExpTime-hard domino problem to the satisfiability problem for SF k (see Lemma 20 in [22]). Hence, the satisfiability problem for GBSR k is k-NExpTime-hard as well.…”

Section: Proofmentioning

confidence: 99%

“…The number of leading existential quantifiers in BSR sentences induces an upper bound on the size of small models-every satisfiable BSR sentences has such a small model. One can adapt the methods applied in [22] to facilitate the derivation of a tight upper on the number of leading existential quantifiers. To this end, the notion of degree of interaction of existential variables used in that paper needs to be extended so that it also covers the interaction of universally and existentially quantified variables caused by joint occurrences in atoms.…”

Section: A Syntactic Approach To Decidability Of Gbsr-satisfiabilitymentioning

confidence: 99%

“…Its defining principle is that universally and existentially quantified variables may not occur together in atoms. In [22] it is shown that the satisfiability problem for SF is non-elementary. Yet, SF as well as GBSR enjoy the finite model property.…”

Section: Introductionmentioning

confidence: 99%

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On Generalizing Decidable Standard Prefix Classes of First-Order Logic

Voigt

2017

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Recently, the separated fragment (SF) of first-order logic has been introduced. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. SF properly generalizes both the Bernays-Schönfinkel-Ramsey (BSR) fragment and the relational monadic fragment. In this paper the restrictions on variable occurrences in SF sentences are relaxed such that universally and existentially quantified variables may occur together in the same atom under certain conditions. Still, satisfiability can be decided. This result is established in two ways: firstly, by an effective equivalence-preserving translation into the BSR fragment, and, secondly, by a model-theoretic argument.Slight modifications to the described concepts facilitate the definition of other decidable classes of first-order sentences. The paper presents a second fragment which is novel, has a decidable satisfiability problem, and properly contains the Ackermann fragment and-once more-the relational monadic fragment. The definition is again characterized by restrictions on the occurrences of variables in atoms. More precisely, after certain transformations, Skolemization yields only unary functions and constants, and every atom contains at most one universally quantified variable. An effective satisfiability-preserving translation into the monadic fragment is devised and employed to prove decidability of the associated satisfiability problem.

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