Thesis (Dr. rer. nat.)--Technische Universität München, 2000. / "Infix"--Cover. Includes bibliographical references (p. [164]-175) and index.

Univ., Diss.--Cambridge, 1991. / Literaturverz. S. [133] - 138.

Thesis (Ph.D.) - University of Queensland, 2006. / Includes bibliography.

Τα «Συστήματα Αυτόματης Απόδειξης Θεωρημάτων-ΣΑΑΘ» (Automatic Theorem Proving Systems-ATP Systems) είναι συστήματα βασισμένα στη λογική πρώτης τάξεως, τα οποία μπορούν από ένα σύνολο λογικών προτάσεων να συνάγουν την αλήθεια μιας δεδομένης λογικής πρότασης με αυτόματο τρόπο. Η διαδικασία της απόδειξης στα περισσότερα ΣΑΑΘ στηρίζεται στην αρχή της επίλυσης, τον ισχυρότερο κανόνα λογικής εξαγωγής συμπερασμάτων, και την αντίφαση της επίλυσης, μια διαδικασία που εξασφαλίζει την ορθότητα των συμπερασμάτων. Ο ACT-P είναι ένα ΣΑΑΘ που στηρίζεται στην αρχή της επίλυσης και την αντίφαση της επίλυσης, γραμμένο στο εργαλείο GCLISP Developer 5.0 της Gold-Hill, και διαθέτει μια βιβλιοθήκη γνωστών στρατηγικών ελέγχου της διαδικασίας απόδειξης, προσφέροντας τη δυνατότητα στον χρήστη να ορίσει κάθε φορά ένα (κατάλληλο) συνδυασμό στρατηγικών. Στην εργασία αυτή έγινε κατ’ αρχήν μεταφορά του ACT-P σε LispWorks, ένα δυναμικότερο εργαλείο ανάπτυξης εφαρμογών σε Lisp. Επιπλέον, ο χρήστης μέσω του νέου παραθυρικού περιβάλλοντος μπορεί να βλέπει δυο διαφορετικές λύσεις του ίδιου προβλήματος, τη συνοπτική και αναλυτική λύση. Στη συνέχεια, έγινε έλεγχος της καλής λειτουργίας του ACT-P και των στρατηγικών του μέσω δοκιμών με προβλήματα που προέρχονται από την TPTP (Thousands of Problems for Theorem Provers), μια γνωστή βιβλιοθήκη προβλημάτων για ΣΣΑΘ συστήματα στο Διαδίκτυο, και έγιναν οι απαραίτητες διορθώσεις έτσι ώστε να επιλύει προβλήματα από διάφορες κατηγορίες προβλημάτων της βιβλιοθήκης TPTP. Τέλος, έγινε μια μελέτη χρήσης διαφόρων συνδυασμών στρατηγικών ελέγχου για διάφορα προβλήματα της TPTP και εξήχθησαν χρήσιμα συμπεράσματα για την καταλληλότητά τους και την αποδοτικότητά τους σε σχέση με το είδος των προβλημάτων. / Automatic Theorem Proving Systems (ATP Systems) are based on First Order Logic (FOL) and are able to automatically prove the truth of logical sentence. The proof procedure in most ATP Systems uses the resolution principle which is the strongest existing inference rule, and the resolution refutation process which ensure soundeness of the conclusion. The ACT-P is an ATP System which uses the resolution principle and the resolution refutation and it is written in GCLISP Developer 5.0 of Gold-Hill. ACT-P has a library of strategies to control the proof process, and gives users the ability to assign to specify a suitable combination of strategies. In this dissertation a new window based interface is developed for ACTP in Lispworks, which is a powerful tool for developing Lisp applications. The interface gives to the user a more thorough view of the solving process. Moreover, the user can see two different solutions of the problem, the brief and the analytic one. In addition, the functionality and the strategies of ACTP were tested on problems from the TPTP (Thousands of Problems for Theorem Provers) which is a known library of problems for ATP Systems on the web. ACTP has been improved so as to solve problems from various categories of the TPTP library. Finally, different strategy combinations for solving problems from various categories of TPTP library were studied, leading to useful conclusions about the suitability and the performance of the different combinations depending on the problems.

