COLORING MAPS AND THE KOWALSKI

DOCTRINE

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

http://www-formal.stanford.edu/jmc/

Abstract

It is attractive to regard an algorithm as composed of the logic

determining what the results are and the control determining how the

result is obtained. Logic programmers like to regard programming as

controlled deduction, and there have been several proposals for con-

trolling the deduction expressed by a Prolog program and not always

using Prolog’s normal backtracking algorithm. The present note dis-

cusses a map coloring program proposed by Pereira and Porto and

two coloring algorithms that can be regarded as control applied to its

logic. However, the control mechanisms required go far beyond those

that have been contemplated in the Prolog literature.

Robert Kowalski (1979) enunciated the doctrine expressed by the

formula

1 ALGORITHM = LOGIC + CONTROL

The formula isn’t precise, and it won’t be precise until someone proposes a

precise and generally accepted notion of how control is to be added to an

expression of the logic of a program. Nevertheless, the idea is attractive, and

I believe it can be made to work for some interesting class of programs. It

is analogous to my comparison of epistemology and heuristics or Chomsky’s

competence and performance.

Pereira and Porto (1980) give a logic program for coloring planar maps

with four colors and discuss how “selective backtracking” can reduce the

search involved in coloring a map from that done by a straightforward PRO-

LOG execution of the same program.

The discussion by Pereira and Porto treats coloring maps purely as an

example of logic programming, and the improvements they discuss apply

to all logic program systems. We shall consider two mathematical ideas

about map coloring that go back to Kempe (1879), the paper containing the

original false proof of the four color theorem. While Kempe’s proof was false,

the ideas are good and were used in almost all subsequent work including

the recent successful proof.

The question is whether an algorithm involving these ideas can be re-

garded as a form of control adjoined to the basic logic program or whether

they necessarily involve a new program. If they are to be regarded as control

structures, it is not yet clear how they are best expressed. Of course, it is not

hard to write a completely new program in PROLOG or any other language

expressing the algorithms, and this has been done. The interpreted programs

color a map of the United States. However, it is also interesting to try to

regard the algorithms as control attached to the Pereira-Porto logic program

for coloring a specific map.

2 THE PEREIRA–PORTO LOGIC PROGRAMThere are two parts. The first expresses that the adjacent countries must

have different colors by listing the pairs of colors that may be adjacent. We

have

a.

next(yellow, blue).

next(yellow, red).

next(yellow, green).

next(blue, yellow).

etc. for all pairs of distinct colors.

The remaining PROLOG statement is distinct for each map. For the map

of Figure 1, which they use as an example, it is

b.

goal(R1, R2, R3, R4, R5, R6) ← next(R1, R2), next(R1, R3), next(R1, R5),next(R1, R6), next(R2, R3), next(R2, R4), next(R2, R5), next(R2, R6),next(R3, R4), next(R3, R6), next(R5, R6),

where each literal expresses the fact that a particular pair of adjacent regions

must be compatibly colored.

Figure 1.

Pereira and Porto give a trace of the execution of the program by standard

depth first PROLOG. They point out that when an attempt to satisfy a literal

fails, because the two adjacent regions mentioned have been assigned the

same color, standard PROLOG will take back the most recent assignment of

a color even if the region most recently colored was neither of those involved

in the incompatibility. Their intelligent backtracking will change the color of

one of the regions giving the incompatibility.

An outsider to logic programming may react unsympathetically and com-

ment that this is just one more example of a logic programming system, with

its standard way of doing searches, tripping over its own feet. However, we

should also recall that brief and easy statement of the PROLOG program

for the coloring and not give up this virtue without a fight.

Nevertheless, “intelligent backtracking” doesn’t make (a) and (b) into a

good coloring program. Indeed we shall argue that it doesn’t even do full

justice to the logic of the program. To see this we need two ideas of Kempe.

3 REDUCING THE MAP

Kempe (or perhaps someone still earlier) noticed that countries with three

or fewer neighbors present no problem. No matter how the rest of the map

is colored, there is always a color available for such a country.1 We use this

in to improve a Pereira-Porto map coloring program by “reducing the map”

by removing such countries and doing our trial-and-error coloring on the

reduced map, confident that once the reduced map is colored, the coloring

can be extended to the omitted countries.

The idea is even more powerful, because eliminating countries with three

or fewer neighbors may remove enough neighbors from some other countries

so that they have three or fewer neighbors and can themselves be removed.

