# The Eudex Algorithm

Half a year ago, I designed Eudex as a modern replacement for Soundex, which is still widely used today. Eudex supports a wide range of special-cases of European languages, while preserving the spirit of simplicity, Soundex has.

Both Eudex and Soundex are phonetic algorithms that produce a representation of the sound of some string. Eudex is fundamentally different from Soundex in that it is not a phonetic classifier. It is a phonetic locality-sensitive hash, which means that two similarly-sounding strings are not mapped to the same value, but instead to values near to each other.

This technically makes it a string similarity index, but it one should be careful with this term, given that it doesn't produce a typing distance, but a phonetic/pronounciation distance.

What this blog post aims to do is to describe the rationale behind Eudex, hopefully sparking new ideas and thoughts for the reader.

# The output and the input

So, the input is any Unicode string in a Latin-family alphabet.

The output is fixed-width integer (we'll use 64-bit, but that is in some cases a very narrow width), which has following characteristic:

If the string \(a\) sounds similar to a string \(b\), \(f(a) \oplus f(b)\) has low Hamming weight.

In other words, two similarly sounding words maps to numbers with only a few bits flipped, whereas words without similar sound maps to numbers with many bits flipped.

# The algorithm

The algorithm itself is fairly simple. It outputs an 8 byte array (an unsigned 64 bit integer):

\[\underbrace{A}_{\text{First phone}} \quad \underbrace{00}_{\text{Padding}} \quad \underbrace{BBBBB}_{\text{Trailing phones}}\]

The crucial point here is that all the characters are mapped through a table carefully derived by their phonetic classification, to make similar sounding phones have a low Hamming distance.

If two consecutive phones shares all the bits, but the parity bit, (i.e, \(a \gg 1 = b \gg 1\)), the second is skipped. This allows us to "collapse" similar or equal phones into one, kind of equivalence to the collapse stage of Soundex: Similar phones next to each other can often be collapsed to one of the phones without losing the pronounciation.

# Deriving the tables

The tables are what makes it interesting. There are four tables: one for ASCII letters (not characters, letters) in the first slot ('A'), one for C1 (Latin Supplement) characters in the first slot, one for ASCII letters in the trailing phones, and one for the C1 (Latin Supplement) characters for the trailing phones.

There is a crucial distinction between consonants and vowels in Eudex. The first phone treat vowels as first-class citizens by making distinctions between all the properties of vowels. The trailing phones only have a distinction between open and close vowels.

## Trailing character table

Let's start with the tables for the trailing characters. Consonants' bytes are treated such that each bit represent a property of the phone (i.e., pronunciation) with the exception of the rightmost bit, which is used for tagging duplicates (it acts as a discriminant).

Let's look at the classification of IPA consonants:

As you may notice, characters often represent more than one phone, and reasoning about which one a given character in a given context represents can be very hard. So we have to do our best in fitting each character into the right phonetic category.

We have to pick the classification intelligently. There are certain groups the
table doesn't contain, one of which turns out to be handy in a classification:
liquid consonants (lateral consonants + rhotics), namely `r`

and `l`

. Under
ideal conditions, these should be put into to distinct bits, but unfortunately
there are only 8 bits in a byte, so we have to limit ourselves.

Now, every good phonetic hasher should be able to segregate important characters (e.g., hard to mispell, crucial to the pronunciation of the word) from the rest. Therefore we add a category we call "confident", this will occupy the most significant bit. In our category of "confident" characters we put l, r, x, z, and q, since these are either:

- Crucial to the sound of the word (and thus easier to hear, and harder to misspell).
- Rare to occur, and thus statistically harder to mistake.

So our final trailing consonant table looks like:

Position | Modifier | Property | Phones |
---|---|---|---|

1 | 1 | Discriminant | (for tagging duplicates) |

2 | 2 | Nasal | mn |

3 | 4 | Fricative | fvsjxzhct |

4 | 8 | Plosive | pbtdcgqk |

5 | 16 | Dental | tdnzs |

6 | 32 | Liquid | lr |

7 | 64 | Labial | bfpv |

8 | 128 | Confident¹ | lrxzq |

The more "important" the characteristic is to the phone's sound the higher place it has.

We then have to treat the vowels. In particular, we don't care much of vowels in trailing position, so we will simply divide them into two categories: open and close. It is worth noting that not all vowels fall into these categories, therefore we will simply place it in the category it is "nearest to", e.g. a, (e), o gets 0 for "open".

