arXiv subjects

Simon Knäuer

Publications and source records attributed to Simon Knäuer.

7 recordsLinked to original sources

arXiv2026

The Network Satisfaction Problem for Relation Algebras with at most 4 Atoms

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.

math.RA↗

arXiv2025

Datalog-Expressibility for Monadic and Guarded Second-Order Logic

We characterise the sentences in Monadic Second-order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class C of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all integers l,k, there exists a canonical Datalog program Pi of width (l,k) in the sense of Feder and Verdi. The same characterisations also hold for Guarded Second-order Logic (GSO), which properly extends MSO. To prove our results, we show that every class C in GSO whose complement is closed under homomorphisms is a finite union of constraint satisfaction problems (CSPs) of countably categorical structures. The intersection of MSO and Datalog is known to contain the class of nested monadically defined queries (Nemodeq); likewise, we show that the intersection of GSO and Datalog contains all problems that can be expressed by the more expressive language of nested guarded queries. Yet, by exploiting our results, we can show that neither of the two query languages can serve as a characterization, as we exhibit a query in the intersection of MSO and Datalog that is not expressible in nested guarded queries.

cs.LO↗

arXiv2025

Network Satisfaction Problems Solved by k-Consistency

We show that the problem of deciding for a given finite relation algebra A whether the network satisfaction problem for A can be solved by the k-consistency procedure, for some natural number k, is undecidable. For the important class of finite relation algebras A with a normal representation, however, the decidability of this problem remains open. We show that if A is symmetric and has a flexible atom, then the question whether NSP(A) can be solved by k-consistency, for some natural number k, is decidable (even in polynomial time in the number of atoms of A). This result follows from a more general sufficient condition for the correctness of the k-consistency procedure for finite symmetric relation algebras. In our proof we make use of a result of Alexandr Kazda about finite binary conservative structures.

math.LO↗

arXiv2023

The Complexity of Network Satisfaction Problems for Symmetric Relation Algebras with a Flexible Atom

Robin Hirsch posed in 1996 the 'Really Big Complexity Problem': classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is symmetric and has a flexible atom; in this case, the problem is NP-complete or in P. The classification task can be reduced to the case where A is integral. If a finite integral relation algebra has a flexible atom, then it has a normal representation B. We can then study the computational complexity of the network satisfaction problem of A using the universal-algebraic approach, via an analysis of the polymorphisms of B. We also use a Ramsey-type result of Nešetřil and Rödl and a complexity dichotomy result of Bulatov for conservative finite-domain constraint satisfaction problems.

math.LO↗

arXiv2021

On Logics and Homomorphism Closure

Predicate logic is the premier choice for specifying classes of relational structures. Homomorphisms are key to describing correspondences between relational structures. Questions concerning the interdependencies between these two means of characterizing (classes of) structures are of fundamental interest and can be highly non-trivial to answer. We investigate several problems regarding the homomorphism closure (homclosure) of the class of all (finite or arbitrary) models of logical sentences: membership of structures in a sentence's homclosure; sentence homclosedness; homclosure characterizability in a logic; normal forms for homclosed sentences in certain logics. For a wide variety of fragments of first- and second-order predicate logic, we clarify these problems' computational properties.

cs.LO↗

arXiv2020

Hardness of Network Satisfaction for Relation Algebras with Normal Representations

We study the computational complexity of the general network satisfaction problem for a finite relation algebra $A$ with a normal representation $B$. If $B$ contains a non-trivial equivalence relation with a finite number of equivalence classes, then the network satisfaction problem for $A$ is NP-hard. As a second result, we prove hardness if $B$ has domain size at least three and contains no non-trivial equivalence relations but a symmetric atom $a$ with a forbidden triple $(a,a,a)$, that is, $a \not\leq a \circ a$. We illustrate how to apply our conditions on two small relation algebras.

cs.LO↗

arXiv2020

ASNP: a tame fragment of existential second-order logic

Amalgamation SNP (ASNP) is a fragment of existential second-order logic that strictly contains binary connected MMSNP of Feder and Vardi and binary guarded monotone SNP of Bienvenu, ten Cate, Lutz, and Wolter; it is a promising candidate for an expressive subclass of NP that exhibits a complexity dichotomy. We show that ASNP has a complexity dichotomy if and only if the infinite-domain dichotomy conjecture holds for constraint satisfaction problems for first-order reducts of binary finitely bounded homogeneous structures. For such CSPs, powerful universal-algebraic hardness conditions are known that are conjectured to describe the border between NP-hard and polynomial-time tractable CSPs. The connection to CSPs also implies that every ASNP sentence can be evaluated in polynomial time on classes of finite structures of bounded treewidth. We show that the syntax of ASNP is decidable. The proof relies on the fact that for classes of finite binary structures given by finitely many forbidden substructures, the amalgamation property is decidable.

cs.LO↗