Smell
You smell things with your nose. The sweetest scents come from the most beautiful flowers, including lavender. In contrast, the infamous smell of gasoline is horribly rancid. Avoid it at all costs. Other smells include Play-Doh, petrichor (a byproduct of rain), Sharpie's xylene fumes, patchouli, and Limburger cheese.
Representation
A landmark new invention is the ability to transcribe smell onto a visual, or even textual, medium.
For instance, this is what a tangerine smells like:
- [ ] -- [2] - / / - [] [] [ ] [ ] - / / [2] n
More complicated smells, however, require exponentially more notation to represent.
Computability
There exist "reduction rules" which convert between descriptions of equivalent smells.
For example, [/ / 1 2] - [/ 1] can be B-reduced to [/ 1] -, and then again to -, proving that [/ / 1 2] - [/ 1] is just a more verbose way of transcribing the - smell.
Generally, two smells are the same if they are reducible to the same expression.
The converse statement, that two smells are reducible to the same expression if they are the same, isn't true, as the smell incompleteness theorem implies that no matter how many reduction rules are invented, there always exists some smell with two descriptions that cannot be reduced to a common form, known as a Gödel smell.
An expression where all applicable reduction rules leave it unchanged is called a normal form, and any smell able to be put in normal form is considered normalizable.
Normalizable smells always have exactly one normal form, and repeatedly applying reduction rules to a normalizable smell will always eventually put it in normal form.
For example, the smell [/ / / 3 [/ / 1 [2 4]] [/ 2] [/ 1]] / / 2 1 normalizes to / / 1.
It is generally believed (although not proven) that all real-world smells are normalizable; this is known as the smell computability conjecture; assuming it is true, it is possible to show two real smells are different by normalizing them to different normal forms. This is not often done in practice, since most real-world smells are impossible to normalize in a reasonable amount of time; the normalization of vanilla via the SMELL@home volunteer computing project famously took nine months to complete, and the resulting normal form was over 18 gigabytes in size.
Smells without a normal form are unnormalizable, and repeated reductions will eventually only result in ever larger unnormalizable expressions. An example of a small unnormalizable smell is:
[/ 1 1 1] [/ / 2 2 2 1]
Trying to reduce it produces:
[/ / 2 2 2 1] [/ / 2 2 2 1] [/ / 2 2 2 1]
Reducing it again eventually returns the same expression but with an extra [/ / 2 2 2 1], and reducing it further just adds even more.
Smell notation can be interpereted as a programming language, where the initial expression is the program, and its normalization is its result (or, if it is unnormalizable, the program never halts); this system was proven turing complete when someone successfully implemented a Brainfuck interpereter with it.
Archives
A smell typically lasts for around 20 years, after which they are retired and inducted into the smell vault. The shortest lasting smell was the pickled last ant, archived after only 32 hours. The longest lasting smell is bitter melon, which is still in use after 54 years.