Q-systems are a method of directed graph transformations according to given grammar rules, developed at the Université de Montréal by Alain Colmerauer in 1967--70 for use in natural language processing. The Université de Montréal's machine translation system, TAUM-73, used the Q-Systems as its language formalism. The data structure manipulated by a Q-system is a Q-graph, which is a directed acyclic graph with one entry node and one exit node, where each arc bears a labelled ordered tree. An input sentence is usually represented by a linear Q-graph where each arc bears a word. After analysis, the Q-graph is usually a bundle of 1-arc paths, each arc bearing a possible analysis tree. After generation, the goal is usually to produce as many paths as desired outputs, with again one word per arc. A Q-System consists of a sequence of Q-treatments, each being a set of Q-rules, of the form . The Q-treatments are applied in sequence, unless one of them produces the empty Q-graph, in which case the result is the last Q-graph obtained. The three parts of a rule can contain variables for labels, trees, and forests. All variables after "" must appear in the part. Variables are local to rules. A Q-treatment works in two steps, addition and cleaning. It first applies all its rules exhaustively, using instantiation, thereby adding new paths to the current Q-graph. If and when this addition process halts, all arcs used in some successful rule application are erased, as well as all unused arcs that are no more on any path from the entry node to the exit node. Hence, the result, if any, is again a Q-graph. That allows several Q-Systems to be chained, each of them performing a specialized task, together forming a complex system. For example, TAUM 73 consisted of fifteen chained Q-Systems. An extension of the basic idea of the Q-Systems, namely to replace instantiation by unification led to Prolog, designed by Alain Colmerauer and Philippe Roussel in 1972. Refinements in the other direction by John Chandioux led to GramR, used for programming METEO from 1985 onward. In 2009, Hong Thai Nguyen of GETALP, Laboratoire d'Informatique de Grenoble. That implementation was corrected, completed and extended by David Cattanéo in 2010-11.
