# Expressive power

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In computer science, the **expressive power** of a language may refer to:

- what can be said in the language (at all)
- how concisely it can be said.

Formal discussions mostly use the term in the first sense, leaving conciseness as a less important factor. This is the case in areas of mathematics that deal with the exact 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^{[1]}. Efforts have been made to formalize these informal uses of the term ^{[2]}.

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.

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## 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 state machines 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 the contents of a database extract certain information from it. In the predominant relational database paradigm, the contents of a database are described as a finite set of finite mathematical relations; Boolean queries, that always yield *true* or *false*, are formulated in first-order logic.

It turns out that first-order logic is lacking in expressive power: it cannot express certain types of Boolean queries, e.g. queries involving transitive closure^{[3]}. However, adding expressive power must be done with care: it must still remain possible to evaluate queries with reasonable efficiency, which is not the case, e.g., for second-order logic. Consequently, a literature sprang up in which many query languages and language constructs were compared on expressive power and efficiency, e.g. various versions of Datalog^{[4]}.

Similar considerations apply for query languages on other types of data, e.g. XML query languages such as XQuery.

## References

- ↑
*Structure and Interpretation of Computer Programs*, by Abelson and Sussman - ↑
*On the Expressive Power of Programming Languages*, by Matthias Felleisen (1990) - ↑ Serge Abiteboul, Richard B. Hull, Victor Vianu: Foundations of Databases. Addison-Wesley, 1995.
- ↑ Evgeny Dantsin, Thomas Eiter, Georg Gottlob, and Andrei Voronkov: Complexity and expressive power of logic programming. ACM Comput. Surv. 33(3): 374-425 (2001).