Therefore, the reduction process should be continued until a completely re-

duced map is obtained in which all countries have at least four neighbors.

The maps of the states of the U.S. and the countries of Europe, Asia, Africa

and South America all reduce to null maps when countries with three or

fewer neighbors are successively eliminated

Likewise the map of Figure 1 reduces to the empty map. Thus we may

remove country 4 with two neighbors and country 5 with three neigbors.

This leaves all the remaining countries with three or fewer neighbors, so the

second cycle of reduction leaves the null map, reduced map. Therefore, when

we colored in the reverse order 1, 2, 3, 6, 4, 5, each country is colored without

changing the color of any previously colored country.

If the programmer performs this reduction before he writes the goal state-

ment, he will write

goa l(R1, R2, R3, R4, R5, R6) ← next(R1, R2), next(R1, R3),

next(R1, R6), next(R2, R3), next(R2, R6), next(R3, R6),

next(R2, R4), next(R3, R4), next(R1, R5), next(R2, R5),

next(R5, R6).

This PROLOG program will run with only the most local backtracking.

Namely, after R1 has been chosen arbitrarily, several values will have to be

tried for each of the variables R1, R2, R3, R6, R4, and R5, but once a value

has been found that is compatible with the previously determined variables,

it won’t have to be changed again.

1Kempe thereby inferred that a minimal uncolorable map would not have any countries

with three or fewer neighbors.

The new PROLOG program is logically equivalent to the previous pro-

gram because it is just a rearrangement of the literals of a conjunction.

However, the programmer has done the control. The interesting question

is whether the reduction can be expressed in some way that can be regarded

as adding control to the original logic, i.e., without changing the original

logic.

4 KEMPE TRANSFORMATIONS

Another idea of Kempe’s can be used to strengthen the reduction process,

but regarding it as mere control added to the original logic program seems

even harder.

The strengthened reduction procedure also removes countries with four

neighbors so that the reduced map contains only countries with five or more

neighbors. The validity of this reduction depends on the following Kempe

proof that if we have colored a partial map and want to add a country with

four neighbors, we can always revise the coloring of the partial map to permit

coloring the four neighbor country.

If fewer than four colors have been used to color the neighbors, there is

no problem, so suppose that the four neighbors have been colored with four

different colors as shown in Figure 2.

D = red

X

B = green

A = blue

C = yellow

Figure 2.

Consider the set S of all countries that can be reached from the blue

country A on top of Figure 2 by a path going through only blue and yellow

countries. S is called the blue-yellow Kempe component of country A. There

are two cases. Either it contains country C or not. If not, we recolor the

partial map by reversing blue and yellow on all countries in S. This still

leaves the partial map properly colored.

Since S does not contain C, C remains yellow while A has become yellow.

This makes blue available to extend the coloring to X.

In the other case, S contains C, i.e, there is a chain of adjacent countries

from A to C each of which is colored blue or yellow. Then there cannot

be a red-green chain from B to D (by the topology of the plane or sphere),

so that a red-green Kempe transformation applied to the red-green Kempe

component of D will make D green, leaving red available to color X.

The fact that a blue-yellow path from A to C blocks a red-green path

from B to D is where we have used the fact that the map is on a plane or

sphere.

This justifies eliminating countries with four neighbors in the reduction

process. If we have colored a partial map and want to add a country with four

neighbors, we can do so, but we may have to modify the previous coloring

by means of a Kempe transformation.2

Our improved coloring algorithm then reduces the map by repeatedly

dropping countries with four or fewer neighbors, colors the reduced map

exhaustively, and then colors the dropped countries in the reverse order using

Kempe transformations when necessary.

5 REALIZING THE REDUCTION ALGO-

RITHM BY CONTROL OF THE PEREIRA–PORTO LOGIC

From the point of view of logic programming, successively reducing the map

by postponing countries with three or fewer neighbors is an example of a

more general notion — that of a postponable variable. A variable in the body

of a clause is postponable if, no matter how the other variables are assigned,

2Kempe mistakenly thought he could extend a coloring to a country with five neighbors

colored with four distinct colors. Had he been able to do that, he would have proved the

Four Color Theorem, because and argument using Euler’s formula E + 2 = F + V enabled

him to show that any minimal uncolorable map must have at least one country with five

or fewer neighbors.

there is a value for this variable that causes all the goals involving that

variable to be satisfied. Clearly any postponable variable can be postponed

to the end. Moreover, just as in the map coloring problem, postponing some

variables may remove enough goals involving other variables so that they in

turn become postponable.

