Information distance

Key takeaways

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

Reference excerpt

Information distance is the distance between two finite objects (represented as computer files) expressed as the number of bits in the shortest program which transforms one object into the other one or vice versa on a universal computer. This is an extension of Kolmogorov complexity. The Kolmogorov complexity of a single finite object is the information in that object; the information distance between a pair of finite objects is the minimum information required to go from one object to the other or vice versa. Information distance was first defined and investigated in based on thermodynamic principles, see also. Subsequently, it achieved final form in. It is applied in the normalized compression distance and the normalized Google distance.

Properties Formally the information distance I D ( x , y ) {\displaystyle ID(x,y)} between x {\displaystyle x} and y {\displaystyle y} is defined by

I D ( x , y ) = min { | p | : p ( x ) = y & p ( y ) = x } , {\displaystyle ID(x,y)=\min\{|p|:p(x)=y\;\&\;p(y)=x\},}

with p {\displaystyle p} a finite binary program for the fixed universal computer with as inputs finite binary strings x , y {\displaystyle x,y} . In it is proven that

I D ( x , y ) = E ( x , y ) + O ( log ⋅ max { K ( x ∣ y ) , K ( y ∣ x ) } ) {\displaystyle ID(x,y)=E(x,y)+O(\log \cdot \max\{K(x\mid y),K(y\mid x)\})} with

E ( x , y ) = max { K ( x ∣ y ) , K ( y ∣ x ) } , {\displaystyle E(x,y)=\max\{K(x\mid y),K(y\mid x)\},}

where K ( ⋅ ∣ ⋅ ) {\displaystyle K(\cdot \mid \cdot )} is the Kolmogorov complexity defined by of the prefix type. This E ( x , y ) {\displaystyle E(x,y)} is the important quantity.

Universality Let Δ {\displaystyle \Delta } be the class of upper semicomputable distances D ( x , y ) {\displaystyle D(x,y)} that satisfy the density condition

∑ x : x ≠ y 2 − D ( x , y ) ≤ 1 , ∑ y : y ≠ x 2 − D ( x , y ) ≤ 1 , {\displaystyle \sum _{x:x\neq y}2^{-D(x,y)}\leq 1,\;\sum _{y:y\neq x}2^{-D(x,y)}\leq 1,}

This excludes irrelevant distances such as D ( x , y ) = 1 2 {\displaystyle D(x,y)={\frac {1}{2}}} for x ≠ y {\displaystyle x\neq y} ; it takes care that if the distance growth then the number of objects within that distance of a given object grows. If D ∈ Δ {\displaystyle D\in \Delta } then E ( x , y ) ≤ D ( x , y ) {\displaystyle E(x,y)\leq D(x,y)} up to a constant additive term. The probabilistic expressions of the distance is the first cohomological class in information symmetric cohomology, which may be conceived as a universality property.

Metricity The distance E ( x , y ) {\displaystyle E(x,y)} is a metric up to an additive O ( log . max { K ( x ∣ y ) , K ( y ∣ x ) } ) {\displaystyle O(\log .\max\{K(x\mid y),K(y\mid x)\})}

term in the metric (in)equalities. The probabilistic version of the metric is indeed unique has shown by Han in 1981.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Information distance

Start with the simplest possible case. Write down what Information distance 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 Information distance 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 Information distance 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 Information distance

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

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

Frequently asked questions

What is Information distance in simple terms?

Information distance is the distance between two finite objects (represented as computer files) expressed as the number of bits in the shortest program which transforms one object into the other one or vice versa on a universal computer. This is an extension of Kolmogorov complexity.

Why does Information distance 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 Information distance?

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 Information distance.