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 ω-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 homogeneous 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.

Abstract 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, Jonušas, Mitchell, Péresse, and Pinsker.

Abstract. For a set [Formula: see text] of finite tournaments, the [Formula: see text]-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 [Formula: see text]. Using the theory of smooth approximations, we give a new shorter proof of the complexity dichotomy for such problems obtained recently by Bodirsky and Guzmán-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.

Guarded Monotone Strict NP (GMSNP) extends Monotone Monadic Strict NP (MMSNP) by guarded existentially quantified predicates of arbitrary arities. We prove that the containment problem for GMSNP is 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 problem, which matches the lower bound for containment of MMSNP due to Bourhis and Lutz. In order to obtain these results, we significantly improve the state of knowledge of the model-theoretic properties of GMSNP. Bodirsky, Knäuer, and Starke previously showed that every GMSNP sentence defines a finite union of CSPs of ω-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 certain map called recolouring, albeit in a more general setting than GMSNP; a careful analysis of this yields said upper bound. As a secondary contribution, we refine the construction of Bodirsky, Knäuer, 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.

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 ω-categorical structures; the strongest lifting results hitherto not going beyond a genera-lisation of the Hell-Nešetřil theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary ω-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.

We study constraint satisfaction problems of non-rigid structures in a finite and omega-categorical setting. We show that not having a binary essential polymorphism is a sufficient criterion for NP-hardness of the constraint satisfaction problem of a (model-complete) core, as long as its automorphism group is not the free action of a Boolean group. To understand the behaviour of low arity polymorphisms, we classify the possible types of minimal operations above an arbitrary permutation group. In this, we generalise a classical theorem of Rosenberg above the trivial group, and significantly improve a result of Bodirsky and Chen above the automorphism groups of omega-categorical structures. Finally, we answer three questions of Bodirsky on binary polymorphisms of infinite templates for constraint satisfaction problems.

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 provide answers to three fundamental questions on the scope of the Bodirsky-Pinsker conjecture. 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 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 generalizes a recent result of Mottet and provides a strong hitherto unknown connection between the Bodirsky-Pinsker conjecture and finite-domain PCSPs.

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 some homogeneous hypergraphs. This forms a vast generalization of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongrácz (ICALP'16) on graph satisfiability.

We introduce the novel machinery of smooth approximations to provide a systematic algebraic approach to the complexity of CSPs over finitely bounded homogeneous structures. We apply smooth approximations to confirm the CSP dichotomy conjecture for first-order reducts of the random tournament and to give new short proofs of the conjecture for various homogeneous graphs including the random graph (STOC’11, ICALP’16, JACM 2015, SICOMP 2019), and for expansions of the order of the rationals (STOC’08, JACM 2009). Apart from obtaining these dichotomy results, we show how our new proof technique allows one to unify and significantly simplify the previous results from the literature. For all but the last structure, we moreover characterize for the first time those CSPs that are solvable by local consistency methods, again using the same machinery.

Abstract. We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities [Formula: see text]) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain constraint satisfaction problems (CSPs) studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of Monotone Monadic SNP (MMSNP) sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al. 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.

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.

We introduce the novel machinery of smooth approximations, and apply it to confirm the CSP dichotomy conjecture for first-order reducts of the random tournament, and to give new short proofs of the conjecture for various homogeneous graphs including the random graph (STOC’11, ICALP’16), and for expansions of the order of the rationals (STOC’08). Apart from obtaining these dichotomy results, we show how our new proof technique allows to unify and significantly simplify the previous results from the literature. For all but the last structure, we moreover characterize for the first time those CSPs which are solvable by local consistency methods, again using the same machinery.

We produce a class of $\omega$-categorical structures with finite signature by applying a model-theoretic construction---a refinement of the Hrushovski-encoding---to $\omega$-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 $\omega$-categorical templates whose CSPs are complete for a variety of complexity classes of arbitrarily high complexity and $\omega$-categorical templates that show that membership in any given complexity class containing AC$^0$ 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 $\omega$-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.

