Editing Operational semantics

{{Short description|Category of formal programming language semantics}}
{{Semantics}}

'''Operational semantics''' is a category of [[Formal language|formal programming language]] [[Semantics (computer science)|semantics]] in which certain desired properties of a [[Computer program|program]], such as correctness, safety or security, are [[formal verification|verified]] by constructing [[Mathematical proof|proof]]s from logical statements about its [[Execution (computing)|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 programming]], 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 algorithm|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.)

Perhaps the first formal incarnation of operational semantics was the use of the [[lambda calculus]] to define the semantics of [[Lisp (programming language)|Lisp]].<ref>{{Cite web |title=Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I |last=McCarthy |first=John |author-link=John McCarthy (computer scientist) |url=http://www-formal.stanford.edu/jmc/recursive.html |access-date=2006-10-13 |url-status=dead |archive-url=https://web.archive.org/web/20131004215327/http://www-formal.stanford.edu/jmc/recursive.html |archive-date=2013-10-04}}</ref> [[Abstract machine]]s in the tradition of the [[SECD machine]] are also closely related.

== History ==
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:

<blockquote>
The meaning of a program in the strict language is explained in terms of a hypothetical computer
which performs the set of actions that constitute the elaboration of that program. ([[#algol68|Algol68]], Section 2)
</blockquote>

The first use of the term "operational semantics" in its present meaning is attributed to
[[Dana Scott]] ([[#plotkin04|Plotkin04]]).
What follows is a quote from Scott's seminal paper on formal semantics,
in which he mentions the "operational" aspects of semantics.

<blockquote>
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. ([[#scott70|Scott70]])
</blockquote>

== Approaches ==
[[Gordon Plotkin]] introduced the structural operational semantics, [[Matthias Felleisen]] and Robert Hieb the reduction semantics,<ref name="felleisen-hieb-92">{{cite journal |title=The Revised Report on the Syntactic Theories of Sequential Control and State |journal=Theoretical Computer Science |last1=Felleisen |first1=M. |last2=Hieb |first2=R. |year=1992 |volume=103 |issue=2 |pages=235–271 |doi=10.1016/0304-3975(92)90014-7|doi-access= }}</ref> and [[Gilles Kahn]] the natural semantics.

=== Small-step semantics ===
==== Structural operational semantics ====
'''Structural operational semantics''' (SOS, also called '''structured operational semantics''' or '''small-step semantics''') was introduced by [[Gordon Plotkin]] in ([[#plotkin81|Plotkin81]]) as a logical means to define operational semantics. The basic idea behind SOS is to define the behavior of a program in terms of the behavior of its parts, thus providing a structural, i.e., syntax-oriented and [[inductive definition|inductive]], view on operational semantics. An SOS specification defines the behavior of a program in terms of a (set of) [[State transition system|transition relation]](s). SOS specifications take the form of a set of [[inference rule]]s that define the valid transitions of a composite piece of syntax in terms of the transitions of its components.

For a simple example, we consider part of the semantics of a simple programming language; proper illustrations are given in [[#plotkin81|Plotkin81]] and [[#hennessybook|Hennessy90]], and other textbooks. Let <math>C_1, C_2</math> range over programs of the language, and let <math>s</math> range over states (e.g. functions from memory locations to values). If we have expressions (ranged over by <math>E</math>), values {{nobreak|(<math>V</math>)}} and locations (<math>L</math>), then a memory update command would have semantics:

<math>
\frac{\langle E,s\rangle \Rightarrow V}{\langle L:=E\,,\,s\rangle\longrightarrow (s\uplus (L\mapsto V))}
</math>
<!-- NOTE: This is only a fragment of a full semantics. I've tried to include enough to illustrate the points but not so much that it takes a disproportionate amount of space.  -->

Informally, the rule says that "'''if''' the expression <math>E</math> in state <math>s</math> reduces to value <math>V</math>, '''then''' the program <math>L:=E</math> will update the state <math>s</math> with the assignment <math>L=V</math>".

