In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called denotations) that describe the meanings of expressions from the languages. Other approaches providing formal semantics of programming languages include axiomatic semantics and operational semantics.
Broadly speaking, denotational semantics is concerned with finding mathematical objects called domains that represent what programs do. For example, programs (or program phrases) might be represented by partial functionsDana S. Scott. Outline of a mathematical theory of computation. Technical Monograph PRG-2, Oxford University Computing Laboratory, Oxford, England, November 1970.Dana Scott and Christopher Strachey. Toward a mathematical semantics for computer languages Oxford Programming Research Group Technical Monograph. PRG-6. 1971. or by gamesJan Jürjens. J. Games In The Semantics Of Programming Languages – An Elementary Introduction. Synthese 133, 131–158 (2002). https://doi.org/10.1023/A:1020883810034 between the environment and the system.
An important tenet of denotational semantics is that semantics should be compositional: the denotation of a program phrase should be built out of the denotations of its subphrases.
Historical development Denotational semantics originated in the work of Christopher Strachey and Dana Scott published in the early 1970s. As originally developed by Strachey and Scott, denotational semantics provided the meaning of a computer program as a function that mapped input into output. To give meanings to recursively defined programs, Scott proposed working with continuous functions between domains, specifically complete partial orders. As described below, work has continued in investigating appropriate denotational semantics for aspects of programming languages such as sequentiality, concurrency, non-determinism and local state.
Denotational semantics has been developed for modern programming languages that use capabilities like concurrency and exceptions, e.g., Concurrent ML,John Reppy "Concurrent ML: Design, Application and Semantics" in Springer-Verlag, Lecture Notes in Computer Science, Vol. 693. 1993 CSP,A. W. Roscoe. "The Theory and Practice of Concurrency" Prentice-Hall. Revised 2005. and Haskell.Simon Peyton Jones, Alastair Reid, Fergus Henderson, Tony Hoare, and Simon Marlow. "A semantics for imprecise exceptions" Conference on Programming Language Design and Implementation. 1999. The semantics of these languages is compositional in that the meaning of a phrase depends on the meanings of its subphrases. For example, the meaning of the applicative expression f(E1,E2) is defined in terms of semantics of its subphrases f, E1 and E2. In a modern programming language, E1 and E2 can be evaluated concurrently and the execution of one of them might affect the other by interacting through shared objects causing their meanings to be defined in terms of each other. Also, E1 or E2 might throw an exception which could terminate the execution of the other one. The sections below describe special cases of the semantics of these modern programming languages.
Meanings of recursive programs Denotational semantics is ascribed to a program phrase as a function from an environment (holding current values of its free variables) to its denotation. For example, the phrase produces a denotation when provided with an environment that has binding for its two free variables: and . If in the environment has the value 3 and has the value 5, then the denotation is 15.
A function can be represented as a set of ordered pairs of argument and corresponding result values. For example, the set {(0,1), (4,3)} denotes a function with result 1 for argument 0, result 3 for the argument 4, and undefined otherwise.
Consider for example the factorial function, which might be defined recursively as: int factorial(int n) { if (n == 0) return 1; else return n * factorial(n - 1); }
To provide a meaning for this recursive definition, the denotation is built up as the limit of approximations, where each approximation limits the number of calls to factorial. At the beginning, we start with no calls - hence nothing is defined. In the next approximation, we can add the ordered pair (0,1), because this doesn't require calling factorial again. Similarly we can add (1,1), (2,2), etc., adding one pair each successive approximation because computing factorial(n) requires n+1 calls. In the limit we get a total function from to defined everywhere in its domain.
Formally we model each approximation as a partial function . Our approximation is then repeatedly applying a function implementing "make a more defined partial factorial function", i.e. , starting with the empty function (empty set). F could be defined in code as follows (using Map<int,int> for ): int factorial_nonrecursive(Map<int,int> factorial_less_defined, int n) { if (n == 0) then return 1; else if (fprev = lookup(factorial_less_defined, n-1)) then return n * fprev; else return NOT_DEFINED; }
Map<int,int> F(Map<int,int> factorial_less_defined) { Map<int,int> new_factorial = Map.empty(); for (int n in all<int>()) { if (f = factorial_nonrecursive(factorial_less_defined, n) != NOT_DEFINED) new_factorial.put(n, f);
return new_factorial; }
Then we can introduce the notation Fn to indicate F applied n times. F0({}) is the totally undefined partial function, represented as the set {}; F1({}) is the partial function represented as the set {(0,1)}: it is defined at 0, to be 1, and undefined elsewhere; F5({}) is the partial function represented as the set {(0,1), (1,1), (2,2), (3,6), (4,24)}: it is defined for arguments 0,1,2,3,4.
