MU puzzle

The MU puzzle is a puzzle stated by Douglas Hofstadter and found in Gödel, Escher, Bach. As stated, it is an example of a Post canonical system and can be reformulated as a string rewriting system.

The puzzle

Suppose there are the symbols M, I, and U which can be combined to produce strings of symbols. The MU puzzle asks one to start with the "axiomatic" string MI and transform it into the string MU using in each step one of the following transformation rules:

Nr.	Formal rule			Informal explanation	Example
1.	xI	→	xIU	Add a U to the end of any string ending in I	MI	to	MIU
2.	Mx	→	Mxx	Double the string after the M	MIU	to	MIUIU
3.	xIIIy	→	xUy	Replace any III with a U	MUIIIU	to	MUUU
4.	xUUy	→	xy	Remove any UU	MUUU	to	MU

Solution

The puzzle cannot be solved: it is impossible to change the string MI into MU by repeatedly applying the given rules.

In order to prove assertions like this, it is often beneficial to look for an invariant, that is some quantity or property that doesn't change while applying the rules.

In this case, one can look at the total number of I in a string. Only the second and third rules change this number. In particular, rule two will double it while rule three will reduce it by 3. Now, the invariant property is that the number of I is not divisible by 3:

• In the beginning, the number of Is is 1 which is not divisible by 3.
• Doubling a number that is not divisible by 3 does not make it divisible by 3.
• Subtracting 3 from a number that is not divisible by 3 does not make it divisible by 3 either.

Thus, the goal of MU with zero I cannot be achieved because 0 is divisible by 3.

In the language of modular arithmetic, the number $n$$n$ of I obeys the congruence

$n\equiv 2^{a}\not \equiv 0{\pmod {3}}.\,$$n\equiv 2^{a}\not \equiv 0{\pmod 3}.\,$

where $a$$a$ counts how often the second rule is applied.

Relationship to logic

The MIU system illustrates several important concepts in logic by means of analogy.

It can be interpreted as an analogy for a formal system — an encapsulation of mathematical and logical concepts using symbols. The MI string is akin to a single axiom, and the four transformation rules are akin to rules of inference.

The MU string and the impossibility of its derivation is then analogous to a statement of mathematical logic which cannot be proven or disproven by the formal system.

It also demonstrates the contrast between interpretation on the "syntactic" level of symbols and on the "semantic" level of meanings. On the syntactic level, there is no knowledge of the MU puzzle's insolubility. The system does not refer to anything: it is simply a game involving meaningless strings. Working within the system, an algorithm could successively generate every valid string of symbols in an attempt to generate MU, and though it would never succeed, it would search forever, never deducing that the quest was futile. For a human player however, after a number of attempts, one soon begins to suspect that the puzzle may be impossible. One then "jumps out of the system" and starts to reason about the system, rather than working within it. Eventually, one realises that the system is in some way about divisibility by three. This is the "semantic" level of the system — a level of meaning that the system naturally attains. On this level, the MU puzzle can be seen to be impossible.

The inability of the MIU system to express or deduce facts about itself, such as the inability to derive MU, is a consequence of its simplicity. However, more complex formal systems, such as systems of mathematical logic, may possess this ability. This is the key idea behind Godel's Incompleteness Theorem.

See also

• Invariant (computer science)
• Unrestricted grammar

External links

• "Hofstadter's MIU System". Archived from the original on 4 March 2016. Retrieved 29 November 2016.  An online interface for producing derivations in the MIU System.