Monadic Datalog over Finite Structures with Bounded Treewidth

TL;DR

This paper introduces monadic datalog as an efficient alternative to MSO logic for fixed-parameter tractability on finite structures with bounded treewidth, enabling linear-time evaluation and practical algorithms.

Contribution

It demonstrates that properties expressible in MSO can also be represented in monadic datalog, avoiding state explosion and enabling linear-time algorithms.

Findings

Monadic datalog can express MSO properties over bounded treewidth structures.

The approach yields linear-time algorithms for 3-Colorability and PRIMALITY problems.

Experimental results show practical efficiency of the proposed methods.

Abstract

Bounded treewidth and Monadic Second Order (MSO) logic have proved to be key concepts in establishing fixed-parameter tractability results. Indeed, by Courcelle's Theorem we know: Any property of finite structures, which is expressible by an MSO sentence, can be decided in linear time (data complexity) if the structures have bounded treewidth. In principle, Courcelle's Theorem can be applied directly to construct concrete algorithms by transforming the MSO evaluation problem into a tree language recognition problem. The latter can then be solved via a finite tree automaton (FTA). However, this approach has turned out to be problematical, since even relatively simple MSO formulae may lead to a ``state explosion'' of the FTA. In this work we propose monadic datalog (i.e., datalog where all intentional predicate symbols are unary) as an alternative method to tackle this class of fixed-parameter tractable problems. We show that if some property of finite structures is expressible in MSO then this property can also be expressed by means of a monadic datalog program over the structure plus the tree decomposition. Moreover, we show that the resulting fragment of datalog can be evaluated in linear time (both w.r.t. the program size and w.r.t. the data size). This new approach is put to work by devising new algorithms for the 3-Colorability problem of graphs and for the PRIMALITY problem of relational schemas (i.e., testing if some attribute in a relational schema is part of a key). We also report on experimental results with a prototype implementation.