computer science

Semi-membership

Semi-membership 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 Semi-membership rather than just read about it. In short: In mathematics and theoretical computer science, the semi-membership problem for a set is the problem of deciding which of two possible elements is logically more likely to belong to that set; alternatively, given two elements of which at least one is in the set, to distinguish the member from the non-member. The semi-membership problem may be significantly easier than the membership problem.

Key takeaways

Semi-membership 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 Semi-membership to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Semi-membership from memory before moving on to harder problems.

Reference excerpt

In mathematics and theoretical computer science, the semi-membership problem for a set is the problem of deciding which of two possible elements is logically more likely to belong to that set; alternatively, given two elements of which at least one is in the set, to distinguish the member from the non-member. The semi-membership problem may be significantly easier than the membership problem. For example, consider the set S(x) of finite-length binary strings representing the dyadic rationals less than some fixed real number x. The semi-membership problem for a pair of strings is solved by taking the string representing the smaller dyadic rational, since if exactly one of the strings is an element, it must be the smaller, irrespective of the value of x. However, the language S(x) may not even be a recursive language, since there are uncountably many such x, but only countably many recursive languages. A function f on ordered pairs (x,y) is a selector for a set S if f(x,y) is equal to either x or y and if f(x,y) is in S whenever at least one of x, y is in S. A set is semi-recursive if it has a recursive selector, and is P-selective or semi-feasible if it is semi-recursive with a polynomial time selector. Semi-feasible sets have small circuits; they are in the extended low hierarchy; and cannot be NP-complete unless P=NP.

References Derek Denny-Brown, "Semi-membership algorithms: some recent advances", Technical report, University of Rochester Dept. of Computer Science, 1994 Lane A. Hemaspaandra, Mitsunori Ogihara, "The complexity theory companion", Texts in theoretical computer science, EATCS series, Springer, 2002, ISBN 3-540-67419-5, page 294 Lane A. Hemaspaandra, Leen Torenvliet, "Theory of semi-feasible algorithms", Monographs in theoretical computer science, Springer, 2003, ISBN 3-540-42200-5, page 1 Ker-I Ko, "Applying techniques of discrete complexity theory to numerical computation" in Ronald V. Book (ed.), "Studies in complexity theory", Research notes in theoretical computer science, Pitman, 1986, ISBN 0-470-20293-9, p. 40 C. Jockusch jr (1968). "Semirecursive sets and positive reducibility" (PDF). Trans. Amer. Math. Soc. 137 (2): 420–436. doi:10.1090/S0002-9947-1968-0220595-7.

Worked examples

Example 1 — a first encounter with Semi-membership

Start with the simplest possible case. Write down what Semi-membership 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 Semi-membership 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 Semi-membership 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 Semi-membership

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

Read the reference excerpt below once, without taking notes.
Close the page and write down what Semi-membership 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 Semi-membership out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Semi-membership in simple terms?

Why does Semi-membership 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 Semi-membership?

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 Semi-membership.