Operational semantics are a category of formal programming language semantics in which certain desired properties of a program, such as correctness, safety or security, are verified by constructing proofs from logical statements about its execution and procedures, rather than by attaching mathematical meanings to its terms (denotational semantics). Operational semantics are classified in two categories: structural operational semantics (or small-step semantics) formally describe how the individual steps of a computation take place in a computer-based system. By opposition natural semantics (or big-step semantics) describe how the overall results of the executions are obtained. Other approaches to providing a formal semantics of programming languages include axiomatic semantics and denotational semantics.
The operational semantics for a programming language describes how a valid program is interpreted as sequences of computational steps. These sequences then are the meaning of the program. In the context of functional programs, the final step in a terminating sequence returns the value of the program. (In general there can be many return values for a single program, because the program could be nondeterministic, and even for a deterministic program there can be many computation sequences since the semantics may not specify exactly what sequence of operations arrives at that value.)
The concept of operational semantics was used for the first time in defining the semantics of Algol 68. The following statement is a quote from the revised ALGOL 68 report:
The meaning of a program in the strict language is explained in terms of a hypothetical computer which performs the set of actions which constitute the elaboration of that program. (, Section 2)
It is all very well to aim for a more ‘abstract’ and a ‘cleaner’ approach to semantics, but if the plan is to be any good, the operational aspects cannot be completely ignored.
Perhaps the first formal incarnation of operational semantics was the use of the lambda calculus to define the semantics of LISP by [John McCarthy. "Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I" ]. Abstract machines in the tradition of the SECD machine are also closely related.
Gordon Plotkin introduced the structural operational semantics, Robert Hieb and Matthias Felleisen the reduction contexts, and Gilles Kahn the natural semantics.