arXiv subjects

Michal Wrona

Publications and source records attributed to Michal Wrona.

3 recordsLinked to original sources

arXiv2024

Model-checking positive equality free logic on a fixed structure (direttissima)

We give a new, direct proof of the tetrachotomy classification for the model-checking problem of positive equality-free logic parameterised by the model. The four complexity classes are Logspace, NP-complete, co-NP-complete and Pspace-complete. The previous proof of this result relied on notions from universal algebra and core-like structures called U-X-cores. This new proof uses only relations, and works for infinite structures also in the distinction between Logspace and NP-hard under Turing reductions. For finite domains, the membership in NP and co-NP follows from a simple argument, which breaks down already over an infinite set with a binary relation. We develop some interesting new algorithms to solve NP and co-NP membership for a variety of infinite structures. We begin with those first-order definable in (Q;=), the so-called equality languages, then move to those first-order definable in (Q;<), the so-called temporal languages. However, it is first-order expansions of the Random Graph (V,E) that provide the most interesting examples. In all of these cases, the derived classification is a tetrachotomy between Logspace, NP-complete, co-NP-complete and Pspace-complete.

cs.LO↗

arXiv2024

Quasi Directed Jonsson Operations Imply Bounded Width (For fo-expansions of symmetric binary cores with free amalgamation)

Every CSP(B) for a finite structure B is either in P or it is NP-complete but the proofs of the finite-domain CSP dichotomy by Andrei Bulatov and Dimitryi Zhuk not only show the computational complexity separation but also confirm the algebraic tractability conjecture stating that tractability origins from a certain system of operations preserving B. The establishment of the dichotomy was in fact preceded by a number of similar results for stronger conditions of this type, i.e. for system of operations covering not necessarily all tractable finite-domain CSPs. A similar, infinite-domain algebraic tractability conjecture is known for first-order reducts of countably infinite finitely bounded homogeneous structures and is currently wide open. In particular, with an exception of a quasi near-unanimity operation there are no known systems of operations implying tractability in this regime. This paper changes the state-of-the-art and provides a proof that a chain of quasi directed Jonsson operations imply tractability and bounded width for a large and natural class of infinite structures.

cs.LO↗

arXiv2021

The complete classification for quantified equality constraints

We prove that QCSP$(\mathbb{N};x=y\rightarrow y=z)$ is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP$(\Gamma)$, for $\Gamma$ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

cs.CC↗