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Transactions of the American Mathematical Society

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$\omega$-categorical structures avoiding height 1 identities


Authors: Manuel Bodirsky, Antoine Mottet, Miroslav Olšák, Jakub Opršal, Michael Pinsker and Ross Willard
Journal: Trans. Amer. Math. Soc. 374 (2021), 327-350
MSC (2010): Primary 08B05, 03C05, 08A70; Secondary 03C10, 03D15
DOI: https://doi.org/10.1090/tran/8179
Published electronically: October 14, 2020
MathSciNet review: 4188185
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Abstract:

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template has a pseudo-Siggers polymorphism, and is NP-complete otherwise.

One of the important questions related to the dichotomy conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each nontrivial set of height 1 identities a structure within the range of the conjecture whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless.

An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of nontrivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of nontrivial height 1 identities differ for $\omega$-categorical structures with less than doubly exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.


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Additional Information

Manuel Bodirsky
Affiliation: Institute of Algebra, Technische Universität Dresden, Dresden, Germany
MR Author ID: 693478
ORCID: 0000-0001-8228-3611
Email: manuel.bodirsky@tu-dresden.de

Antoine Mottet
Affiliation: Department of Algebra, Charles University in Prague, Czech Republic
MR Author ID: 1135698
ORCID: 0000-0002-3517-1745
Email: mottet@karlin.mff.cuni.cz

Miroslav Olšák
Affiliation: Department of Algebra, Charles University in Prague, Czech Republic
ORCID: 0000-0002-9361-1921
Email: mirek@olsak.net

Jakub Opršal
Affiliation: Department of Computer Science, Durham University, Durham, United Kingdom
MR Author ID: 1104311
ORCID: 0000-0003-1245-3456
Email: jakub.oprsal@durham.ac.uk

Michael Pinsker
Affiliation: Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Austria; and Department of Algebra, Charles University in Prague, Czech Republic
MR Author ID: 742015
ORCID: 0000-0002-4727-918X
Email: marula@gmx.at

Ross Willard
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada
MR Author ID: 183075
ORCID: 0000-0002-3297-0453
Email: ross.willard@uwaterloo.ca

Keywords: Mal’cev condition, nonnested identity, pointwise convergence topology, $\omega$-categoricity, orbit growth, homogeneous structure, finite boundedness, constraint satisfaction problem, complexity dichotomy.
Received by editor(s): February 5, 2020
Received by editor(s) in revised form: April 7, 2020
Published electronically: October 14, 2020
Additional Notes: The first and fourth authors were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 681988, CSP-Infinity).
The second author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 771005, CoCoSym).
The third and fifth authors received funding from the Czech Science Foundation (grant No 13-01832S)
The fourth author also received funding from the UK EPSRC (grant No EP/R034516/1).
The fifth author received funding from the Austrian Science Fund (FWF) through project No P32337.
The sixth author was supported by the Natural Sciences and Engineering Research Council of Canada.
A conference version of this material appeared at the Thirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 2019 \cite{TopologyIsRelevant}.
Article copyright: © Copyright 2020 American Mathematical Society