Undercomputing

Hypercomputing is about the hypothetical extension of the capabilities of traditional Turing-machines. I.e. how to go beyond Turing-compatibility? How to calculate infinitely many steps within a finite amount of time (these are supertasks, see i.e. the Thomson lamp) and how to solve a mathematical problem which is unsolvable by Turing machines? Obviously, these are mainly thought experiments, but the main point is to exceed certain limits of Turing-computing.
But we can turn into another direction and we can examine the limits of physically existing computers. A Turing machine is an abstract construction originally with to aims: to model how a “human computer” can solve a problem (in the year of publication of Turing’s paper about Turing machines (1936) the computational processes were performed by humans with paper and pen); and to create a mathematical construction to describe it–this was adopted to electromechanical/digital computers around the end of WWII. In other worlds: it was based on an abstract conception about mathematics.
According to this approach, every Turing machine which are capable to solve a problem are equal, since they give the same answer using the same logic and it is indifferent if one of them is a super fast computer while the other one is a slow, mechanical model made of tinker toy.
But–opposite to the physical reality–mathematics does not contain time as parameter and it explain why one of the central problems of Turing machines is the “halting problem”. The question is whether the computer halts because of a given input sometime at all, but it is irrelevant whether we have to wait either a second or one trillion years.
Penrose makes a distinction between hardware and software (Emperor’s new Mind, p. 24.). The previous one is “the actual machinery involved in a computer” (circuits, processors, etc.) while the software “refers to the various programs which can be run on the machine”. So Turing machines are equal regarding their software–after all, they are based on the same logic. But they aren’t equal in the real world, since it is not the same to wait for a result a second or to a longer time than the earth would exist. It is indifferent from a practical point of view that we don’t know the answer since a Turing machine unable to solve the problem or because it takes too long time.
But making a distinction between the abstract mathematics of computing and the real, physically existing machines makes possible to introduce a neologism called undercomputing. Similarly in a certain way to the concept of the Landauer number, it is about the limits determined by the physical reality, and while hypercomputing is to exceed the traditional Turing machine, undercomputing is to take into consideration the physical limits of every physically existing computer.
Bennett and Landauer [The fundamental physical limits of computation. Scientific American 253(1), 1985] have studied some fundamental physical limits of information processing pointing out that, for example, PI doesn’t have an exact value in reality, since it cannot be calculable to the last decimal. So we can interpret computers as machines with performing capacities influenced by their hardware that are determined by the physical laws.

A new kind of infinite machines

Since David Hilbert’s thought experiment, it is popular to demonstrate the strangeness of infinities describing a hotel with infinite rooms where new and new tourists/tourist groups arrives (even in a countably infinite number). The trick is that although all the rooms are full, the management always can find free ones – after rearranging the reservations. I.e. if only one tourist wants to check in, then the person occupying room 1 is can be moved to room 2; and the occupier of room 2 to room 3 etc. (and the occupier of room n moves to room n+1). If a countably infinite amount of new guest arrives, then the person from room 1 moves to room 2; the person from room 2 to room 4 (from room n to room 2n). After all, there as many odd as even numbers and the new visitors can occupy the odd-numbered rooms that are free now. This method works even if countably infinitely many buses arrives with countably infinitely many passengers on each.
The infinite hotel is misleading in a certain way, since it suggests that these algorithms are the simplest solutions for pairing the rooms and visitors. But there is a simpler method: at the time of the arriving of a new group with even countably infinitely visitors, we can ask every occupier to leave their room – the result is infinitely many free room with infinitely many persons (including the newly arrived ones) without room. Then we ask everybody to go into a still free places – and that’s all: we paired infinitely many persons with infinitely many rooms.
Keeping in mind the lesson of the infinite hotel, we can introduce a new kind of infinite machines with a new typology.
From our point of view, there are two fundamental parameters to determine these machines: the number of steps of the process to reach infinity and the needed time.
It’s obvious that there are impossible machines. You cannot build a machine that solve a problem in zero time even it infinitely fast; and similarly impossible that version that takes only finite number of steps in an infinitely long period – but not because it halts at a certain point in the process (i.e. since it is prescribed that it has to stop after a certain number of steps or reaching a number), but because – as a reversed Thomson lamp – its algorithm prescribes it.
So the simplest infinite machine is a Turing machine with an infinite tape – it can take infinitely many steps over an infinitely long period (and every step can be paired with the moment of the step).
Opposite to it, a Tomson's lamp takes infinitely many steps within a finite period of time. The solution is that 1+1/2+1/4…=2 so if we can press the Thomson lamp’s button two times faster at the n+1st step than at the nth step, then we can finish the process within 2 unit of time (i.e. within two seconds, if it took 1 second to press the button for the first time).
But it is possible a third type of infinite machine. Obviously, the last time we press the Thompson lamp’s button we have to do it infinitely fast, and the pressing process is infinitely short. It means on the one hand, that we handle (at least mathematically) an infinitely small amount. On the other hand: Why should we vary the pressing time to reach our aim? It is possible theoretically to press the button infinitely fast even for the first time; and even an infinitely small time is enough to do it infinitely many times. So this infinite machine finish its process not in infinite time (as a Turing machine) and not in a finite time (as a Thomson lamp), but in an infinitely short time.

machine type	number of steps	time
Impossible zero	infinite	zero
Impossible finite	finite	infinite
Turing	infinite	infinite
Tomson lamp	infinite	finite
Third type	infinite	infinitely small