May 23, · Likes, 2 Comments - Dr Raymond C Lee MD (@drrayleemd) on Instagram: “What an amazing virtual aats. Congratulations to my chairman Dr Vaughn Starnes th AATS ” In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. Type theory is the academic study of type systems.. Some type theories serve as alternatives to set theory as a foundation of blogger.com well-known such theories are Alonzo Church's typed λ-calculus and In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. Type theory is the academic study of type systems.. Some type theories serve as alternatives to set theory as a foundation of blogger.com well-known such theories are Alonzo Church's typed λ-calculus and
Type theory - Wikipedia
In mathematicslogicand computer sciencea type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. Type theory is the academic study of type systems. Some type theories serve rob dekkers phd thesis alternatives to set theory as a foundation of mathematics.
Two well-known such theories are Alonzo Church 's typed λ-calculus and Per Martin-Löf 's intuitionistic type theory. Type theory was created to avoid paradoxes in previous foundations such as naive set theoryformal logics and rewrite systems. Type theory is closely related to, and in some cases overlaps with, computational type systemswhich are a programming language feature used to reduce bugs.
Between and Bertrand Russell proposed various "theories of type" in response to his discovery that Gottlob Frege 's version of naive set theory was afflicted with Russell's paradox. By Russell arrived at a "ramified" theory of types together with an " axiom of reducibility " both of which featured prominently in Whitehead and Russell 's Principia Mathematica published between and They attempted to resolve Russell's paradox by first creating a hierarchy of types, then assigning each concrete mathematical and possibly other entity to a type.
Entities of a given type are built exclusively from entities of those types that are lower in their hierarchy, thus preventing an entity from being assigned to itself. In the s, Leon Chwistek and Frank P. Ramsey proposed an unramified type theory, now known as the "theory of simple types" or simple type theorywhich collapsed the hierarchy of the types in the earlier ramified theory and as such did not require the axiom of reducibility.
The common usage of "type theory" is when those types are used with a term rewrite system, rob dekkers phd thesis.
The most famous early example is Alonzo Church 's simply typed lambda calculus. Church's theory of types [1] helped the formal system avoid the Kleene—Rosser paradox that afflicted the original untyped lambda calculus. Church demonstrated that it could serve as a foundation of mathematics and it was referred to as a higher-order logic.
Some other type theories include Per Martin-Löf 's intuitionistic type theoryrob dekkers phd thesis, which has been the foundation used in some areas of constructive mathematics. Thierry Coquand 's calculus of constructions and its derivatives are the foundation used by CoqLeanand others. The field is an area of active research, as demonstrated by homotopy type theory.
The contemporary presentation of type systems in the context of type theory has been made systematic by a conceptual framework introduced by Henk Barendregt. In a system of type theory, a term is opposed to a type.
Traditionally, the term is followed by a colon and its type, such as 2 : nat - this means that the number 2 is of type nat, rob dekkers phd thesis. Beyond this opposition and syntax, little can be said about types in this generality, but often, they are interpreted as some kind of collection not necessarily sets of the values that the term might evaluate to. It is usual to denote terms by e and types by τ.
How terms and types are shaped depends on the particular type system and is made precise by some syntax and additional restrictions of well-formedness, rob dekkers phd thesis. This pair is sometimes called an assignment.
The context completes the above opposition. Type theories have explicit computation and it is encoded in rules for rewriting terms. These are called conversion rules or, if the rule only works in one direction, a reduction rule. Various systems of typed lambda calculus including the simply typed lambda calculusJean-Yves Girard's System Fand Thierry Coquand's calculus of constructions are strongly normalizing. In such systems, rob dekkers phd thesis, a successful type check implies a termination proof of the term.
Based on the judgements and equivalences type inference rules can be used to describe how a type system assigns a type to a syntactic constructions termsmuch like in natural deduction. To be meaningful, conversion and type rules are usually closely related as in e. by a subject reduction property, which might establish a part of the soundness of a type system.
A type system is naturally associated with the decision problems of type checkingtypabilityand type inhabitation. Decidability of type checking means that type safety of any given program text source code can be verified. A variant of typability is typability wrt, rob dekkers phd thesis.
If the given term does not contain external references such as free term variablesthen typability coincides with typability wrt. the empty type environment. Typability is closely related to type inference. Whereas typability as a decision problem addresses the existence of a type for a given term, type inference as a computation rob dekkers phd thesis requires an actual type to be computed, rob dekkers phd thesis.
Girard's paradox shows that type inhabitation is strongly related to the consistency of a type system with Curry—Howard correspondence. To be sound, such a system must have uninhabited types, rob dekkers phd thesis. The opposition of terms and types can also be views as one of implementation and specification. By program synthesis the computational counterpart of type inhabitation see below can be used to construct all or parts of programs from specification given in form of rob dekkers phd thesis information.
