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Lambda calculus is a formalism introduced by Church in 1932 that was intended to be used as a foundation for mathematics, including its computational aspects. Supported by his students Kleene and Rosser – who showed that the prototype system was inconsistent – Church distilled a consistent computational part and ventured in 1936 the Thesis that exactly the intuitively computable functions could be captured by it. He also presented a function that could not be captured by the λ-calculus. In that same year Turing introduced another formalism, describing what are now called Turing Machines, and formulated the related Thesis that exactly the mechanically computable functions are able to be captured by these machines. Turing also showed in the same paper that the question of whether a given statement could be proved(from a given setofaxioms) using the rules of any reasonable system of logic is not computable in this mechanical way. Finally Turing showed that the formalism of λ-calculus and Turing Machines define the same class of functions.
Together Church's Thesis, concerning computability by homo sapiens, and Turing's Thesis, concerning computability by mechanical devices, using formalisms that are equally powerful and that have their computational limitations, made a deep impact on the 20th century philosophy of the power and limitations of the human mind. So far, cognitive neuropsychology has not been able to refute the combined Church-Turing Thesis. On the contrary, that discipline also shows the limitation of human capacities.
This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.
This book is about lambda terms typed using simple, recursive and intersection types. In some sense it is a sequel to Barendregt (1984).That book is about untyped lambda calculus. Types give the untyped terms more structure: function applications are allowed only in some cases. In this way one can single out untyped terms having special properties. But there is more to it. The extra structure makes the theory of typed terms quite different from the untyped ones.
The emphasis of the book is on syntax. Models are introduced only insofar as they give useful information about terms and types or if the theory can be applied to them.
The writing of this book has been different from the one on untyped lambda calculus. First of all, since many researchers are working on typed lambda calculus, we were aiming at a moving target. Moreover there has been a wealth of material to work with. For these reasons the book was written by several authors. Several long-term open problems have been solved during the period the book was written, notably the undecidability of lambda definability in finite models, the undecidability of second-order typability, the decidability of the unique maximal theory extending βη-conversion and the fact that the collection of closed terms of not every simple type is finitely generated, and the decidability of matching at arbitrary types of order higher than 4. The book has not been written as an encyclopedic monograph: many topics are only partially treated; for example, reducibility among types is analyzed only for simple types built up from only one atom.