A Non-Dichotomy for CSPs of Stable $ω$-Categorical Structures

Abstract

We construct countable, stable, $ω$-categorical structures with at most two binary relations whose CSPs exhibit three complexity regimes: NP-complete, decidable but not in NP, and undecidable. The templates arise from a controlled Hrushovski construction of stable $ω$-categorical graphs $M_α$ for suitable irrational parameters $α>1$. Some of these graphs have NP-complete CSPs. Expanding the graphs $M_α$ by the non-edge relation turns age membership into a restriction of the CSP. For every time-constructible bound $t$, there is such an expansion whose CSP is decidable but not in DTIME$(t)$. For an explicit parameter $α$, the restriction to complete instances belongs to $\mathrm{coNP}$ and the full CSP belongs to $Σ_2^{\mathrm P}$. Separating the ages of the $M_α$ using graphs with large girth and prescribed density yields $2^{\aleph_0}$ distinct CSPs of these expansions, all but countably many of which are undecidable.