The semantics of sequencing can be given by the following three rules:

<math>
\frac{\langle C_1,s\rangle \longrightarrow s'}
{\langle C_1;C_2 \,,s\rangle\longrightarrow \langle C_2, s'\rangle}
\quad\quad
\frac{\langle C_1,s\rangle \longrightarrow \langle C_1',s'\rangle}
{\langle C_1;C_2 \,,s\rangle\longrightarrow \langle C_1';C_2\,, s'\rangle}
\quad\quad
\frac{}
{\langle \mathbf{skip} ,s\rangle\longrightarrow s}
</math>

Informally, the first rule says that,
if program <math>C_1</math> in state <math>s</math> finishes in state <math>s'</math>, then the program <math>C_1;C_2</math> in state <math>s</math> will reduce to the program <math>C_2</math> in state <math>s'</math>.
(You can think of this as formalizing "You can run <math>C_1</math>, and then run <math>C_2</math>
using the resulting memory store.)
The second rule says that
if the program <math>C_1</math> in state <math>s</math> can reduce to the program <math>C_1'</math> with state <math>s'</math>, then the program <math>C_1;C_2</math> in state <math>s</math> will reduce to the program <math>C_1';C_2</math> in state <math>s'</math>.
(You can think of this as formalizing the principle for an optimizing compiler:
"You are allowed to transform <math>C_1</math> as if it were stand-alone, even if it is just the
first part of a program.")
The semantics is structural, because the meaning of the sequential program <math>C_1;C_2</math>, is defined by the meaning of <math>C_1</math> and the meaning of <math>C_2</math>.

If we also have Boolean expressions over the state, ranged over by <math>B</math>, then we can define the semantics of the '''while''' command:
<math>
\frac{\langle B,s\rangle \Rightarrow \mathbf{true}}{\langle\mathbf{while}\ B\ \mathbf{ do }\ C,s\rangle\longrightarrow \langle C;\mathbf{while}\ B\ \mathbf{do}\ C,s\rangle}
\quad
\frac{\langle B,s\rangle \Rightarrow \mathbf{false}}{\langle\mathbf{while}\ B\ \mathbf{ do }\ C,s\rangle\longrightarrow s}
</math>

Such a definition allows formal analysis of the behavior of programs, permitting the study of [[Relation (mathematics)|relations]] between programs. Important relations include [[simulation preorder]]s and [[bisimulation]].
These are especially useful in the context of [[Concurrency (computer science)|concurrency theory]].

Thanks to its intuitive look and easy-to-follow structure,
SOS has gained great popularity and has become a de facto standard in defining
operational semantics. As a sign of success, the original report (so-called Aarhus
report) on SOS ([[#plotkin81|Plotkin81]]) has attracted more than 1000 citations according to the CiteSeer [http://citeseer.ist.psu.edu/673965.html],
making it one of the most cited technical reports in [[Computer Science]].

==== Reduction semantics ====
'''Reduction semantics''' is an alternative presentation of operational semantics.  Its key ideas were first applied to purely functional [[call by name]] and [[call by value]] variants of the [[lambda calculus]] by [[Gordon Plotkin]] in 1975<ref>{{cite journal|last=Plotkin|first=Gordon|date=1975|title=Call-by-name, call-by-value and the λ-calculus|journal=Theoretical Computer Science|volume=1|issue=2|pages=125–159|doi=10.1016/0304-3975(75)90017-1|url=https://www.sciencedirect.com/science/article/pii/0304397575900171/pdf?md5=db2e67c1ada7163a28f124934b483f3a&pid=1-s2.0-0304397575900171-main.pdf|access-date=July 22, 2021|doi-access=free}}</ref> and generalized to higher-order functional languages with imperative features by [[Matthias Felleisen]] in his 1987 dissertation.<ref>{{cite thesis|type=PhD|last=Felleisen|first=Matthias|date=1987|title=The calculi of Lambda-v-CS conversion: a syntactic theory of control and state in imperative higher-order programming languages|publisher=Indiana University|url=https://www2.ccs.neu.edu/racket/pubs/dissertation-felleisen.pdf|access-date=July 22, 2021}}</ref> The method was further elaborated by Matthias Felleisen and Robert Hieb in 1992 into a fully [[equational theory]] for [[control flow|control]] and [[program state|state]].<ref name="felleisen-hieb-92" /> The phrase “reduction semantics” itself was first coined by Felleisen and [[Daniel P. Friedman]] in a PARLE 1987 paper.<ref>{{cite conference|last1=Felleisen|first1=Matthias|last2=Friedman|first2=Daniel P.|date=1987|title=A Reduction Semantics for Imperative Higher-Order Languages|book-title=Proceedings of the Parallel Architectures and Languages Europe|volume=1|pages=206–223|conference=International Conference on Parallel Architectures and Languages Europe|publisher=Springer-Verlag|doi=10.1007/3-540-17945-3_12}}</ref>

