Hrushovski’s Encoding and ω-Categorical CSP Monsters
We produce a class of ω-categorical structures with finite signature by applying a model-theoretic construction - a refinement of an encoding due to Hrushosvki - to ω-categorical structures in a possibly infinite signature. We show that the encoded structures retain desirable algebraic properties of the original structures, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity. This method allows us to systematically generate ω-categorical templates whose CSPs are complete for a variety of complexity classes of arbitrarily high complexity, and ω-categorical templates that show that membership in any given complexity class cannot be expressed by a set of identities on the polymorphisms. It moreover enables us to prove that recent results about the relevance of topology on polymorphism clones of ω-categorical structures also apply for CSP templates, i.e., structures in a finite language. Finally, we obtain a concrete algebraic criterion which could constitute a description of the delineation between tractability and NP-hardness in the dichotomy conjecture for first-order reducts of finitely bounded homogeneous structures.
Constraint satisfaction problem
complexity
polymorphism
pointwise convergence topology
height 1 identity
ω-categoricity
orbit growth
Mathematics of computing~Combinatoric problems
131:1-131:17
Track B: Automata, Logic, Semantics, and Theory of Programming
A long version of this abstract is available on arXiv [Pierre Gillibert et al., 2020], https://arxiv.org/abs/2002.07054.
Pierre
Gillibert
Pierre Gillibert
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria
https://orcid.org/0000-0003-0981-4652
received funding from the Austrian Science Fund (FWF) through projects No P27600 and P32337.
Julius
Jonušas
Julius Jonušas
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria
https://orcid.org/0000-0003-3279-5939
received funding from the Austrian Science Fund (FWF) through Lise Meitner grant No M 2555.
Michael
Kompatscher
Michael Kompatscher
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
https://orcid.org/0000-0002-0163-6604
supported by the grants PRIMUS/SCI/12 and UNCE/SCI/022 of Charles University Research Centre programs, as well as grant No 18-20123S of the Czech Science Foundation.
Antoine
Mottet
Antoine Mottet
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
https://orcid.org/0000-0002-3517-1745
received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 771005, CoCoSym).
Michael
Pinsker
Michael Pinsker
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Czech Republic
https://orcid.org/0000-0002-4727-918X
received funding from the Austrian Science Fund (FWF) through projects No P27600 and P32337 and from the Czech Science Foundation (grant No 18-20123S).
10.4230/LIPIcs.ICALP.2020.131
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Pierre Gillibert, Julius Jonušas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker
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