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ≥ 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 et al.. In particular, the bounded width hierarchy collapses in those cases as well.

Abstract It is known that a countable $\omega $ -categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

Abstract We prove that if an $\omega $ -categorical structure has an $\omega $ -categorical homogeneous Ramsey expansion, then so does its model-complete core.

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.

The tractability conjecture for finite domain constraint satisfaction problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise technical sense, from the 3-SAT problem; otherwise, they are NP-complete. Its recent resolution draws on an algebraic characterization of the conjectured borderline: the CSP of a finite structure permits no natural reduction from 3-SAT if and only if the stabilizer of the polymorphism clone of the core of the structure satisfies some nontrivial system of identities, and such satisfaction is always witnessed by several specific nontrivial systems of identities which do not depend on the structure. The tractability conjecture has been generalized in the above formulation to a certain class of infinite domain CSPs, namely, CSPs of reducts of finitely bounded homogeneous structures. It was subsequently shown that the conjectured borderline between hardness and tractability, i.e., a natural reduction from 3-SAT, can be characterized for this class by a combination of algebraic and topological properties. However, it was not known whether the topological component is essential in this characterization. We provide a negative answer to this question by proving that the borderline is characterized by one specific algebraic identity, namely, the pseudo-Siggers identity $\alpha s(x,y,x,z,y,z) \approx \beta s(y,x,z,x,z,y)$. This accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our main theorem is also of independent mathematical interest, characterizing a topological property of any $\omega$-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).

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 when the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to this 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 non-trivial set of height 1 identities a structure 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 ω -categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

For $n\geq 3$, let $(H_n, E)$ denote the $n$th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on $n$ vertices. We show that for all structures $\Gamma$ with domain $H_n$ whose relations are first-order definable in $(H_n,E)$ the constraint satisfaction problem for $\Gamma$ either is in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.

A partial order is called semilinear if the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by |$(\mathbb{S}_{2};\leq )$|⁠. We study the reducts of |$(\mathbb{S}_{2};\leq )$|⁠, that is, the relational structures with domain |$\mathbb{S}_{2}$|⁠, all of whose relations are first-order definable in |$(\mathbb{S}_{2};\leq )$|⁠. Our main result is a classification of the model-complete cores of the reducts of |$\mathbb{S}_{2}$|⁠. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all subgroups of the full symmetric group on |$\mathbb{S}_{2}$| that contain the automorphism group of |$(\mathbb{S}_{2};\leq )$| and are closed with respect to the pointwise convergence topology.

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being non-trivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that ω-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking a different approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a non-trivial system of linear identities, and obtain non-trivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every ω-categorical structure is homomorphically equivalent to a model-complete core.

We prove that an ω-categorical core structure primitively positively interprets all finite structures with parameters if and only if some stabilizer of its polymorphism clone has a homomorphism to the clone of projections, and that this happens if and only if its polymorphism clone does not contain operations α, β, s satisfying the identity αs(x, y, x, z, y, z) ≈ βs(y, x, z, x, z, y).

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems : it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Φ of statements (“constraints”) about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifier-free first-order formulas; the question is whether Φ is satisfiable in a graph. We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method for classifying the computational complexity of those CSPs is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.

Abstract For a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ0, ϕ1, …, ϕn that define relations R0, R1, …, Rn over Γ. The question is whether the relation R0 is primitive positive definable from R1, …, Rn, i.e., definable by a first-order formula that uses only relation symbols for R1, …, Rn, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden). We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

For a fixed infinite structure Γ with finite signature τ, we study the following computational problem: input are quantifier-free first-order τ-formulas φ0, φ1,..., φnthat define relations R0, R1,..., Rnover Γ. The question is whether the relation R0is primitive positive definable from R1,..., Rn, i.e., definable by a first-order formula that uses only relation symbols for R1,..., Rn, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden). We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite language whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples for structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Phi of statements ("constraints") about these variables in the language of graphs, where each statement is taken from a fixed finite set Psi of allowed quantifier-free first-order formulas; the question is whether Phi is satisfiable in a graph.

Abstract We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.