Mechanical Theorem Proving in Tarski's geometry.

International audienceThis paper describes the mechanization of the proofs of the first height chapters of Schwabäuser, Szmielew and Tarski's book: Metamathematische Methoden in der Geometrie. The goal of this development is to provide foundations for other formalizations of geometry and implementations of decision procedures. We compare the mechanized proofs with the informal proofs. We also compare this piece of formalization with the previous work done about Hilbert's Gründlagen der Geometrie. We analyze the differences between the two axiom systems from the formalization point of view

Information enforcement in learning with graphics : improving syllogistic reasoning skills

This thesis is an investigation into the factors that contribute to good choices among graphical systems used in teaching, and the feasibility of implementing teaching software that uses this knowledge.The thesis describes a mathematical metric derived from a cognitive theory of human diagram processing. The theory characterises differences among representations by their ability to express information. The theory provides the factors and relationships needed to build the metric. It says that good representations are easily processed because they are more vivid, more tractable and less expressive, than poor representations.The metric is applied to abstract systems for teaching and learning syllogistic reasoning, TARSKI'S WORLD, EULER CIRCLES, VENN DIAGRAMS and CARROLL'S GAME OF LOGIC. A rank ordering reflects the value of each system predicted by the theory and the metric. The theory, the metric and the systems are then tested in empirical studies. Five studies involving sixty-eight learners, examined the benefit of software based on these abstract systems.Studies showed the theory correctly predicted learners' success with the circle systems and poorer performance with TARSKI'S WORLD. The metric showed small but clear differences in expressivity between the circle systems. Differences between results of the learners using the circle systems contradicted the predictions of the metric.Learners with mathematical training were better equipped and more successful at learning syllogistic reasoning with the systems. Performance of learners without mathematical training declined after using the software systems. Diagrams drawn by learners together with video footage collected during problem solving, led to a catalogue of errors, misconceptions and some helpful strategies for learning from graphical systems.A cognitive style test investigated the poor performance of non-mathematically trained learners. Learners with mathematics training showed serialist and versatile learning styles while learners without this training showed a holist learning style. This is consistent with the hypothesis that non-mathematically trained learners emphasise the use of semantic cues during learning and problem solving.A card-sorting task investigated learners' preferences for parts of the graphical lexicon used in the diagram systems. Preferences for the EULER lexicon increased difficulty in explaining the system's poor results in earlier studies. Video footage of learners using the systems in the final study illustrated useful learning strategies and improved performance with EULER while individual instruction was available.Further work describes a preliminary design for an adaptive syllogism tutor and other related work

A Graphical User Interface for Formal Proofs in Geometry.

International audienceWe present in this paper the design of a graphical user interface to deal with proofs in geometry. The software developed combines three tools: a dynamic geometry software to explore, measure and invent conjectures, an automatic theorem prover to check facts and an interactive proof system (Coq) to mechanically check proofs built interactively by the user

Formalization and automation of Euclidean geometry

Напредак геометрије кроз векове се може разматрати кроз развој различитих аксиоматских система који је описују. Употреба аксиоматских система започиње са Хилбертом и Тарским али се ту не завршава. Чак и данас се развијају нови аксиоматски ситеми за рад са еуклидском геометријом...The advance of geometry over the centuries can be observed through the development of dierent axiomatic systems that describe it. The use of axiomatic systems begins with Euclid, continues with Hilbert and Tarski, but it doesn't end there. Even today, new axiomatic systems for Euclidean geometry are developed..

An axiomatic approach for solving geometric problems symbolically

technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design

The Algebra of Logic Tradition

The algebra of logic, as an explicit algebraic system showing the underlying mathematical structure of logic, was introduced by George Boole (1815-1864) in his book The Mathematical Analysis of Logic (1847). The methodology initiated by Boole was successfully continued in the 19th century in the work of William Stanley Jevons (1835-1882), Charles Sanders Peirce (1839-1914), Ernst Schröder (1841-1902), among many others, thereby establishing a tradition in (mathematical) logic. From Boole's first book until the influence after WWI of the monumental work Principia Mathematica (1910 1913) by Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970), versions of thealgebra of logic were the most developed form of mathematical above allthrough Schröder's three volumes Vorlesungen über die Algebra der Logik(1890-1905). Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. Inaddition, in 1941, Alfred Tarski (1901-1983) in his paper On the calculus of relations returned to Peirce's relation algebra as presented in Schröder's Algebra der Logik. The tradition of the algebra of logic played a key role in thenotion of Logic as Calculus as opposed to the notion of Logic as Universal Language . Beyond Tarski's algebra of relations, the influence of the algebraic tradition in logic can be found in other mathematical theories, such as category theory. However this influence lies outside the scope of this entry, which is divided into 10 sections.Fil: Burris, Stanley. University of Waterloo; CanadáFil: Legris, Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Interdisciplinario de Economía Politica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Económicas. Instituto Interdisciplinario de Economía Politica de Buenos Aires; Argentin

Relation algebras and their application in temporal and spatial reasoning

Abstract Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only nitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is pointless. Relation algebras were introduced into temporal reasoning by Allen [1] and into spatial reasoning by Egenhofer and Sharm

Visualisation and manipulation tools for Modal logic

In this thesis, an investigation into how visualisation and manipulation tools can provide better support for learners of Modal logic is described. Problems associated with learning Modal logic are also researched.Seven areas topics in Modal logic are investigated, as is the influence of domain independent factors (e. g. motivation) on learning. Studies show that students find concepts such as Modal proofs and systems difficult to learn, whilst possible worlds and Modes are fairly straightforward. Areas such as reference, belief and accessibility relations fall between these extremes.Two roles for representations in reasoning are identified: providing a concrete domain for students to reason about, and supporting the process of reasoning. Systems which make use of these complementary representations were found to be more effective for learners than either the syntactic or the diagrammatic representations traditionally used to teach Modal logic.A review of software used to support students learning logic highlights two important features: the use of examples, and automation of routine tasks. A learning environment for Modal logic was designed which incorporated these. The environment was developed using an adapted version of Smalltalk's Model-View-Controller mechanism, and incorporates complementary representations, enhance by direct manipulation.A further study investigates the added benefits of using this tool, as opposed to using the same representation but working with pen and paper. This confirms the importance of using 'concrete' content representations and minimising learners' cognitive load. Performance measures show that software users learnt more, had a deeper style of learning, and found the topics less abstract than their counterparts working with pen & paper.This research shows that complementary representations are an effective way of supporting students studying Modal logic, and that visualisation and manipulation tools which incorporate these systems will provide additional benefits for learners

Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki