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Michael Pinsker

Publications and source records attributed to Michael Pinsker.

At least 19 recordsLinked to original sources
arXiv2026

All mixed identities are singular in groups with no algebraicity

We show that if a group admits an action with no algebraicity then all of its mixed identities are singular. Previously, such groups were only known to be lawless by a theorem of Ab\'{e}rt. Our result confirms, in particular, a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups. It thereby not only subsumes numerous results from the literature in a simple uniform theorem, but also settles the question for prominent groups for which the conjecture was an open problem, such as the automorphism group of $(\mathbb{Q}; <)$. It also applies outside the oligomorphic context, e.g. to much-investigated groups such as Thompson's groups $F, T$, and $V$, to Grigorchuk's group, and to the homeomorphism groups of any manifold of dimension $\geq 1$. More generally, we prove that all mixed identities of a group are singular as long as it has an action satisfying certain geometric conditions. This additionally covers, for example, the infinite-dimensional general and projective linear groups.

math.GR↗
arXiv2026

Minimal and intrinsic topologies on monoids of elementary embeddings

To every $\omega$-categorical structure $M$ one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group $\mathrm{Aut}(M)$ of its automorphisms and the topological monoid $\mathrm{EEmb}(M)$ of its elementary embeddings, both equipped with the topology of pointwise convergence $\tau_{\mathrm{pw}}$. We investigate the relation of $\tau_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $\tau_{\mathrm{pw}}$ is minimal, i.e. does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of $\tau_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $\tau_{\mathrm{Z}}$. We show that $\tau_{\mathrm{pw}}$ differs from $\tau_{\mathrm{Z}}$ on $\mathrm{EEmb}(M)$ whenever $\mathrm{Aut}(M)$ has a non-trivial centre. In spite of this, we then prove that whenever algebraic closure on $M$ is modular, then $\tau_{\mathrm{pw}}$ is minimal on $\mathrm{EEmb}(M)$. This covers, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{Aut}(M)$, we describe the semigroup topologies coarser than $\tau_{\mathrm{pw}}$ on the automorphism groups of structures for which algebraic independence satisfies independent 3-amalgamation. We conclude by proving that for the real and the rational Urysohn space and sphere, the metric pointwise topology $\tau_{\mathrm{mp}}$ is minimal on $\mathrm{EEmb}(M)$, equals $\tau_{\mathrm{Z}}$, and is strictly coarser than $\tau_{\mathrm{pw}}$.

math.LO↗
arXiv2026

Decidability of Interpretability

The Bodirsky-Pinsker conjecture asserts a P vs. NP-complete dichotomy for the computational complexity of Constraint Satisfaction Problems (CSPs) of first-order reducts of finitely bounded homogeneous structures. Prominently, two structures in the scope of the conjecture have log-space equivalent CSPs if they are pp-bi-interpretable, or equivalently, if their polymorphism clones are topologically isomorphic. The latter gives rise to the algebraic approach which regards structures with topologically isomorphic polymorphism clones as equivalent and seeks to identify structural reasons for hardness or tractability in topological clones. We establish that the equivalence relation of pp-bi-interpretability underlying this approach is reasonable: On the one hand, we show that it is decidable under mild conditions on the templates; this improves a theorem of Bodirsky, Pinsker and Tsankov (LICS'11) on decidability of equality of polymorphism clones. On the other hand, we show that within the much larger class of transitive $\omega$-categorical structures without algebraicity, the equivalence relation is of lowest possible complexity in terms of descriptive set theory: namely, it is smooth, i.e., Borel-reduces to equality on the real numbers. On our way to showing the first result, we establish that the model-complete core of a structure that has a finitely bounded Ramsey expansion (which might include all structures of the Bodirsky-Pinsker conjecture) is computable, thereby providing a constructive alternative to previous non-constructive proofs of its existence.

math.LO↗
arXiv2025

A guide to topological reconstruction on endomorphism monoids and polymorphism clones

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the topology of pointwise convergence. In some cases, the algebraic structure of such space alone is sufficiently rich to determine its topology (under some requirements on the topology). For automorphism groups, the problem of when this happens has been actively pursued over the last 40 years. With the exception of some early work of Lascar, the analogue of this problem for endomorphism monoids and polymorphism clones has only received attention in the past 15 years. In this guide, we survey the current state of affairs in this relatively young line of research. We moreover use this opportunity to polish several existing results and to extend them beyond what was hitherto known.

