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From: ok@goanna.cs.rmit.oz.au (Richard A. O'Keefe)
Newsgroups: comp.lang.prolog
Subject: Re: origin of "arity"
Message-ID: <6402@goanna.cs.rmit.oz.au>
Date: 20 Jun 91 11:06:32 GMT
References: <3405@shodha.enet.dec.com>
Organization: Comp Sci, RMIT, Melbourne, Australia
Lines: 44

In article <3405@shodha.enet.dec.com>, wahl@shodha.enet.dec.com (David Wahl) writes:
> The word "arity" came up during lunch today in our group and our
> resident mathematician asked whether the word was invented by
> logic programming people or whether it was borrowed from another
> field.  We looked it up in both a standard college English dictionary
> and in a scientific and technical English dictionary and couldn't 
> find the word "arity" in either of them.
> Does anyone know where the origin of the word "arity"?

I _wonder_ what field your resident mathematician works in, and I
suggest that you look around for another dictionary.
Neither ``arity'' nor ``functor'' was invented by logic programmers.
You can find ``functor'' in Robinson's book.

On p16 of Gallier's "Logic for Computer Science" we find
    A ranked alphabet is a set \Sigma together with a rank function
    r: \Sigma -> N.  Every symbol f \in \Sigma has a rank (or arity)
    r(f) indicating the fixed number of arguments of f.       ^^^^^
    Symbols of arity 0 are also called constants.

On p96 of Boyer & Moore's "A Computational Logic Handbook", we find
    Associated with every function symbol is a nonnegative integer
    called the arity of the symbol.  The arity indicates how many
    arguments must follow each application of the function symbol.

Buchberge & Loos, in a paper on Algebraic Simplification, write
    A signature (or arity-function) is a family of non-negative
    integers (i.e. a function a: Def(a) -> N_0.)  The index set
    of a (i.e. Def(a)) may be called the set of function symbols
    of a, in short F(a).  Furthermore, F(a,n) := {f\in F(a)|a(f)=n}
    (a set of n-ary function symbols).

The word comes from mathematical logic and ``higher algebra''.
It comes from talking about un-ary, bin-ary, tern-ary, ... n-ary functions.

The point is that (Edinburgh) Prolog borrowed its *entire* set of
data structures from mathematical logic, in which a _term_ is a
_variable_ or a _function symbol_ together with 0 or more _arguments_
each of which is a term, function symbols of arity 0 being identified
with constants.  Prolog borrowed much of its vocabulary from the same
source.

-- 
I agree with Jim Giles about many of the deficiencies of present UNIX.