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From: Jacob Errington <nictytan@gmail.com>
Date: Sat, 16 Aug 2014 11:32:38 -0400
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Subject: [lojban] A recursively-defined predicate for building chains (was:
 Lojban for great-great-grandfather)
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Our motivation is the following: we want a predicate that takes two
arbitrary sumti of the same type, a binary predicate, and a number, to
construct a chain of applications of that predicate along intermediate
values to "connect" the two sumti. This is the nature of constructs such as
"great-great-grandfather" and "three days from now" and "the friend of a
friend of a friend of mine", which as we have seen through usage, are
difficult to concisely express. However, the underlying principle of all
the words is the same.

We propose a predicate to construct such chains, of arbitrary length,
defined as follows in Lojban:

.i lo ka broda cu ka ko'a ko'e fo'a fo'e ce'ai lo du'u fo'a li pa dubmau
kei goi dy zo'u ge ganai me'au dy gi ko'a me'au fo'e lo broda be ko'e bei
li mo'e fo'a vu'u pa bei fo'e gi ga me'au dy gi ko'a ko'e me'au fo'e

Note that the definition is primitive recursive in its third argument. The
predicate yields a non-terminating computation for values in its x3 that
are less than or equal to zero. From this predicate, we have the following
relation.

.i go ko'a brode lo brode be lo brode be ko'a gi ko'a ko'e broda li ci lo
ka [ce'u] [ce'u] brode

A tentative translation of the definition to English would be difficult; I
will simply list how the sumti places interact without giving a definition
as one would expect in a dictionary.

* x1, x2: arbitrary sumti of the same type
* x3: the number of applications separating these sumti in the chain
* x4: a binary predicate used to build the chain connecting the x1 and x2.

Of course, we can apply the usual technique of the flipping principle to
make concise and easily-interpretable tanru: ko'a ko'e brode broda li ci

Examples:
"the day three days from now" -> "lo bavlamdei broda be fi li ci"
"my great-great-grandfather" (a parent of a parent of a parent of a parent
of a parent of mine) -> "lo rirni broda be mi bei li mu"
"a friend of a friend of a friend of mine" -> "lo pendo broda be mi bei li
vo"

The idea of the food chain is naturally evoked by {lo si'o broda fo lo ka
citka}.

The definition uses the experimental cmavo {ce'ai} and {me'au}, but they
are used only for convenience here. I leave it as an exercise to the reader
to rewrite the definition to use {ce'u goi} and {ckini}, respectively, to
rewrite it if they so choose.

All that remains is to choose an adequate word for this predicate. I would
expect us to use a zi'evla, at least until we realize that this word is so
useful that we choose to make it a gismu.

.i mi'e la tsani mu'o

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<div dir=3D"ltr">Our motivation is the following: we want a predicate that =
takes two arbitrary sumti of the same type, a binary predicate, and a numbe=
r, to construct a chain of applications of that predicate along intermediat=
e values to &quot;connect&quot; the two sumti. This is the nature of constr=
ucts such as &quot;great-great-grandfather&quot; and &quot;three days from =
now&quot; and &quot;the friend of a friend of a friend of mine&quot;, which=
 as we have seen through usage, are difficult to concisely express. However=
, the underlying principle of all the words is the same.<br>

<br>We propose a predicate to construct such chains, of arbitrary length, d=
efined as follows in Lojban:<br><br>.i lo ka broda cu ka ko&#39;a ko&#39;e =
fo&#39;a fo&#39;e ce&#39;ai lo du&#39;u fo&#39;a li pa dubmau kei goi dy zo=
&#39;u ge ganai me&#39;au dy gi ko&#39;a me&#39;au fo&#39;e lo broda be ko&=
#39;e bei li mo&#39;e fo&#39;a vu&#39;u pa bei fo&#39;e gi ga me&#39;au dy =
gi ko&#39;a ko&#39;e me&#39;au fo&#39;e<br>

<br>Note that the definition is primitive recursive in its third argument. =
The predicate yields a non-terminating computation for values in its x3 tha=
t are less than or equal to zero. From this predicate, we have the followin=
g relation.<br>

<br>.i go ko&#39;a brode lo brode be lo brode be ko&#39;a gi ko&#39;a ko=
9;e broda li ci lo ka [ce&#39;u] [ce&#39;u] brode<br><br>A tentative transl=
ation of the definition to English would be difficult; I will simply list h=
ow the sumti places interact without giving a definition as one would expec=
t in a dictionary.<br>

<br>* x1, x2: arbitrary sumti of the same type<br>* x3: the number of appli=
cations separating these sumti in the chain<br>* x4: a binary predicate use=
d to build the chain connecting the x1 and x2.<br><br>Of course, we can app=
ly the usual technique of the flipping principle to make concise and easily=
-interpretable tanru: ko&#39;a ko&#39;e brode broda li ci<br>

<br>Examples: <br>&quot;the day three days from now&quot; -&gt; &quot;lo ba=
vlamdei broda be fi li ci&quot;<br>&quot;my great-great-grandfather&quot; (=
a parent of a parent of a parent of a parent of a parent of mine) -&gt; &qu=
ot;lo rirni broda be mi bei li mu&quot;<br>

&quot;a friend of a friend of a friend of mine&quot; -&gt; &quot;lo pendo b=
roda be mi bei li vo&quot;<br><br>The idea of the food chain is naturally e=
voked by {lo si&#39;o broda fo lo ka citka}.<br><br>The definition uses the=
 experimental cmavo {ce&#39;ai} and {me&#39;au}, but they are used only for=
 convenience here. I leave it as an exercise to the reader to rewrite the d=
efinition to use {ce&#39;u goi} and {ckini}, respectively, to rewrite it if=
 they so choose.<br>

<br>All that remains is to choose an adequate word for this predicate. I wo=
uld expect us to use a zi&#39;evla, at least until we realize that this wor=
d is so useful that we choose to make it a gismu.<br><br>.i mi&#39;e la tsa=
ni mu&#39;o<br>

<br><br></div>

<p></p>

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