This iterative process builds a sequence of partial functions from to . Partial functions form a chain-complete partial order using ⊆ as the ordering. Furthermore, this iterative process of better approximations of the factorial function forms an expansive (also called progressive) mapping because each using ⊆ as the ordering. So by a fixed-point theorem (specifically Bourbaki–Witt theorem), there exists a fixed point for this iterative process.
In this case, the fixed point is the least upper bound of this chain, which is the full function, which can be expressed as the union
The fixed point we found is the least fixed point of F, because our iteration started with the smallest element in the domain (the empty set). To prove this we need a more complex fixed point theorem such as the Knaster–Tarski theorem.
Denotational semantics of non-deterministic programs The concept of power domains has been developed to give a denotational semantics to non-deterministic sequential programs. Writing P for a power-domain constructor, the domain P(D) is the domain of non-deterministic computations of type denoted by D.
There are difficulties with fairness and unboundedness in domain-theoretic models of non-determinism.
Denotational semantics of concurrency Many researchers have argued that the domain-theoretic models given above do not suffice for the more general case of concurrent computation. For this reason various new models have been introduced. In the early 1980s, people began using the style of denotational semantics to give semantics for concurrent languages. Examples include Will Clinger's work with the actor model; Glynn Winskel's work with event structures and Petri nets;Event Structure Semantics for CCS and Related Languages. DAIMI Research Report, University of Aarhus, 67 pp., April 1983. and the work by Francez, Hoare, Lehmann, and de Roever (1979) on trace semantics for CSP.Nissim Francez, C. A. R. Hoare, Daniel Lehmann, and Willem-Paul de Roever. "Semantics of nondeterminism, concurrency, and communication", Journal of Computer and System Sciences. December 1979. All these lines of inquiry remain under investigation (see e.g. the various denotational models for CSP).
Recently, Winskel and others have proposed the category of profunctors as a domain theory for concurrency.
Denotational semantics of state State (such as a heap) and simple imperative features can be straightforwardly modeled in the denotational semantics described above. The key idea is to consider a command as a partial function on some domain of states. The meaning of "" is then the function that takes a state to the state with assigned to . The sequencing operator "" is denoted by composition of functions. Fixed-point constructions are then used to give a semantics to looping constructs, such as "".
Things become more difficult in modelling programs with local variables. One approach is to no longer work with domains, but instead to interpret types as functors from some category of worlds to a category of domains. Programs are then denoted by natural continuous functions between these functors.Peter W. O'Hearn, John Power, Robert D. Tennent, Makoto Takeyama. Syntactic control of interference revisited. Electron. Notes Theor. Comput. Sci. 1. 1995.Frank J. Oles. A Category-Theoretic Approach to the Semantics of Programming. PhD thesis, Syracuse University, New York, USA. 1982.
Denotations of data types Many programming languages allow users to define recursive data types. For example, the type of lists of numbers can be specified by datatype list = Cons of nat * list | Empty This section deals only with functional data structures that cannot change. Conventional imperative programming languages would typically allow the elements of such a recursive list to be changed.
For another example: the type of denotations of the untyped lambda calculus is datatype D = D of (D → D) The problem of solving domain equations is concerned with finding domains that model these kinds of datatypes. One approach, roughly speaking, is to consider the collection of all domains as a domain itself, and then solve the recursive definition there.
Polymorphic data types are data types that are defined with a parameter. For example, the type of α s is defined by datatype α list = Cons of α * α list | Empty Lists of natural numbers, then, are of type , while lists of strings are of .
Some researchers have developed domain theoretic models of polymorphism. Other researchers have also modeled parametric polymorphism within constructive set theories.
A recent research area has involved denotational semantics for object and class based programming languages.
Denotational semantics for programs of restricted complexity Following the development of programming languages based on linear logic, denotational semantics have been given to languages for linear usage (see e.g. proof nets, coherence spaces) and also polynomial time complexity.
Denotational semantics of sequentiality The problem of full abstraction for the sequential programming language PCF was, for a long time, a big open question in denotational semantics. The difficulty with PCF is that it is a very sequential language. For example, there is no way to define the parallel-or function in PCF. It is for this reason that the approach using domains, as introduced above, yields a denotational semantics that is not fully abstract.
This open question was mostly resolved in the 1990s with the development of game semantics and also with techniques involving logical relations. For more details, see the page on PCF.
Denotational semantics as source-to-source translation It is often useful to translate one programming language into another. For example, a concurrent programming language might be translated into a process calculus; a high-level programming language might be translated into byte-code. (Indeed, conventional denotational semantics can be seen as the interpretation of programming languages into the internal language of the category of domains.)