MU puzzle

The MU puzzle^[6] is a good example of a logical problem where determining an invariant is of use for an impossibility proof. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:

1. If a string ends with an I, a U may be appended (xI → xIU)
2. The string after the M may be completely duplicated (Mx → Mxx)
3. Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy)
4. Any two consecutive U's may be removed (xUUy → xy)

An example derivation (with superscripts indicating the applied rules) is

MI →² MII →² MIIII →³ MUI →² MUIUI →¹ MUIUIU →² MUIUIUUIUIU →⁴ MUIUIIUIU → ...

In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a property that is invariant to all rules (i.e. that isn't changed by any of them), and demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:

The number of I's in the string is not a multiple of 3.

This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:

Rule	#I's	#U's	Effect on invariant
1	+0	+1	Number of I's is unchanged. If the invariant held, it still does.
2	×2	×2	If n is not a multiple of 3, then 2×n isn't either. The invariant still holds.
3	−3	+1	If n is not a multiple of 3, n−3 isn't either. The invariant still holds.
4	+0	−2	Number of I's is unchanged. If the invariant held, it still does.

The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it won't be afterwards either.

Given that there is a single I in the starting string MI, and one that is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).

Unchanged under group action

Firstly, if one has a group G acting on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group.

Frequently one will have a group acting on a set X, which leaves one to determine which objects in an associated set F(X) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.

More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions.

Dual to the notion of invariants are coinvariants, also known as orbits, which formalizes the notion of congruence: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the perimeter of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.

These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In classification problems, one might seek to find a complete set of invariants, such that if two objects have the same values for this set of invariants, then they are congruent.

For example, triangles such that all three sides are equal are congruent under rigid motions, via SSS congruence, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants.

Independent of presentation

Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.

The most common examples are:

• The presentation of a manifold in terms of coordinate charts – invariants must be unchanged under change of coordinates.
• Various manifold decompositions, as discussed for Euler characteristic.
• Invariants of a presentation of a group.

Unchanged under perturbation

Thirdly, if one is studying an object which varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).

Automatic invariant detection in imperative programs

Abstract interpretation tools can compute simple invariants of given imperative computer programs. The kind of properties that can be found depend on the abstract domains used. Typical example properties are single integer variable ranges like 0<=x<1024, relations between several variables like 0<=i-j<2*n-1, and modulus information like y%4==0. Academic research prototypes also consider simple properties of pointer structures.^[12]

More sophisticated invariants generally have to be provided manually. In particular, when verifying an imperative program using the Hoare calculus,^[13] a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs.

In the context of the above MU puzzle example, there is currently no general automated tool that can detect that a derivation from MI to MU is impossible using only the rules 1–4. However, once the abstraction from the string to the number of its "I"s has been made by hand, leading, for example, to the following C program, an abstract interpretation tool will be able to detect that ICount%3 can't be 0, and hence the "while"-loop will never terminate.

void MUPuzzle(void) {
    volatile int RandomRule;
    int ICount = 1, UCount = 0;
    while (ICount % 3 != 0)                         // non-terminating loop
        switch(RandomRule) {
        case 1:                  UCount += 1;   break;
        case 2:   ICount *= 2;   UCount *= 2;   break;
        case 3:   ICount -= 3;   UCount += 1;   break;
        case 4:                  UCount -= 2;   break;
        }                                          // computed invariant: ICount % 3 == 1 || ICount % 3 == 2
}