Reduction semantics are given as a set of ''reduction rules'' that each specify a single potential reduction step. For example, the following reduction rule states that an assignment statement can be reduced if it sits immediately beside its variable declaration:

<math>\mathbf{let\ rec}\ x = v_1\ \mathbf{in}\ x \leftarrow v_2;\ e\ \ \longrightarrow\ \ \mathbf{let\ rec}\ x = v_2\ \mathbf{in}\ e</math>

To get an assignment statement into such a position it is “bubbled up” through function applications and the right-hand side of assignment statements until it reaches the proper point. Since intervening <math>\mathbf{let}</math> expressions may declare distinct variables, the calculus also demands an extrusion rule for <math>\mathbf{let}</math> expressions. Most published uses of reduction semantics define such “bubble rules” with the convenience of evaluation contexts. For example, the grammar of evaluation contexts in a simple [[call by value]] language can be given as

<math>
  E ::= [\,]\ \big|\ v\ E\ \big|\ E\ e\ \big|\ x \leftarrow E
  \ \big|\ \mathbf{let\ rec}\ x = v\ \mathbf{in}\ E\ \big|\ E;\ e
</math>

where <math>e</math> denotes arbitrary expressions and <math>v</math> denotes fully-reduced values. Each evaluation context includes exactly one hole <math>[\,]</math> into which a term is plugged in a capturing fashion. The shape of the context indicates with this hole where reduction may occur. To describe “bubbling” with the aid of evaluation contexts, a single axiom suffices:

<math>E[\,x \leftarrow v;\ e\,]\ \ \longrightarrow\ \ x \leftarrow v;\ E[\,e\,]   \qquad   \text{(lift assignments)}</math>

This single reduction rule is the lift rule from Felleisen and Hieb's lambda calculus for assignment statements. The evaluation contexts restrict this rule to certain terms, but it is freely applicable in any term, including under lambdas.

Following Plotkin, showing the usefulness of a calculus derived from a set of reduction rules demands (1) a Church-Rosser lemma for the single-step relation, which induces an evaluation function, and (2) a Curry-Feys standardization lemma for the transitive-reflexive closure of the single-step relation, which replaces the non-deterministic search in the evaluation function with a deterministic left-most/outermost search. Felleisen showed that imperative extensions of this calculus satisfy these theorems. Consequences of these theorems are that the equational theory—the symmetric-transitive-reflexive closure—is a sound reasoning principle for these languages. However, in practice, most applications of reduction semantics dispense with the calculus and use the standard reduction only (and the evaluator that can be derived from it).

Reduction semantics are particularly useful given the ease by which evaluation contexts can model state or unusual control constructs (e.g., [[first-class continuations]]). In addition, reduction semantics have been used to model [[object-oriented]] languages,<ref>{{cite book|title=A Theory of Objects|last1=Abadi|first1=M.|last2=Cardelli|first2=L.|date=8 September 2012|publisher=Springer |isbn=9781441985989|url=https://books.google.com/books?id=AgzSBwAAQBAJ&q=%22operational+semantics%22}}</ref> [[design by contract|contract systems]], exceptions, futures, call-by-need, and many other language features. A thorough, modern treatment of reduction semantics that discusses several such applications at length is given by Matthias Felleisen, Robert Bruce Findler and Matthew Flatt in ''Semantics Engineering with PLT Redex''.<ref>{{cite book|last1=Felleisen|first1=Matthias|last2=Findler|first2=Robert Bruce|last3=Flatt|first3=Matthew|title=Semantics Engineering with PLT Redex|year=2009|publisher=The MIT Press|isbn=978-0-262-06275-6|url=https://mitpress.mit.edu/9780262062756/semantics-engineering-with-plt-redex/}}</ref>

===Big-step semantics===
====Natural semantics====
Big-step structural operational semantics is also known under the names '''natural semantics''', '''relational semantics''' and '''evaluation semantics'''.<ref>[https://web.archive.org/web/20131019133339/https://fsl.cs.illinois.edu/images/6/63/CS422-Spring-2010-BigStep.pdf University of Illinois CS422]</ref> Big-step operational semantics was introduced under the name ''natural semantics'' by [[Gilles Kahn]] when presenting Mini-ML, a pure dialect of [[ML (programming language)|ML]].