math.LO↗
arXiv2025

When Darwin met Ianus: dichotomies of expressivity

The classifications of temporal and phylogeny constraint languages stand among the most seminal complexity classifications within infinite-domain Constraint Satisfaction Problems (CSPs), yet remain the most mysterious in terms of algorithms and algebraic invariants for the tractable cases. We show that those languages which do not pp-construct EVERYTHING (and thus by the classifications are solvable in polynomial time) have, in fact, very limited expressive power as measured by the graphs and hypergraphs they can pp-interpret. This limitation yields many previously unknown algebraic consequences, while also providing new, uniform proofs for known invariance properties. In particular, we show that such temporal and phylogeny constraint languages admit $4$-ary pseudo-Siggers polymorphisms -- a result that sustains the possibility that the existence of such polymorphisms extends to the much broader context of the Bodirsky-Pinsker conjecture. Although temporal and phylogeny constraint languages appear to follow fundamentally different algorithmic principles, our proofs reveal a common core and proceed along strikingly similar lines.

cs.LO↗
arXiv2025

Three Fundamental Questions in Modern Infinite-Domain Constraint Satisfaction

The Feder-Vardi dichotomy conjecture for Constraint Satisfaction Problems (CSPs) with finite templates, confirmed independently by Bulatov and Zhuk, has an extension to certain well-behaved infinite templates due to Bodirsky and Pinsker which remains wide open. We formulate three fundamental questions on the scope of the Bodirsky-Pinsker conjecture and provide positive answers to them. Our first two main results provide two simplifications of this scope, one of structural, and the other one of algebraic nature. The former simplification implies that the conjecture is equivalent to its restriction to templates without algebraicity, a crucial assumption in the most powerful classification methods. The latter yields that the higher-arity invariants of any template within its scope can be assumed to be essentially injective, and any algebraic condition characterizing any complexity class within the conjecture closed under Datalog reductions must be satisfiable by injections, thus lifting the mystery of the better applicability of certain algebraic conditions over others. Our third main result uses the first one to show that any non-trivially tractable template within the scope serves, up to a Datalog-computable modification of it, as the witness of the tractability of a non-finitely tractable finite-domain Promise Constraint Satisfaction Problem (PCSP) by the so-called sandwich method. This provides a particularly strong connection between the Bodirsky-Pinsker conjecture and finite-domain PCSPs. In the light of the third main result, we initiate a new case study-of phylogeny CSPs-which we investigate from the perspective of descriptive complexity. Within this study, we show that there exists a tractable phylogeny CSP that pp-constructs a finite-domain PCSP inexpressible in fixed-point logic with counting but does not pp-construct any finite-domain CSP with this property.

math.LO↗
arXiv2025

New Sufficient Algebraic Conditions for Local Consistency over Homogeneous Structures of Finite Duality

The path to the solution of Feder-Vardi dichotomy conjecture by Bulatov and Zhuk led through showing that more and more general algebraic conditions imply polynomial-time algorithms for the finite-domain Constraint Satisfaction Problems (CSPs) whose templates satisfy them. These investigations resulted in the discovery of the appropriate height 1 Maltsev conditions characterizing bounded strict width, bounded width, the applicability of the few-subpowers algorithm, and many others. For problems in the range of the similar Bodirsky-Pinsker conjecture on infinite-domain CSPs, one can only find such a characterization for the notion of bounded strict width, with a proof essentially the same as in the finite case. In this paper, we provide the first non-trivial results showing that certain height 1 Maltsev conditions imply bounded width, and in consequence tractability, for a natural subclass of templates within the Bodirsky-Pinsker conjecture which includes many templates in the literature as well as templates for which no complexity classification is known.

cs.CC↗
arXiv2025

The sorrows of a smooth digraph: the first hardness criterion for infinite directed graph-colouring problems

Two major milestones on the road to the full complexity dichotomy for finite-domain constraint satisfaction problems were Bulatov's proof of the dichotomy for conservative templates, and the structural dichotomy for smooth digraphs of algebraic length 1 due to Barto, Kozik, and Niven. We lift the combined scenario to the infinite, and prove that any smooth digraph of algebraic length 1 pp-constructs, together with pairs of orbits of an oligomorphic subgroup of its automorphism group, every finite structure -- and hence its conservative graph-colouring problem is NP-hard -- unless the digraph has a pseudo-loop, i.e. an edge within an orbit. We thereby overcome, for the first time, previous obstacles to lifting structural results for digraphs in this context from finite to $\omega$-categorical structures; the strongest lifting results hitherto not going beyond a generalisation of the Hell-Ne\v{s}et\v{r}il theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary $\omega$-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.

cs.LO↗
arXiv2024

Minimal operations over permutation groups

We classify the possible types of minimal operations above an arbitrary permutation group. Above the trivial group, a theorem of Rosenberg yields that there are five types of minimal operations. We show that above any non-trivial permutation group there are at most four such types. Indeed, except above Boolean groups acting freely on a set, there are only three. In particular, this is the case for oligomorphic permutation groups, for which we improve a result of Bodirsky and Chen by showing one of the types in their classification does not exist. Building on these results, we answer three questions of Bodirsky that were previously open.

math.RA↗
arXiv2024

Collapsing the bounded width hierarchy for infinite-domain CSPs: when symmetries are enough

