From Constraints to Resolution Rules, Part II: chains, braids, confluence and T&E

TL;DR

This paper extends the theory of resolution rules for CSPs, introducing chains and braids, analyzing Sudoku puzzles, and establishing confluence and solving capacity equivalence with Trial-and-Error methods.

Contribution

It introduces the notions of chains and braids for CSP resolution, proves confluence of certain theories, and shows braids are as powerful as T&E without guessing.

Findings

Confluence property of resolution theories based on braids.

Braid resolution methods are equivalent in power to Trial-and-Error.

Empirical analysis classifies Sudoku puzzles by complexity.

Abstract

In this Part II, we apply the general theory developed in Part I to a detailed analysis of the Constraint Satisfaction Problem (CSP). We show how specific types of resolution rules can be defined. In particular, we introduce the general notions of a chain and a braid. As in Part I, these notions are illustrated in detail with the Sudoku example - a problem known to be NP-complete and which is therefore typical of a broad class of hard problems. For Sudoku, we also show how far one can go in 'approximating' a CSP with a resolution theory and we give an empirical statistical analysis of how the various puzzles, corresponding to different sets of entries, can be classified along a natural scale of complexity. For any CSP, we also prove the confluence property of some Resolution Theories based on braids and we show how it can be used to define different resolution strategies. Finally, we prove that, in any CSP, braids have the same solving capacity as Trial-and-Error (T&E) with no guessing and we comment this result in the Sudoku case.