Type Theory and Formal Proof: An Introduction 1st Edition by Rob Nederpelt (PDF)

Description

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As someone who is not a math major, but studies math in my spare time, this book changed me.Before reading this book, as an amateur, I struggled often with simple proofs on an existential level. I’d read some books on formal logic and foundational mathematics before, and I could understand proofs and prove things, but thinking about these proofs *formally* always made me dizzy. The language and proof concepts used by working mathematicians just didn’t seem to match up with what is actually formal.One such example of the mismatch between practical proofs and formal proofs is our use of definitions and “named” theorems. In a formal system of mathematics, how can we *really* know that we are allowed to use *this* definition or theorem *here*? How do we even know the definition is valid?There were many other difficulties I was having other than that one example, but I’ve found that, after reading this book, these difficulties disappeared. I didn’t realize that they’d disappeared while I was reading the book, or even while I was doing the exercises. The book (at points) felt somewhat routine and repetitive, but easy to get through and understand. It sometimes felt like what the content of the book couldn’t be *that* unique or interesting. “The book isn’t going into depth about any of the cool paradoxes and curiosities I’ve come to associate with foundational mathematics” I thought. “How does this relate to the turing machine? Where are Godel’s theorems? When will I learn the crane kick?” All I got were a set of instructions on how to derive things in lambda calculus, essentially repeated over and over again with slight variations, “Wax on right hand, Wax off left hand, Wax on, Wax Off”It wasn’t until I started doing exercises from other books on various (unrelated) topics, that I began having real “Mr. Miyagi” moments. Proofs which previously took me a considerable amount of thought, now take almost no thought at all. When I write proofs now, my mind naturally imagines the flag style lambda D derivations which are introduced in this book. I can look at a definition or theorem and make it formal. I can also formulate several methods of how I might attack the proof, and moreover, I can know which proofs are “constructive” proofs and which ones are “classical” almost without thinking. I also now know generally how to prove consistency and inconsistency, and have a better grasp on undecidability (granted, the book doesn’t go into much depth about either topic, but it seems to lay the right foundation and provides in its “further reading” sections ample resources to look into these topics on your own)I am sure that this profound change is in large part because of the subject matter itself, and not necessarily because of this particular book. However, I am equally sure that the clear, concise writing, the choice of subjects and examples covered, the order in which said subjects and examples are introduced, and the bountiful, well crafted exercises was the only reason that I internalized the concepts as quickly as I did, and began to naturally apply them to the “real world” of mathematics.As others have mentioned, I think this book is a must read for all students of mathematics.

⭐Before consulting this text, I examined Turner/Handbook of Logic and Language, Hindley + Seldin, Girard, Michaelson, and Revesz on Lambda calculus. On type theory I examined Alonzo Church’s student Peter B. Andrews. I found Michaelson (“An Introduction to Functional Programming through Lambda Calculus”) to be useful, as many do, and I think the best intro to the subject of type theory for NLP practitioners remains L.T.F. Gamut vol. 2 chapter 4, which should be consulted before the work you are reading the review for.Needless to say, N+G is the best resource I’ve encountered on type theory and Lambda calculus. The level of exposition is far more detailed, and many needlessly rebarbative proofs are omitted (if you are more grounded in proof theory than me you may want to consider Andrews). This is the right resource to move onto after Gamut and Michaelson. N+G is still not for the faint-hearted, but it’s far friendlier than the existing literature on type theory and Lambda calculus.

⭐This is the most accessible book on type theory I have found so far. It starts with the untyped lambda calculus and incrementally increases the sophistication of the type system. At every step everything is worked out with simple example and plenty of details. The authors don’t pull the typical math trick of leaving the details to be filled in as exercises for the reader. The book does have exercises and they’re also accessible and build up on each other incrementally.In short, if you want an accessible introduction to type theory then this is the book to get.

⭐The content of the book is good. But reading the hardcover is an exercise in patience.The binding material used to hold pages is rigid and is little more than cheap adhesive. Trying to stay on a page without the book closing on you or flipping and turning the pages requires a firm hold. I have good quality books of the same dimensions as this one, but have cloth and threading and good adhesive for the binding and I can turn to any page I want and it sits still. This book requires you have your hand pressed on the pages at all times.

⭐I read the first part of the book and was enlightened on the importance of the type theory. Every programmer should have look to this book.

⭐Greate book written by great people. One of the best mathematical book I have ever read.I am very grateful to the authors.But the theme is very specific. It is for fans of FP and the theory of computations.

⭐If you lose yourself in this book, you will manage to forget, if only for a moment, the cruelty and malicious notations of the outside world.

⭐Brilliant book, it helped me to finally understand and enjoy lambda calculus

⭐If one aspires to be a working type-theorist then that’s your book.

⭐In the first six chapters, the authors introduce several variations of type theory, and they classify such theories through the concept of the lambda cube. Those chapters present a great introduction to the subject. The approach of the book really helped me to understand the ideas behind each kind of type theory, and their relationship.Throughout the rest of the book, the authors develop the system of deduction that they will use for formal proofs: A calculus of constructions with definitions. They present the advantages of this kind of lambda calculus and work some examples of proofs using it. They conclude with a complete formal proof of Bezout’s lemma. I believe that one could write their proof in a proof assistant. However, wisely, the authors do not commit to a particular software.Of course, this book assume that the reader is interested in lambda calculus and type theory in the context of using it for formal proofs. So I would not recommend it for those interested in computer science applications. The book assumes some familiarity with classical logic (at the level of van Dalen, or any other similar introductory book). If you have some familiarity with basic logic, and you are interested on an easy to read introduction to type theory, this is the book for you.

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