mathematics

Pattern calculus

Pattern calculus is a mathematics 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 Pattern calculus rather than just read about it. In short: Pattern calculus bases all computation on pattern matching of a very general kind. Like lambda calculus, it supports a uniform treatment of function evaluation.

Key takeaways

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

Reference excerpt

Pattern calculus bases all computation on pattern matching of a very general kind. Like lambda calculus, it supports a uniform treatment of function evaluation. Also, it allows functions to be passed as arguments and returned as results. In addition, pattern calculus supports uniform access to the internal structure of arguments, be they pairs or lists or trees. Also, it allows patterns to be passed as arguments and returned as results. Uniform access is illustrated by a pattern-matching function size that computes the size of an arbitrary data structure. In the notation of the programming language bondi, it is given by the recursive function

The second, or default case x -> 1 matches the pattern x against the argument and returns 1. This case is used only if the matching failed in the first case. The first, or special case matches against any compound, such as a non-empty list, or pair. Matching binds x to the left component and y to the right component. Then the body of the case adds the sizes of these components together. Similar techniques yield generic queries for searching and updating. Combining recursion and decomposition in this way yields path polymorphism. The ability to pass patterns as parameters (pattern polymorphism) is illustrated by defining a generic eliminator. Suppose given constructors Leaf for creating the leaves of a tree, and Count for converting numbers into counters. The corresponding eliminators are then

For example, elimLeaf (Leaf 3) evaluates to 3 as does elimCount (Count 3). These examples can be produced by applying the generic eliminator elim to the constructors in question. It is defined by

Now elim Leaf evaluates to | {y} Leaf y -> y which is equivalent to elimLeaf. Also elim Count is equivalent to elimCount. In general, the curly braces {} contain the bound variables of the pattern, so that x is free and y is bound in | {y} x y -> y.

External links

Worked examples

Example 1 — a first encounter with Pattern calculus

Start with the simplest possible case. Write down what Pattern calculus claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Pattern calculus 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 Pattern calculus 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 Pattern calculus

In research: Pattern calculus appears in mathematics 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 Pattern calculus 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: Pattern calculus is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lambda calculus, so understanding it makes those chapters shorter.
In everyday life: Look for Pattern calculus 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 Pattern calculus in 20 minutes

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

Frequently asked questions

What is Pattern calculus in simple terms?

Pattern calculus bases all computation on pattern matching of a very general kind. Like lambda calculus, it supports a uniform treatment of function evaluation.

Why does Pattern calculus matter?

Because it connects several mathematics 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 Pattern calculus?

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 Pattern calculus.