One can view big-step definitions as definitions of functions, or more generally of relations, interpreting each language construct in an appropriate domain. Its intuitiveness makes it a popular choice for semantics specification in programming languages, but it has some drawbacks that make it inconvenient or impossible to use in many situations, such as languages with control-intensive features or concurrency.<ref>{{cite book|last1=Nipkow|first1=Tobias|last2=Klein|first2=Gerwin|date=2014|title=Concrete Semantics|pages=101–102|doi=10.1007/978-3-319-10542-0|url=http://concrete-semantics.org/concrete-semantics.pdf|access-date=Mar 13, 2024|doi-access=free|isbn=978-3-319-10541-3 }}</ref>

A big-step semantics describes in a divide-and-conquer manner how final evaluation results of language constructs can be obtained by combining the evaluation results of their syntactic counterparts (subexpressions, substatements, etc.).

== Comparison ==
There are a number of distinctions between small-step and big-step semantics that influence whether one or the other forms a more suitable basis for specifying the semantics of a programming language.

Big-step semantics have the advantage of often being simpler (needing fewer inference rules) and often directly correspond to an efficient implementation of an interpreter for the language (hence Kahn calling them "natural".) Both can lead to simpler proofs, for example when proving the preservation of correctness under some [[program transformation]].<ref name="leroy-coinductivebigstep">[[Xavier Leroy]]. "Coinductive big-step operational semantics".</ref>

The main disadvantage of big-step semantics is that non-terminating ([[divergence (computer science)|diverging]]) computations do not have an inference tree, making it impossible to state and prove properties about such computations.<ref name="leroy-coinductivebigstep" />

Small-step semantics give more control over the details and order of evaluation. In the case of instrumented operational semantics, this allows the operational semantics to track and the semanticist to state and prove more accurate theorems about the run-time behaviour of the language. These properties make small-step semantics more convenient when proving [[type soundness]] of a type system against an operational semantics.<ref name="leroy-coinductivebigstep" />

== See also ==
* [[Algebraic semantics (computer science)|Algebraic semantics]]
* [[Axiomatic semantics]]
* [[Denotational semantics]]
* [[Formal semantics of programming languages]]

== References ==
{{Reflist}}

== Further reading ==
*[[Gilles Kahn]]. "Natural Semantics". ''Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science''. Springer-Verlag. London. 1987.
*<cite id=plotkin81>[[Gordon Plotkin|Gordon D. Plotkin.]] [http://citeseer.ist.psu.edu/673965.html A Structural Approach to Operational Semantics]. (1981) Tech. Rep. DAIMI FN-19, Computer Science Department, Aarhus University, Aarhus, Denmark. (Reprinted with corrections in J. Log. Algebr. Program. 60-61: 17-139 (2004), ([http://homepages.inf.ed.ac.uk/gdp/publications/sos_jlap.pdf preprint]). </cite>
*<cite id=plotkin04>[[Gordon Plotkin|Gordon D. Plotkin.]] The Origins of Structural Operational Semantics.  J. Log. Algebr. Program. 60-61:3-15, 2004.  ([http://homepages.inf.ed.ac.uk/gdp/publications/Origins_SOS.pdf preprint]). </cite>
*<cite id=scott70>[[Dana Scott|Dana S. Scott.]] Outline of a Mathematical Theory of Computation, Programming Research Group, Technical Monograph PRG–2, Oxford University, 1970.</cite>
*<cite id=algol68> [[Adriaan van Wijngaarden]] et al. Revised Report on the Algorithmic Language [[ALGOL 68]]. IFIP. 1968. [https://jemarch.net/algol68-revised-report.pdf]</cite>
*<cite id=hennessybook>[[Matthew Hennessy]]. Semantics of Programming Languages. Wiley, 1990. [https://www.cs.tcd.ie/matthew.hennessy/splexternal2015/resources/sembookWiley.pdf available online].</cite>

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