The simply typed lambda calculus (), a form
of type theory, is a typed interpretation of the lambda calculus with only one type constructor: that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties.
The term simple type is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The former are still considered simple because the Church encodings of such structures can be done using only and suitable type variables, while polymorphism and dependency cannot.
Lambda calculus may be typed and untyped. In typed lambda calculus functions can be applied only if they are capable of accepting the given input's "type" of data.
A typed lambda calculus is a typed formalism that uses the lambda-symbol () to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type.
Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here typability usually captures desirable properties of the program, e.g. the program will not cause a memory access violation.
CONTENT
This video is part of the playlist "Lambda Cube Unboxed", a series of 13 videos that explores and exposes the basics of (un-)typed λ-calculus including higher-order parametric polymorphism and dependent types. The resources, from which this video series extracts the offered material, are listed in the description of the playlist.
This video (2.1):
We now take a step back to discuss some of the shortcomings of the untyped lambda calculus and present types as a way of fixing some of these issues. In particular we look at the simply typed lambda calculus as a way of adding types to the untyped lambda calculus.
AUTHORS / OWNERS
The slides and script of the videos were made by Michelle Döring, Felix Moebius and Falk Schimweg. The video was narrated and edited by Douglas Rouse. (Und...
published: 18 Nov 2021
Simply typed lambda calculus demo
https://github.com/Pieter-JanLavaerts/lambda
published: 21 Apr 2021
Simply-Typed Lambda Calculus, part 1: Types
published: 24 Mar 2020
Astra Kolomatskaia --- The Objective Metatheory of Simply Typed Lambda Calculus.
Talk given on November 2, 2022, 7:00 - 8:30 PM. in The Graduate Center.
Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds b...
published: 03 Nov 2022
The Lambda Calculus, part 1 2 Applied lambda calculi; the simply typed lambda calculus
published: 27 Sep 2016
Semantics: Lambda Calculus and Types
We introduce lambda calculus but mainly focus on type theory in relation to syntactic trees. We introduce the type e (entities) and type t (truth values) and how they can be combined to account for V, VP, and S. #semantics #linguistics
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published: 15 Feb 2023
Backpropagation in the Simply Typed Lambda-calculus with Linear Negation
Presenter: Damiano Mazza
Presented at POPL'20.
published: 05 Feb 2020
013 Simply Typed Lambda Calculus
In this lecture I introduce simply typed lambda calculus and its typing rules.
published: 21 Oct 2020
The simply typed lambda calculus
Rules for the simply typed lambda calculus. Introduction to the Curry-Howard isomorphism.
CONTENT
This video is part of the playlist "Lambda Cube Unboxed", a series of 13 videos that explores and exposes the basics of (un-)typed λ-calculus including...
Talk given on November 2, 2022, 7:00 - 8:30 PM. in The Graduate Center.
Abstract: Lambda calculus is the language of functions. One reduces the application of...
Talk given on November 2, 2022, 7:00 - 8:30 PM. in The Graduate Center.
Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.
Slides:
http://www.sci.brooklyn.cuny.edu/~noson/CTslides/NYCCTS26Oct22.pdf
Talk given on November 2, 2022, 7:00 - 8:30 PM. in The Graduate Center.
Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.
Slides:
http://www.sci.brooklyn.cuny.edu/~noson/CTslides/NYCCTS26Oct22.pdf
We introduce lambda calculus but mainly focus on type theory in relation to syntactic trees. We introduce the type e (entities) and type t (truth values) and ho...
We introduce lambda calculus but mainly focus on type theory in relation to syntactic trees. We introduce the type e (entities) and type t (truth values) and how they can be combined to account for V, VP, and S. #semantics #linguistics
Join this channel to get access to perks:
https://www.youtube.com/channel/UCGYSfZbPp3BiAFs531PBY7g/join
Instagram: http://instagram.com/TrevTutorOfficial
Website: http://TrevTutor.com
Subscribe: http://bit.ly/1vWiRxW
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible.
We introduce lambda calculus but mainly focus on type theory in relation to syntactic trees. We introduce the type e (entities) and type t (truth values) and how they can be combined to account for V, VP, and S. #semantics #linguistics
Join this channel to get access to perks:
https://www.youtube.com/channel/UCGYSfZbPp3BiAFs531PBY7g/join
Instagram: http://instagram.com/TrevTutorOfficial
Website: http://TrevTutor.com
Subscribe: http://bit.ly/1vWiRxW
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible.
Talk given on November 2, 2022, 7:00 - 8:30 PM. in The Graduate Center.
Abstract: Lambda calculus is the language of functions. One reduces the application of a function to an argument by substituting the argument for the function's formal parameter inside of the function's body. The result of such a reduction may have further instances of function application. We can write down expressions, such as ((λ f. f f) (λ f. f f)), in which this process does not terminate. In the presence of types, however, one has a normalisation theorem, which effectively states that "programs can be run". One proof of this theorem, which only works for the most elementary of type theories, is to assign some monotone well-founded invariant to a given reduction algorithm. A much more surprising proof proceeds by constructing the normal form of a term by structural recursion on the term's syntactic representation, without ever performing reduction. Such normalisation algorithms fall under the class of Normalisation by Evaluation. Since the accidental discovery of the first such algorithm, it was clear that NbE had some underlying categorical content, and, in 1995, Altenkirch, Hofmann, and Streicher published the first categorical normalisation proof. Discovering this content requires first asking the question “What is STLC?”, perhaps preceded by the question “What is a type theory?”. In this talk we will lay out the details of Altenkirch's seminal paper and explore conceptual refinements discovered in the process of its formalisation in Cubical Agda.
Slides:
http://www.sci.brooklyn.cuny.edu/~noson/CTslides/NYCCTS26Oct22.pdf
We introduce lambda calculus but mainly focus on type theory in relation to syntactic trees. We introduce the type e (entities) and type t (truth values) and how they can be combined to account for V, VP, and S. #semantics #linguistics
Join this channel to get access to perks:
https://www.youtube.com/channel/UCGYSfZbPp3BiAFs531PBY7g/join
Instagram: http://instagram.com/TrevTutorOfficial
Website: http://TrevTutor.com
Subscribe: http://bit.ly/1vWiRxW
Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible.
The simply typed lambda calculus (), a form
of type theory, is a typed interpretation of the lambda calculus with only one type constructor: that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties.
The term simple type is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The former are still considered simple because the Church encodings of such structures can be done using only and suitable type variables, while polymorphism and dependency cannot.