A Contemporary Look at Vacillia's Strangest Game

by B. Culver, M. Ersgang, and Levanota

Proposed for the Journal of Recovered Culture, Winter 729

This is an early version of a paper which, due to our current situation, probably won't ever be published in its entirety. Some parts are missing or incomplete, but there are some raw notes and dialogue in their place--maybe you can piece things together. Good luck.

Abstract

The Levanota Gadgets are fascinating game pieces recovered from the remains of Old Vacillia. Their purpose, rules, and importance have been hotly debated for centuries, and while interest has waned in recent years, the conversation may benefit from a modern perspective. In this paper we'll review the Levanota gadgets, what we understand and how our knowledge developed, and we'll consider a promising case study which reveals the incredible versatility of these mysterious toys.
Levanota gadgets are arranged in patterns which resemble their own little language. And in order to understand the full range of a language, we need to hear it spoken by many people, in different contexts and dialects. From there, we would assemble a set of statistics: which words are most common, which associate commonly with others, which convey recognizable ideas or motives. The more voices we would add to the conversation, the better. Tragically, this method no longer applies. The sacking of Old Vacillia wasn't kind to the country's literature, and history wasn't kind to its oral traditions; and so the vibrant conversations in this great language have been silenced. The thousands of voices speaking Levanota have quieted down to just one.
Our best source of information about the Levanota Gadgets is the game's matron deity: we simply call her Levanota. Though she died along with the culture she represents, she still has the intuition to read a stack of game pieces and write an appropriate response. She shares our goal of rediscovering the game and its purpose, hence her coauthorship.

Notation

For the sake of concision, and because actual Levanota gadgets are difficult to come by today, we use the following syntax to represent stacks of pieces. Each piece is represented by two characters: a letter or word, representing the symbol written on the piece, and an arrow, representing the piece's orientation forward ("nota") or backward ("leva"). We use capitalization to distinguish the front and back sides of a piece.

<a  leva piece with "a" symbol
b>  nota piece with "b" symbol
<C  back-leva piece with "c" symbol
D>  back-nota piece with "d" symbol
e:  doubled piece with "e" symbol
F:  doubled-back ("lord") piece with "f" symbol

Composing and Parsing Sentences

We stated above that Levanota can read and translate stacks of Levanota gadgets, but she can only make sense of certain patterns. This is because a valid stack is like a "sentence:" though a wide variety of sentences exists, a random string of words is likely meaningless. From Levanota's perspective, the early scholars were sending her messages like "i market the to went," reacting with frustration when she failed to translate them.
It's also important to know that Levanota parses sentences using a very strict set of rules. Even if a sentence almost makes sense, like "I went the market," it won't register. Therefore, we'll be very careful to establish exactly which sentences are valid.

Leva and Nota

When we restrict ourselves to front-side pieces, the rules are fairly simple. A piece facing backward ("leva") is atomic, able to stand on its own. A piece facing forward ("nota") indicates that another piece should come after it.

p>
"I don't understand that."


<p
"That makes sense."
"What does it mean?"
"I'm not sure I understand the question."


p> <p
"That makes sense."


<p p> <p
"I don't understand that."
"Same reason as before?"
"It seemed like there were pieces missing before. Now there are too many."


a> b> c> d> e> f> g> <h
"That makes sense."


a> b> c> d> <e <f <g <h
"I don't--"
"--understand it, yeah. Now we're getting somewhere."

Front and Back

Since each piece has a distinct front and back side, a single gadget actually has four different orientations. The back-side pieces are interesting in that each piece considers more than just the piece directly after it. When a piece is on its back, it indicates that more pieces will follow after the sentence would normally be finished.

a> <B <c
"That makes sense."
"What's the difference between this and 'a> b> <c?'"
"When you say 'a> b> <c,' it sounds like 'A...and, B and C.'
 But this one is more like 'A and B...and C.'"
"This may be a silly question, but what's 'and' in this context?"
"It's a binary operator."
"Thanks. I actually think it's impossible to give less information than you just gave."


