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Author: Hantao Zhang
Publisher: Springer Science & Business Media
ISBN: 9400916752
Category : Computers
Languages : en
Pages : 223
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Book Description
It has been shown how the common structure that defines a family of proofs can be expressed as a proof plan [5]. This common structure can be exploited in the search for particular proofs. A proof plan has two complementary components: a proof method and a proof tactic. By prescribing the structure of a proof at the level of primitive inferences, a tactic [11] provides the guarantee part of the proof. In contrast, a method provides a more declarative explanation of the proof by means of preconditions. Each method has associated effects. The execution of the effects simulates the application of the corresponding tactic. Theorem proving in the proof planning framework is a two-phase process: 1. Tactic construction is by a process of method composition: Given a goal, an applicable method is selected. The applicability of a method is determined by evaluating the method's preconditions. The method effects are then used to calculate subgoals. This process is applied recursively until no more subgoals remain. Because of the one-to-one correspondence between methods and tactics, the output from this process is a composite tactic tailored to the given goal. 2. Tactic execution generates a proof in the object-level logic. Note that no search is involved in the execution of the tactic. All the search is taken care of during the planning process. The real benefits of having separate planning and execution phases become appar ent when a proof attempt fails.
Author: Hantao Zhang
Publisher: Springer Science & Business Media
ISBN: 9400916752
Category : Computers
Languages : en
Pages : 223
Get Book Here
Book Description
It has been shown how the common structure that defines a family of proofs can be expressed as a proof plan [5]. This common structure can be exploited in the search for particular proofs. A proof plan has two complementary components: a proof method and a proof tactic. By prescribing the structure of a proof at the level of primitive inferences, a tactic [11] provides the guarantee part of the proof. In contrast, a method provides a more declarative explanation of the proof by means of preconditions. Each method has associated effects. The execution of the effects simulates the application of the corresponding tactic. Theorem proving in the proof planning framework is a two-phase process: 1. Tactic construction is by a process of method composition: Given a goal, an applicable method is selected. The applicability of a method is determined by evaluating the method's preconditions. The method effects are then used to calculate subgoals. This process is applied recursively until no more subgoals remain. Because of the one-to-one correspondence between methods and tactics, the output from this process is a composite tactic tailored to the given goal. 2. Tactic execution generates a proof in the object-level logic. Note that no search is involved in the execution of the tactic. All the search is taken care of during the planning process. The real benefits of having separate planning and execution phases become appar ent when a proof attempt fails.
Author: David S. Gunderson
Publisher: Chapman & Hall/CRC
ISBN: 9781138199019
Category : Induction (Mathematics)
Languages : en
Pages : 921
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Book Description
Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn's lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process.
Author: Alan Bundy
Publisher: Cambridge University Press
ISBN: 9780521834490
Category : Computers
Languages : en
Pages : 224
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Book Description
Rippling is a radically new technique for the automation of mathematical reasoning. It is widely applicable whenever a goal is to be proved from one or more syntactically similar givens. It was originally developed for inductive proofs, where the goal was the induction conclusion and the givens were the induction hypotheses. It has proved to be applicable to a much wider class of tasks, from summing series via analysis to general equational reasoning. The application to induction has especially important practical implications in the building of dependable IT systems, and provides solutions to issues such as the problem of combinatorial explosion. Rippling is the first of many new search control techniques based on formula annotation; some additional annotated reasoning techniques are also described here. This systematic and comprehensive introduction to rippling, and to the wider subject of automated inductive theorem proving, will be welcomed by researchers and graduate students alike.
Author: Hantao Zhang
Publisher: Springer
ISBN:
Category : Computers
Languages : en
Pages : 240
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Book Description
Two decades ago, Boyer and Moore built one of the first automated theorem provers that was capable of proofs by mathematical induction. Today, the Boyer-Moore theorem prover remains the most successful in the field. For a long time, the research on automated mathematical induction was confined to very few people. In recent years, as more people realize the importance of automated inductive reasoning to the use of formal methods of software and hardware development, more automated inductive proof systems have been built. Three years ago, the interested researchers in the field formed two consortia on automated inductive reasoning - the MInd consortium in Europe and the IndUS consortium in the United States. The two consortia organized three joint workshops in 1992-1995. There will be another one in 1996. Following the suggestions of Alan Bundy and Deepak Kapur, this book documents advances in the understanding of the field and in the power of the theorem provers that can be built. In the first of six papers, the reader is provided with a tutorial study of the Boyer-Moore theorem prover. The other five papers present novel ideas that could be used to build theorem provers more powerful than the Boyer-Moore prover.