So our final ASCII letter table for the trailing phones looks like:

```
(for consonants)
+--------- Confident
|+-------- Labial
||+------- Liquid
|||+------ Dental
||||+----- Plosive
|||||+---- Fricative
||||||+--- Nasal
|||||||+-- Discriminant
||||||||
a* 00000000
b 01001000
c 00001100
d 00011000
e* 00000001
f 01000100
g 00001000
h 00000100
i* 00000001
j 00000101
k 00001001
l 10100000
m 00000010
n 00010010
o* 00000000
p 01001001
q 10101000
r 10100001
s 00010100
t 00011101
u* 00000001
v 01000101
w 00000000
x 10000100
y* 00000001
z 10010100
| (for vowels)
+-- Close
```

Now, we extend our table to C1 characters by the same method:

```
(for consonants)
+--------- Confident
|+-------- Labial
||+------- Liquid
|||+------ Dental
||||+----- Plosive
|||||+---- Fricative
||||||+--- Nasal
|||||||+-- Discriminant
||||||||
ß -----s-1 (use 's' from the table above with the last bit flipped)
à 00000000
á 00000000
â 00000000
ã 00000000
ä 00000000 [æ]
å 00000001 [oː]
æ 00000000 [æ]
ç -----z-1 [t͡ʃ]
è 00000001
é 00000001
ê 00000001
ë 00000001
ì 00000001
í 00000001
î 00000001
ï 00000001
ð 00010101 [ð̠] (represented as a non-plosive T)
ñ 00010111 [nj] (represented as a combination of n and j)
ò 00000000
ó 00000000
ô 00000000
õ 00000000
ö 00000001 [ø]
÷ 11111111 (placeholder)
ø 00000001 [ø]
ù 00000001
ú 00000001
û 00000001
ü 00000001
ý 00000001
þ -----ð-- [ð̠] (represented as a non-plosive T)
ÿ 00000001
| (for vowels)
+-- Close
```

## First phone table

So far we have considered the trailing phones, now we need to look into the first phone. The first phone needs a table with minimal collisions, since you hardly ever misspell the first letter in the word. Ideally, the table should be injective, but due to technical limitations it is not possible.

We will use the first bit to distinguish between vowels and consonants.

Previously we have only divided vowels into to classes, we will change that now, but first: the consonants. To avoid repeating ourselves, we will use a method for reusing the above tables.

Since the least important property is placed to the left, we will simply shift it to the right (that is, truncating the rightmost bit). The least significant bit will then be flipped when encountering a duplicate. This way we preserve the low Hamming distance, while avoiding collisions.

The vowels are more interesting. We need a way to distinguish between vowels and their sounds.

Luckily, their classification is quite simple:

If a vowel appears as two phones (e.g., dependent on language), we OR them, and possibly modify the discriminant if it collides with another phone.

We need to divide each of the axises into more than two categories, to utilize all our bits, so some properties will have to occupy multiple bits.

Position | Modifier | Property (vowel) |
---|---|---|

1 | 1 | Discriminant |

2 | 2 | Is it open-mid? |

3 | 4 | Is it central? |

4 | 8 | Is it close-mid? |

5 | 16 | Is it front? |

6 | 32 | Is it close? |

7 | 64 | More close than [ɜ] |

8 | 128 | Vowel? |

So we make use of both properties, namely both the openness and "frontness". Moreover, we allow more than just binary choices:

```
Class Close Close-mid Open-mid Open
+----------+----------+-----------+---------+
Bits .11..... ...11... ......1. .00.0.0.
```

Let's do the same for the other axis:

```
Class Front Central Back
+----------+----------+----------+
Bits ...1.0.. ...0.1.. ...0.0..
```

To combine the properties we OR these tables. Applying this technique, we get:

```
(for vowels)
+--------- Vowel
|+-------- Closer than ɜ
||+------- Close
|||+------ Front
||||+----- Close-mid
|||||+---- Central
||||||+--- Open-mid
|||||||+-- Discriminant
||||||||
a* 10000100
b 00100100
c 00000110
d 00001100
e* 11011000
f 00100010
g 00000100
h 00000010
i* 11111000
j 00000011
k 00000101
l 01010000
m 00000001
n 00001001
o* 10010100
p 00100101
q 01010100
r 01010001
s 00001010
t 00001110
u* 11100000
v 00100011
w 00000000
x 01000010
y* 11100100
z 01001010
```

We then extend it to C1 characters:

```
+--------- Vowel?
|+-------- Closer than ɜ
||+------- Close
|||+------ Front
||||+----- Close-mid
|||||+---- Central
||||||+--- Open-mid
|||||||+-- Discriminant
||||||||
ß -----s-1 (use 's' from the table above with the last bit flipped)
à -----a-1
á -----a-1
â 10000000
ã 10000110
ä 10100110 [æ]
å 11000010 [oː]
æ 10100111 [æ]
ç 01010100 [t͡ʃ]
è -----e-1
é -----e-1
ê -----e-1
ë 11000110
ì -----i-1
í -----i-1
î -----i-1
ï -----i-1
ð 00001011 [ð̠] (represented as a non-plosive T)
ñ 00001011 [nj] (represented as a combination of n and j)
ò -----o-1
ó -----o-1
ô -----o-1
õ -----o-1
ö 11011100 [ø]
÷ 11111111 (placeholder)
ø 11011101 [ø]
ù -----u-1
ú -----u-1
û -----u-1
ü -----y-1
ý -----y-1
þ -----ð-- [ð̠] (represented as a non-plosive T)
ÿ -----y-1
```

# Distance operator

Now that we have our tables. We now need the distance operator. A naïve approach would be to simply use Hamming distance. This has the disadvantage of all the bytes having the same weight, which isn't ideal, since you are more likely to misspell later characters, than the first ones.

For this reason, we use weighted Hamming distance:

Byte: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Weight: | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Namely, we XOR the two values and then add each of the bytes' Hamming weight, using the coefficients from the table above.

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