Symmetric Linear Arc Monadic Datalog and Gadget Reductions

2024-07-06

AI Summary

This work characterizes the class of finite-domain constraint satisfaction problems (CSPs) solvable by symmetric linear arc-monotone Datalog (slam Datalog). We establish, for the first time, a complete expressibility criterion for this Datalog fragment—jointly imposing symmetry, linearity, and arc monotonicity—by proving that a CSP is slam Datalog-definable if and only if it admits either (i) a gadget reduction to a Boolean CSP, (ii) an *unfolding caterpillar duality* (a novel combinatorial structure introduced herein), or (iii) a quasi-Maltsev or $k$-absorbing operation (an algebraic characterization). This yields a unified logical, combinatorial, and algebraic account of slam Datalog’s expressive power. Crucially, we thereby show that the problem of deciding whether a given finite-domain CSP is expressible in slam Datalog is decidable—a first for any nontrivial multi-constrained Datalog fragment. Key innovations include the definition and application of unfolding caterpillar duality and the first necessary and sufficient algebraic conditions for a Datalog fragment subject to three simultaneous syntactic restrictions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Finite-domain
• Constraint Problems
• SLAM Datalog

Innovation

• SLAM Datalog
• Unfolding Caterpillar Duality
• Finite Choice Constraint Problems