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] × ℚ;<1,…,<k,∼) where <i := {((i,x),(i,x')) | x<x'} and ∼ := {((i,x),(i,x')) | i ∈ [k], x,x' ∈ ℚ}, i.e., k copies of (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 ℚ. In this case, we prove that the CSP is even expressible in the less expressive logic FP*, 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.
Related
- Finite BM-algebras and fixed-point combinators
- Discrete Temporal CSPs et al
- Optimal Control Under Signal Temporal Logic Constraints: A Robust Reformulation via Augmented Dynamics
- Finite Spectra of Syllogistic Logic with Cardinality Comparisons
- Completeness of the Gödel--Löb Provability Logic for Finite Derived Ordinal Topologies
BibTeX
@misc{bodirsky2026finiteontopoftemporal,
title = {{Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic}},
author = {Manuel Bodirsky and Philipp A. Grzywaczyk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18287},
eprint = {2610.18287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18287},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic
AU - Manuel Bodirsky
AU - Philipp A. Grzywaczyk
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18287
UR - https://arxiv.org/abs/2610.18287
AB - 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] × ℚ;<_1,…,<_k,∼) where <_i := {((i,x),(i,x')) | x<x'} and ∼ := {((i,x),(i,x')) | i ∈ [k], x,x' ∈ ℚ}, i.e., k copies of (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 ℚ. In this case, we prove that the CSP is even expressible in the less expressive logic FP^*, 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.
ER -