Symmetric Linear Arc Monadic Datalog and Gadget Reductions

TL;DR

This paper characterizes the class of finite-domain CSPs solvable by symmetric linear arc monadic Datalog, linking it to gadget reductions, duality theories, and algebraic conditions, and shows the problem is decidable.

Contribution

It introduces the symmetric linear arc monadic Datalog fragment, provides exact characterizations via gadget reductions and algebraic conditions, and establishes decidability of expressibility.

Findings

Characterization of CSPs solvable by slam Datalog via gadget reductions.

Introduction of unfolded caterpillar duality for CSPs.

Decidability of whether a CSP can be expressed by slam Datalog.

Abstract

A Datalog program solves a constraint satisfaction problem (CSP) if and only if it derives the goal predicate precisely on the unsatisfiable instances of the CSP. There are three Datalog fragments that are particularly important for finite-domain constraint satisfaction: arc monadic Datalog, linear Datalog, and symmetric linear Datalog, each having good computational properties. We consider the fragment of Datalog where we impose all of these restrictions simultaneously, i.e., we study \emph{symmetric linear arc monadic (slam) Datalog}. We characterise the CSPs that can be solved by a slam Datalog program as those that have a gadget reduction to a particular Boolean constraint satisfaction problem. We also present exact characterisations in terms of a homomorphism duality (which we call \emph{unfolded caterpillar duality}), and in universal-algebraic terms (using known minor conditions, namely the existence of quasi Maltsev operations and $k$-absorptive operations of arity $nk$}, for all $n,k \geq 1$). Our characterisations also imply that the question whether a given finite-domain CSP can be expressed by a slam Datalog program is decidable.