Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for

Control has to do with the intelligent, adaptive execution of a piece of a task, or an action, and with its interaction with the environment; at the same time, it copes with the disturbances coming from the external world (i.e., with the “struggle of the world” against our intentions). Planning has to do with the definition of sequences of actions, or tasks, to attain complex goals. The relation between planning and control is traditionally considered hierarchical: planning is performed at a higher level of abstraction as compared with control. Artificial intelligence (AI) planning systems are calculi that operate on explicit, declarative representations of both actions and states of the world. Their primitive terms denote actions, constraints, events, situations, scheduling relations, temporal entities and their relations, and so on. Paradigmatic of this approach is traditional, symbolic AI, according to which planning consists of a specific form of logical inference. Similar in this respect are other approaches to planning based on forms of representation such as graphs, Bayesian networks, Petri nets, and so on (see [1] for a review). In most cases, however, planning and control run concurrently, influencing each other on the same time scale. As a consequence, the problem arises of devising models of reasoning and kinds of representations that