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.

http://types10.mimuw.edu.pl/content/slides/day1/HenkBarendregt.pdf

Lambda Calculus with Types

Publisher Summary The theory of types is intended to be a full-scale system for formalizing intuitionistic mathematics as developed. The language of the theory is richer than the languages of traditional intuitionistic systems in permitting proofs to appear as parts of propositions so that the propositions of the theory can express properties of proofs. There are axioms for universes that link the generation of objects and types and play somewhat the same role for the present theory as does the replacement axiom for Zermelo–Fraenkel set theory. The present theory is based on a strongly impredicative axiom that there is a type of all types in symbols. This axiom has to be abandoned, however, after it has been shown to lead to a contraction. This chapter discusses Normalization theorem, which can be strengthened in two ways: it can be made to cover open terms and it can be proved that every reduction sequence starting from an arbitrary term leads to a unique normal term after a finite number of steps. The definition of the notion of convertibility and the proof that an arbitrary term is convertible can no longer be separated because the type symbols and the terms are generated simultaneously.

An Intuitionistic Theory of Types: Predicative Part

We present an order-extensional, order (or inequationally) fully abstract model for Scott's language pcf. The approach we have taken is very concrete and in nature goes back to S. C Kleene (1978, in “General Recursion Theory II, Proceedings of the 1977 Oslo Symposium,” North-Holland, Amsterdam) and R. O. Gandy (1993, “Dialogues, Blass Games, Sequentiality for Objects of Finite Type,” unpublished manuscript) in one tradition, and to G. Kahn and G. D. Plotkin (1993, Theoret. Comput. Sci.121, 187?278) and G. Berry and P.-L. Curien (1982, Theoret. Comput. Sci.20, 265?321) in another. Our model of computation is based on a kind of game in which each play consists of a dialogue of questions and answers between two players who observe the following principles of civil conversation: 1.Justification. A question is asked only if the dialogue at that point warrants it. An answer is proffered only if a question expecting it has already been asked. 2.Priority. Questions pending in a dialogue are answered on a last-asked-first-answered basis. This is equivalent to Gandy's no-dangling-question-mark condition. We analyze pcf-style computations directly in terms of partial strategies based on the information available to each player when he or she is about to move. Our players are required to play an innocent strategy: they play on the basis of their view which is that part of the history that interests them currently. Views are continually updated as the play unfolds. Hence our games are neither history-sensitive nor history-free. Rather they are view-dependent. These considerations give expression to what seems to us to be the nub of pcf-style higher-type sequentiality in a (dialogue) game-semantical setting.

/pdf/on-full-abstraction-for-pcf-3zs5sgq4zx.pdf

On Full Abstraction for PCF

T0 will denote Godel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schemaof choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.

Intensional Interpretations of Functionals of Finite Type I

Publisher Summary This chapter presents an exposition of certain themes in proof theory. The chapter discusses ideas behind what may be called Gentzen-type analysis in proof theory, where in particular, one wants to draw attention to the fact that they constitute the embryo to a general proof theory; extensions of the results obtained by Gentzen to more powerful theories; and the connection between proofs and the terms used in functional interpretations of intuitionistic logic, in particular, the connection between Gentzen-type analysis and the Godel-type analysis that originated with Godel's socalled Dialectica interpretation. A notion of validity of derivations is presented that may be contemplated as a possible explication of Gentzen's ideas about an operational interpretation of the logical constants. Proofs of the results have shown how this notion of validity may be used as a convenient tool to establish the main result about strong normalization in first order logic.

Ideas and Results in Proof Theory

Zusammenfassung

P. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulasst, Die abstrakten Begriffe, die bisher fur diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs uber den naturlichen Zahler benutzen kann, wobei keine anderen Konstruktionsverfahren fur solche Funktionen notig sind, als einfache Rekursion nach einer Zahlvariablen und Einsetzung von Funktionen ineinander (mit trivialen Funktionen als Ausgangspunkt).



Abstract

P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary stand-point by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other (starting with trivial functions).

https://www.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1746-8361.1958.tb01464.x

Von Kurt Gödel

Papers

Über eine bisher noch nicht benützte erweiterung Des finiten standpunktes