- / -

Capes / A partir dos teoremas encontrados na publicação Geometriae Pars Universalis de 1668 são apresentadas, pela primeira vez em português, as demonstrações dos teoremas de Pappus para os sólidos de revolução. Essa publicação, escrita originalmente em latim, foi feita pelo matemático escocês James Gregory (1638-1675) e é anterior ao desenvolvimento do Cálculo. Além disso, alguns conceitos de Cálculo e de centro de gravidade são revistos a fim de também apresentarumademonstraçãodessesteoremasapartirdessasferramentas. Ainda são feitas algumas aplicações dos teoremas de Pappus para os casos diretos, onde o eixo de rotação ou revolução é representado por um dos eixos coordenados ou ainda, por uma reta paralela a eles. Também são mostrados casos onde o eixo de rotação é dado por uma reta inclinada no plano cartesiano, deixando claro a abrangência, eficiência e a relativa simplicidade de aplicação desses teoremas. / From the theorems found in the publication Geometriae Pars Universalisof 1668 are presented, for the first time in portuguese, the proof of Pappus’s theorems for solids of revolution. This publication , originally written in latin, is due to the scottish mathematician James Gregory (1638-1675) and is prior to the development of Calculus. Furthermore some concepts of Calculus and center of gravity are also revised to present a proof of these theorems from these tools. Some direct cases for Pappus’s theorems are presented, where the axis of rotation or revolution is represented by one of the coordinate axes or by a straight line parallel to them. Also shown are cases where the axis of rotation is given by a straight tilted in the cartesian plane, showing the scope, efficiency and relative simplicity of applying these theorems.

La pâte feuilletée cuite se caractérise par une texture singulière aérée et friable, qui est générée par une alternance de fines couches laminées de matière grasse et de détrempe. L'objectif de la présente étude est d'observer les couches et de quantifier leur structure dans la pâte levée feuilletée. L'effet du nombre de couches sera étudié afin de faire varier leur structure. Deux techniques d'imagerie permettant d'accéder à des résolutions spatiales différentes (1.24µm et 0.5mm/pixel) ont été combinées : la MCBL pour caractériser les couches de matière grasse après laminage, et l'IRM pour étudier en dynamique l'évolution des couches pendant la fermentation. La méthode de traitement développée à partir des images MCBL a permis de quantifier l'épaisseur de chaque couche, et la rétractation des couches de détrempe après le laminage. Deux échelles de taille de ruptures dans les couches de matière grasse ont été identifiées grâce à la complémentarité des techniques d'imagerie. Une nouvelle méthode de quantification des proportions de trois composants (gaz, détrempe sans air et matière grasse) dans chaque pixel, appliquée aux images IRM pendant la fermentation a été utilisée. Elle a permis de suivre l'évolution de chaque couche de détrempe. Le nombre de couches (4,8 et 12), aussi bien que la position de chaque couche de détrempe, n'ont pas d'effet sur la vitesse d'expansion. L'étape de fermentation contribue le plus à l'expansion de la pâte levée feuilletée mais les couches de matière grasse n'ont pas de rôle à cette étape. Pendant la cuisson l'expansion relative de la pâte augmente de 20 à 65% de 4 à 32 couches de matière grasse et reste stable au-delà de 48 couc / Laminated dough is characterized after baking by a unique light and flaky texture which is the result of alternating thin sheeted layers of dough and fat. The aim of the present study was to visualize the layers and quantify their feature in Danish pastry like laminated dough. The effect of the number of layers was studied in order to have different structures of the layers. Two imaging techniques offering different spatial resolutions (1.24µm and 0.5mm/pixel) were combined: CLSM to characterize the fat layers after sheeting and MRI to study dynamically the layers during proving. The developed method from CLSM images allowed quantification of layers thickness and of the elastic recoil of dough layers. Two scales of fat breaches after sheeting were measured by complementary of the two imaging methods. A new method of quantification of the three components (gas, paste and fat) on each pixel in MRI images with partial volume during proving, was used. It made possible to monitor expansion of individual dough layers. Neither the number of fat layers (4, 8 and 12), nor the height position of dough layer, had an effect on the expansion rate. The proving step contributed to most of the expansion of Danish pastry, but with no specific role of fat layers. During baking, relative expansion of Danish pastry ranged from 20 to 65% for 4 to 32 fat layers and remained unchanged above 48 layers.

The thesis describes a rational reconstruction of Sowa's theory of Conceptual Graphs. The reconstruction produces a theory with a firmer logical foundation than was previously the case and which is suitable for computation whilst retaining the expressiveness of the original theory. Also, several areas of incompleteness are addressed. These mainly concern the scope of operations on conceptual graphs of different types but include extensions for logics of higher orders than first order. An important innovation is the placing of negation onto a sound representational basis. A comparison of theorem proving techniques is made from which the principles of theorem proving in Peirce logic are identified. As a result, a set of derived inference rules, suitable for a goal driven approach to theorem proving, is developed from Peirce's beta rules. These derived rules, the first of their kind for Peirce logic and conceptual graphs, allow the development of a novel theorem proving approach which has some similarities to a combined semantic tableau and resolution methodology. With this methodology it is shown that a logically complete yet tractable system is possible. An important result is the identification of domain independent heuristics which follow directly from the methodology. In addition to the theorem prover, an efficient system for the detection of selectional constraint violations is developed. The proof techniques are used to build a working knowledge base system in Prolog which can accept arbitrary statements represented by conceptual graphs and test their semantic and logical consistency against a dynamic knowledge base. The same proof techniques are used to find solutions to arbitrary queries. Since the system is logically complete it can maintain the integrity of its knowledge base and answer queries in a fully automated manner. Thus the system is completely declarative and does not require any programming whatever by a user with the result that all interaction with a user is conversational. Finally, the system is compared with other theorem proving systems which are based upon Conceptual Graphs and conclusions about the effectiveness of the methodology are drawn.

In 2016, Hamoon Mousavi published Walnut which is a program that implements automated theorem proving of propositions about automatic sequences. The main purpose of this thesis was to show the theoretical functi- onality of Walnut on the basis of the relation between automatic sequences and Presburger (resp. B¨uchi) arithmetic that is a decidable theory. Another goal was to describe adequately how the decision procedure of Walnut really works, and finally, to show the practical use of Walnut on several particular problems. One of these particular problems that are solved in the thesis is computation of the critical exponent of the Rudin-Shapiro sequence - this exercise was presented as an open problem in a book of 2003 (however, this exercise does not belong among open problems any more since Shallit proved in 2011 that the critical exponent is computable for all automatic sequences.) The last chapter itself can be also used as a brief manual for newcomers to Walnut that want to use this program for their own applications. 1

[pt] Prova automática de teoremas consiste na prova de teoremas matemáticos por intermédio de programas de computador. Dependendo da linguagem lógica em uso, o processo de provar uma determinada fórmula pode não ser computável. Além disso, dependendo do cálculo dedutivo empregado, a busca por uma prova envolve lidar com a possibilidade de aplicação de longas sequências de axiomas e regras de inferência. Tudo isso reforça a necessidade da intervenção humana no processo de prova em sistemas denominados provadores interativos de teoremas ou assistentes de prova. Em um cenário típico, um usuário interage com a máquina de prova através de uma interface gráfica, normalmente implementada como um aplicativo desktop. Recentemente, porém, muitos aplicativos deste tipo passaram a ser oferecidos para seus usuários através da web. Esta forma de disponibilizar software evita que o usuário final se preocupe com questões de instalação e configuração e possibilita o acesso ao sistema de qualquer computador, com qualquer sistema operacional, bastando ter disponível uma conexão com a Internet. Nesta dissertação, estudamos possibilidades de uso da web como plataforma para a construção de ambientes interativos para prova de teoremas. Nossa proposta é estudar os diferentes modelos de interação entre usuário e ambientes de prova automatizados e verificar como estes modelos podem ser adaptados para a web. Como resultado, apresentamos uma ferramenta gráfica para visualização e manipulação direta de provas formais na web como uma interface alternativa entre usuários e provadores. / [en] Automatic theorem proving consists of proving mathematical theorems by means of computer programs. Depending on the logic used, the process of proving a formula is not computable. Moreover, depending of the deductive system applied to, the search for a proof can involve the application of long sequences of axioms and inference rules, reinforcing the need of human intervention in the proof process. Such systems are known as interactive theorem provers or proof assistants. In a typical scenario, the user interacts with the prover through a graphical interface, usually a desktop application. Recently, however, applications like those started to be delivered to users through the web. This way of software deployment avoids that final users have to deal with complex activities like prover installation and configuration and allows this user to access the system from different machines with a simple Internet connection. In this research we study the use of web as a platform for interactive theorem proving environments construction. Our purpose is to study some interaction models between user and automated proof environments and verify how these models can be adapted to work as a web application. As a result we show a graphical tool for visualization and direct manipulation of formal proofs on web to work as an alternative interface between user and proving machines.