computer science

Indexed language

Indexed language is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Indexed language rather than just read about it. In short: Indexed languages are a class of formal languages discovered by Alfred Aho; they are described by indexed grammars and can be recognized by nested stack automata. Indexed languages are a proper subset of context-sensitive languages.

Key takeaways

Indexed language belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Indexed language to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Indexed language from memory before moving on to harder problems.

Reference excerpt

Indexed languages are a class of formal languages discovered by Alfred Aho; they are described by indexed grammars and can be recognized by nested stack automata. Indexed languages are a proper subset of context-sensitive languages. They qualify as an abstract family of languages (furthermore a full AFL) and hence satisfy many closure properties. However, they are not closed under intersection or complement. The class of indexed languages has practical importance in natural language processing as a computationally affordable generalization of context-free languages, since indexed grammars can describe many of the nonlocal constraints occurring in natural languages. Gerald Gazdar (1988) and Vijayashanker (1987) introduced a mildly context-sensitive language class now known as linear indexed grammars (LIG). Linear indexed grammars have additional restrictions relative to indexed grammars. LIGs are weakly equivalent (generate the same language class) as tree adjoining grammars.

Examples The following languages are indexed, but are not context-free:

{ a n b n c n d n | n ≥ 1 } {\displaystyle \{a^{n}b^{n}c^{n}d^{n}|n\geq 1\}}

{ a n b m c n d m | m , n ≥ 0 } {\displaystyle \{a^{n}b^{m}c^{n}d^{m}|m,n\geq 0\}} These two languages are also indexed, but are not even mildly context sensitive under Gazdar's characterization:

{ a 2 n | n ≥ 0 } {\displaystyle \{a^{2^{n}}|n\geq 0\}}

{ w w w | w ∈ { a , b } + } {\displaystyle \{www|w\in \{a,b\}^{+}\}} On the other hand, the following language is not indexed:

{ ( a b n ) n | n ≥ 0 } {\displaystyle \{(ab^{n})^{n}|n\geq 0\}}

Properties Hopcroft and Ullman tend to consider indexed languages as a "natural" class, since they are generated by several formalisms, such as:

Aho's indexed grammars Aho's one-way nested stack automata Fischer's macro grammars Greibach's automata with stacks of stacks Maibaum's algebraic characterization Hayashi generalized the pumping lemma to indexed grammars. Conversely, Gilman gives a "shrinking lemma" for indexed languages.

Worked examples

Example 1 — a first encounter with Indexed language

Start with the simplest possible case. Write down what Indexed language claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Indexed language before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Indexed language ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Indexed language

In research: Indexed language appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Indexed language in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Indexed language is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational linguistics, Formal languages, so understanding it makes those chapters shorter.
In everyday life: Look for Indexed language outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Indexed language in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Indexed language means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Indexed language out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Indexed language in simple terms?

Why does Indexed language matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Indexed language?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Indexed language.