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Any examples of computers (old or modern) that do not utilize directories or directory paths? If so, what architecture was/is used instead? Just curious.

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PDP-7 Unix used a different form of directories (not the regular tree hierarchy):

The Evolution of the Unix Time-sharing System

Dennis M. Ritchie

[...]

In spite of its considerable similarity to the current file system, the PDP-7 file system was in one way remarkably different: there were no path names, and each file-name argument to the system was a simple name (without `/') taken relative to the current directory. Links, in the usual Unix sense, did exist. Together with an elaborate set of conventions, they were the principal means by which the lack of path names became acceptable.

The link call took the form

link(dir, file, newname)

where dir was a directory file in the current directory, file an existing entry in that directory, and newname the name of the link, which was added to the current directory. Because dir needed to be in the current directory, it is evident that today's prohibition against links to directories was not enforced; the PDP-7 Unix file system had the shape of a general directed graph. So that every user did not need to maintain a link to all directories of interest, there existed a directory called dd that contained entries for the directory of each user. Thus, to make a link to file x in directory ken, I might do

ln dd ken ken
ln ken x x
rm ken

This scheme rendered subdirectories sufficiently hard to use as to make them unused in practice. Another important barrier was that there was no way to create a directory while the system was running; all were made during recreation of the file system from paper tape, so that directories were in effect a nonrenewable resource.

The dd convention made the chdir command relatively convenient. It took multiple arguments, and switched the current directory to each named directory in turn. Thus

chdir dd ken

would move to directory ken. (Incidentally, chdir was spelled ch; why this was expanded when we went to the PDP-11 I don't remember.)

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