constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
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.
Under the Curry-Howard correspondence, also proofs can be seen as programs and thus one can apply denotational semantics to proofs. In this way proofs are interpreted as functions or more generally as morphisms of some category. This is used to interpret proofs of intuitionistic logic and linear logic as morphisms of respectively cartesian closed categories and star-autonomous categories, as discussed at relation between category theory and type theory. For more general such semantics see at categorical semantics of dependent type theory.
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:
Robert D. Tennent, The denotational semantics of programming languages, Communications of the ACM 19 8 (1976) 437–453 [doi:10.1145/360303.360308, pdf]
Robert D. Tennent, Denotational semantic, in: Handbook of Logic in Computer Science 3, Oxford University Press (1995) [ISBN:9780198537625]
Andrew M. Pitts, Lecture Notes on Denotational Semantics (2012) [pdf, pdf]
Textbook accounts:
Glynn Winskel, §5 of: The Formal Semantics of Programming Languages, MIT Press (1993) [ISBN:9780262731034, pdf]
Kenneth Slonneger, Barry Kurtz, Denotational semantics [pdf], Chapter 9 of: Formal Syntax and Semantics of Programming Languages, Addison-Wesley (1995) [webpage, pdf]
David A. Schmidt, Denotational Semantics – A methodology for language development, Allyn and Bacon (1986) [pdf, webpage]
Thomas Streicher, Domain-Theoretic Foundations of Functional Programming, World Scientific (2006) [pdf, doi:10.1142/6284]
See also:
Discussion of denotational semantics for Haskell:
Last revised on March 5, 2023 at 06:15:34. See the history of this page for a list of all contributions to it.