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"Automatic theorem proving."
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Proof technology and computation
Proof technology will become an established field in software engineering. It generally aims at integrating proof processing into industrial design and verifications tools. The origins of this technology lie in the systematic understanding of a fully-fledged, precise notion of proof by mathematics and logics. Using this profound understanding, computer scientists are able to implement proofs, to check and create them automatically and to connect the concepts of proof and programs in a deep way. Via this, connection proofs are used to support the development of reliable software systems. Software engineers have integrated proof processing into industrial development tools, and these implementations are now getting very efficient. The chapters deal with: The benefits and technical challenges of sharing formal mathematics among interactive theorem provers; proof normalization for various axiomatic theories; abstraction-refinement framework of temporal logic model checking; formal verification in industrial hardware design; readable machine-checked proofs and semantics and more.
eBook
Recent Progress in the Boolean Domain
In todays world, people are using more and more digital systems in daily life. Such systems utilize the elementariness of Boolean values. A Boolean variable can carry only two different Boolean values: FALSE or TRUE (0 or 1), and has the best interference resistance in technical systems. However, a Boolean function exponentially depends on the number of its variables. This exponential complexity is the cause of major problems in the process of design and realization of circuits. According to.
eBook
Automated Theorem Proving in GeoGebra: Current Achievements
GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way.
Journal Article
First-order logic and automated theorem proving
There are many kinds of books on formal logic. Some have philosophers as their intended audience, some mathematicians, some computer scien tists. Although there is a common core to all such books, they will be very different in emphasis, methods, and even appearance. This book is intended for computer scientists. But even this is not precise. Within computer science formal logic turns up in a number of areas, from pro gram verification to logic programming to artificial intelligence. This book is intended for computer scientists interested in automated theo rem proving in classical logic. To be more precise yet, it is essentially a theoretical treatment, not a how-to book, although how-to issues are not neglected. This does not mean, of course, that the book will be of no interest to philosophers or mathematicians. It does contain a thorough presentation of formal logic and many proof techniques, and as such it contains all the material one would expect to find in a course in formal logic covering completeness but, not incompleteness issues. The first item to be addressed is, What are we talking about and why are we interested in it? We are primarily talking about truth as used in mathematical discourse, and our interest in it is, or should be, self evident. Truth is a semantic concept, so we begin with models and their properties. These are used to define our subject.
eBook
Rewriting input expressions in complex algebraic geometry provers
We present an algorithm to help converting expressions having non-negative quantities (like distances) in Euclidean geometry theorems to be usable in a complex algebraic geometry prover. The algorithm helps in refining the output of an existing prover, therefore it supports immediate deployment in high level prover systems. We prove that the algorithm may take doubly exponential time to produce the output in polynomial form, but in many cases it is still computable and useful.
Journal Article
Machine Learning for First-Order Theorem Proving
We applied two state-of-the-art machine learning techniques to the problem of selecting a good heuristic in a first-order theorem prover. Our aim was to demonstrate that sufficient information is available from simple feature measurements of a conjecture and axioms to determine a good choice of heuristic, and that the choice process can be automatically learned. Selecting from a set of 5 heuristics, the learned results are better than any single heuristic. The same results are also comparable to the prover’s own heuristic selection method, which has access to 82 heuristics including the 5 used by our method, and which required additional human expertise to guide its design. One version of our system is able to decline proof attempts. This achieves a significant reduction in total time required, while at the same time causing only a moderate reduction in the number of theorems proved. To our knowledge no earlier system has had this capability.
Journal Article
Conditional and Preferential Logics
This volume contains a revised and updated version of the author's Ph.D. dissertation, and is focused on proof methods and theorem proving for Conditional and Preferential logics. Conditional logics are extensions of classical logic by means of a conditional operator, usually denoted as =>. Conditional logics have a long history, and recently they have found application in several areas of AI, including belief revision and update, the representation of causal inferences in action planning and the formalization of hypothetical queries in deductive databases. Conditional logics have also been applied in order to formalize nonmonotonic reasoning. The study of the relations between conditional logics and nonmonotonic reasoning has led to the seminal work by Kraus, Lehmann and Magidor, who have introduced the so called KLM framework. According to this framework, a defeasible knowledge base is represented by a finite set of conditional assertions of the form A |~ B, whose intuitive reading is \"typically (normally), the A's are B's\". The operator |~ is nonmonotonic in the sense that A |~ B does not imply A and C |~ B. The logics of the KLM framework, also known as preferential logics, allow to infer new conditional assertion from a given knowledge base. In spite of their significance, very few deductive mechanisms have been developed for conditional and preferential logics. In this book the author tries to (partially) fill the existing gap by introducing proof methods (sequent and tableau calculi) for conditional and preferential logics, as well as theorem provers obtained by implementing the proposed calculi.
eBook
Mathematical Foundation of a Functional Implementation of the CNF Algorithm
The conjunctive normal form (CNF) algorithm is one of the best known and most widely used algorithms in classical logic and its applications. In its algebraic approach, it makes use in a loop of a certain well-defined operation related to the “distributivity” of logical disjunction versus conjunction. For those types of implementations, the loop iteration runs a comparison between formulas to decide when to stop. In this article, we explain how to pre-calculate the exact number of loop iterations, thus avoiding the work involved in the above-mentioned comparison. After that, it is possible to concatenate another loop focused now on the “associativity” of conjunction and disjunction. Also for that loop, we explain how to calculate the optimal number of rounds, so that the decisional comparison phase for stopping can be also avoided.
Journal Article