Rediscovering Physics With BACON.3*
Pat Langley
Department of Psychology
Carnegie-Mellon University
Pittsburgh, Pennsylvania 152:3
BACON.3 is a production system that discovers empirical laws. The program uses a few simple heuristics to
solve a broad range of tasks. These rules detect constancies and trends in data, and lead to the formulation
of hypotheses and the definition of theoretical terms. BACON.3 represents data at varying levels of
d e s c r i p t i o n , where the lowest have been directly observed and the highest correspond to hypotheses that
explain e v e r y t h i n g so far observed. The system can also run and relate multiple experiments, collapse
h y p o t h e s e s w i t h identical conditions, ignore differences between similar concepts, and discover and ignore
i r r e l e v a n t variables. BACON.3 has shown its generality by rediscovering versions of the Ideal gas law,
Kepler's t h i r d law, Coulomb's law, Ohm's law, and Galileo's laws for the pendulum and constant acceleration.
505
3. LEVELS OF DESCRIPTION
t h e line relating the t w o variables. Otherwise, a new
t e r m is d e f i n e d as the product or the ratio of the
v a r i a b l e s , depending on the numbers involved.
As t h e reader may have guessed, BACON.3 rediscovers
t h e ideal gas law in a manner much like the above.
BACON.3 uses strategies very similar to those used by
its p r e c u r s o r , BACON. 1 [1] yet BACON.3 can discover
t h e gas law while its predecessor could not. BACON. 1
made a sharp distinction between the data it had
o b s e r v e d and the hypotheses which explained those
d a t a . BACON.3 blurs the distinction between data and
h y p o t h e s e s by allowing various levels of description.
In t h e new program, regularities in one level of
d e s c r i p t i o n lead to the creation of a higher level of
description.
A f t e r a theoretical term has been defined, a third set
of 3 productions calculates the values of this term.
Once calculated, these values are fair game for the
regularity
detectors.
Defined terms are not
d i s t i n g u i s h e d from direct observables when noting
r e g u l a r i t i e s ; it is this recursive ability to apply the
same heuristics to concepts of increasing complexity
w h i c h gives BACON.3 its power.
B e f o r e calculating the values of a new theoretical
t e r m , BACON.3 must make sure that the term is not
e q u i v a l e n t to an existing concept.
Accordingly, a
f o u r t h set of 22 productions decomposes the new
t e r m into its primitive components. If the definition of
t h e new t e r m is identical with an existing definition,
t h e t e r m is t h r o w n out and other relations are
considered.
Like the earlier program, BACON.3 is implemented as
an OPS2 [ 2 ] production system.
BACON.3 shares a
n u m b e r of heuristics with BACON. 1, though these have
b e e n generalized to deal with any level of description.
Like BACON. 1, the new system defines theoretical
t e r m s like pV, pV/T, and pV/nT to describe its data
parsimoniously.
These heuristics and others are
discussed in the following section.
Suppose BACON.3 has defined two intercept concepts
f o r the ideal gas data.
The values of the first,
i n t e r c e p t p v , t , 1 are 0 when the number of moles is 1,
w h i l e the values of the second, intercept p v , t , 2 0
w h e n the number of moles is 2.
One would like
BACON.3 to generalize at this point, but because the
t w o i n t e r c e p t s are different terms, the constancy
d e t e c t o r cannot be applied.
BACON.3 notes such
similar terms, and defines an abstracted term which
i g n o r e s their differences. The values of the new term
a r e c o p i e d from the originals, and the constancy
d e t e c t o r is applied to the new data.
4. THE HEURISTICS OF BACON3
The BACON.3 program consists of some 86 0PS2
p r o d u c t i o n s . These can be divided into seven major
s e t s , w h i c h I discuss below. The first four sets are
h e l d in common w i t h the BACON. 1 system; the final
three
are
additions
required
by
the
new
r e p r e s e n t a t i o n and the tasks BACON.3 must handle.
The f i r s t set of productions is responsible for
g a t h e r i n g d i r e c t l y observable data.
Seven of these
a r e responsible for gathering information from the
user about the task to be considered. The remaining
10 p r o d u c t i o n s gather data through a standard
f a c t o r i a l design, v a r y i n g first one independent term,
t h e n another.
BACON.3
generates
different
descriptions
to
summarize different constancies. If two descriptions
a r e f o u n d to have identical conditions, they are
c o m b i n e d into a single structure; only 3 productions
a r e d e v o t e d to this process. Once this has happened
to a number of descriptions, the values of the
d e p e n d e n t terms can be compared and regularities
may e m e r g e .
T h e second set of 16 productions is responsible for
n o t i n g regularities in the data collected by the first
set.
BAC0N.3's constancy detectors can deal with
e i t h e r symbolic or numerical data, and lead to the
c r e a t i o n of higher level descriptions.
