In programming language theory, semantics is the rigorous mathematical study of the meaning of programming languages.[1] Semantics assigns computational meaning to valid strings in a programming language syntax. It is closely related to, and often crosses over with, the semantics of mathematical proofs.
Semantics describes the processes a computer follows when executing a program in that specific language. This can be done by describing the relationship between the input and output of a program, or giving an explanation of how the program will be executed on a certain platform, thereby creating a model of computation.
In 1967, Robert W. Floyd published the paper Assigning meanings to programs; his chief aim was "a rigorous standard for proofs about computer programs, including proofs of correctness, equivalence, and termination".[2][3] Floyd further wrote:[2]
A semantic definition of a programming language, in our approach, is founded on a syntactic definition. It must specify which of the phrases in a syntactically correct program represent commands, and what conditions must be imposed on an interpretation in the neighborhood of each command.
In 1969, Tony Hoare published a paper on Hoare logic seeded by Floyd's ideas, now sometimes collectively called axiomatic semantics.[4][5]
In the 1970s, the terms operational semantics and denotational semantics emerged.[5]
The field of formal semantics encompasses all of the following:
- The definition of semantic models
- The relations between different semantic models
- The relations between different approaches to meaning
- The relation between computation and the underlying mathematical structures from fields such as logic, set theory, model theory, category theory, etc.
It has close links with other areas of computer science such as programming language design, type theory, compilers and interpreters, program verification and model checking.
There are many approaches to formal semantics; these belong to three major classes:
- Denotational semantics,[6] whereby each phrase in the language is interpreted as a denotation, i.e. a conceptual meaning that can be thought of abstractly. Such denotations are often mathematical objects inhabiting a mathematical space, but it is not a requirement that they should be so. As a practical necessity, denotations are described using some form of mathematical notation, which can in turn be formalized as a denotational metalanguage. For example, denotational semantics of functional languages often translate the language into domain theory. Denotational semantic descriptions can also serve as compositional translations from a programming language into the denotational metalanguage and used as a basis for designing compilers.
- Operational semantics,[7] whereby the execution of the language is described directly (rather than by translation). Operational semantics loosely corresponds to interpretation, although again the "implementation language" of the interpreter is generally a mathematical formalism. Operational semantics may define an abstract machine (such as the SECD machine), and give meaning to phrases by describing the transitions they induce on states of the machine. Alternatively, as with the pure lambda calculus, operational semantics can be defined via syntactic transformations on phrases of the language itself;
- Axiomatic semantics,[8] whereby one gives meaning to phrases by describing the axioms that apply to them. Axiomatic semantics makes no distinction between a phrase's meaning and the logical formulas that describe it; its meaning is exactly what can be proven about it in some logic. The canonical example of axiomatic semantics is Hoare logic.
Apart from the choice between denotational, operational, or axiomatic approaches, most variations in formal semantic systems arise from the choice of supporting mathematical formalism.[citation needed]
Some variations of formal semantics include the following:
For a variety of reasons, one might wish to describe the relationships between different formal semantics. For example:
- To prove that a particular operational semantics for a language satisfies the logical formulas of an axiomatic semantics for that language. Such a proof demonstrates that it is "sound" to reason about a particular (operational) interpretation strategy using a particular (axiomatic) proof system.
- To prove that operational semantics over a high-level machine is related by a simulation with the semantics over a low-level machine, whereby the low-level abstract machine contains more primitive operations than the high-level abstract machine definition of a given language. Such a proof demonstrates that the low-level machine "faithfully implements" the high-level machine.
It is also possible to relate multiple semantics through abstractions via the theory of abstract interpretation.[citation needed]
Schmidt, David A. (1986). Denotational Semantics: A Methodology for Language Development. William C. Brown Publishers. ISBN 9780205104505. Deransart, Pierre; Jourdan, Martin; Lorho, Bernard (1988). "Attribute Grammars: Definitions, Systems and Bibliography. Lecture Notes in Computer Science 323. Springer-Verlag. ISBN 9780387500560.
- Textbooks
- Floyd, Robert W. (1967). "Assigning Meanings to Programs" (PDF). In Schwartz, J.T. (ed.). Mathematical Aspects of Computer Science. Proceedings of Symposium on Applied Mathematics. Vol. 19. American Mathematical Society. pp. 19–32. ISBN 0821867288.
- Hennessy, M. (1990). The semantics of programming languages: an elementary introduction using structural operational semantics. Wiley. ISBN 978-0-471-92772-3.
- Tennent, Robert D. (1991). Semantics of Programming Languages. Prentice Hall. ISBN 978-0-13-805599-8.
- Gunter, Carl (1992). Semantics of Programming Languages. MIT Press. ISBN 0-262-07143-6.
- Nielson, H. R.; Nielson, Flemming (1992). Semantics With Applications: A Formal Introduction (PDF). Wiley. ISBN 978-0-471-92980-2. Archived from the original (PDF) on 2012-04-17. Retrieved 2011-05-27.
- Winskel, Glynn (1993). The Formal Semantics of Programming Languages: An Introduction. MIT Press. ISBN 0-262-73103-7.
- Mitchell, John C. (1995). Foundations for Programming Languages (Postscript).
- Slonneger, Kenneth; Kurtz, Barry L. (1995). Formal Syntax and Semantics of Programming Languages. Addison-Wesley. ISBN 0-201-65697-3.
- Reynolds, John C. (1998). Theories of Programming Languages. Cambridge University Press. ISBN 0-521-59414-6.
- Harper, Robert (2006). Practical Foundations for Programming Languages (PDF). Archived from the original (PDF) on 2007-06-27. (Working draft)
- Nielson, H. R.; Nielson, Flemming (2007). Semantics with Applications: An Appetizer. Springer. ISBN 978-1-84628-692-6.
- Stump, Aaron (2014). Programming Language Foundations. Wiley. ISBN 978-1-118-00747-1.
- Krishnamurthi, Shriram (2012). "Programming Languages: Application and Interpretation" (2nd ed.).
- Lecture notes