A> <b <c
"That makes sense, I guess."
"You guess? Isn't this the same as 'a> <B <c?'"
"Yes, but I don't like starting a sentence with a back piece. It may cause trouble."
"But A> <b <c and a> <B <c are equivalent, otherwise."
"Yes."

In general, our sentences will always begin and end with a front piece. Under this standard, every sentence has a unique representation.

x> a> <B <c
x> A> <b <c
"These are, no, these are different."
"Why?"
"X, and A-and-B...and C.
 X...and A-and-B, and C."
"Sorry, can you repeat that? A few times?"

With these rules in mind, we can list all the distinct sentences which may be constructed (ignoring symbols and doubled pieces).

Sentences with one piece
1: <a


Sentences with two pieces
1: a> <b


Sentences with three pieces
1: a> b> <c
2: a> <B <c


Sentences with four pieces
1: a> b> c> <d
2: a> b> <C <d
3: a> B> <c <d
4: a> <B c> <d
5: a> <B <C <d


Sentences with five pieces
(This part is missing)

Solution

1: a> b> c> d> <e
2: a> b> c> <D <e
3: a> b> C> <d <e
4: a> B> c> <d <e
5: a> b> <C d> <e
6: a> b> <C <D <e
7: a> B> <C <d <e
8: a> B> <c d> <e
9: a> B> <c <D <e
10: a> <B c> d> <e
11: a> <B c> <D <e
12: a> <B C> <d <e
13: a> <B <C d> <e
14: a> <B <C <D <e

Symbols as Variables

It took decades to unpack the meaning of the symbols written on each piece. They were sometimes used as decorations on ancient buildings and curios, but they rarely matched any Old Vacillian script. Some critics implied that any sacred purpose these symbols once held no longer exists; it isn't language, but numerology, meaningless in today's world. As Levanota herself was happy to point out, these critics were correct in only the most technical sense.
In the 3rd century, scholars began to read the Levanota gadgets from a mathematical angle. Instead of looking for the implicit meaning of each symbol, they treated each piece as a "variable." After all, a variable is a symbol whose meaning is first ascribed, then applied; this matches neatly the concepts of "nota" and "leva." In this way, we've established the second rule for manipulating Levanota gadgets: a gadget with a special symbol may be defined as shorthand for a longer phrase.

var = a> <a
"Okay."


x> X> <x <X <var
"This is equivalent to x> X> <x <X a> <a."


x> X> <x <Var <x
"This is equivalent to x> X> <x A> <a <x."


x> Var> <x <X <x
"This is equivalent to x> A> <A <x <X <x."


var> X> <x <X <x
"This is equivalent to a> <A X> <x <X <x."

Note that when a variable is on its back side ("Var"), the first piece in its definition must also be on its back side. When a variable is nota ("var>"), the last piece in its definition must be on its back side.

var = A> <b <c
"Okay, but that's going to cause problems."


x> <Var <y
"I can't translate this."


var = a> <B <c
"Okay."


x> <Var <y
"This is equivalent to x> A> <B <c <y."

Treating symbols as variables can be helpful when understanding how sentences are parsed. Recall the difficulty we had before:

A> <b <c  and
a> <B <c
"These are equivalent."


x> A> <b <c  and
x> a> <B <c
"These are different."

How does the "x>" piece convert equivalent sentences into different ones? With a clever variable assignment, we can justify what's happening here and understand the structure of 4-piece sentences.

This part is missing.

Solution

var = a> <b
"Okay."


x> var> <c
"This is equivalent to x> a> <B <c."


x> <Var <c
"This is equivalent to x> A> <b <c."

Symbols as functions

Of course, Levanota can do much more than parse sentences. When a sentence contains a "lord," a doubled piece on its back, she dissolves it by the first rule of Levanota gadgets.

A: <a <b
"This is equivalent to <b."


A: a> <a <b
"This is equivalent to b> <b."


A: a> <X <a b> <b
"This is equivalent to b> <B <X b> <b."

