Lower Bounds and Faster Algorithms for Equality Constraints

We study the fine-grained complexity of NP-complete, infinite-domain constraint satisfaction problems (CSPs) parameterised by a set of first-order definable relations (with equality). Such CSPs are of central importance since they form a subclass of any infinite-domain CSP parameterised by a set of first-order definable relations. We prove that under the randomised exponential-time hypothesis it is not possible to find c > 1 such that a CSP over an arbitrary finite equality language is solvable in O(c^n) time (n is the number of variables). Stronger lower bounds are possible for infinite equality languages where we rule out the existence of 2^o(n log n) time algorithms; a lower bound which also extends to satisfiability modulo theories solving for an arbitrary background theory. Despite these lower bounds we prove that for each c > 1 there exists an NP-hard equality CSP solvable in O(c^n) time. Lower bounds like these immediately ask for closely matching upper bounds, and we prove that a CSP over a finite equality language is always solvable in O(c^n) time for a fixed c.

RAMSEY GROWTH IN SOME NIP STRUCTURES

We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal 163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$ -minimal expansions of $\mathbb{R}$ , and show that it does not hold in $\mathbb{R}_{\exp }$ . This provides a new combinatorial characterization of polynomial boundedness for $o$ -minimal structures. We also prove an analog for relations definable in $P$ -minimal structures, in particular for the field of the $p$ -adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society 366(9) (2014), 5043–5065), we show that in distal structures the upper bound for $k$ -ary definable relations is given by the exponential tower of height $k-1$ .

PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS

Given two structures${\cal M}$and${\cal N}$on the same domain, we say that${\cal N}$is a reduct of${\cal M}$if all$\emptyset$-definable relations of${\cal N}$are$\emptyset$-definable in${\cal M}$. 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${\aleph _0}$-categorical, determining their reducts is equivalent to determining the closed supergroupsG≤ Sym(ℕ) of their automorphism groups.A consequence of the classification is that there are${2^{{\aleph _0}}}$pairwise noninterdefinable Henson digraphs which have no proper nontrivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are${2^{{\aleph _0}}}$pairwise nonconjugate maximal-closed subgroups of Sym(ℕ). By the reconstruction results of Rubin, these groups are also nonisomorphic as abstract groups.

Rice sequences of relations

We propose a framework to study the computational complexity of definable relations on a structure. Many of the notions we discuss are old, but the viewpoint is new. We believe that all the pieces fit together smoothly under this new point of view. We also survey related results in the area. More concretely, we study the space of sequences of relations over a given structure. On this space, we develop notions of c.e.-ness, reducibility, join and jump. These notions are equivalent to other notions studied in other settings. We explain the equivalences and differences between these notions.