Author: Boro Sitnikovski
Publisher: Apress
ISBN: 9781484279779
Category : Computers
Languages : en
Pages : 131
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Book Description
Get introduced to software verification and proving correctness using the Microsoft Research-backed programming language, Dafny. While some other books on this topic are quite mathematically rigorous, this book will use as little mathematical symbols and rigor as possible, and explain every concept using plain English. It's the perfect primer for software programmers and developers with C# and other programming language skills. Writing correct software can be hard, so you'll learn the concept of computation and software verification. Then, apply these concepts and techniques to confidently write bug-free code that is easy to understand. Source code will be available throughout the book and freely available via GitHub. After reading and using this book you'll be able write correct, big free software source code applicable no matter which platform and programming language you use. What You Will Learn Discover the Microsoft Research-backed Dafny programming language Explore Hoare logic, imperative and functional programs Work with pre- and post-conditions Use data types, pattern matching, and classes Dive into verification examples for potential re-use for your own projects Who This Book Is For Software developers and programmers with at least prior, basic programming experience. No specific language needed. It is also for those with very basic mathematical experience (function, variables).
Author: Alan J.A. Robinson
Publisher: Gulf Professional Publishing
ISBN: 9780444829498
Category : Computers
Languages : en
Pages : 1004
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Book Description
Handbook of Automated Reasoning.
Author: André Platzer
Publisher: Springer Nature
ISBN: 3030798763
Category : Artificial intelligence
Languages : en
Pages : 655
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Book Description
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions.
Author: Tobias Nipkow
Publisher: Springer
ISBN: 3319105426
Category : Computers
Languages : en
Pages : 304
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Book Description
Part I of this book is a practical introduction to working with the Isabelle proof assistant. It teaches you how to write functional programs and inductive definitions and how to prove properties about them in Isabelle’s structured proof language. Part II is an introduction to the semantics of imperative languages with an emphasis on applications like compilers and program analysers. The distinguishing feature is that all the mathematics has been formalised in Isabelle and much of it is executable. Part I focusses on the details of proofs in Isabelle; Part II can be read even without familiarity with Isabelle’s proof language, all proofs are described in detail but informally. The book teaches the reader the art of precise logical reasoning and the practical use of a proof assistant as a surgical tool for formal proofs about computer science artefacts. In this sense it represents a formal approach to computer science, not just semantics. The Isabelle formalisation, including the proofs and accompanying slides, are freely available online, and the book is suitable for graduate students, advanced undergraduate students, and researchers in theoretical computer science and logic.
Author: Martin Aigner
Publisher: Springer Science & Business Media
ISBN: 3662223430
Category : Mathematics
Languages : en
Pages : 194
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Book Description
According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.
Author: Frank van Harmelen
Publisher: Elsevier
ISBN: 0080557023
Category : Computers
Languages : en
Pages : 1035
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Book Description
Handbook of Knowledge Representation describes the essential foundations of Knowledge Representation, which lies at the core of Artificial Intelligence (AI). The book provides an up-to-date review of twenty-five key topics in knowledge representation, written by the leaders of each field. It includes a tutorial background and cutting-edge developments, as well as applications of Knowledge Representation in a variety of AI systems. This handbook is organized into three parts. Part I deals with general methods in Knowledge Representation and reasoning and covers such topics as classical logic in Knowledge Representation; satisfiability solvers; description logics; constraint programming; conceptual graphs; nonmonotonic reasoning; model-based problem solving; and Bayesian networks. Part II focuses on classes of knowledge and specialized representations, with chapters on temporal representation and reasoning; spatial and physical reasoning; reasoning about knowledge and belief; temporal action logics; and nonmonotonic causal logic. Part III discusses Knowledge Representation in applications such as question answering; the semantic web; automated planning; cognitive robotics; multi-agent systems; and knowledge engineering. This book is an essential resource for graduate students, researchers, and practitioners in knowledge representation and AI. * Make your computer smarter* Handle qualitative and uncertain information* Improve computational tractability to solve your problems easily
