The distinction between parallel and serial computers has also generated some confusion. A common claim is that the brain cannot be a “serial digital computer” because it is “parallel” (Churchland et al. 1990, p. 47; Churchland and Sejnowski 1992, p. 7). In evaluating this claim, we should keep in mind that digital computers can be parallel too, in several senses of the term. There are many distinctions to be made, and our understanding of computers can only improve if we make at least the main ones explicit.
There is the question of whether in a computing mechanism only one computationally relevant event—for instance, the transmission of a signal between components—can occur during any relevant time interval. In this sense, most complex computing mechanisms are parallel. For example, in most computing mechanisms, the existence of many communication lines between different components allows data to be transmitted in parallel. Even Turing Machines do at least two things at a time; namely, they act on their tape and update their internal state.
A separate question is whether a computing mechanism can perform only one or more than one computational operation during any relevant time interval. Analog computers are parallel in this sense. This is the sense that connectionist computationalists appeal to when they say that the brain is “massively parallel.” But again, most complex computing mechanisms, such as Boolean circuits, are parallel in this sense. Since most (digital) computers are made out of Boolean circuits and other computing components that are parallel in the same sense, they are parallel in this sense too. In this respect, then, there is no principled difference between ordinary computers and connectionist computing mechanisms.
A third question is whether a computing mechanism executes one or more than one instruction at a time. There have been attempts to design parallel processors, which perform many operations at once by employing many executive units (such as datapaths) in parallel. Another way to achieve the same goal is to connect many processors together within one supercomputer. There are now in operation supercomputers that include thousands of processors working in parallel. The difficulty in using these parallel computers consists of organizing computational tasks so that they can be modularized, i.e. divided into sub-problems that can be solved independently by different processors. Instructions must be organized so that executing one (set of) instruction(s) is not a prerequisite for executing other (sets of) instructions in parallel to it (them), and does not interfere with their execution. This is sometimes possible and sometimes not, depending on which part of which computational problem is being solved. Some parts of some problems can be solved in parallel, but others can’t, and some problems must be solved serially. Another difficulty is that in order to obtain the benefits of parallelism, typically the size of the hardware that must be employed in a computation grows (linearly) with the size of the input. This makes for prohibitively large (and expensive) hardware as soon as the problem instances to be solved by parallel computation become nontrivial in size. This is the notion of parallelism that is most relevant to computability theory. Strictly speaking, it applies only to computing mechanisms that execute instructions. Hence, it is irrelevant to discussing ordinary analog computers and connectionist computing mechanisms, which do not execute instructions.
In the connectionist literature, there is a persistent tendency to call neurons “processors” (e.g., Siegelmann 2003). Presumably, the intended analogy is between a brain and a parallel computer that contains many processors, so that every neuron corresponds to one processor. This is misleading, because there is no useful sense in which a neuron can be said to execute instructions in the way that a computer processor can. In terms of their functional role within a computing mechanism, neurons compare more to logic gates than to the processors of digital computers; in fact, modeling neurons as logic gates was the basis for the first formulation of a computational theory of the brain (McCulloch and Pitts 1943). Nevertheless, it may be worth noticing that if the comparison between neurons and processors is taken seriously, then in this sense—the most important for computability theory—neurons are serial processors, because they do only one functionally relevant thing (either they fire or they don’t) during any relevant time interval. Even if one looks at an entire connectionist network as a processor, one gets the same result; namely, that the network is a serial processor (it turns one input string into one output string). However, neither neurons nor ordinary connectionist computing mechanisms are really comparable to computer processors in terms of their organization and function—they certainly don’t execute instructions. So, to compare connectionist systems and computer processors in this respect is inappropriate.
Finally, there is the question of whether a processor starts executing an instruction only after the end of the execution of the preceding instruction, or whether different instructions are executed in an overlapping way. This is possible when the processor is organized so that the different activities that are necessary to execute instructions (e.g., fetching an instruction from memory, executing the corresponding operation, and writing the result in memory) can all be performed at the same time on different instructions by the same processor. This kind of parallelism in the execution of instructions diminishes the global computation time for a given program, and it applies only to processors that execute instructions. It is a common feature of contemporary computers, where it is called pipelining.26
The above parallel-serial distinctions apply clearly to computing mechanisms, but the parallel-serial distinction is obviously broader than a distinction between modes of computing. Many things other than computations can be done in parallel. For example, instead of digging ten holes by yourself, you can get ten people to dig ten holes at the same time. Whether a process is serial or parallel is a different question from whether it is digital or analog (in various senses of the term), computational or non-computational.