The meaning of words
Words, Words, Words

As a general principle, we could define a word found in the dictionary as more meaningful than a jumble of letters; so "king" is more meaningful than "kxjz". Two words forming a valid phrase are, of course, more meaningful than a single word; so "the king" is more meaningful than simply "king". Consequently, a single word with a leading or trailing space is more meaningful than just a single word as it implies a connection to another word; so " king" is more meaningful than simply "king". The addition of special symbols can also add meaning; so "the king!" is more meaningful, and of a somewhat different meaning, than "the king". 

Though this be madness, yet there is method in ’t.

So how do we determine, for the purposes of Phrasenation, what constitutes a valid word or phrase? Well, we will take a miniscule sample of English literature, and compare our phrase to that sample! If the sample is found, then it will be considered a valid phrase. One proviso: All words must be complete. No half words.

 (As a technical matter, we will count certain symbols, such as "!" and "?" as separate words.) 

Our Phrase Book
Within the book and volume of my brain.

For our Phrase Book, we will use The Tragedy of Hamlet, Prince of Denmark by William Shakespeare. I'm sure most everyone can agree that just about anything the Bard said is in some sense meaningful. After all, he practically invented modern English single-handedly! Keep in mind that this will exclude the vast majority of valid phrases, including even most of Shakespeare. However, you can add phrases to the Phrase Book, if you choose. 

Finally, all valid phrasenations are ranked by numbers of characters. Longer is "better". 

Hamletations
If circumstances lead me, I will find
Where truth is hid.

When generating a phrasagen (a mutant phrase), we will use random mutation and recombination. Starting from just two words, "the" and "question", some not-so-valid phrasagens might look like these:

Phrasenation allows one to adjust the relative frequency of each type of mutation. Once having generated a phrasagen, we must compare it to our Phrase Book in order to determine its meaningfulness. If it is not found, we will ruthlessly eliminate it. To be honest, as this world goes, is to be one man pick'd out of ten thousand.

Indexing the Bard
Yea, from the table of my memory
I’ll wipe away all trivial fond records.

To make this a practical matter, we must index every valid phrase in our Phrase Book. But it isn't enough to index just the first word in a phrase. We must index every single word. 

But what if the first words are the same? Well, then we will compare the word that follows, and if necessary each succeeding word until we find a word which is different. For instance, "to be" is not a unique phrase, as it could be found as "to be, or not" or as "to be- that is the question". 

In fact, the phrase "to be" shows up 34 times as first words, including "to be your valentine", but "to be," (note the comma) only once, in "to be, or not to be".  

The index includes 36,176 words, including symbols. 

When testing the index, the Phrasenator outputted every indexed word followed by a specified number of words. Then the Phrasenator counted the number of unique phrases. For one word phrases, there were 4,801 unique phrases. But what about other numbers of words? 

For large numbers of words, the answer is surely 36,176, but what is a "large" number? Somewhat surprisingly, if you select any four words in series, the vast majority will constitute a unique phrase! And for that small percentage which are not unique, those are nearly all purposefully repeated phrases, such as "a pit of clay for to be made" from the singing Clown's refrain, 

Definitions:
You must translate; 'tis fit we understand them.

• L = Length of phrase or collection of phrases.

• N = Number of valid phrases of length L. N is generally difficult to define. In Phrasenation, we explicitly define it as the contents of The Tragedy of Hamlet, Prince of Denmark, by William Shakespeare.

• M = Total number of Mutations, the number of Phrasagens whether valid or not.

• C = Number of allowable characters (letters or symbols).

• C^L = Sequence space. Number of possible combinations of letters in a string of length L.

• (C^L)/N = Pitman's Number, the number of possible combinations of characters divided by the number of valid phrases. Zillions and zillions.

• N/(C^L) = Pitman's Ratio, the probability of evolving a valid sequence through Pitman's Random Walk across the Neutral Gap. 

With "O Sean Pitman", we introduced our modest project with simple concatenation and point mutation. Then, in accordance with Dr. Pitman's wishes, we added Insertion. Now we add Exchange and Complex Recombination. Each of these categories are approximately related to powers of L.

That he is mad, ’t is true: ’t is true ’t is pity;
And pity ’t is ’t is true.

• L^0 Concatenation: There are only two ways to join two strings end-to-end. The number of ways to choose is not dependent on L.

• L^1 Point Mutation: There are L+2 possible locations for each such mutation, including adding one at either end. The number of ways to choose is proportional to L.

• L^2 Exchange: Exchanging same-length snippets between strings. We must choose the beginning and end of snippet to be exchanged.

• L^3 Insertion: Inserting snippets from one string into another string. We must choose the beginning and ending of each snippet, plus the insertion point.

• L^4 Complex: Exchanging snippets of possibly different lengths between strings. We must choose the beginning and ending of the snippet, and the beginning and ending of the section to be replaced. 

Dr. Pitman's assertions relate to the behavior of Phrasenation as L increases.  Pitman's Vastness Assertion Pitman's Ratio implies the Neutral Gap N/(C^L) --> 0  Pitman's Principle of Zillions Pitman's Number approaches Zillions and Zillions M --> (C^L)/N

As L increases, Dr. Pitman claims that the ratio of valid phrases to the totality of sequence space, approaches zero. Valid sequences get lost in the vastness of sequence space. He concludes that it is impossible to evolve sequences beyond the "lowest level of complexity". However, Dr. Pitman has failed to provide a method of calculating N, much less a map of how valid sequences are distributed in sequence space. Generally, any collection of valid phrases and sentences have some validity. Language can ramble somewhat and still be valid. We could start by talking about Pitman's handwaving and suddenly, for no particular reason, change the subject to ghosts and the murder of kings. Imagine that!

 Results Matter
More matter, with less art.

However, even using a tiny sliver of the English language—just one play by one playwright—it can be shown that phrases of substantial length can be easily evolved. 

Our experiments have shown that words and phrases appear to have some underlying connection related to their own evolution. As such, words and phrases make ideal subjects for evolutionary algorithms. 

Zachriel's Phrasenation brought to you by 

Moods in Music
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