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Circuit equivalence in 2-nilpotent algebras [article]

Piotr Kawałek, Michael Kompatscher, Jacek Krzaczkowski
2019 arXiv   pre-print
The circuit equivalence problem of a finite algebra A is the computational problem of deciding whether two circuits over A define the same function or not. This problem not just generalises the equivalence problem for Boolean circuits, but is also of high interest in universal algebra, as it models the problems of checking identities in A. In this paper we discuss the complexity for algebras from congruence modular varieties. A partial classification was already given by Idziak and
more » ... leaving essentially only a gap for nilpotent but not supernilpotent algebras. We start a systematic study of this open case, proving that the circuit equivalence problem is in P for 2-nilpotent such algebras.
arXiv:1909.12256v1 fatcat:5t5ttzgblvazjlsn3m3ntte62u

Forbidden cycles in metrically homogeneous graphs [article]

Jan Hubička, Michael Kompatscher, Matěj Konečný
2018 arXiv   pre-print
Aranda, Bradley-Williams, Hubička, Karamanlis, Kompatscher, Konečný and Pawliuk recently proved that for every primitive 3-constrained space Γ of finite diameter δ from Cherlin's catalogue of metrically  ...  Michael Kompatscher was supported by Charles University Research Centre programs PRIMUS/SCI/12 and UNCE/SCI/022. Lemma 3. 3 ( 3 Lemma 4.4 in [ABWH + 17c]).  ... 
arXiv:1808.05177v1 fatcat:axjmnjrntzabvj4w2h67u62y2q

Some notes on extended equation solvability and identity checking for groups [article]

Michael Kompatscher
2018 arXiv   pre-print
Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horv\'ath and Szab\'o; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hardness. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices.
arXiv:1808.07776v1 fatcat:beqzd7aq3beancaozuozfmigg4

The local-global property for G-invariant terms [article]

Alexandr Kazda, Michael Kompatscher
2021 arXiv   pre-print
Kompatscher Assume without loss of generality that = (1, 2, . . . , ) (otherwise we reorder the direct factors of ).  ...  Kompatscher Another example of -term conditions equivalent to cyclic terms are the terms given by even dihedral groups: Lemma 9.  ... 
arXiv:2109.02065v2 fatcat:sgqbk5xrw5ekldtopmf3wtgl34

Ramsey expansions of metrically homogeneous graphs [article]

Andrés Aranda, David Bradley-Williams, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk
2017 arXiv   pre-print
We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the
more » ... xistence of a stationary independence relation and the coherent EPPA. Our results can be seen as a new contribution to Nešetřil's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a "completion algorithm" then allows us to apply several strong results in the areas that imply EPPA and respectively the Ramsey property. The main results have numerous corollaries on the automorphism groups of the Fraïssé limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).
arXiv:1707.02612v2 fatcat:b5qkd5fhijdxnoiwlus4zlkztm

A complexity dichotomy for poset constraint satisfaction [article]

Michael Kompatscher, Trung Van Pham
2016 arXiv   pre-print
In this paper we determine the complexity of a broad class of problems that extends the temporal constraint satisfaction problems. To be more precise we study the problems Poset-SAT(Φ), where Φ is a given set of quantifier-free ≤-formulas. An instance of Poset-SAT(Φ) consists of finitely many variables x_1,...,x_n and formulas ϕ_i(x_i_1,...,x_i_k) with ϕ_i ∈Φ; the question is whether this input is satisfied by any partial order on x_1,...,x_n or not. We show that every such problem is
more » ... e or can be solved in polynomial time, depending on Φ. All Poset-SAT problems can be formalized as constraint satisfaction problems on reducts of the random partial order. We use model-theoretic concepts and techniques from universal algebra to study these reducts. In the course of this analysis we establish a dichotomy that we believe is of independent interest in universal algebra and model theory.
arXiv:1603.00082v3 fatcat:jdc6iljhhzgfnplp7ypckjgsee

When symmetries are not enough: a hierarchy of hard Constraint Satisfaction Problems [article]

Pierre Gillibert, Julius Jonušas, Michael Kompatscher, Antoine Mottet, Michael Pinsker
2021 arXiv   pre-print
We produce a class of ω-categorical structures with finite signature by applying a model-theoretic construction – a refinement of the Hrushosvki-encoding – 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
more » ... ally 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.
arXiv:2002.07054v2 fatcat:7gjieagt2rdthb7imxs535e3a4

A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoids [article]

Manuel Bodirsky, David Evans, Michael Kompatscher, Michael Pinsker
2016 arXiv   pre-print
We present an example of two countable ω-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids -- in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable ω-categorical structure in a finite relational
more » ... uage which can neither be reconstructed up to first-order bi-interpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.
arXiv:1510.00356v2 fatcat:p7zs4kfwyfcjvkg4m2j7kc673a

