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Low and high hierarchies

Low and high hierarchies 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 Low and high hierarchies rather than just read about it. In short: In the computational complexity theory, the low hierarchy and high hierarchy of complexity levels were introduced in 1983 by Uwe Schöning to describe the internal structure of the complexity class NP. The low hierarchy starts from complexity class P and grows "upwards", while the high hierarchy starts from class NP and grows "downwards".

Key takeaways

  • Low and high hierarchies 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 Low and high hierarchies to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Low and high hierarchies from memory before moving on to harder problems.

Reference excerpt

In the computational complexity theory, the low hierarchy and high hierarchy of complexity levels were introduced in 1983 by Uwe Schöning to describe the internal structure of the complexity class NP. The low hierarchy starts from complexity class P and grows "upwards", while the high hierarchy starts from class NP and grows "downwards". Later these hierarchies were extended to sets outside NP. The framework of high/low hierarchies makes sense only under the assumption that P is not NP. On the other hand, if the low hierarchy consists of at least two levels, then P is not NP. It is not known whether these hierarchies cover all NP.

References

Worked examples

Example 1 — a first encounter with Low and high hierarchies

Start with the simplest possible case. Write down what Low and high hierarchies 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 Low and high hierarchies 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 Low and high hierarchies 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 Low and high hierarchies

In research
Low and high hierarchies 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 Low and high hierarchies 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
Low and high hierarchies is common in secondary-school and first-year university syllabi. It links to neighbouring topics Structural complexity theory, Theoretical computer science stubs, so understanding it makes those chapters shorter.
In everyday life
Look for Low and high hierarchies 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 Low and high hierarchies in 20 minutes

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

Frequently asked questions

What is Low and high hierarchies in simple terms?

In the computational complexity theory, the low hierarchy and high hierarchy of complexity levels were introduced in 1983 by Uwe Schöning to describe the internal structure of the complexity class NP. The low hierarchy starts from complexity class P and grows "upwards", while the high hierarchy sta…

Why does Low and high hierarchies 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 Low and high hierarchies?

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 Low and high hierarchies.

Tags

  • Structural complexity theory
  • Theoretical computer science stubs

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