The Exact Complexity of the First-Order Logic Definability Problem

Arenas, Marcelo; Diaz, Gonzalo I.

doi:10.1145/2886095

“…For instance, in reverse engineering of queries, the goal is typically to decide whether there is a query that fits (or separates) a set of positive and negative examples. Relevant work under the closed world assumption include (Arenas and Diaz 2016;Barceló and Romero 2017) and under the open world assumption (Gutiérrez-Basulto, Jung, and Sabellek 2018;Funk et al 2019). Related work on active learning not yet discussed include the identification of EL-queries (Funk, Jung, and Lutz 2021) and ontologies (Konev, Ozaki, and Wolter 2016;Konev et al 2017), and of schema-mappings (ten Cate, Dalmau, and Kolaitis 2013;ten Cate et al 2018).…”

Section: Related Workmentioning

confidence: 99%

Unique Characterisability and Learnability of Temporal Instance Queries

Fortin

¹

,

Konev

²

,

Ryzhikov

³

et al. 2022

Proceedings of the Nineteenth International Conference on Principles of Knowledge Representation and Reasoning

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We aim to determine which temporal instance queries can be uniquely characterised by a (polynomial-size) set of positive and negative temporal data examples. We start by considering queries formulated in fragments of propositional linear temporal logic LTL that correspond to conjunctive queries (CQs) or extensions thereof induced by the until operator. Not all of these queries admit polynomial characterisations, but by imposing a further restriction to path-shaped queries we identify natural classes that do. We then investigate how far the obtained characterisations can be lifted to temporal knowledge graphs queried by 2D languages combining LTL with concepts in description logics EL or ELI (i.e., tree-shaped CQs). While temporal operators in the scope of description logic constructors can destroy polynomial characterisability, we obtain general transfer results for the case when description logic constructors are within the scope of temporal operators. Finally, we apply our characterisations to establish (polynomial) learnability of temporal instance queries using membership queries in the active learning framework.

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“…For instance, in reverse engineering of queries, the goal is typically to decide whether there is a query that fits (or separates) a set of positive and negative examples. Relevant work under the closed world assumption include (Arenas and Diaz 2016;Barceló and Romero 2017) and under the open world assumption (Gutiérrez-Basulto, Jung, and Sabellek 2018;Funk et al 2019).…”

Section: Related Workmentioning

confidence: 99%

Unique Characterisability and Learnability of Temporal Instance Queries

Fortin¹,

Konev²,

Ryzhikov³

et al. 2022

Preprint

0

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We aim to determine which temporal instance queries can be uniquely characterised by a (polynomial-size) set of positive and negative temporal data examples. We start by considering queries formulated in fragments of propositional linear temporal logic LTL that correspond to conjunctive queries (CQs) or extensions thereof induced by the until operator. Not all of these queries admit polynomial characterisations but by restricting them further to path-shaped queries we identify natural classes that do. We then investigate how far the obtained characterisations can be lifted to temporal knowledge graphs queried by 2D languages combining LTL with concepts in description logics E L or E LI (i.e., tree-shaped CQs). While temporal operators in the scope of description logic constructors can destroy polynomial characterisability, we obtain general transfer results for the case when description logic constructors are within the scope of temporal operators. Finally, we apply our characterisations to establish (polynomial) learnability of temporal instance queries using membership queries in the active learning framework. PreliminariesBy a signature we mean any finite set σ = ∅ of atomic concepts A, B, C, . . . representing observations, measurements, events, etc. A σ-data instance is any finite sequenceWe do not distinguish between D and its variants of the form (δ 0 , . . . , δ n , ∅, . . . , ∅).We access data by means of queries, q, constructed from atoms, ⊥ and ⊤ using ∧ and the temporal operators , ♦, ♦ r and U. The set of atomic concepts occurring in q is denoted by sig(q). The set of queries that only use the operators from Φ ⊆ { , ♦, ♦ r , U} is denoted byto a signature σ. The size |q| of q is the number of symbols in q, and the temporal depth tdp(q) of q is the maximum number of nested temporal operators in q.Q[ , ♦, ♦ r ]-queries can be equivalently defined as treeshaped conjunctive queries (CQs) with the binary predicates suc, <, ≤ over N, and atomic concepts as unary predicates. Each such CQ is a set Q(t 0 ) of assertions of the form A(t), suc(t, t ′ ), t < t ′ , and t ≤ t ′ , with a distinguished variable t 0 , such that, for every variable t in Q(t 0 ), there exists exactly one path from t 0 to t along the binary predicates suc, <, ≤.The set of Q[ , ♦, ♦ r ]-queries with path-shaped CQ counterparts is denoted by Q p [ , ♦, ♦ r ]. Such queries q take the form (1), where o i ∈ { , ♦, ♦ r } and ρ i is a conjunction of atoms (the empty conjunction is ⊤). Similarly, Q p [U]-queries take the form (2).Given a data instance D = (δ 0 , . . . , δ n ), the truth-relation D, ℓ |= q, for ℓ < ω, is defined as follows:Note that D, n |= ♦⊤ ∧ ⊤ ∧ (q U ⊤) as (δ 0 , . . . , δ n , ∅) is a variant of D. We write q |= q ′ if D, ℓ |= q implies D, ℓ |= q ′ for any D and ℓ. If q |= q ′ and q ′ |= q, we call q and q ′ equivalent and write q ≡ q ′ . Since q ≡ ⊥Uq, ♦q ≡ ⊤Uq and ♦q ≡ ♦ r q, one can assume that Q

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“…We observe that FO has the dimension-collapse property, which means that every training database that is FO-separable is also separable by a statistics with a single FO feature. This allows us to show that FO-separability has the same complexity as the QBE problem for FO, which is known to be GI-complete [4]. We also provide a characterization based on a definability condition of when a query language has the dimension collapse property.…”

Section: Introductionmentioning

confidence: 99%