On the philosophical basis of computational theories

In this short note we exemplify the problems that may arise when presenting computational theories based on theories of natural phenomena that lack computational complexity components. These problems may arise when the latter theories refers to environments of unbounded size. In contrast, the standard theory of computation (e.g., based on the model of Turing machines) refer implicitly to theories about bounded portions of the environment (e.g., the mechanics or electricity of possible implementations of a Turing machine). We start by recalling the postulates that underly all reasonable computational models, as presented in standard computational classes and texts. We then focus on exemplifying what happens when these postulates are ignored. The metaphysical foundations of all reasonable computational models A reasonable model of computation consists of large environments that are transformed according to simple and/or local rules (having small description). The vague dichotomy between “large” and “small” is typically modeled by the formal dichotomy between infinite and finite. The environments are bit strings of arbitrary, a priori unbounded, length, where bits represent binary attributes. In contrast, the rules-of-transformation are local and describe how an a priori bounded part of the environment changes, or more precisely how the value of each bit in the environment is updated as a function of a fixed (bounded) number of bits. The computation consists of the sequential (or parallel) applications of such rules. Indeed, the models of Turing machine and Boolean circuits fit the foregoing paradigm. In contrast, applying arbitrary linear transformation to the entire environment does not fit We comment, in passing, that the notion of a binary attribute is one of the most basic philosophical notions (cf., from the most ancient philosophies to Structuralism). We refer to Boolean circuits of bounded fan-in. Boolean circuits of unbounded fan-in are defined in order to facilitate the introduction of a complexity measure that reflects alternation of gate types in the former circuits.