Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

Abstract

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.

Pseudo‐loop conditions

“…here, u and v are unary functional symbols, and s is 6-ary. For a certain subclass of ω-categorical structures, the converse implication holds as well [3,4], and it has been conjectured that satisfaction of this identity is the delineation of tractability and hardness of a large class of infinite-domain Constraint Satisfaction Problems [8,14]. It is known that the pseudo-Siggers identity, even in this restricted setting, does not imply any non-trivial set of h1 identities [11], and hence it is indeed necessary to consider pseudo-conditions (or other non-h1 alternatives) in this context.…”

“…Some important Mal'cev conditions, for example the abovementioned Siggers' identities (3) and any of the cyclic identities, are given by a single height 1 (in short h1) identity…”

“…In this case, smallness or regularity assumptions on the relational structure which come from outside the field of universal algebra appear naturally. Such conditions might be, for example, model‐theoretic (for example, ‐categoricity , homogeneity , finite boundedness , model‐completeness , or coreness ), combinatorial (for example, Ramsey‐type properties ), or group‐theoretic (for examnple, oligomorphicity or a certain orbit growth of the automorphism group of ) — see, for example, . One recent prime source of algebras satisfying such conditions are infinite domain Constraint Satisfaction Problems, where the understanding of the structure of polymorphism algebras has led to a great number of results of classifying the computational complexity for various classes of ‐categorical relational structures (for example, in ).…”

Discrete Temporal Constraint Satisfaction Problems

“…∈ X, a contradiction. In all three cases 1-3) above, we get a contradiction, thus there is no tuple t / [3], t [4]). It is easy to see that the auxiliary formula…”

“…as auxiliary relations. We first show that both R MX {1},3 and R MX [3],4 are pp-definable in (Q; X), then we proceed to treat more complicated temporal relations. We claim that the following is a pp-definition of R MX [3],4 in (Q; X):…”