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A Geometric Constraint Solver
, 1995
"... We report on the development of a twodimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a highlevel geometry representation. The solver uses a graphreduction directed algebraic approach, and achieves interact ..."
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We report on the development of a twodimensional geometric constraint solver. The solver is a major component of a new generation of CAD systems that we are developing based on a highlevel geometry representation. The solver uses a graphreduction 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 N0001490J...
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... 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 appro ..."
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Cited by 66 (2 self)
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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 fullyexpansive 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 ...
Geometric constraint solving
 Computing in Euclidean Geometry
, 1995
"... We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solving is considered, and different approaches are characterized. ..."
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Cited by 44 (7 self)
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We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solving is considered, and different approaches are characterized.
Editable Representations For 2D Geometric Design
, 1993
"... : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii 1. INTRODUCTION AND RELATED WORK : : : : : : : : : : : : : : : : 1 1.1 Trends in two dimensional sketching : : : : : : : : : : : : : : : : : : : 2 1.1.1 The descriptive approach : : : : : : : : : : : : : : : : : : : : : 2 1. ..."
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Cited by 13 (4 self)
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: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 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...
Two computersupported proofs in metric space topology
 Notices of the American Mathematical Society
, 1991
"... Every mathematician will agree that the discovery, analysis, and communication ..."
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Cited by 8 (3 self)
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Every mathematician will agree that the discovery, analysis, and communication
A short survey of automated reasoning
"... 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 f ..."
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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
Basics on Geometric Constraint Solving
"... We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solving is considered, and different approaches are characterized. ..."
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We survey the current state of the art in geometric constraint solving. Both 2D and 3D constraint solving is considered, and different approaches are characterized.
Second Reader
"... Abstract We survey models and theories of geometric structures of parallelism, orthogonality, incidence, betweenness and order, thus gradually building towards full elementary geometry of Euclidean spaces, in Tarski’s sense. Besides the geometric aspects of such structures we look at their logical ..."
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Abstract We survey models and theories of geometric structures of parallelism, orthogonality, incidence, betweenness and order, thus gradually building towards full elementary geometry of Euclidean spaces, in Tarski’s sense. Besides the geometric aspects of such structures we look at their logical (firstorder and modal) theories and discuss logical issues such as: expressiveness and definability, axiomatizations and representation results, completeness and decidability, and interpretations between structures and theories. 21. Introduction and historical overview In ancient Babylon and Egypt geometry was just a set of empirical observations and practical skills and methods for measuring land and designing irrigation systems, although it already had a degree of sophistication (e.g., Pythagoras ’ theorem was already known and the triangle
Abstract
, 2003
"... This paper deals with linear and integer programming problems in which the constraint matrix is a binet matrix. Binet matrices are pivoted versions of the nodeedge incidence matrices of bidirected graphs. It is shown that efficient methods are available to solve such optimization problems. Linear p ..."
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This paper deals with linear and integer programming problems in which the constraint matrix is a binet matrix. Binet matrices are pivoted versions of the nodeedge incidence matrices of bidirected graphs. It is shown that efficient methods are available to solve such optimization problems. Linear programs can be solved with the generalized network simplex method, while integer programs are converted to a matching problem. It is also proved that an integral binet matrix has strong Chvátal rank 1. An example of binet matrices, namely matrices with at most three nonzeros per row, is given.
The Logical Validation of Mathematical Diagrammatic Proofs
, 2000
"... Diagrams have been used for problem solving for thousands of years but have only recently had a resurgence into mainstream science with applications in cognitive science, artificial intelligence, computer science, physics, mathematics, and other disciplines. Diagrammatic reasoning is %he understandi ..."
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Diagrams have been used for problem solving for thousands of years but have only recently had a resurgence into mainstream science with applications in cognitive science, artificial intelligence, computer science, physics, mathematics, and other disciplines. Diagrammatic reasoning is %he understanding of concepts and ideas by the use of diagrams and imagery, as opposed to linguistic or algebraic representations" [GCN95, on back cover]. This paper aims to introduce the reader to diagrammatic reasoning, specifically in the area of diagrammatic proofs, logically validate the soundness of the construction steps in a diagrammatic proof, as well as help develop a theoretical basis for computing directly with diagrammatic representations. In doing this, the work of Jamnik (in [Jam99]) and of Foo, et al. (in [FNP99]) will be extended. This will be done through an analysis of diagrammatic proofs of geometric theorems and a study of some problematic proofs in this area. In addition, a proof showing the equivalence of the two solutions to the problem of generalization presented in [FNP99] and a link between traditional theories of computation, such as fixed points, invariants, and continuations, with diagrammatic proofs is shown. In essence, this paper intends to help advance the understanding of what is involved in diagrammatic proofs, why they work, and why they sometimes do not work as well as show that diagrams alone can be regarded as legitimate (or even desirable) proofs in the area of geometric theorems. Hopefully, this will help to open new opportunities for study and development in the justification and in later work on the automation of diagrammatic proofs.