wahl@shodha.enet.dec.com (David Wahl) (06/20/91)
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"? Thanks, Dave Wahl =================================================================== Digital Equipment Corporation Database Systems Research (CXN1/2) 1175 Chapel Hills Drive Colorado Springs, CO 80920-2080 Tel 719-260-2758 Email: wahl@cookie.enet.dec.com %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The opinions expressed are my own, not Digital's. % % "I suppose you would have to be very well educated to get that % % kind of job." % % "Extremely well educated. Typing, everything." % % -- Donald Barthelme, "The Emerald" % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
mcovingt@athena.cs.uga.edu (Michael A. Covington) (06/20/91)
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"? > Self-evidently, it's a clipped form. "binarity" = "property of being binary (taking 2 arguments)" "ternarity" = "property of being ternary (taking 3 arguments)" "quaternarity" etc. So, "arity" = "property of taking whatever given number of arguments". It's odd because it is derived from two suffixes with no preceding root. To find out who first used it, you might try the New Oxford English Dictionary, which I don't have at hand. -- ------------------------------------------------------- Michael A. Covington | Artificial Intelligence Programs The University of Georgia | Athens, GA 30602 U.S.A. -------------------------------------------------------
ok@goanna.cs.rmit.oz.au (Richard A. O'Keefe) (06/20/91)
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.
weiss@theory.lcs.mit.edu (Paul G. Weiss) (06/21/91)
I also don't know who first coined it (I'm ashamed to admit), but I do remember back in college I took a combinatorics course and we were discussing polynomials. A polynomial was classified as an n-ary p-ic, where n was the number of variables and p was the degree, for example a binary cubic (2-ary 3-ic) has 2 variables and degree 3. The nouns arity and degree were used to refer to n and p, i.e. "an n-ary p-ic polynomial" = "a polynomial of arity n and degree p". This branch of mathematics and (I believe) this terminology goes back to the nineteenth century. A related question is when did the noun "degree" receive the above meaning? -Paul Weiss -Arity Corp.
ada612@csc.anu.edu.au (06/21/91)
It's my impression that `n-ary' is *much* commoner in math discourse that ternary, etc. (one isn't often interested in the special case), so would be the most likely source for `arity'. Howbout `n-adic' -> `adicity'? A model that the formation might be based on is `n ton' -> `tonnage', e.g. numeral+measure_noun -> measure_noun+nominalizer, forming an expression designating the amount present of whatever the measure noun measures. The hyphen in `n-ary' would make it considerably easier for people to treat `ar' as if it were in effect a noun, more or less synonymous with `place' in `n-place', placedness (??)). Philologically, I guess the next step would be finding the first attestation. The date to beat seems to be 1979 (Boyer and Moore), from ROK's posting. Avery Andrews (ada612@csc.anu.edu.au)
eggert@twinsun.com (Paul Eggert) (06/22/91)
ada612@csc.anu.edu.au writes: >Philologically, I guess the next step would be finding the first >attestation. The date to beat seems to be 1979 (Boyer and Moore), >from ROK's posting. A nit: ROK cited the 1988 ``A Computational Logic Handbook'', not the 1979 ``A Computational Logic''.
citrin@csn.org (Wayne Citrin) (06/22/91)
In article <1991Jun21.173147.27862@twinsun.com> eggert@twinsun.com (Paul Eggert) writes: >ada612@csc.anu.edu.au writes: > >>Philologically, I guess the next step would be finding the first >>attestation. The date to beat seems to be 1979 (Boyer and Moore), >>from ROK's posting. > >A nit: ROK cited the 1988 ``A Computational Logic Handbook'', >not the 1979 ``A Computational Logic''. I find a use of the term "arity" in Stanat and McAllister, "Discrete Mathematics in Computer Science" (Prentice-Hall, 1977). Wayne -- Wayne Citrin citrin@soglio.colorado.edu citrin@boulder.colorado.edu
pereira@alice.att.com (Fernando Pereira) (06/23/91)
In article <6402@goanna.cs.rmit.oz.au> ok@goanna.cs.rmit.oz.au (Richard A. O'Keefe) writes: >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. >Neither ``arity'' nor ``functor'' was invented by logic programmers. >You can find ``functor'' in Robinson's book. > > [various references] Richard's references are relatively recent. I first encountered the term in books on universal algebra, eg. P. M. Cohn's ``Universal Algebra'' (1965). Mac Lane's ``Categories for the Working Mathematician'' (1971) uses it too. This in just a quick survey of my home library. Fernando Pereira AT&T Bell Laboratories, Murray Hill pereira@research.att.com
rim@csadfa.cs.adfa.oz.au (Bob McKay) (06/24/91)
From article <1991Jun21.200810.1@csc.anu.edu.au>, by ada612@csc.anu.edu.au: > Philologically, I guess the next step would be finding the first > attestation [ of arity]. The date to beat seems to be 1979 (Boyer and Moore), > from ROK's posting. Mac Lane, S 'Categories for the Working Mathematician',Springer,NY,1971,P120: "....called the arity of omega." My own impression is that it's a much earlier term in universal algebra/category theory. Mac Lane is just the oldest text I have that happens to record it in the index. Bob McKay Phone: +61 6 268 8169 fax: +61 6 268 8581 Dept. Computer Science ACSNET,CSNET: rim@csadfa.cs.adfa.oz Aust. Defence Force Academy UUCP: ...!uunet!munnari!csadfa.cs.adfa.oz!rim Canberra ACT 2600 AUSTRALIA ARPA: rim%csadfa.cs.adfa.oz@uunet.uu.net
sjmz@otter.hpl.hp.com (Stefek Zaba) (06/26/91)
pereira@alice.att.com (Fernando Pereira) writes: >In article <6402@goanna.cs.rmit.oz.au> ok@goanna.cs.rmit.oz.au (Richard A. O'Keefe) writes: >>Neither ``arity'' nor ``functor'' was invented by logic programmers. >>You can find ``functor'' in Robinson's book. >> >> [various references] > >Richard's references are relatively recent. I first encountered the term >in books on universal algebra, eg. P. M. Cohn's ``Universal Algebra'' (1965). >Mac Lane's ``Categories for the Working Mathematician'' (1971) uses it too. >This in just a quick survey of my home library. > >Fernando Pereira >AT&T Bell Laboratories, Murray Hill >pereira@research.att.com >---------- Having been perhipherally peripherally involved with the compilation of the New OED, I asked a spy (Tim Bray at Waterloo) and got the following reply: |Date: Tue, 25 Jun 91 07:45:59 -0400 |From: Tim Bray <tbray@watsol.waterloo.edu> |Message-Id: <9106251145.AA12407@watsol.waterloo.edu> |To: sjmz@hplb.hpl.hp.com |Subject: Re: NOED has "arity"? | |I checked, and yes, the OED2 has 'arity', but the earliest citation |is from 'Fundamenta Mathematicae' in 1968, so Pereira is ahead there; |he should drop a note to Oxford (or to me, I'm in touch with the right |people). | |Cheers, Tim Bray So, netters, Fernando Pereira's 1965 reference is already ahead of the New OED; any earlier citations out there? (Now that the source is held in electronic SGML-tagged form and published on CD-ROM as well as paper, the potential exists for the update cycle to be substantially shorter than the previous 50 years :-). A reference including the publisher's name and location is optimal. Cheers, Stefek