Preface

Abstract

Magic squares, chess-like problems, cryptarithmetic puzzles, and similar classes of problems have been extensively used to challenge human reasoning capabilities. Lo Shu magic square can be traced back to 650 B.C., the eight-queens problem was proposed in 1848 by the chess player Max Bazzel TWO × TWO = THREE puzzle appeared in Strand Magazine in 1924. These puzzles are nowadays widely used in constraint programming courses. The first programming language provided with constraint modelling primitives (Sketchpad) has been proposed by the Turing award winner Ivan Sutherland in his PhD thesis (1963). Logemann and Loveland, when implementing the Davis–Putnam procedure (Davis and Putnam 1960) for testing the satisfiability of a propositional formula (SAT), devised an algorithm (Davis–Putnam–Logemann–Loveland (DPLL)) that has become the core of all SAT/and Answer Set Programming solvers (50 years later). It consists in choosing an un-assigned variable, assigning it a value 0 or 1, propagating the chosen value (unit propagation), and proceeding with the alternative value, if the original assignment leads to a contradiction (backtracking). Some years later Waltz (1975) introduced the notion of domain filtering (arc-consistency-based constraint propagation). With this idea the same DPLL scheme can be used for verifying the satisfiability of a constraint satisfaction problem, where the assignment is no longer 0/1 and the unit propagation is replaced by constraint propagation. For a detailed history of these early years achievements, we refer the reader to the works by Loveland et al. (2017), Jaffar and Maher (1994), and Freuder and Mackworth (2006).