Type theory is closely linked to many fields of active research. Most particular, the Curry—Howard correspondence rob dekkers phd thesis a deep isomorphism between intuitionistic logic, typed rob dekkers phd thesis calculus and cartesian closed categories. Beside the view of types as collection of values of a term, type theory offers a second interpretation of the opposition of term and types.
Types can be seen as propositions and terms as proofs, rob dekkers phd thesis. The internal language of the cartesian closed category is the simply typed lambda calculus. This view can be extend to other typed lambda calculi, rob dekkers phd thesis. Certain Cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional set theory.
There are many different set theories and many different systems of type theory, so what follows are generalizations. A dependent type is a type that depends on a term or another type. Thus, the type returned by a function may depend on the argument to the function.
In a type theory with dependent types, it is possible to define a function that takes a parameter "n" and returns a list containing "n" zeros. Calling the function rob dekkers phd thesis 4 would produce a term with a different type than if the function was called with 5. See Curry-Howard Correspondence, rob dekkers phd thesis. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like IdrisATSAgda and Epigram.
Many systems of type theory have a type that represents equality of types and of terms. Rob dekkers phd thesis type is different from convertibility, and is often denoted propositional equality.
In intuitionistic type theory, new terms of equality start with reflexivity. More complicated equalities can rob dekkers phd thesis created by creating a reflexive term and then doing a reduction on one side, rob dekkers phd thesis.
Thus, in this system, the equality type denotes that two values of the same type are rob dekkers phd thesis by reductions. Having a type for equality is important because it can be manipulated inside the system. Homotopy type theory differs from intuitionistic type theory mostly by its handling of the equality type. A system of type theory requires some basic terms and types to operate on.
Some systems build them out of functions using Church encoding. Other systems have inductive types : a set of base types and a set of type constructors that generate types with well-behaved properties.
For example, certain recursive functions called on inductive types are guaranteed to terminate, rob dekkers phd thesis. Coinductive types are infinite data types created by giving a function that generates the next element s. See Coinduction and Corecursion. Induction-induction is a feature for declaring an inductive type and a family of types which depends on the inductive type. Induction recursion allows a wider range of well-behaved types, allowing the type and recursive functions operating on it to be defined at the same time.
Types were created to prevent paradoxes, such as Russell's paradox. However, the motives that lead to those paradoxes—being able to say things about all types—still exist. So, many type theories have a "universe type", which contains all other types and not itself.
In systems where you might want to say something about universe types, there is a hierarchy of universe types, each containing the one below it in the hierarchy, rob dekkers phd thesis. The hierarchy is defined as being infinite, but statements must only refer to a finite number rob dekkers phd thesis universe levels. Type universes are particularly tricky in type theory. The initial proposal of intuitionistic type theory suffered from Girard's paradox.
Many systems of type theory, such as the simply-typed lambda calculusintuitionistic type theoryand the calculus of constructionsare also programming languages. That is, they are said to have a "computational component". The computation is the reduction of terms of the language using rewriting rules. A system of type theory that has a well-behaved computational component also has a simple connection to constructive mathematics through the BHK interpretation.
Non-constructive mathematics rob dekkers phd thesis these systems is possible by adding operators on continuations such as call with current continuation. However, these operators tend to break desirable properties such as canonicity and parametricity. There is extensive overlap and interaction between the fields of type theory and type systems. Type systems are a programming language feature designed to identify bugs.
Any static program analysissuch as the type checking algorithms in the semantic analysis phase of compilerhas a connection to type theory. A prime example is Agdaa programming language which uses UTT Luo's Unified Theory of dependent Types for its type system. The programming language ML was developed for manipulating type theories see LCF and its own type system was heavily influenced by them. The first computer proof assistant, called Automathused type theory to encode mathematics on a computer.
Martin-Löf specifically developed intuitionistic type theory to encode all mathematics to serve as a new foundation for mathematics. There is ongoing research into mathematical foundations using homotopy type theory. Mathematicians working in category theory already had difficulty working with the widely accepted foundation of Zermelo—Fraenkel set theory.
This led to proposals such as Lawvere's Elementary Theory of the Category of Sets ETCS. Researchers are exploring connections between dependent types especially the identity type and algebraic topology specifically homotopy.
Much of the current research into type theory is driven by proof checkersinteractive proof assistantsand automated theorem provers.
PhD Final Stages - Preparing your Thesis for Examination
, time: 10:11
May 23, · Likes, 2 Comments - Dr Raymond C Lee MD (@drrayleemd) on Instagram: “What an amazing virtual aats. Congratulations to my chairman Dr Vaughn Starnes th AATS ” In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. Type theory is the academic study of type systems.. Some type theories serve as alternatives to set theory as a foundation of blogger.com well-known such theories are Alonzo Church's typed λ-calculus and May 23, · Likes, 2 Comments - Dr Raymond C Lee MD (@drrayleemd) on Instagram: “What an amazing virtual aats. Congratulations to my chairman Dr Vaughn Starnes th AATS ”
No comments:
Post a Comment