Axiomatisability and hardness for universal Horn classes of hypergraphs

We characterise finite axiomatisability and intractability of deciding membership for universal Horn classes generated by finite loop-free hypergraphs. A universal Horn class is a class of model-theoretic structures of the same signature, closed under taking ultraproducts (Pu), direct products over nonempty families (P) and isomorphic copies of substructures (S); see [9, 17, 30, 32] for example. Equivalently they are classes axiomatisable by way of universal Horn sentences : universally quantified disjunctions α1 ∨ · · · ∨ αk, where each αi is either an atomic formula of the language, or a negated atomic formula, and all but at most one of the αi are negated. Quasivarieties are very closely related classes, differing from the universal Horn class definition only in that the trivial one-element structure (in which all relations are total) is automatically included; this corresponds to allowing the degenerate direct product over an empty family of structures. Problems of axiomatisability for universal Horn classes and quasivarieties have a relatively long history. The starting point is perhaps Maltsev’s characterisation of semigroups embeddable in groups [28, 29], with subsequent developments in semigroup theory including Sapir [36], Margolis and Sapir [31], Jackson and Volkov [25]. There is also a wealth of literature within universal algebra and relational structures; see Gorbunov’s book [17], or the Studia Logica special issue [2] for many examples. An extra impetus for investigation of universal Horn classes comes from computational complexity. For example, the fixed template constraint satisfaction problem over a finite relational structure is the problem of deciding membership of relational structures in a certain universal Horn class [23]. Computational issues for universal Horn classes of relational structures also play a hidden role behind a number of examples demonstrating intractability of deciding membership of finite algebras in a finitely generated pseudovariety. Indeed, several of the relatively few known examples involve encoding a NP-complete universal Horn class membership problem into a pseudovariety membership problems. This is true for Szekely [37], Jackson and McKenzie [24] and [21] for example. The present article concerns both axiomatisability and computational complexity for universal Horn classes of loop-free hypergraphs, and we are able to extend all of the known results for simple graphs. The characterisation of finitely axiomatisable universal Horn classes of finite simple graphs was given by Caicedo [10], by combining a probabilistic result of Erdős [14] with work of Nešetřil and Pultr [34]. In fact, Caicedo’s work covers any universal Horn class whose members have bounded chromatic number. After fixing a reasonable model-theoretic meaning to “hypergraph” we show that Caicedo’s classification may be extended to arbitrary loop-free 2010 Mathematics Subject Classification. Primary: 08C15, 05C65; Secondary: 05C60, 68Q17. The second author was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666.