[Om] Multistep "equation" symbol?

[Lars Hellström]

Bruce Miller skrev:
> Lars Hellström wrote:
>> Is there an established OM symbol for "multistep equations" (see 
>> example below)? If not, would it make sense as part of some official 
>> content dictionary?
> 
> When I've asked such questions in the past,
> I generally got James' (essentially correct) response,
> that it is equivalent to a conjunction of relations.

But things can be different even if logically equivalent; there is 
certainly no shortage of OM symbols which can produce results 
equivalent to those some other symbol can produce.

I suspect more pretinent arguments against the n-ary relations would be 
that the semantics start to get non-obvious once one goes beyond the 
familiar relations. "Not equal" is a classical headache, but it is 
certainly not the only one.


> I shared your concern that such a multi-something
> is easily converted to the conjunction form, but
> less reliably converted back.  Perhaps <OMR> would be
> useful here?
> <OMA>
>   <OMS cd="relation1" name="eq"/>
>   <OMV name="a"/>
>   <OMwhatever id="rhs1"/>
> </OMA>
> 
> <OMA>
>   <OMS cd="relation1" name="eq"/>
>   <OMR xref="rhs1"/>
>   <OMwhatever id="rhs2"/>
> </OMA>
> ...

Not really. While it may change the size overhead introduced by a 
transformation from

<OMA><OMS name="multistep">
    x0 R1 x1 R2 x2 ... Rn xn
</OMA>

to

<OMA><OMS cd="logic1" name="and"/>
    <OMA> R1 x0 x1 </OMA>
    <OMA> R2 x1 x2 </OMA>
    ...
    <OMA> Rn xn-1 xn </OMA>
</OMA>

from being proportional to the size of the original expression (as most 
steps appear twice) to being proportional to the number of steps, I 
don't think it is of much help for detecting the structure, unless this 
was the /only/ thing references were used for.

> While the issue is mainly about a short-hand
> that you don't want to loose, 

No, the shorthand aspect is not the issue; had that been all there was 
then I could just as well have gotten by by using references (even 
though I generally dislike them). The issue is instead that a 
"multistep" is as much a proof as a collection of facts, and this is a 
piece of information that I'd really hate to loose.

> there are,
> I think, some subtle semantics sneaking around
> behind such a common notation.
> 
> As James points out, your case seems to
> be sort of a proof, or derivation. Each
> rhs follows from, can be derived from, is implied
> by, is the asymptotic expansion of, ...
> it's associated rhs.  The set of
> relation(-like) operators, and their sensible
> sequences is tricky, and in general there isn't
> a trivial transitivity.

Which is why I didn't require anything of that sort; direct claims are 
only made about neighbouring sides[*], drawing conclusions from them is 
outside the scope of the construct (even though its use certainly hints 
that there is a conclusion that can be drawn).

[*] Side issue: Is there an English term for the parts in this kind of 
multistep construction? "Side" seems awkward for anything but the two 
extremes, and "step" feels more like it is referring to a relation 
(possibly together with its two operands). Swedish has "led", which is 
beautifully simple.

> The same notation is used as, eg. a < b = c <= d << e,
> where there is (seemingly) less hidden semantic,
> just a shorthand.
> 
> I suspect there are cases where there other
> hidden semantics.
> 
> My main point being, if one _were_ to define
> such a symbol, it would seem that more than one
> "Multi<something>" would be called for.

No, this one suffices fine. One could certainly invent specialisations 
of it, which were guaranteed to always imply a relation between the 
extremes provided the construction as a whole fits some syntactic 
pattern, but that's a bit like asking for a language in which you may 
only express truths. The necessary theorems that a specific sequence of 
relations combine to some relation between the extremes can equally 
well be applied to the basic "multistep" as to a specialisation thereof.

Lars Hellström