A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoid

Manuel Bodirsky, David Evans, Michael Kompatscher, Michael Pinsker
2018 Israel Journal of Mathematics  
KOMPATSCHER, AND M. PINSKER are stable under bi-interpretability.  ... 
doi:10.1007/s11856-018-1645-9 fatcat:i2v7fa3ei5fazm4rb2swkqjwh4

The equation solvability problem over nilpotent Mal'cev algebras [article]

Michael Kompatscher
2018 arXiv   pre-print
By a result of Horv\'ath the equation solvability problem over finite nilpotent groups and rings is in P. We generalize his result, showing that the equation solvability over every finite supernilpotent Mal'cev algebra is in P. We also give an example of a nilpotent, but not supernilpotent Mal'cev algebra, whose identity checking problem is coNP-complete.
arXiv:1710.03083v3 fatcat:3wvd6t6whngstczgkpxible3yy

Hrushovski's Encoding and ω-Categorical CSP Monsters

Pierre Gillibert, Julius Jonušas, Michael Kompatscher, Antoine Mottet, Michael Pinsker, Emanuela Merelli, Artur Czumaj, Anuj Dawar
2020 International Colloquium on Automata, Languages and Programming  
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
more » ... ematically 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.
doi:10.4230/lipics.icalp.2020.131 dblp:conf/icalp/GillibertJKMP20 fatcat:l7wkhxwupvghrgceqfd55y64tm

Completing graphs to metric spaces [article]

Andrés Aranda, David Bradley-Williams, Eng Keat Hng, Jan Hubička, Miltiadis Karamanlis, Michael Kompatscher, Matěj Konečný, Micheal Pawliuk
2019 arXiv   pre-print
We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.
arXiv:1706.00295v3 fatcat:sfxputmtkve5xeg26qtkf5jwl4

2^_0 pairwise non-isomorphic maximal-closed subgroups of Sym(N) via the classification of the reducts of the Henson digraphs [article]

Lovkush Agarwal, Michael Kompatscher
2015 arXiv   pre-print
Given two structures M and N on the same domain, we say that N is a reduct of M if all ∅-definable relations of N are ∅-definable in M. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are _0-categorical, determining their reducts is equivalent to determining all closed supergroups G< Sym(N) of their automorphism groups. A consequence of the classification is
more » ... there are 2^_0 pairwise non-isomorphic Henson digraphs which have no proper non-trivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are 2^_0 pairwise non-conjugate maximal-closed subgroups of Sym(N). By the reconstruction results of Rubin, these groups are also non-isomorphic as abstract groups.
arXiv:1509.07674v1 fatcat:4ebw7uzecfc6dagqntd62ttovu

Snorkelling and trampling in shallow-water fringing reefs: Risk assessment and proposed management strategy

Judith S. Hannak, Sarah Kompatscher, Michael Stachowitsch, Jürgen Herler
2011 Journal of Environmental Management  
Shallow reefs (reef flats <1.5 m) in the northern Red Sea are impacted by growing tourism that includes swimmers, snorkellers and reef walkers but have largely been neglected in past studies. We selected a fringing reef along the lagoon of Dahab (Sinai, Egypt) as a model for a management strategy. Pointintercept line transects were used to determine substrate composition, coral community and condition, and the coral damage index (CDI) was applied. Approximately 84% of the coral colonies showed
more » ... igns of damage such as breakage, partial mortality or algal overgrowth, especially affecting the most frequent coral genus Acropora. Questionnaires were used to determine the visitors' socio-economic background and personal attitudes regarding snorkelling, SCUBA-diving and interest in visiting a prospective snorkelling trail. Experiencing nature (97%) was by far the strongest motivation, and interest in further education about reef ecology and skill training was high. Less experienced snorkellers and divers e the target group for further education and skill training e were those most prepared to financially support such a trail. We therefore recommend a guided underwater snorkelling trail and restricting recreational use to a less sensitive 'ecotourism zone' while protecting the shallow reef flat. Artificial structures can complete the trail and offer the opportunity to snorkel over deeper areas at unfavourable tide or wind conditions. This approach provides a strategy for the management and conservation of shallow-water reefs, which are facing increasing human impact here and elsewhere.
doi:10.1016/j.jenvman.2011.06.012 pmid:21708420 pmcid:PMC3150666 fatcat:cfam4zt7dvbbnpa6mgzkvbgkhm

Equations in oligomorphic clones and the Constraint Satisfaction Problem for ω-categorical structures [article]

Libor Barto, Michael Kompatscher, Miroslav Olšák, Trung Van Pham, Michael Pinsker
2018 arXiv   pre-print
The research of Michael Kompatscher, Trung Van Pham, and Michael Pinsker has been funded through project P27600 of the Austrian Science Fund (FWF). present article we first establish their equivalence,  ... 
arXiv:1612.07551v3 fatcat:nipnkh3cj5fsxezuqcmvuwjxwi
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