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Testing a Guess: an Opera in Two Acts

a.1 When we have (somehow) formed a guess about a yy-gram that might fit the puzzle, we test its input-output behavior against the behavior represented in the outcome catalog.

Yy-gram behavior

a.2 Behavior of a yy-gram means input-output behavior. "I/o behavior" is shorthand for the question, "What are the value outcomes after I send every there-list through it for a valuate-evaluate cycle?"

a.3 Valuating means assigning a value to each name in the yy-gram. Values will be the simple yy-grams yiN and Yang.

On a valuation, if a name appears more than once in the yy-gram, it is to be given the same value wherever it appears.

a.3.1 The rule just mentioned, about "same value at each appearance," is known as Law Zero, or "the Zero-th Law of Algebra."

a.4 The values we assign in a valuation come from just one there-list at a time. In a puzzle we have a complete catalog of there-lists, in two parts—one for those that provoke the value outcome Yang, and the other, those that provoke the value outcome yiN.

That two-part listing spells out the i/o behavior of the mystery yy-gram. If our guess is correct, the value outcomes we develop in the Twopenny Opera will all match up.

a.5 There are shortcuts to evaluating a valuated yy-gram. Shortcuts will occur to you as we go. A list of shortcuts awaits your pleasure. For now, though, we plod along without shortcuts.

a.6 We replay this two-act opera for every there-list in the puzzle.

Act One: Valuation

Scene 1: Discard

b.1 Here is a yy-gram guess to test: (k).

b.2 Perhaps (k) is your first guess at a solution to a puzzle or a game. To see whether it is a solution indeed, you make the substitutions for which you have value outcomes in hand. Substition, followed by zang-fu disappearance if necessary to complete an evaluation, tells you whether your guess matches the hidden yy-gram known to the person who sent you the puzzle.

b.3 If a name in the yy-gram is not on the there-list, that name is temporarily removed from the yy-gram. This is what we're calling Scene 1: Discard.

Scene 2: Ignore

b.4 If a name on the there-list at hand is not present in a yy-gram, that name is ignored for the valuation.

Scene 3: Substitute

b.5 If a name on the there-list at hand is found present in the yy-gram, substitute the value Yang for it.

b.6 There-lists might also be called Yang-lists, but we don't, so as not to seem to refer to value outcomes.

b.7 To substitute the there-list {g} into (k), first eliminate any name not on the there-list.

b.8 This is a quick task for the yy-gram (k). Eliminating k leaves just the enclosing parentheses as a yy-gram-in-process:

( ).

b.9 To complete the substitution, look for all there-listed names in the yy-gram and change them to ( ), "there", yang.

b.10 In yy-gram (k) we find no appearance of the name g, so we have completed the valuation (k) | {g}.

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Act Two: Evaluation

Scene 1: Is it simple already?

b.13 By valuating (k) | {g} we transformed yy-gram (k) to ( ), Yang, one of the simple yy-grams: (k) | {g} = ( ).

b.14 Accordingly, there-list {g} belongs in the y-outcome column for yy-gram (k).

        (k)

yy col heads

{g}

Another valuation there-list

c.1 Here comes another there-list, {}. We now make the valuation (k) | {}.

As before, step one is to eliminate all appearances of names not on the there-list from the yy-gram (k).

And as before, this results in yy-gram-in-process ( ).

Final step is to replace every name in the yy-gram that is on there-list {} with Yang, ( ). This results in yy-gram ( ).


c.1.1 So (k) | {} = ( )

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Is it simple yet?

c.2 ( ) is one of the simple yy-grams, so we can now add there-list {} to the y-outcome column:

        (k)

yy col heads

{g}

{}

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Another there-list to substitute

d.1 Now comes another there-list, {g,k}. To make the substitution (k) | {g,k}, first eliminate all names not on the there-list:

(k)

Then substitute Yang for there-listed names:

(( ))

Is it simple yet?

d.2 Substitution alone has not arrived at a simple yy-gram. For that another step is required, a zang-fu good luck step.

(( ))

=        , zang

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Is it simple yet?

d.3 The substitution (k) | {g,k} results in yin, simple.

(k) | {g,k} =     , so we add there-list {g,k} to the n-outcome column:

        (k)

yy col heads

{g}

{}

{g,k}

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One remaining there-list to substitute

e.1 There is a there-list we have not yet substituted. Do you see what it would be?

We seem to have a list of names g, k. Then there is one more there-list, one with k alone listed: {k}.

e.2 Make the substitution (k) | {k} and proceed with zang-fu good luck:

(k) | {k}

=(( )) , subst.

=       , zang; simple: yin.

        (k)

yy col heads

{g}

{}

{g,k}

{k}



zang good luck: (( ))=    

fu good luck: ( )( )=( )

catalog (of value outcomes)

— A yyGrams™ puzzle consists mainly in a catalog of value outcomes. The catalog is in two lists of there-lists, one for yang or "Yes" outcomes and another for yiN or "No" outcomes.

simple

— There are exactly two simple yy-grams: ( ), called Yang, and      , called yiN.

Yin and yang are values. To valuate a name yiN, you simply remove it temporarily from the yy-gram. To valuate a name Yang, you substitute ( ) for it.

there-list (substitution list; valuation)

— Names in a there-list are substituted in the yy-gram with Yang, ( ). All other variables in the yy-gram are removed for the valuation (in effect, substituting the value yiN). Then the yy-gram is evaluated by zang-fu good luck chops.

Variables in the there-list not appearing in the yy-gram are ignored.

value

— The simple yy-grams are values.

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©2012 David Zethmayr
2012.3.31 = 12ff