I suppose = the idea is to have a function which is always reflexive, i.e., one with on= e fewer argument by identifying two originally separate places: da srbroda&= nbsp; =3D da{ce'u de broda de} (using {ce'u}sensibly as a lambda operator, = not as a variable). Why we need this seems less clear.

Fr= om: MorphemeAddict <lytlesw@gmail.com>
To: lojban@googlegroups.com
Sent: Saturday, February 23, 2013= 5:59 PM
Subject: Re: [lojban] Com= positional Self-Reflexive Lujvo

Why have a self-reflexive lujvo at all? Can't the= various sumti be made equal to (or refer to the same thing as) any other s= umti in a non-reflexive lujvo? How is self-reflexive different from reflexi= ve?

stevo

On Sat, Feb 23, 2= 013 at 3:32 PM, Jacob Errington <nictytan@gmail.com> wrote:

These are impossible to make, due to the lack of a selbri meaning "x1 is in= all the ce'u-places of function x2". So I decided today, in IRC, to make t= hat selbri. This is the documentation of my quest.

There are two types of reflexive selbri right now. Lujvo of the = first kind are created compositionally whereas lujvo of the second kind are= created non-compositionally. Non-self-reflexive selbri are produced with {= simxu}, e.g. {.i mi joi do simxu lo ka ce'u cinba ce'u} "You and I kiss eac= h other," which in a lujvo looks like {mi joi do cinbysi'u}, whereas the se= lf-reflexive counterpart is {mi cinba mi/vo'a} which in a lujvo looks like = this {mi sezycinba}. The full Lojban definition of {sezycinba} is {.i lo ka= sezycinba cu ka ko'a ce'ai ko'a cinba lo sevzi be ko'a} where {lo sevzi be= ko'a} is taken to reduce into plain {ko'a}. This seems pretty hackish to m= e and is naljvajvo due to the invocation of {lo sevzi}.

Making up a selbri to mean "x1 does x2 to itself" isn't= as easy as it seems. Well, I'm not telling the whole truth when I say that= . It's easy to make up a selbri meaning "x1 does x2 to itself" as long as t= he x2 is a function with fixed arity. This is probably a high percentage of= the cases: {mi mi tavla} -> {.i mi broda lo ka ce'u tavla ce'u}, where = broda is that selbri. However, talking about yourself to yourself can't be = expressed with that same selbri, due to its restriction on the arity of the= function. The arity restriction arises from the naive lojban definition:

let ko'e =3D lo ka ce'u broda ce'u
#1 {.i ka = ko'a ko'a me'au ko'e} =3D {.i ka ko'a ko'a me'au lo ka ce'u broda ce'u} =3D= {.i ka ko'a broda ko'a}

Very straightforward. Make new selbri for every new {ko'a} you throw i= n. Very poorly extensible system.

This brought me = to the very important side-quest of figuring out currying in Lojban. {be} a= s we all know basically curries: {lo broda be ko'a} =3D {zo'e noi ke'a brod= a ko'a} =3D {zo'e noi ke'a ckaji lo ka ce'u broda ko'a}. Also as demonstrat= ed right there, we can use {ckaji} (and therefore {me'au}) to simulate the = effect of {be}. This led me to produce the following selbri which produces = a function with the x1 is filled by a definite parameter. I've called it ka= mni'oi for now, based on -kam- from {ka} and -ni'oi- from -ni'o- from {cnin= o}.

#2 {.i lo ka kamni'oi cu ka ko'a ko'e ko'i ce'ai ko'a k= a ko'e me'au ko'i} "x1 is the function derived from filling the x1 of x3 wi= th x2."
e.g. {lo ka mi citka ce'u cu kamni'oi mi lo ka ce'u citka= ce'u}

e.g. {lo du'u mi citka lo plise cu kamni'oi lo plise lo ka mi citka ce= 'u}

Then I needed a selbri to get the arity of a f= unction. This is possible by converting bridi3 into a set and they using zi= lkancu. This selbri could arguably be a lujvo, but I've decided to call it = ka'ance'u, from {zilkancu} and {ce'u}:

#3 {.i lo ka ka'ance'u cu ka ko'a ko'e ce'ai ko'a se zi= lkancu lu'i lo te bridi be ko'e} "x1 (li) is the arity of function x2."
e.g. {.i li mu ka'ance'u me'ei klama}

e.g. {.i li no ka'ance'u lo du'u mi citka lo plise}

Finally, these are all the tools required to build the definition of= the self-reflexive selbri for arbitrary-arity functions. Call it {sevzike}= for now.

#4 {.i lo ka sevzike cu ka ko'a ko'e ce'ai ge ganai lo = ka'ance'u be ko'e cu zmadu li no (fi dubu) gi sevzike lo kamni'oi be ko'e b= ei ko'a gi ganai lo ka'ance'u be ko'e cu du li no (fi dubu) gi me'au ko'e}<= /div>
e.g. {.i mi sevzike lo ka ce'u lumci ce'u} =3D {.i mi mi lumci}
<= div>e.g. {.i mi sevzike lo ka ce'u ce'u ce'u tavia} =3D {.i mi mi mi tavla}=
e.g. {.i mi sevzike lo ka ce'u ce'u ce'u tavla da poi ce'u finti= ke'a ca lo nu ce'u citno mutce} =3D {.i mi mi mi tavla da poi mi finti ke'= a ca lo nu mi citno mutce}

An important identity of this function is that when app= lied to unary functions, it becomes equivalent to plain {ckaji} (or {me'au}= ).
{mi sevzike lo ka ce'u citka lo plise} =3D {mi ckaji lo ka ce'= u citka lo plise} =3D {.i mi citka lo plise}

The function has an interesting quirk due to its defini= tion, namely that if a du'u is supplied in the x2, the x1 becomes completel= y irrelevant to the relationship.
{sevzike lo du'u broda} has the= same truth value regardless of the sumti supplied in the x1.

Now for an explanation of the definition. (The definiti= on is recursive, which I think is a first for Lojban definitions. Still, th= is is a proof of concept of the sheer power that can be achieved with Lojba= n definitions.) The function checks the arity of the x2. If it is gre= ater than zero, it curries the x1 into the x2 to yield an intermediate func= tion which it then passes back to itself. If the arity is zero, which will = occur when all the ce'u-places have had x1s curried into them, the function= simply evaluates the x2.

With this selbri, it is now possible to create composit= ional self-reflexive lujvo:
{.i mi cinba zei sevzike} =3D {.i mi = sevzike lo ka ce'u ce'u cinba} =3D {.i mi mi cinba}.

If sevzike were a gismu and had a CVV rafsi, it would be possibl= e to neatly create shorter self-reflexive luvjo.

N= ow that I understand how recursion can be used in lujvo and how to create t= hese arity conditions, it may be possible to define n-ary simxu, which we h= ad decided some time ago was not possible.

.i mi'e la tsani mu'o
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