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Computer Science Department
Stanford, CA 94305
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
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
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
goal(R1, R2, R3, R4, R5, R6) ←
where each literal expresses the fact that a particular pair of adjacent regions
must be compatibly colored.
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
l(R1, R2, R3, R4, 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
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
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
B = green
A = blue
C = yellow
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
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-
The postponability of a variable is expressed by a postponement lemma.
For example, the postponability of R4 is expressed by the formula
next(R2, 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, 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
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-
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.
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
This research was supported by the National Science Foundation under re-
search grant no. MCS81-04877.
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-
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.