Question

Explain what denotational semantics are with an example. Provide the denotational semantics for x = x + 1 along with the justification.

Answer 1

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.

There are two main aspects to a programming language, the syntax and its semantics.
The syntax defines the correct form for valid programs and the semantics determines what they do.

The most successful system for describing the semantics is the denotational semantics which describes all the features found in imperative programming languages and has a sound mathematical basis.

The denotational semantics method maps a program directly to its meaning, called its denotation. The denotation is usually a mathematical value, such as a number or a function. No interpreters are used; a valuation function maps a program directly to its meaning. A denotational definition is more abstract than an operational definition, for it does not specify computation steps. Its high-level, modular structure makes it especially useful to language designers and users, for the individual parts of a language can be studied without having to examine the entire definition. On the other hand, the implementor of a language is left with more work. The numbers and functions must be represented as objects in a physical machine, and the valuation function must be implemented as the processor. This is an ongoing area of study.

Here are some basic examples of denotational semantics:

Boolean Expressions

Denotational semantics for X = X + 1