Thomson’s infinite lamp as a mathematical monster

Imagine that we have a lamp – it is switched off at its initial state and this state can be changed by pressing a button. Having been an hour to play with this lamp, we switch the lamp on after 30 minutes. Then after waiting for 15 minutes, we switch it off – and then we switch it on again exactly 7.5 minutes later – and so on. I think that the end of the story is self-evident: after one hour (and neglecting that it is physically impossible) we pressed the button infinitely many times.
But what will be the result? Will the lamp light? Or will not?
It seems to be an unanswerable question – after all, we can regard the switch off state as an “odd” and the switch on as an "even" number (or vice versa). The source of this problem is that only a natural number is either odd or even – but infinite is not a number in a traditional way.
But there is another analogy and it can help. Nobody knows PI’s exact value since it is an irrational number. What is more, according to our actual knowledge, its digits are randomly distributed. But if we would be able to compute all of its digits, would the last digit be an even number?
Perhaps it seems to be an acceptable answer that there is no a last digit of PI, so it is neither odd nor even. But PI is nothing more than the ratio of the circumference of a circle to its diameter and although we do not know exactly the numerical value of this ratio, it is a certain, existing value. Computing more and more digits of PI, we’ll know it more and more accurately – and computing it to the infinity, we’ll know it exactly.
Ad analogiam: if we press the button of the lamp infinitely many times, then the lamp will be necessarily either switched on or switched off – although we cannot predict the lamp’s state.
At this point we can distinguish to different types in math: random and compressible strings. The previous one means that we cannot find a representation of the given string which is shorter than the original one. Heads and tails is a good example for it: you won't know the result without tossing the coin in reality.
Or onecould mention the cellular automatons (CAs). The state of their cells depend on the neighboring cells’ states and a CA changes in discrete steps. The result is that although the system is absolute deterministic (certain starting configurations always results the same next phases), cellular automation is an incompressible process. We cannot compute the next phase without executing the program itself.
Opposite to these above mentioned examples, a compressible string can be regarded to be “regular” in a sense that if we know the rule, then we can find the nth digit without computing others.
Our lamp represents a totally different solution. We can compute its every stage and its algorithm is ridiculously simple, so it is compressible - except for its endpoint. We cannot answer whether the lamp is switched on at its final stage – unless we de facto pressed that button for infinitely many times.
I wonder whether there are other, strange categories – for example, who could imagine a string which is compressible only at its endpoint? Perhaps other mathematical monsters lurking somewhere.

Compressibility, law and known facts to memorize

David Deutsch begins his book entitled The Fabric of Reality stating that the meaning of “to know everything is known” is equal “to understand” the world, but it doesn’t mean the knowledge of every isolated fact. We are far from a theory of everything which would be able to explain how our World works, but it “distinctly less fantastic than the idea that one person could memorize every known fact.” [p. 3]
It is not a surprise to found ourselves in a Universe which can be at least partially interpretable by laws –perhaps a primitive form of life could be able to evolve without a kind of predictability, but an intelligence able to interpret the World surely not. A kind of compressibility is the feature which allows to use shorthand to calculate the reality from certain pieces of information (i.e. a planetary orbits). Opposite to it, a string is random if we cannot present a representation of it which is shorter than the original sequence (Barrow: New Theories of Everything 2007, p. 11.).
Having studied a special musical instrument named lyre the Pythagoreans concluded that the reality ruled by numbers, and because of their intellectual heritage we regard self-evident that the description of the nature is compressible thanks to the numbers. It is not questionable that this mathematics based method proved to be really efficient to describe some aspects of reality.
On the other hand, there are incompressible things: i.e. it is trivial that there is no a shorter version for a poem which is equal to it, and it is a particular feature of the Universe that it can be divided into a compressible and an incompressible part.
Obviously, not “every known fact” mentioned by Deutsch is mathematically non-accessible, but our World’s another feature the proportion of the rules and the data we should memorize is unequal.
There are so few rules that some scientists hope to reduce all of them to a super formula. As it was mentioned earlier, the existence of intelligent life prescribes the existence of a very huge amount of data, since the evolution of thinking required billion years, and a long period of time means many events.
But on the one hand, why was it a so slow process? On the other hand: why do not exist many-many more laws even comparable in amount to the “every know facts?” Is it a requirement of a biofil Universe or only an accident?

And a concluding remark: we can draw up a categorization based on the relationship between "rules" (circa compressibility) and "data". Obviously the fifth subcategory is the most interesting where the efficiency of rules is determined either by a nonlinear function or changes randomly.

	efficiency	example
1.	perfect (theoretically)	compressible macro physical laws
2.	zero	poem, human society's evolution, etc.
3.	less and less efficient over time	meteorology
4.	more and more efficient	calculation by iterations
5.	different patterns determines the effectifiency in time than the third or fourth subcategory	?