GEOTHER: A Geometry Theorem Prover

title: Automated Generation of Diagrams with Maple and Java
abstract: This note shows how to draw diagrams automatically from the predicate specification of a given set of geometric relations among a set of points in the plane. It is done first in Maple by translating the geometric relations into polynomial equations, decomposing the obtained system of polynomials into irreducible representative triangular sets, and finding an adequate numerical solution from each triangular set. A Java class coding the solution and the polynomials in each triangular set is generated, compiled, and then executed with the main Java programs to draw a diagram. The whole process combining symbolic elimination in Maple with numerical computation, graphic drawing, and letter labeling in Java is fully automatic. The drawn diagrams may be animated and fine-tuned by mouse click and dragging and saved as PostScript files. We present the drawing method, discuss some techniques of implementation, and give several sample diagrams drawn by our program.

title: On Wu's Method for Proving Constructive Geometric Theorems
abstract: In 1977 Wu Wen-tsiin discovered an efficient method for mechanical theorem proving. This method has been applied to prove and discover theorems in various kinds of geometries. Based on an extension of one of Wu's algorithmic procedures, the author implemented a prover CPS for proving so-called constructive geometric theorems, as well as a general theorem prover PS. Using this prover we proved more than one hundred non-trivial theorems and discovered also a number of interesting ones. This paper is a summarization of our early work, as a abbreviation of the chapter 2 from my Chinese doctoral dissertation.

title: A simple method for implicitizing rational curves and surfaces
abstract: This paper presents a simple method for converting rational parametric equations of curves and surfaces into implicit equations. The method proceeds via writing out the implicit polynomial F of estimated degree with indeterminate coefficients u i , substituting the rational expressions for the given parametric curve or surface into F to yield a rational expression g / h in the parameter s (or s and t ), equating the coefficients of g in terms of s (and t ) to 0 to generate a sparse, partially triangular system of linear equations in u i with constant coefficients, and finally solving the linear system for u i . If a nontrivial solution is found, then an implicit polynomial is obtained; otherwise, one repeats the same process, increasing the degree of F . Our experiments show that this simple method is efficient. It performs particularly well in the presence of base points and may detect the dependency of parameters incidentally.

title: A Strategy for Speeding-up the Computation of Characteristic Sets
abstract: In this note a strategy is presented that speeds up the computation of characteristic sets several times in many cases. We test the possibility of success for this strategy in the quasi-sense by 50 examples, where the strategy succeeds in 58 and fails in 23 of the 81 completed cases. The speed-up of the strategy, while successful, ranges from a few to hundred times, whereas in the failure case the computation may be slightly slowed down (not less than 0.6 times).

title: Clifford Term Rewriting for Geometric Reasoning in 3D
abstract: Clifford algebra formalism has been used recently in combination with standard term-rewriting techniques for proving many non-trivial geometric theorems. The key issue in this approach consists in verifying whether two Clifford expressions are equal. This paper is concerned with the generalization of the work to 3D geometric problems. A rewriting system is proposed and its theoretical properties are investigated. Some examples and potential applications are also presented.

title: Integrating ecoinformatics resources on the semantic web
abstract: We describe ELVIS (the Ecosystem Location Visualization and Information System), a suite of tools for constructing food webs for a given location. We express both ELVIS input and output data in OWL, thereby enabling its integration with other semantic web resources. In particular, we describe using a Triple Shop application to answer SPARQL queries from a collection of semantic web documents. This is an end-to-end case study of the semantic web's utility for ecological and environmental research.

title: Enhancing the estimation of plant Jacobian for adaptive neural inverse control
abstract: To implement the specialized learning of the inverse dynamic neuro-controller for controlling nonlinear plants with noise, it is strongly desirable that an on-line trained neural plant emluator may provide a reasonably good estimation of the plant Jacobian under noise environments. This paper presents an approach for enhancing the estimation of the plant Jacobian which is on-line used in direct adaptive neural inverse control schemes. The estimated teaching signals are obtained by using the input–output data available at each time step, and then they are used to train the neural plant emluator by a new cost function introduced in this work for training the plant emluator. Convergence theorem for the adaptive back-propagation algorithm and stability of the closed-loop control system are established by using the Lyapunov theory. Simulations are conducted for demonstrating the effectiveness of the proposed strategy.

title: An Edge-Preserving Image Reconstruction Using Neural Network
abstract: This paper presents an image restoration model based on the implicit function theorem and edge-preserving regularization. We then apply the model on the subband-coded images using the artificial neural network. The edge information is extracted from the source image as a priori nowledge to recover the details and reduce the ringing artifact of the subband-coded image. The multilayer perceptron model is employed to implement the restoration process. The main merit of the presented approach is that the neural network model is massively parallel with strong robustness for the transmission noise and parameter or structure perturbation, and it can be realized by VLSI technologies for real-time applications. To evaluate the performance of the proposed approach, a comparative study with the set partitioning in hierarchical tree (SPIHT) has been made by using a set of gray-scale digital images. The experimental results showed that the proposed approach could result in compatible performances compared with SPIHT on both objective and subjective quality for lower compression ratio subband coded image.

title: GLIF3: a representation format for sharable computer-interpretable clinical practice guidelines
abstract: The Guideline Interchange Format (GLIF) is a model for representation of sharable computer-interpretable guidelines. The current version of GLIF (GLIF3) is a substantial update and enhancement of the model since the previous version (GLIF2). GLIF3 enables encoding of a guideline at three levels: a conceptual flowchart, a computable specification that can be verified for logical consistency and completeness, and an implementable specification that is intended to be incorporated into particular institutional information systems. The representation has been tested on a wide variety of guidelines that are typical of the range of guidelines in clinical use. It builds upon GLIF2 by adding several constructs that enable interpretation of encoded guidelines in computer-based decision-support systems. GLIF3 leverages standards being developed in Health Level 7 in order to allow integration of guidelines with clinical information systems. The GLIF3 specification consists of an extensible object-oriented model and a structured syntax based on the resource description framework (RDF). Empirical validation of the ability to generate appropriate recommendations using GLIF3 has been tested by executing encoded guidelines against actual patient data. GLIF3 is accordingly ready for broader experimentation and prototype use by organizations that wish to evaluate its ability to capture the logic of clinical guidelines, to implement them in clinical systems, and thereby to provide integrated decision support to assist clinicians.

title: On Connectedness: A Solution Based on Oscillatory Correlation
abstract: A long-standing problem in neural computation has been the problem of connectedness, first identified by Minsky and Papert (1969). This problem served as the cornerstone for them to establish analytically that perceptrons are fundamentally limited in computing geometrical (topological) properties. A solution to this problem is offered by a different class of neural networks: oscillator networks. To solve the problem, the representation of oscillatory correlation is employed, whereby one pattern is represented as a synchronized block of oscillators and different patterns are represented by distinct blocks that desynchronize from each other. Oscillatory correlation emerges from LEGION (locally excitatory globally inhibitory oscillator network), whose architecture consists of local excitation and global inhibition among neural oscillators. It is further shown that these oscillator networks exhibit sensitivity to topological structure, which may lay a neurocomputational foundation for explaining the psychophysical phenomenon of topological perception.