ASNP: A Tame Fragment of Existential Second-Order Logic

Bodirsky, Manuel; Knäuer, Simon; Starke, Florian

doi:10.1007/978-3-030-51466-2_13

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…By Theorem 5, this is the case iff there is a homogeneous structure D such that Age(D) = Forb e (N ). A decision procedure for this problem was also described recently in [29]. As mentioned in the proof of Theorem 4 in [29], it is enough to show AP restricted to triples A, B 1 , B 2 such that e i : A → B i is the identity map and…”

Section: Theorem 15mentioning

confidence: 99%

“…Given a finite set of bounds N , we guess a triple A, B 1 , B 2 satisfying (#) and check whether this triple witnesses that Forb e (N ) does not have AP. According to the proof of Theorem 4 in [29], the size of a smallest counterexample to AP for Forb e (N ) is bounded by a polynomial in m • where m := max F∈N |F| and := 2 |τ | . Thus, by what we have shown above for the size of N , we may assume that the size of A, B 1 , B 2 is polynomial in the size of the input N , which shows that this triple can be guessed within NP.…”

Section: Theorem 15mentioning

confidence: 99%

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Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Baader

Rydval

2022

J Autom Reasoning

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Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL $$\mathcal {ALC}$$ ALC in the presence of GCIs, quite strong restrictions, in sum called $$\omega $$ ω -admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of $$\omega $$ ω -admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate $$\omega $$ ω -admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield $$\omega $$ ω -admissible concrete domains. This allows us to show $$\omega $$ ω -admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the $$\mathcal {EL}$$ EL family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although $$\omega $$ ω -admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into $$\mathcal {ALC}$$ ALC yields decidable DLs.

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Section: Theorem 15mentioning

confidence: 99%

Section: Theorem 15mentioning

confidence: 99%

Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Baader

Rydval

2022

J Autom Reasoning

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“…An important open question about universal sentences over relational signatures is the decidability of the amalgamation property (AP), which is a strong form of the joint embedding property, and which is of fundamental importance in constraint satisfaction [Bod21]. Unlike the JEP, the AP is decidable if all relation symbols in the signature are at most binary (see, e.g., [KL87,BKS20]). For general relational signatures, we ask the following question.…”

Section: Open Problemsmentioning

confidence: 99%

Universal Horn Sentences and the Joint Embedding Property

Bodirsky¹,

Rydval²,

Schrottenloher³

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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The finite models of a universal sentence $\Phi$ in a finite relational signature are the age of a structure if and only if $\Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $\Phi$ has the joint embedding property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most two.

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“…An important open question about universal sentences over relational signatures is the decidability of the amalgamation property (AP), which is a strong form of the joint embedding property, and which is of fundamental importance in constraint satisfaction [Bod21]. Unlike the JEP, the AP is decidable if all relation symbols in the signature are at most binary (see, e.g., [BKS20]). For general relational signatures, we ask the following question.…”

Section: Open Problemsmentioning

confidence: 99%