Hb lambda calculus with types types10, october, 2010 new book cambridge university press asl perspectives in logic, 2011 lambda calculus with types 698. Weve got the basic simplytyped monomorphic lambda calculus now lets make it more complicated. Would adding recursive named functions to simply typed lambda. In typed lambda calculus, functions can be applied only if they are capable of. Jarvi is in turn based on pierce as the underlying textbook.
Typed lambda calculus for syntacticians carl pollard department of linguistics ohio state university january 12, 2012 carl pollard typed lambda calculus for syntacticians. They are languages that express both computational and logical information. One must be able to repre sent and efficiently reason about complex recursivelydefined expressions, define arbitrary induction schemes. Fixed points overview recursion and the fixedpoint combinator. The syntax of the simplytyped lambda calculus is similar to that of untyped lambda calculus, with the exception of abstractions. If by without the use of recursion you mean without general recursion and hence without fixpoints or self applications, we can simply observe that the factorial function is primitive recursive that is, iterative, in essence, and there is a very general and simple encoding of primitive recursion by means of iteration provided by church numerals and pairs.
Say i have simply typed lambda calculus, and add an assignment rule. This paper presents an alternative called the calculus of dependent. Modeling basic programming constructs references h. Where is the name of the function, is the function type and is the abstraction to be assigned to the identifier then i add a typing rule that says that when you see an assignment such as the above, you use a temporary type context, in which the. The lambda calculus is an abstract mathematical theory of computation, involving functions, and can be thought of as being the theoretical foundation of functional programming. If we denote the encoding of a term \t\ by \\lceil t\rceil\, then we can recursively encode any term with the following three rules for variables, applications, and lambda abstractions, respectively. The introduction to untyped lambda calculus, combinatory logic and simply typed lambda calculus provides a very principled coverage of those topics, discussing the subtle differences between the calculi in much gory detail, and being richly adorned with many lemmas one would take for granted, but which are discussed here in full formal glory. Schmidt, denotational semantics a methodology for language. An important property of reduction is a stability property. Through lambda calculus, is pointed at from both \readings and \resources links and the course schedule. Free and bound variables, reductions, church numerals, ordered pairs, relational operators, recursion, computability, typed lambda calculus.
Csc173 lambda calculus exercises 1 practice and practical help. The trick is to benefit from list comprehension and pythons name scope. Fixedpoint combinators may also be easily defined in other functional and imperative languages. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a strong theoretical foundation. The idea is due to sch on nkel 1924 but is often called currying, after h. Well, not exactly pure lambda recursion, but its applicable in places, where you can only use lambdas, e. It is a universal model of computation that can be used to simulate any turing machine. Typed lambda calculi are closely related to mathematical logic and proof theory via the curryhoward isomorphism and they can be considered as the internal language of classes of categories, e.
The simply typed lambda calculus ralf lammel programming paradigms and formal semantics 1h resources. Lambda calculus lambda calculus stanford university. I the simply typed lambda calculus i a basic and natural type system. The implementation in lambda calculus is more difficult due to limitations in lambda calculus. Computer science stack exchange is a question and answer site for students, researchers and practitioners of computer science. Pdf soundness of the simply typed lambda calculus in acl2. The simply typed lambda calculus was originally introduced by alonzo church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda. Introduction to the lambda calculus iowa state university. Notes on simply typed lambda calculus ralph loadery february, 1998 the purpose of this course is to provide an introduction to calculi, speci cally the simply typed lambda calculus. Barendregt, the lambda calculus its syntax and semantics, northholland, 1984.
As for any typed language, type safety is a fundamental property of all of these evaluation strategies. A short introduction to the lambda calculus achim jung. If e1 and e2 are values, then, it must be the case that they. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can, but on the other hand typed lambda calculi allow more things to be proved. Typed closure conversion for recursivelydefined functions extended abstract article pdf available in electronic notes in theoretical computer science 10 december 1998 with 23 reads. Can a lambda function call itself recursively in python. For typed calculi with sufficient computing power this is immediately seen. Does a definition of primitive recursion exist for the untyped lambda calculus. The simply typed lambda calculus uni koblenzlandau. But are there programs that do not get stuck that are not well typed. Likewise, the operational semantics of simply typed lambda calculus can be fixed as for the untyped lambda calculus, using call by name, call by value, or other evaluation strategies. This restriction comes with the simply typed lambda calculus.
