The Curry Howard Isomorphism

The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.The isomorphism has many aspects, even at the syntactic level:formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc.But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transformsproofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features- The Curry-Howard Isomorphism treated as common theme- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics- Thorough study of the connection between calculi and logics- Elaborate study of classical logics and control operators- Account of dialogue games for classical and intuitionistic logic- Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme.· Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics.· Elaborate study of classical logics and control operators.· Account of dialogue games for classical and intuitionistic logic.· Theoretical foundations of computer-assisted reasoning

An introduction to simple type theory, containing 200 exercises with complete solutions.

This monograph details several important advances in the direction of a practical proofs-as-programs paradigm, which constitutes a set of approaches to developing programs from proofs in constructive logic with applications to industrial-scale, complex software engineering problems. One of the books central themes is a general, abstract framework for developing new systems of programs synthesis by adapting proofs-as-programs to new contexts.

This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will be essential reading for all those working in logic & computer science.

Despite using them every day, most software engineers know little about how programming languages are designed and implemented. For many, their only experience with that corner of computer science was a terrifying "compilers" class that they suffered through in undergrad and tried to blot from their memory as soon as they had scribbled their last NFA to DFA conversion on the final exam. That fearsome reputation belies a field that is rich with useful techniques and not so difficult as some of its practitioners might have you believe. A better understanding of how programming languages are built will make you a stronger software engineer and teach you concepts and data structures you'll use the rest of your coding days. You might even have fun. This book teaches you everything you need to know to implement a full-featured, efficient scripting language. You'll learn both high-level concepts around parsing and semantics and gritty details like bytecode representation and garbage collection. Your brain will light up with new ideas, and your hands will get dirty and calloused. Starting from main(), you will build a language that features rich syntax, dynamic typing, garbage collection, lexical scope, first-class functions, closures, classes, and inheritance. All packed into a few thousand lines of clean, fast code that you thoroughly understand because you wrote each one yourself.

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999.