We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B. This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP(B). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete. ACM Reference Format:Madelaine, F. and Martin, B. 2012. The complexity of positive first-order logic without equality. ACM Trans.

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Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idempotent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [1], we prove that partially reflexive paths bequeath a set of idempotent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP); or the exponentially generated powers property (EGP). Similarly, we build on [2] to prove that semicomplete digraphs have the same property.These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we effect also a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [3], and show that collapsibility is equivalent to its Π 2 -restriction. We also give a decision procedure for k-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures).Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [1], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete.

Abstract. The quantified constraint satisfaction problem QCSP(A) is the problem to decide whether a positive Horn sentence, involving nothing more than the two quantifiers and conjunction, is true on some fixed structure A. We study two containment problems related to the QCSP.Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) ⊆ QCSP(B). We prove that QCSP(A) ⊆ QCSP(B), that is all sentences of positive Horn logic true on A are true on B, iff there is a surjective homomorphism from A |A| |B| to B. This can be seen as improving an old result of Keisler that shows the former equivalent to there being a surjective homomorphism from A ω to B. We note that this condition is already necessary to guarantee containment of the Π2 restriction of the QCSP, that is Π2-CSP(A) ⊆ Π2-CSP(B). The exponent's bound of |A| |B| places the decision procedure for the model containment problem in non-deterministic double-exponential time complexity. We further show the exponent's bound |A| |B| to be close to tight by giving a sequence of structures A together with a fixed B, |B| = 2, such that there is a surjective homomorphism from A r to B only when r ≥ |A|. Secondly, we prove that the entailment problem for positive Horn fragment of first-order logic is decidable. That is, given two sentences ϕ and ψ of positive Horn, we give an algorithm that determines whether ϕ → ψ is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive first-order logic (i.e. positive Horn plus disjunction) is undecidable.In the final part of the paper we ponder a notion of Q-core that is some canonical representative among the class of templates that engender the same QCSP. Although the Q-core is not as well-behaved as its better known cousin the core, we demonstrate that it is still a useful notion in the realm of QCSP complexity classifications.