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Authors: Douglas R. Hofstadter
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game of any formal system can become something of an art. The major point, which almost doesn't need stating, is that you must not do anything which is outside the rules. We might call this restriction the "Requirement of Formality". In the present Chapter, it probably won't need to be stressed at all. Strange though it may sound, though, I predict that when you play around with some of the formal systems of Chapters to come, you will find yourself violating the Requirement of Formality over and over again, unless you have worked with formal systems before.
The first thing to say about our formal system-the MIU - system -is that it utilizes only three letters of the alphabet: M, I, U . That means that the only strings of the MIU -
system are strings which are composed of those three letters. Below are some strings of the MIU -system:
MU
UIM
MUUMUU
UIIUMIUUIMUIIUMIUUIMUIIU
* In this book, we shall employ the following conventions when we refer to strings. When the string is in the same typeface as the text, then it will be enclosed in single or double quotes.
Punctuation which belongs to the sentence and not to the string under discussion will go outside of the quotes, as logic dictates. For example, the first letter of this sentence is 'F', while the first letter of 'this ‘sentence’.is 't'. When the string is in Quadrata Roman , however, quotes will usually be left off, unless clarity demands them. For example, the first letter of Quadrata is Q.
But although all of these are legitimate strings, they are not strings which are "in your possession". In fact, the only string in your possession so far is MI . Only by using the rules, about to be introduced, can you enlarge your private collection. Here is the first rule:
RULE I: If you possess a string whose last letter is I, you can add on a U at the end.
By the way, if up to this point you had not guessed it, a fact about the meaning of "string"
is that the letters are in a fixed order. For example, MI and IM are two different strings.
A string of symbols is not just a "bag" of symbols, in which the order doesn't make any difference.
Here is the second rule:
RULE II: Suppose you have Mx . Then you may add Mxx to your collection.
What I mean by this is shown below, in a few examples.
From MIU , you may get MIUIU .
From MUM , you may get MUMUM .
From MU , you may get MUU .
So the letter `x' in the rule simply stands for any string; but once you have decided which string it stands for, you have to stick with your choice (until you use the rule again, at which point you may make a new choice). Notice the third example above. It shows how, once you possess MU , you can add another string to your collection; but you have to get MU first! I want to add one last comment about the letter `x': it is not part of the formal system in the same way as the three letters ` M' , ` I' , and ` U' are. It is useful for us, though, to have some way to talk in general about strings of the system, symbolically-and that is the function of thèx': to stand for an arbitrary string. If you ever add a string containing an 'x' to your "collection", you have done something wrong, because strings of the MIU -system never contain "x" “s”!
Here is the third rule:
RULE III: If III occurs in one of the strings in your collection, you may make a new string with U in place of III .
Examples:
From UMIIIMU , you could make UMUMU .
From MII11 , you could make MIU (also MUI ).
From IIMII , you can't get anywhere using this rule.
(The three I's have to be consecutive.)
From MIII , make MU .
Don't, under any circumstances, think you can run this rule backwards, as in the following example:
From MU , make MIII <- This is wrong.
Rules are one-way.
Here is the final rule.
RULE IV: If UU occurs inside one of your strings, you can drop it.
From UUU , get U .
From MUUUIII , get MUIII .
There you have it. Now you may begin trying to make MU . Don't worry you don't
The first thing to say about our formal system-the MIU - system -is that it utilizes only three letters of the alphabet: M, I, U . That means that the only strings of the MIU -
system are strings which are composed of those three letters. Below are some strings of the MIU -system:
MU
UIM
MUUMUU
UIIUMIUUIMUIIUMIUUIMUIIU
* In this book, we shall employ the following conventions when we refer to strings. When the string is in the same typeface as the text, then it will be enclosed in single or double quotes.
Punctuation which belongs to the sentence and not to the string under discussion will go outside of the quotes, as logic dictates. For example, the first letter of this sentence is 'F', while the first letter of 'this ‘sentence’.is 't'. When the string is in Quadrata Roman , however, quotes will usually be left off, unless clarity demands them. For example, the first letter of Quadrata is Q.
But although all of these are legitimate strings, they are not strings which are "in your possession". In fact, the only string in your possession so far is MI . Only by using the rules, about to be introduced, can you enlarge your private collection. Here is the first rule:
RULE I: If you possess a string whose last letter is I, you can add on a U at the end.
By the way, if up to this point you had not guessed it, a fact about the meaning of "string"
is that the letters are in a fixed order. For example, MI and IM are two different strings.
A string of symbols is not just a "bag" of symbols, in which the order doesn't make any difference.
Here is the second rule:
RULE II: Suppose you have Mx . Then you may add Mxx to your collection.
What I mean by this is shown below, in a few examples.
From MIU , you may get MIUIU .
From MUM , you may get MUMUM .
From MU , you may get MUU .
So the letter `x' in the rule simply stands for any string; but once you have decided which string it stands for, you have to stick with your choice (until you use the rule again, at which point you may make a new choice). Notice the third example above. It shows how, once you possess MU , you can add another string to your collection; but you have to get MU first! I want to add one last comment about the letter `x': it is not part of the formal system in the same way as the three letters ` M' , ` I' , and ` U' are. It is useful for us, though, to have some way to talk in general about strings of the system, symbolically-and that is the function of thèx': to stand for an arbitrary string. If you ever add a string containing an 'x' to your "collection", you have done something wrong, because strings of the MIU -system never contain "x" “s”!
Here is the third rule:
RULE III: If III occurs in one of the strings in your collection, you may make a new string with U in place of III .
Examples:
From UMIIIMU , you could make UMUMU .
From MII11 , you could make MIU (also MUI ).
From IIMII , you can't get anywhere using this rule.
(The three I's have to be consecutive.)
From MIII , make MU .
Don't, under any circumstances, think you can run this rule backwards, as in the following example:
From MU , make MIII <- This is wrong.
Rules are one-way.
Here is the final rule.
RULE IV: If UU occurs inside one of your strings, you can drop it.
From UUU , get U .
From MUUUIII , get MUIII .
There you have it. Now you may begin trying to make MU . Don't worry you don't
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