If there were only one stage of postponement, we could regard postpone-

ment as a case of selecting the first goal to be attempted, the postponable

variables being rejected for selection. However, this wouldn’t prevent the

selection of a variable postponable in the second stage. Therefore, the post-

ponement process should be completed before any goals are selected for at-

tempt.

The postponability of a variable is expressed by a postponement lemma.

For example, the postponability of R4 is expressed by the formula

∀R2R3.∃R4.(next(R2, R4) ∧ next(R3, R4)).

Notice that our quick recognition of the postponability of R4 is based

on the symmetry. We say that whatever colors are assigned to R2 and R3,

a compatible color can be found for R4. We don’t have to enumerate the

possible assignements to R2 and R3. A program would have to do more work

unless it also discovered or was told that coloring problems are invariant when

the names of the colors are permuted.

We can imagine several combinations of programmer and computer effort

in postponing variables. We already discussed the case in which the pro-

grammer himself re-ordered the goals in the clause. The other extreme is

that the PROLOG compiler attempt to prove postponability lemmas. Since

some cases of postponability may depend on some variables already having

values, additional postponements can be accomplished by a suitable inter-

preter. Since most variables in most programs are not postponable, it seems

wasteful to have the interpreter always try for postponement. Therefore, it is

also possible for the user to specify that the compiler and/or run-time system

look for postponable variables, perhaps by enclosing the clause or part of it

within which postponable variables may be expected within a macro. Thus

the above program might be written

goal(R1, R2, R3, R4, R5, R6) ← postpone(next(R1, R2), next(R1, R3), . . . , etc.).The most powerful way of achieving postponement is for the programmer

to use the full power of PROLOG to transform the body. Alain Colmer-

auer (1981) wrote such a program for rewriting the Pereira-Porto coloring

program.

If the programmer can arbitrarily rewrite the program, he may

change the logic as well as the control. However, we can imagine that a re-

stricted set of re-arrangement operators are used that is guaranteed to only

affect the control.

I was informed by Herv´e Gallaire that the system for specifying control

described in (Gallaire 1981) could not express the postponement heuristic

for the coloring problem, but that a small modification to the system would

make it possible.

6 REALIZING THE KEMPE TRANSFOR-

MATION ALGORITHM

Realizing the Kempe transformation algorithm as control of the Pereira-Porto

logic presents a more difficult challenge to the designers of control languages

for logic programming. Of course, the postponement part of the algorithm

is the same as before; the difficulty comes when it is necessary to color a

country with four differently colored neighbors.

The first step is to identify opposite neighbors of the four neighbor coun-

try. This depends not merely on the fact that the map is planar but on the

actual imbedding in the plane. This information has been discarded when

the map is represented as a graph. If the graph is described by giving for

each country a list of its neighbors, the imbedding information can be includ-

ing by listing the neighbors in cyclic order — clockwise or counterclockwise.

Otherwise, it can be restored in general only with difficulty. Figure 3 shows

cases 3 where this isn’t trivial. Of course, we can modify the algorithm to

try every pair of vertices to see if they are unconnected by a path of their

two colors. The above argument shows that this is guaranteed to succeeed

but presumably at somewhat greater cost than if the cyclic order is known.

31996: I no longer understand how figure 3 shows such cases.

Figure 3.

Looking for a changeable country is a process of search whereby only

certain values are allowed for certain variables and goals that become unsat-

isfied are re-satisfied by changing only certain variables in certain ways. A

good control system for logic programs should permit the expression of such

strategies.

7 Acknowledgements

This research was supported by the National Science Foundation under re-

search grant no. MCS81-04877.

8 References

Colmerauer, Alain (1981). Personal communication.

Kempe, A.B. (1879). On the geometrical problem of four colors. Amer. J.

Math. 2, 193–204.

Gallaire, Herv´e (1981). Personal communication.

Kowalski, Robert (1979). Logic for Problem Solving, New York: North-

Holland.

Pereira, Luis Moniz and Antonio Porto (1980). Selective Backtracking for

Logic Programs, Departamento de Informatica, Faculdade de Ciencias e Tec-

nologia, Universidade Nova de Lisboa, Lisboa, Portugal.

/@steam.stanford.edu:/u/jmc/s96/coloring.tex: begun 1996 May 14, latexed 2000 Oct 31 at 3:47 p.m.