Learning and memory, for example, can effectively occur in any system that has structures that form in response to input, and that can persist for a long time and affect the behavior of the system. … Indeed, as soon as one thinks of a system as performing computations one can immediately view features of those computations as being like abstract representations of input to the system.

So is it possible to set up forms of hashing that will in fact keep similar pieces of data together? … In actual brains it is fairly clear that input received by all the various sensory systems is first processed by assemblies of nerve cells that in effect extract certain specific features. And it seems likely that especially in lower organisms it is often representations formed quite directly from such features that are what is stored in memory.

But other criteria can equally well be used—say the head reaching a particular position (see page 759 ), or a certain pattern of colors being formed on the tape. … User interface and operating system programs are not normally intended to halt in an explicit sense, although without external input they often reach states that do not change. Mathematica works by taking its input and repeatedly applying transformation rules—a process which normally reaches a fixed point that is returned as the answer, but with definitions like x = x + 1 ( x having no value) formally does not.

In the first few layers of the visual cortex about half the cells respond to elongated versions of similar stimuli, while others seem sensitive to various forms of change or motion. In the fovea at the center of the retina, a single center-surround cell seems to get input from just a few nearby photoreceptors. In successive layers of the visual cortex cells seem to get input from progressively larger regions.

Another common approach to data compression is based on forming blocks of fixed length, and then representing whatever distinct blocks occur by specific codewords. … In each case the input is taken to be broken into blocks of length 3.

Higher Forms of Perception and Analysis…For if one gives a piece of data as the input to the program, then the output one gets—whatever it may be—can be viewed as corresponding to some kind of description of the data. … And potentially therefore our lack of higher forms of perception and analysis might in the end have nothing to do with any difficulty in implementing such forms, but instead may just be a reflection of the fact that we only have enough context to make descriptions of data useful when these descriptions are fairly close to the ones we get from our own built-in human methods of perception.

Yet often it seemed inevitable just from the syntactic structure of statements (say as well-formed formulas) that each of them must at some level be either true or false. … In computational systems, showing that it is unprovable that a given Turing machine halts with given input immediately implies that in fact it must not halt. But showing that it is unprovable whether a Turing machine halts with every input (a Π 2 statement in the notation of page 1139 ) does not immediately imply anything about whether this is in fact true or false.

For even in the very best case any block of cells in the input can never be compressed to less than one cell in the output. … After this comes a specification of the length of sequence represented by this section of output, with the number given in the form used for run-length encoding above.

Boolean formulas A Boolean function of n variables can always be specified by an explicit table giving values for all 2 n possible inputs. … Those on pages 616 and 618 use so-called disjunctive normal form (DNF) And[…] ∨ And[…] ∨ … , which is common in practice in programmable logic arrays (PLAs). … Conjunctive normal form (CNF) Or[…] ∧ Or[…] ∧ … is the rough analog of applying Factor .

But while Leibniz considered the possibility of checking his descriptions by machine, he apparently did not imagine setting up the analog of a computation in which something is explicitly generated from input that has been given. … But the notion of abstract functions in mathematics reached its modern form only near the end of the 1800s. At the beginning of the 1800s practical devices such as the player pianos and the Jacquard loom were invented that could in effect be fed different inputs using analogs of punched cards.