FRP

When we view a function as a set of ordered pairs and the whole functional space as a result of a Cartesian Product of Domain and Codomain we have a fundamental problem - the function is not total, it is not an one-to-one relation, so it is not a function at all.

One cannot use a ruler before one gets an “object”.

The function of a -> (Time, a) is just a log. it tags an event with a time-tag. This also means that the event a is already observed (by those who tags it with a timestamp - who used the ruler) - it is already emerged from a source.

Again, Time -> a is just wrong.

If partial function from an abstract Time is what they want, that would mean getting Nothing most of the time, which is what a busy-waiting is.

Nature does not use a busy-waiting because it is enery-inefficient, grossly inefficient. It uses tresholds and triggers (which is what a nervous system is - a signal-propagation network).

Maybe monad comes to the mind, which propagates Nothing and distributes f over its Just x.

A Number Line is an imaginary mental construction (a generalized abstraction of the mind of an external observer) which is used to visualize (and imagine) the generalized abstract concept of a Number.

Originally it was just a primitive ruler - a stick with equally-spaced notches or marks, which is, it turns out, a common but decissive pattern of the mind. An external observer is required to use a ruler, and without such an observer the whole notion does not exits anywhere outside his head.

This is how we (intelligent observers) measure lengths, including lengths of time, which itself is an non-existent, abstract notion derived by an external observer. You will be unable to find any time anywhere outside your head.

Again, that is it - just equally-spaced marks superimposed onto observed phenomena by an external (relative to the phenomena) observer.

This concrete (or abstract) ruler is superimposed against (onto) something, and the number of notches are counted (summed up or added together).

In abstract mathematics the notion of counting is exactly this - superimpose an imaginary Number Line and associate each “whatever” with a Number.

We can think of it as a Set of ordered pairs (and a Cartesian Product) and, yes, length is the same process as count (how many)

count xs = fst . last $ zip [1 ..] xs
  where
    zip (n : ns) (x : xs) = (n, x) : zip ns xs
    zip _ _               = []

Time is modeled as discrete (ticks) or continuous (a Real line).

Continuous time is mathematically more convenient and derivatives and differential calculus arise from this notion.

No matter how one model it - as discrete or continuous - it does not make it more real.

The assumed implication that continuois time can vary is just a play of conditioned (by shared culture) mind - pure imagination, however sophisticated.

Once you have superimposed time you have a change over time, or \[\frac{dy}{dt}\]

We associate a continuous time-line with a discrete event “flow” (stream).

So we get a “log” (ledger) - such event occurred at such timestamp.

And that is it.

The better name for “Behavior” is “Signal”.

Semantically it is a relation over continuous (Real) time.

The relation is many-to-one which means that many timestamps result in the same a.

Obviously, a time-line, just as a number-line, have associative “addition” or “concatenation”.

Associativity of concatenation and distribution of operations (maps) over concatenation are Universal patterns on linear sequences.

RNA and DNA are such sequences and enzymes “distribute” over them.

The universal Algebraic notions (structures) are, yes, generalized abstractions of… guess whom?

So, all the familiar /abstract algebraic structures (with corresponding “laws”) are there.

Length is counting of “ticks”

length xs = sum [1 | _ <- xs]
  where
    sum = foldr (+) 0

and, by the way

count x = length . filter (== x)

Busy-waiting is energy-inefficient, unless it is “sleeping”.

Arrival of a necessary condition (in a locality). A required molecular structure within a reach (same locality). A threshold (temperature, energy, etc).

A time marker (tag) implies that a value (event) has already been observed by someone who superimposed a ruler and put a tag.

The “entering into a system” (crossing a cell-membrane) event.

A sequence of tagging (modifications)