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).

Information Description

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

• regardless of ease (theoretical expressivity)
• concisely and readily (practical expressivity)

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.

Examples

Expressive power in formal language theory

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.

An important yardstick for describing the relative expressive power of formalisms in this area is the Chomsky hierarchy. It says, for instance, that regular expressions, nondeterministic finite automatons and regular grammars have equal expressive power, while that of context-free grammars is greater; what this means is that the sets of sets of strings described by the first three formalisms are equal, and a proper subset of the set of sets of strings described by context-free grammars.

In this area, the cost of expressive power is a central topic of study. It is known, for instance, that deciding whether two arbitrary regular expressions describe the same set of strings is hard, while doing the same for arbitrary context-free grammars is completely impossible. However, it can still be efficiently decided whether any given string is in the set.

For more expressive formalisms, this problem can be harder, or even undecidable. For a Turing complete formalism, such as arbitrary formal grammars, not only this problem, but every nontrivial property regarding the set of strings they describe is undecidable, a fact known as Rice's Theorem.

There are some results on conciseness as well; for instance, nondeterministic state machines and regular grammars are more concise than regular expressions, in the sense that the latter can be translated to the former without a blowup in size (i.e. in O(1)), while the reverse is not possible.

Similar considerations apply to formalisms that describe not sets of strings, but sets of trees (e.g. XML schema languages), of graphs, or other structures.

Expressive power in database theory

Database theory is concerned, among other things, with database queries, e.g. formulas that given a <query predicate>, e.g.: Name = 'Smith', in a <query statement> as "Select distinct * from

where <query sentence>;", the query engine returns the set of rows of the matching the Boolean conditions of the <query sentence> (:= <query predicate> [<AND|OR> <query predicate>]). Our simple query could be "Select distinct * from EMPLOYEE where Name = 'Smith';" what will give us the names of the columns of the queried table as header of a set of rows of the people called Smith (if any) with a value per column: in the column called "Name" all its cells shall say "Smith". The expressive power of the query language....

See also

• Turing tarpit
• semantic spectrum