On the Expression Complexity of Equivalence and Isomorphism of Primitive Positive Formulas

Abstract

Abstract. We study the complexity of equivalence and isomorphism on primitive positive formulas with respect to a given structure. We study these problems for various fixed structures; we present generic hardness and complexity class containment results, and give classification theorems for the case of two-element (boolean) structures.

Generic expression hardness results for primitive positive formula comparison

“…In the case that B is a subalgebra or homomorphic image of A, the result is proved in [5,Proposition 4] for PPEQ(·), and from the argumentation there it is clear that exactly the same reduction works for PPCON(·). 2…”

“…The G-set conjecture, which is typically phrased on idempotent algebras, yields a prediction on the CSP complexity of all structures via a theorem [7] showing that each structure B has the same CSP complexity as a structure B whose associated algebra is idempotent. The mapping from B to B does not preserve the complexity of the problems studied here, and indeed, there are examples of two-element structures B such that our hardness result applies to B -the equivalence and containment problems on B are Π p 2 -complete -but B does not admit the unary type and indeed has a polynomial-time tractable CSP [5]. Our new result requires establishing a deeper understanding of the identified algebras' structure, some of which admit a tractable CSP, in order to obtain hardness.…”

“…Then (Section 3.2), we establish the desired hardness result by reducing from the containment problem over a boolean structure having constant polymorphisms, known to be Π p 2 -complete [5], to the containment problem of interest (Theorem 4).…”

“…This correspondence provides a way to pass from a relational structure B to an algebra A B whose set of operations is the mentioned clone, in such a way that two structures having the same algebra have the same complexity (for each of the mentioned problems). In a previous paper by the present authors [5], we developed this correspondence and presented some basic complexity results for the problems at hand, including a classification of the complexity of the problems on all two-element structures.…”

“…Previous work identified one most general condition for the tractability of the equivalence and containment problems: if the algebra has few subpowers -a combinatorial condition [13,9] involving the number of subalgebras of powers of an algebra -then these problems are polynomial-time tractable [5,Theorem 7]. This second hardness result appears to perfectly complement this tractability result: there are no known examples of algebras A B (of structures B having finitely many relations) that are not covered by one of these results, and in fact the Edinburgh conjecture predicts that none exist, stating that every such algebra A B that generates a congruence modular variety also has few subpowers.…”