This rule is a little strange, but it's actually quite similar to the "symbols as variables" rule. When a sentence looks like this,

X: <p <y  or
x: <P <y

it indicates that "x" is a variable which stands for "y." Thus, whenever an "x" appears within this structure, it may be replaced by "y." If no such substitution can be made, we can replace the "<y" with "<p" and remove everything else.

x: <P <y
"This is equivalent to <p."


x: P> <q <y
"This is equivalent to p> <q."


x: <P <Y <z
"This is equivalent to p> <z."

This rule was once Levanota's greatest mystery: why is it important, why is it so oddly specific, and how is it used?
If our hypotheses about the Levanota gadgets are right, this ancient game has a remarkably modern sensibility. Essentially, any sentence which begins with a doubled piece is like a function, describing an operation which is applied to another piece.

nochange = x: <x
left = x: y: <x
right = x: y: <y
sub = a: b: x: a> <X b> <x
compose = a: b: x: a> b> <x
swap = a: b: c: a> <C <b
copy = a: x: a> <X <x
"Okay."


right> <P <q
"This is
x: y: <Y <P <q, or
y: <Y <q, or
<q."

Using Levanota Gadgets in Arithmetic

There have been numerous case studies which attempt to make sense and meaning out of the Levanota gadgets. The unique structures formed by these sentences have surprising applications in a variety of fields. In this paper we'll discuss one of the most promising cases: the use of Levanota gadgets to simulate arithmetic. The following encoding is used to represent the nonnegative integers, in a way that preserves the ordinal relationships between them.

zero  = a: x: <x
one   = a: x: a> <x
two   = a: x: a> a> <x
three = a: x: a> a> a> <x
etc.

This simple representation is surprisingly good at capturing the elements of number theory. Each number can be used to repeatedly apply a function,

three> A: x> <a <y
"This is equivalent to x> x> x> <y."

and numbers can be concatenated in an elegant way:

sum = p: q: a: x: p> <A q> <A <x
"Okay."


sum> <Two <three
"This is equivalent to <five."

We can even simulate the acts of multiplication and exponentiation.

(This part is missing)

Solution

product = p: q: a: x: p> Q> <a <x
exp = p: q: a: x: p> <q

Thus, these simple but intricate gadgets can not only be used to build interesting structures, but also to demonstrate the connections between ancient and modern fields of study.

Remaining Questions

How many sentences are possible?

Given a finite number of pieces, we evaluated the number of distinct sentences:

2 pieces - 1 sentence
3 pieces - 2 sentences
4 pieces - 5 sentences
5 pieces - 14 sentences
6 pieces - 42 sentences

The sequence 1, 2, 5, 14, 42, et cetera, is actually quite famous. For one thing, it describes the number of ways to partition a d-sided polygon into d-2 triangles. Does this pattern continue, and is it significant?

Which other arithmetic functions are possible?

We've learned how to perform addition, multiplication, and exponentiation using Levanota gadgets. But these are all cumulative functions which combine numbers to form larger ones. What about subtraction, division, and the rest of the arithmetic spectrum? Since our system can only represent nonnegative integers, it may be impossible to write a Levanota gadget which decrements or otherwise reduces a number.

How do we handle paradoxical sentences?

Finally, we have yet to unpack the "glitch" discovered in 630, a stack which may be reduced infinitely. Levanota sometimes takes a while to evaluate the sentences we send her, but this particular phrase seems impossible to translate (it's even been left overnight with no success).

paradox = a: X: a> x> <x y: a> y> <y

(Most writers tend to use "x" instead of "y." We decided to introduce a third variable in order to distinguish the two branches of the sentence.)
While it's a shame that this "paradox" piece has no apparent translation, we don't see this as a fatal flaw. Some believe that, in a well-defined language (or whatever the Levanota Gadgets are trying to be), every finite sequence must be resolvable in a finite number of steps. And since the 7th century, our understanding of the gadgets has stubbornly refused to expand beyond this idea. But in today's world, we can go a little further.

paradox> <a
"I'm going to run out of pieces again, aren't I?"
"Don't go as far this time. Just make a couple passes and then stop."
"This is equivalent to 'a> a> a> a> a> a> paradox> <a.'"
"There's our pattern. And it should continue that way, no matter now many pieces there are."
"Are you sure? That kind of deduction isn't part of the rules."

Bibliography

"TODO" - Culver