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities $n \geq 3$) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu, ten Cate, Lutz, and Wolter. In particular, the bounded width hierarchy collapses in those cases as well. Our results extend the scope of theorems of Barto and Kozik characterizing bounded width for finite structures, and show the applicability of infinite-domain CSPs to other fields.

cs.LO↗
arXiv2024

An algebraic proof of the dichotomy for graph orientation problems with forbidden tournaments

For a set F of finite tournaments, the F-free orientation problem is the problem of deciding if a given finite undirected graph can be oriented in such a way that the resulting oriented graph does not contain any member of F. Using the theory of smooth approximations, we give a new shorter proof of the complexity dichotomy for such problems obtained recently by Bodirsky and Guzm\'{a}n-Pro. In fact, our approach yields a complexity dichotomy for a considerably larger class of computational problems where one is given an undirected graph along with additional local constraints on the allowed orientations. Moreover, the border between tractable and hard problems is also described by a decidable algebraic condition.

math.CO↗
arXiv2024

Strict width for Constraint Satisfaction Problems over homogeneous strucures of finite duality

We investigate the `local consistency implies global consistency' principle of strict width among structures within the scope of the Bodirsky-Pinsker dichotomy conjecture for infinite-domain Constraint Satisfaction Problems (CSPs). Our main result implies that for certain CSP templates within the scope of that conjecture, having bounded strict width has a concrete consequence on the expressive power of the template called implicational simplicity. This in turn yields an explicit bound on the relational width of the CSP, i.e., the amount of local consistency needed to ensure the satisfiability of any instance. Our result applies to first-order expansions of any homogeneous $k$-uniform hypergraph, but more generally to any CSP template under the assumption of finite duality and general abstract conditions mainly on its automorphism group. In particular, it overcomes the restriction to binary signatures in the pioneering work of Wrona.

cs.LO↗
arXiv2023

The Golden Path to Guarded Monotone Strict NP

Guarded Monotone Strict NP (GMSNP) extends Monotone Monadic Strict NP (MMSNP) by guarded existentially quantified predicates of arbitrary arities. We prove that the containment and the FO-rewritability problems for GMSNP are decidable, thereby settling an open question of Bienvenu, ten Cate, Lutz, and Wolter, later restated by Bourhis and Lutz. Our proof also comes with a 2NEXPTIME upper bound on the complexity of the two problems, which matches the lower bounds for MMSNP due to Bourhis and Lutz. To obtain these results, we significantly improve the state of knowledge of the model-theoretic properties of GMSNP. Bodirsky, Kn\"{a}uer, and Starke previously showed that every GMSNP sentence defines a finite union of CSPs of $\omega$-categorical structures. We show that these structures can be used to obtain a reduction from the containment problem for GMSNP to the much simpler problem of testing the existence of a recolouring; a careful analysis of this yields said upper bound for containment. The upper bound for FO-rewritability is subsequently obtained by an application of several standard techniques from the theory of infinite-domain CSPs. As our secondary contribution, we refine the construction of Bodirsky, Kn\"{a}uer, and Starke by adding a restricted form of homogeneity to the properties of these structures, making the logic amenable to future complexity classifications for query evaluation using techniques developed for infinite-domain CSPs.

cs.LO↗
arXiv2023

On the Zariski topology on endomorphism monoids of omega-categorical structures

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of $\omega$-categorical structures on which the Zariski topology has been analysed thus far, the two topologies were shown to coincide, in turn yielding that the pointwise topology is the coarsest Hausdorff semigroup topology on those endomorphism monoids. We establish two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, we give an example of an $\omega$-categorical structure on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonu\v{s}as, Mitchell, P\'eresse and Pinsker.

math.LO↗
arXiv2023

The semigroup of increasing functions on the rational numbers has a unique Polish topology

The set of increasing functions on the rational numbers, equipped with the composition operation, naturally forms a topological semigroup with respect to the topology of pointwise convergence in which a sequence of increasing functions converges if and only if it is eventually constant at every argument. We develop new techniques to prove there is no other Polish topology turning this semigroup into a topological one, and show that previous techniques are insufficient for this matter.

math.RA↗
arXiv2023

Symmetries of structures that fail to interpret something finite

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize several theorems from the literature and yield further algebraic invariance properties that must be satisfied in every such graph. Algebraic properties of this kind are tightly connected to the tractability of constraint satisfaction problems, and we obtain new such properties even for infinite countably categorical graphs. We balance these positive results by showing the existence of a countably categorical hypergraph that fails to interpret some finite structure, while still lacking some of the most essential algebraic invariance properties known to hold for finite structures.

cs.LO↗
arXiv2023

An order out of nowhere: a new algorithm for infinite-domain CSPs

We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of many homogeneous hypergraphs including the random hypergraphs and hypergraphs omitting a generalised clique. This forms a vast generalisation of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongr\'acz (ICALP'16) on graph satisfiability.

cs.LO↗