Picture language

Key takeaways

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

Reference excerpt

In formal language theory, a picture language is a set of pictures, where a picture is a 2D array of characters over some alphabet. For example, the language L = { a n , n + 1 ∣ n > 0 } {\displaystyle L=\left\{a^{n,n+1}\mid n>0\right\}} defines the language of rectangles composed of the character a {\displaystyle a} . This language L {\displaystyle L} contains pictures such as:

( a a ) , ( a a a a a a ) , ( a a a a a a a a a a a a ) ∈ L {\displaystyle {\begin{pmatrix}a\\a\end{pmatrix}},{\begin{pmatrix}a&a\\a&a\\a&a\end{pmatrix}},{\begin{pmatrix}a&a&a\\a&a&a\\a&a&a\\a&a&a\end{pmatrix}}\in L}

The study of picture languages was initially motivated by the problems of pattern recognition and image processing, but two-dimensional patterns also appear in the study of cellular automata and other parallel computing models. Some formal systems have been created to define picture languages, such as array grammars and tiling systems.

References D. Giammaresi, A. Restivo. Two-dimensional languages. In A. Salomaa, G. Rozenberg (Eds.), Handbook of Formal Languages, Vol. 3, Beyond Words, Springer, Berlin, 1997, pp. 215–267.

Worked examples

Example 1 — a first encounter with Picture language

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

In research: Picture language appears in 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 Picture 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: Picture language is common in secondary-school and first-year university syllabi. It links to neighbouring topics Formal languages, Formal methods stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Picture 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.

How to study Picture language in 20 minutes

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

Frequently asked questions

What is Picture language in simple terms?

In formal language theory, a picture language is a set of pictures, where a picture is a 2D array of characters over some alphabet. For example, the language L = { a n , n + 1 ∣ n > 0 } {\displaystyle L=\left\{a^{n,n+1}\mid n>0\right\}} defines the language of rectangles composed of the character a…

Why does Picture language matter?

Because it connects several 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 Picture 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 Picture language.