Semantics (Computer Science)

Semantics (computer science) In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages. It does so by evaluating the meaning of syntactically legal strings defined by a specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically illegal strings, the result would be noncomputation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will execute on a certain platform, hence creating a model of computation.

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.

Formal semantics, for instance, helps to write compilers, better understand what a program is doing and to prove, e.g., that the following if statement

• Operational semantics, 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;

if 1 = 1 then S1 else S2 has the same effect as S1 alone.

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Overview

The field of formal semantics encompasses all of the following:

• Axiomatic semantics, whereby one gives meaning to phrases by describing the logical 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.

• 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.

The distinctions between the three broad classes of approaches can sometimes be vague, but all known apIt has close links with other areas of computer sci- proaches to formal semantics use the above techniques, ence such as programming language design, type the- or some combination thereof. ory, compilers and interpreters, program verification and Apart from the choice between denotational, operational, model checking. or axiomatic approaches, most variation in formal semantic systems arises from the choice of supporting mathematical formalism.

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Approaches

There are many approaches to formal semantics; these belong to three major classes:

3 Variations

• Denotational semantics, whereby each phrase in Some variations of formal semantics include the followthe language is interpreted as a denotation, i.e. ing: 1

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8 FURTHER READING • Action semantics is an approach that tries to modlow-level machine “faithfully implements” the highularize denotational semantics, splitting the formallevel machine. ization process in two layers (macro and microsemantics) and predefining three semantic entities (ac- It is also possible to relate multiple semantics through tions, data and yielders) to simplify the specification; abstractions via the theory of abstract interpretation. • Algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner;

5 History

• Attribute grammars define systems that systematiRobert W. Floyd is credited with founding the field of cally compute "metadata" (called attributes) for the programming language semantics in Floyd (1967).[1] various cases of the language’s syntax. Attribute grammars can be understood as a denotational semantics where the target language is simply the original language enriched with attribute annota- 6 See also tions. Aside from formal semantics, attribute grammars have also been used for code generation in • Formal semantics (logic) compilers, and to augment regular or context-free • Formal semantics (linguistics) grammars with context-sensitive conditions; • Categorical (or “functorial”) semantics uses category theory as the core mathematical formalism; • Concurrency semantics is a catch-all term for any formal semantics that describes concurrent computations. Historically important concurrent formalisms have included the Actor model and process calculi; • Game semantics uses a metaphor inspired by game theory. • Predicate transformer semantics, developed by Edsger W. Dijkstra, describes the meaning of a program fragment as the function transforming a postcondition to the precondition needed to establish it.

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Describing relationships

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

7 References [1] Knuth, Donald E. “Memorial Resolution: Robert W. Floyd (1936-2001)" (PDF). Stanford University Faculty Memorials. Stanford Historical Society.

• Floyd, Robert W. (1967). “Assigning Meanings to Programs” (PDF). In Schwartz, J.T. Mathematical Aspects of Computer Science. Proceedings of Symposium on Applied Mathematics 19. American Mathematical Society. pp. 19–32. ISBN 0821867288.

8 Further reading Textbooks • Aaron Stump, Programming Language Foundations. Wiley, 2014 (ISBN 978-1-118-00747-1) • Carl Gunter. Semantics of Programming Languages. MIT Press, 1992. (ISBN 0-262-07143-6) • Robert Harper. Practical Foundations for Programming Languages. Working draft, 2006. (online, as PDF) • Shriram Krishnamurthi. Programming Languages: Application and Interpretation. (online, as PDF) • Mitchell, John C.. Foundations for Programming Languages. • John C. Reynolds. Theories of Programming Languages. Cambridge University Press, 1998. (ISBN 0-521-59414-6)

3 • Kenneth Slonneger and Barry L. Kurtz. Formal Syntax and Semantics of Programming Languages. Addison-Wesley. • Glynn Winskel. The Formal Semantics of Programming Languages: An Introduction. MIT Press, 1993 (paperback ISBN 0-262-73103-7) • Robert D. Tennent (1991). Semantics of Programming Languages. Prentice-Hall. • M. Hennessy (1990) The Semantics of Programming Languages: An Elementary Introduction. Wiley. • H. Nielson and F. Nielson (1993) Semantics with Applications. A formal Introduction. Wiley. • H. Nielson and F. Nielson (2007) Semantics with Applications: An Appetizer. Undergraduate Texts in Computer Science. Springer. Lecture notes • Glynn Winskel. Denotational Semantics. University of Cambridge.

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External links • Aaby, Anthony (2004). Introduction to Programming Languages. Semantics.

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