What is type theory in programming?
A “type” in type theory has a role similar to a “type” in a programming language: it dictates the operations that can be performed on a term and, for variables, the possible values it might be replaced with. Some type theories serve as alternatives to set theory as a foundation of mathematics.
What is the purpose of homotopy type theory?
An equation such as “a=b” is a mathematical proposition in which two different symbols have the same value. In homotopy type theory, this is taken to mean that the two shapes which represent the values of the symbols are topologically equivalent.
Can type theory replace set theory?
Thus, type theory is not an alternative to set theory built on the same “sub-foundations”; instead it has re-excavated those sub-foundations and incorporated them into the foundational theory itself.
Does homotopy type theory provide a foundation for mathematics?
Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory.
What are higher inductive types?
Higher inductive types (HITs) are a generalization of inductive types which allow the constructors to produce, not just points of the type being defined, but also elements of its iterated identity types.
Who created type theory?
When the philosopher Bertrand Russell invented type theory at the beginning of the 20th century, he could hardly have imagined that his solution to a simple logic paradoxdefining the set of all sets not in themselveswould one day shape the trajectory of 21st century computer science.
Is type theory consistent?
any theory with a model is consistent. This is not the case. For instance, every algebraic theory has an initial and a terminal model, but neither of these have a bearing on consistency. The initial model is the syntax of a theory, and consistency is a statement about definability in the syntax.
What is cubical type theory?
Cubical type theory is a version of homotopy type theory in which univalence is not just an axiom but a theorem, hence, since this is constructive, has “computational content”. Cubical type theory models the infinity-groupoid-structure implied by Martin-Löf identity types on constructive cubical sets, whence the name.
What are inductive data types?
Inductive data type may refer to: Algebraic data type, a datatype each of whose values is data from other datatypes wrapped in one of the constructors of the datatype. Inductive family, a family of inductive data types indexed by another type or value.
Is recursion an induction?
Recursion is the process in which a function is called again and again until some base condition is met. Induction is the way of proving a mathematical statement. 2. It is the way of defining in a repetitive manner.
What is simple type theory?
Church’s type theory, aka simple type theory, is a formal logical language which includes classical first-order and propositional logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory.
What is the concept in type theory in which a single name?
| Q. | What is that concept in type theory in which a single name may denote objects of manydifferent classes that are related by some common super class referred to ______ |
|---|---|
| B. | type checking |
| C. | polymorphism |
| D. | generalization |
| Answer» c. polymorphism |
What are the 3 basic theories of learning?
Although there are many different approaches to learning, there are three basic types of learning theory: behaviorist, cognitive constructivist, and social constructivist.
What is homotopy type theory?
In mathematical logic and computer science, homotopy type theory (HoTT /hɒt/) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.
When were Homotopy models of intensional type theory first presented in public?
These results were first presented in public at the conference FMCS 2006 at which Warren gave a talk titled “Homotopy models of intensional type theory”, which also served as his thesis prospectus (the dissertation committee present were Awodey, Nicola Gambino and Alex Simpson). A summary is contained in Warren’s thesis prospectus abstract.
What is the IAS book on homotopy type theory?
The book produced by participants in the IAS program was titled “Homotopy type theory: Univalent foundations of mathematics”; although this could refer to either usage, since the book only discusses HoTT as a mathematical foundation.
What are some good books on homotopy?
Homotopy Type Theory (PDF) (Master’s). Utrecht University. Warren, Michael A. (2008). Homotopy Theoretic Aspects of Constructive Type Theory (PDF) (Ph.D.). Carnegie Mellon University.