2023/05/03 by Jakub Bulín, Bulín, Jakub, Michael Kompatscher +1
Computer Science · #08A70 #F.4.1 #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems #Rings and Algebras (math.RA) #semigroups and automata theory
paper · doi:10.48550/arxiv.2305.01984
openalex publication_date 2023/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP(Γ) can be viewed as the problem of deciding the primitive positive theory of Γ, and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages Γ is characterized by having few subpowers, that is, the number of n-ary relations pp-definable from Γ is bounded by 2p(n) for some polynomial p(n). In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to Γ having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers.