Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic

Abstract

We study definability in fixed-point logic with counting (FPC) for a class of constraint satisfaction problems (CSPs) that simultaneously generalises finite-domain CSPs and temporal CSPs. The templates for the CSPs are first-order expansions of the structure $([k]\times \mathbb{Q};{<}_1,\dots ,{<}_k,\sim )$ where ${<}_i:=\{((i,x),(i,x^{\prime }))\mid x<x^{\prime }\}$ and $\sim :=\{((i,x),(i,x^{\prime }))\mid i\in [k],x,x^{\prime }\in \mathbb{Q}\}$, i.e., $k$ copies of $(\mathbb{Q};<)$. Such a CSP is definable in FPC precisely when the template pp-constructs neither the ternary temporal relation $X$, which holds on three elements if the minimum is attained exactly twice, nor a nontrivial finite affine structure. Equivalently, the template has global polymorphisms that witness the tractability of the temporal CSPs on the copies, and that provide 3-4 weak near-unanimity operations on the set of copies of $\mathbb{Q}$. In this case, we prove that the CSP is even expressible in the less expressive logic FP${}^{\ast }$, which is the restriction of FPC without counting terms (i.e., the extension of first-order logic by a fixed point operator and an ordered auxiliary number sort). Our algorithm samples finitely many values in each copy and then performs nested constraint-singleton consistency tests (which is a uniform polynomial-time algorithm for finite-domain CSPs). The correctness proof uses finite-domain CSP theory to combine consistency certificates by a single polymorphism term, fixing each variable to one copy without destroying the certificates needed for the temporal algorithm. We then use a known normal form for algebraic products to control the remaining interactions between different copies.