Recursive types marco kuhlmann innite but regular data structures are indispensable tools in all modern program ming languages. Simplytyped lambda calculus lecture tuesday, march 9, 2010 a type is a collection of computational entities that share some common property. I cant figure out how to do it with a lambda function though for the simple reason that the lambda function has no. It has relevant and representational problems at the end of the chapters, all with answers in the back. Analyse the following lambda expression to clarify its structure. I cant figure out how to do it with a lambda function though for the simple reason that the lambda function has no n. It was introduced by the mathematician alonzo church in the 1930s as part of his. Typed lambda calculus answers to exercises 1 overview of logic and computation australian national university exercise 1. In turn, ptxx asks to check ptx and ptx, which provides the following axiom rules. We start by introducing the untyped lambda calculus. Indeed, type soundness implies that any lambda calculus program that gets stuck is not well typed. That is, the reduction of a welltyped term either diverges, or terminates in a value of the expected type. Soundness of the simply typed lambda calculus in acl2. By modifying the above construction using meyers the typed a calculus is not elementary recursive 81 result for the monadic predicate calculus instead of,2.
I continue to use cbv lambda caluclus as our core model i but will soon enrich with other common primitives this lecture. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the. Topics covered in these notes include the untyped lambda calculus, the churchrosser theorem, combinatory algebras. It is also possible to describe recursively the substitution operation using the following equations. For example, the type int represents all expressions that evaluate to an integer, and the type int. Given a term of the untyped lambda calculus, find all terms of the typed lambda calculus, which yield the given term when the type information on bound variables is deleted since such terms can differ only in their types, this problem is sometimes referred to as finding the. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. The previous exams at the \resources link might be helpful. March 18, 2004 abstract the lambda calculus can appear arcane on. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a. We define the syntax and reduction relation of a recursively typed lambda calculus with a parallel casefunction a parallel conditional. Comparison with the case of the simply typed lambda calculus the.
From lambda calculus to cartesian closed categories. Does the definition of primitive recursion require typing for natural numbers. Pdf typed closure conversion for recursivelydefined. The proof is based on the reducibility theorem in 16 see also theorem 3. Schmidt, the structure of typed programming languages, mit press, 1994. Recursive types are nite representations of such innite struc tures, yielding a natural extension of the simply typed lambdacalculus. Lambda calculus examples in this handout, we look at several examples of lambda terms in order to provide a. So, for example, if one of the base types of a lambda calculus was the natural numbers, the lambda theory would need to include rules to define. The fixedpoint combinator may be used in a number of different areas. The following example traverses the dictionary by the given chain of keys. These names will be either written in bold such as name or underlines such as name.
A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. Full abstraction for a recursively typed lambda calculus. I the simplytyped lambda calculus i a basic and natural type system. This gives the lambda calculus the power of turing machine computations. A denotational semantics is defined with this interpretation. Programming languages and lambda calculi utah cs7520 version matthias felleisen matthew flatt draft. The y combinator is an implementation of a fixedpoint combinator in lambda calculus. If we denote the encoding of a term \t\ by \\lceil t\rceil\, then we can recursively encode any term with the following three rules for variables, applications, and. Csc173 lambda calculus exercises 1 practice and practical.
Our proof proceeds in a similar manner to statmans proof. A regular function can contain a call to itself in its definition, no problem. One must be able to repre sent and efficiently reason about complex recursively defined expressions, define arbitrary induction schemes. The two sides of typed lambda calculus a typed lambda calculus tlc can be viewed in two complementary ways. Lambda calculus with lambda calculus mogensen describes a delightful encoding of lambda terms with lambda terms. If you added polymorphism which in itself does not allow recursive functions, for example system f is normalizing, you could get a polymorphic fixpoint combinator with a suitably generalized recursive let construct. You dont need to expand any operators except y and add, and those only when necessary. Many typed languages have informal descriptions of the type systems e. Soundness welltyped terms in stlc never go wrong theorem type safety. The simply typed lambda calculus, a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor.
The course is entitled lambda calculus and types and it appears in various forms. It is the canonical and simplest example of a typed lambda calculus. Strong typed bohm theorem and functional completeness. Typed lambda calculus answers to exercises 1 overview of logic and computation. This is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of ottawa in 2001 and at dalhousie university in 2007 and 20.
We interpret the recursive types as information systems in a restricted form, which we call prime systems. Type systems i continue to use cbv lambda caluclus as our core model i but will soon enrich with other common primitives this lecture. Lambda calculus is turing complete, that is, it is a universal model of computation that can be used to simulate any turing machine. Full abstraction for a recursively typed lambda calculus with. M need the same type soundness type safety is straightforward.
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