# nLab denotational semantics

Denotational semantics

### Context

#### Computing

intuitionistic mathematics

# Denotational semantics

## Idea

In computer science and formal logic, denotational semantics refers semantics based on the idea that programs and the data they manipulate are symbolic realizations of abstract mathematical objects.

For example, the denotational semantics

The idea of denotational semantics is thus to associate an appropriate mathematical object, such as a number, a tuple, or a function, with each term of the given programming language.

A key requirement on denotational semantics is that it respects the compositionality of programming languages, hence that the semantics of terms constructed from sub-terms is correspondingly built from the semantics of these sub-terms.

## References

Denotational semantics originates with the proposal of domain theory to regard data types as posets ((0,1)-categories):

• Dana S. Scott, Outline of a mathematical theory of computation, in: Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems (1970) 169–176. [pdf, pdf]

• Dana S. Scott, Christopher Strachey, Toward a Mathematical Semantics for Computer Languages, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [pdf, pdf]

• Dana Scott, Data types as lattices. SIAM Journal of Computing 5 3 (1976) 522–587 [doi:10.1137/0205037, pdf]

Lectures and introductions:

Textbook accounts: