Expressive power (computer science)

In computer science, the expressive power (also called expressiveness or expressivity) of a language is the breadth of ideas that can be represented and communicated in that language. The more expressive a language is, the greater the variety and quantity of ideas it can be used to represent.

For example, the Web Ontology Language expression language profile (OWL2 EL) lacks ideas (such as negation) which can be expressed in OWL2 RL (rule language). OWL2 EL may therefore be said to have less expressive power than OWL2 RL. These restrictions allow more efficient (polynomial time) reasoning in OWL2 EL than in OWL2 RL. So OWL2 EL trades some expressive power for more efficient reasoning (processing of the knowledge representation language).

The term expressive power may be used with a range of meaning. It may mean a measure of the ideas expressible in that language:

The first sense dominates in areas of mathematics and logic that deal with the formal description of languages and their meaning, such as formal language theory, mathematical logic and process algebra.

In informal discussions, the term often refers to the second sense, or to both. This is often the case when discussing programming languages. Efforts have been made to formalize these informal uses of the term.

The notion of expressive power is always relative to a particular kind of thing that the language in question can describe, and the term is normally used when comparing languages that describe the same kind of things, or at least comparable kinds of things.

The design of languages and formalisms involves a trade-off between expressive power and analyzability. The more a formalism can express, the harder it becomes to understand what instances of the formalism say. Decision problems become harder to answer or completely undecidable.

Formal language theory mostly studies formalisms to describe sets of strings, such as context-free grammars and regular expressions. Each instance of a formalism, e.g. each grammar and each regular expression, describes a particular set of strings. In this context, the expressive power of a formalism is the set of sets of strings its instances describe, and comparing expressive power is a matter of comparing these sets.

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