Explicit finitism

model of computation rather than to actually compute things. By adding the reals we accomplish two main goals: enable direct modeling of deterministic physical theories (Montague[3]), and make it possible to deploy Blum-Shub-Smale complexity theory to analyze the symbolic (as opposed to numeric) techniques available on the machine. In all other respects our computing model is a straightforward extrapolation of what is happening in any modern computer – here we confine ourselves to a few remarks about memory. To simplify matters, we assume that a cache of 2 bytes (4 gigabytes) is available on chip (currently unrealistic, but easy enough to simulate). We can think of this cache as the innermost (0th) layer of memory, composed of registers, whose contents are directly accessible (in a single cycle) to primitive operations such as addition. At the next (1st) layer, we assume that 2 bytes (16 exabytes) memory is directly addressable – this will be called the core. To perform (arithmetic or logical) operations on numbers stored in the core requires a few CPU cycles for bringing them to the 0th layer, and some care in programming to make sure still valuable parts of the cache are not overwritten. Finally there is an outer (2nd) layer of memory in the 2 byte range, called the disk. Since this requires the whole universe, we do not follow the usual assumption that fetching data from this layer can be done in a constant number of cycles. Rather, we assume that this is limited by the speed of light, so that if memory is arranged linearly, the time required for reading or writing the nth byte is proportional to n. If memory is arranged in concentric circles, the time required is proportional to n, if it is arranged spherically, to n, which is the best we can do. These non-random access characteristics call for a whole set of unusual memory management techniques. First, we wish to be able to seed far parts of the disk with colonies of computing agents, who will use certain parts (local to them, but not to us) as core and cache. Second, we need to make sure that different colonies, who may themselves be engaged in their own secondary or n-th generational colonization efforts, recognize different parts of the disk as being already in use, and don’t step on it. Third, we can not simply assume a generally shared system of coordinates, or a master plan that each colony will abide by, for the simple reason that a system the size of the disk can not be kept noise-free. Therefore, allocating a large segment will actually consume some overhead space to mark the segment as being in active use, and possibly some constant drain on time as well, for running a process that defends the segment from corruption by other processes. Finally, note that at this level of abstraction we do not need to consider parallelism, since all computers under the control of our civilization can be thought of as being part of the same large J-machine.