We report on the development of a two-dimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a high-level geometry representation. The solver uses a graph-reduction directed algebraic approach, and achieves interactive speed. We describe the architecture of the solver and its basic capabilities. Then, we discuss in detail how to extend the scope of the solver, with special emphasis placed on the theoretical and human factors involved in finding a solution --- in an exponentially large search space --- so that the solution is appropriate to the application and the way of finding it is intuitive to an untrained user. 1 Introduction Solving a system of geometric constraints is a problem that has been considered by several communities, and using different approaches. For example, the symbolic computation community has considered the general problem, in the Supported in part by ONR contract N00014-90-J-...

One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fully-expansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii 1. INTRODUCTION AND RELATED WORK : : : : : : : : : : : : : : : : 1 1.1 Trends in two dimensional sketching : : : : : : : : : : : : : : : : : : : 2 1.1.1 The descriptive approach : : : : : : : : : : : : : : : : : : : : : 2 1.1.2 The constructive approach : : : : : : : : : : : : : : : : : : : : 2 1.1.3 The declarative approach : : : : : : : : : : : : : : : : : : : : : 3 1.2 Constraint solving methods : : : : : : : : : : : : : : : : : : : : : : : 5 1.2.1 Numerical constraint solvers : : : : : : : : : : : : : : : : : : : 5 1.2.2 Constructive constraint solvers : : : : : : : : : : : : : : : : : : 6 1.2.3 Propagation methods : : : : : : : : : : : : : : : : : : : : : : : 7 1.2.4 Symbolic constraint solvers : : : : : : : : : : : : : : : : : : : : 9 1.2.5 Solvers using hybrid methods : : : : : : : : : : : : : : : : : : 9 1.2.6 Other methods : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.3 The repertoire of con...

Every mathematician will agree that the discovery, analysis, and communication

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