yyGrams logoyyGrams > yyGrams logoworked > example

zang: (( ))=    

fu: ( )( )=( )

Puzzle Example 12ed1

<#yypuz 5><t,w,f,x 3><n:{f,x} {t,f} {t} {f} {x} { }><y:*>auth 12ed1</yy>

Put the outcome catalog into columns. The pool of names—t,w,f,x—has 24 = 16 combinations, and only those yielding yiN on evaluating the hidden yy-gram are given explicity. The rest yield Yang, as the catalog says—<y:*>—so we reconstruct them for listing in our Yang column.

—some unseen yy-gram—

yy col heads
—whole-pool catalog of outcomes—

{f,x}
{t,f}
{t}
{f}
{x}
{ }

{w}
{t,w,f,x}
{t,w,f}
{t,w,x}
{t,f,x}
{w,f,x}
{w,f}
{t,x}
{w,x}
{t,w}

Group the outcomes into pairs where you detect a deep or possibly bogus name. For example, the pair
{w,f,x}
{w,f}
shows that x being present or absent does not effect the outcome when the other names are present.

A similar pair is
{t,w,f,x}
{t,w,f}

Among the yiN outcomes are more pairs where we see no effect—that is, different outcome—from x, namely

{f}, {f,x} and {x}, { }

Let's make a fresh catalog listing with pairs differing only in x:

—some unseen yy-gram—

yy col heads
—valuation pairs exposing x

{f,x}, {f}
{x}, { }
---
{t,f}
{t}

{w,x}, {w}
{t,w,f,x}, {t,w,f}
{w,f,x}, {w,f}
---
{t,w,x}
{t,f,x}
{t,x}
{t,w}

Below the tick --- we fail to convict x bogus. In particular, the pair {t}, {t,x} shows x affecting the outcome.

Our solution yy-gram must therefore include x.

This interpretation on the catalog information suggests treating another name to discriminatory pairing. To choose a candidate, give attention below the tick.

There we see outcomes where f often fails to make a difference:

—some unseen yy-gram—

yy col heads
— pairs exposing f

{f,x}, {x}
{t,f}, {t}
{ }, {f}
---

{t,w,f,x}, {t,w,x}
{t,f,x}, {t,x}
{w,f,x}, {w,x}
{t,w,f}, {t,w}
{w,f}, {w}
---

The new cataloguing shows no case where f affects the outcome. This is what we understand by a bogus name. A yy-gram solution to this outcome catalog does not have to involve f.

Accordingly we make a fresh catalog listing of only those valuation there-lists that do not mention f:

—some unseen yy-gram—

yy col heads
—catalog ignoring f

{x}
{t}
{ }

{t,w,x}
{t,x}
{w,x}
{t,w}
{w}

The task ahead is to infer a yy-gram with the catalogued and now clarified i/o behavior.

We note that x alone nor t alone is insufficient for a yang outcome. This puts those two names deeper than the outer milieu in any solution yy-gram.

Another analysis on these two names is to ask, is presence of either sufficient to force a yin outcome?

Evidently not, from an inspection of the yang column. This in turn allows the conclusion that both are deeper in any solution yy-gram than depth 1, counting the outermost space as depth 0.

This brings to mind the AND form, conjunction, in logic interpretation: a AND b <—> ((a)(b)), where the names in conjunction are at depth 2.

Now we address the yang column. Let's group the outcomes, segregating the valuations according to the remaining name in the clarified pool, w:

—some unseen yy-gram—

yy col heads
—catalog segregating w

{x}
{t}
{ }

{t,w,x}
{w,x}
{t,w}
{w}
---
{t,x}

Manifestly, yang-presence of w is sufficient to force a ( ) outcome .

Independently of that, [t AND x] is likewise sufficient to force a ( ) outcome.

We are familiar enough with interpreting yy-grams for logic that we can now read the yang column as

"[ t AND x ] OR w""

—grouping moieties with square brackets [ ] so as not to presume the artificial boolean rules for binding.

Expressing these conclusions together in a yy-gram:

((t)(x))w

This solution might appear in tweet format as

<#yygram 5> <ans: ((t)(x))w > auth 12ed1</yy>

or as

<#yygram 5> <ans: w((x)(t)) > auth 12ed1</yy>

or some other equivalent differing only in order within or outside enclosers.

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©2012 David Zethmayr
update 2012.4.10