The basic
c o n s t a n c y detector is a simple restatement of the
traditional
inductive
inference
rule
for
making
g e n e r a l i z a t i o n s . Similar rules add conditions to newly
c r e a t e d hypotheses.
The program has primitive
f a c i l i t i e s f o r dealing w i t h near constancies in noisy
d a t a ; this is accomplished by redefining the LISP equal
f u n c t i o n to ignore small differences.
In r e d i s c o v e r i n g Galileo's law for pendulums, BACON.3
b e g i n s by v a r y i n g the weight of the suspended object
and the initial angle of the string. These variables are
i r r e l e v a n t to the period of the pendulum, but this is
not o b v i o u s from the outset.
BACON.3 draws on a
f i n a l set of 8 productions for noting irrelevant terms.
These modify the data gathering scheme so the values
of t h e i r r e l e v a n t terms are no longer varied.
BACON.3'* t r e n d detectors operate only on numerical
d a t a . Some of these notice increasing and decreasing
monotonic trends between variables. These heuristics
w o r k in conjunction w i t h other trend detectors that
f u r t h e r analyze the data. One of these applies if the
e l o p e is constant, and leads to the definition of two
n e w t h e o r e t i c a l terms, the slope and the intercept of
506
5. THE GENERALITY OF BACON.3
BACON.3 successfully rediscovered the five laws
summarized in Table 4.
These equations do not
e n t i r e l y do justice to BAC0N.3's discoveries.
Along
w i t h omitting the conditions placed on some of the
l a w s , only one equation is shown for each task, while a
n u m b e r w e r e formulated for some. However, they do
suggest the diversity of the laws the program
g e n e r a t e d from its data. The ability to define ratios
and p r o d u c t s leads to terms taken to a power, as in
Coulomb's and Galileo's laws. The abstraction strategy
allows the use of linear combinations in new terms, as
In Ohm's law.
Taken together, these two strategies
lead to a v e r s i o n of Kepler's third law, in which the
s q u a r e of a linear combination plays a role.
Table 5 presents statistics on the relative complexity
of t h e laws found by BACON.3. Three measures are
u s e d - the number of productions fired, the average
size of w o r k i n g memory, and the average size of the
c o n f l i c t set.
These measures are not completely
c o r r e l a t e d , but one trend is clear; the discovery of
Ohm's law r e q u i r e d much more computation than did
t h e o t h e r tasks.
TABLE 5 RELATIVE COMPLEXITY OF THE FIVE TASKS
Closer analysis reveals some of the reasons for the
c o m p l e x i t y of this task.
Table 6 presents some
c h a r a c t e r i s t i c s of the problem spaces for the five
tasks.
Since more terms were related in this law,
m o r e levels of description were needed to arrive at it.
M o r e o v e r , t w o completely independent sets of laws
w e r e d i s c o v e r e d In this run. These required a large
n u m b e r of theoretical terms, and a still greater number
of t e r m s w h i c h w e r e considered and rejected.
IDEAL GAS KEPLER COULOMB GALILEO OHM
How g e n e r a l w e r e the heuristics
h e u r i s t i c s b o r r o w e d from BACON. 1
of t h e tasks. Table 7 shows those
of BACON.3'5 new heuristics were
heuristics w e r e used in multiple tasks, suggesting
considerable generality for the rules.
The single
e x c e p t i o n is misleading, since irrelevant variables
could be added to each of the tasks.
IDEAL GAS KEPLER COULOMB GALILEO OHM TOTAL
TABLE 7 USE OF BACON 3'S HEURISTICS
A general discovery system should be sensitive to the
o r d e r in which it observes its data, but robust enough
to a r r i v e at equivalent laws regardless of the order.
In the Galilean run reported in Table 4, the irrelevant
v a r i a b l e s weight and angle were varied first and
immediately found to have no influence. In a second
r u n w h e r e these variables were varied last, identical
laws w e r e eventually reached but the computation
r e q u i r e d was greater.
M o d i f y i n g the order of relevant variables also affected
t h e behavior of the system. In a second run on the
ideal gas law, the number of moles was varied first,
f o l l o w e d by the temperature, followed by the pressure.
In the initial r u n , three major theoretical terms were
g e n e r a t e d - pV at level 1, pV/T at level 2, and pV/nT
at level 3. In the second run, a different path was
t a k e n to the same conclusion - V / n was defined at the
f i r s t l e v e l , V/nT at the second level, and pV/nT at the
t h i r d level.
In summary, BACON.3 is a production system that can
r e d i s c o v e r a number of laws from the history of
physics.
The system draws on a small number of
s t r a t e g i e s for finding regularities, defining terms,
ignoring
differences, collapsing hypotheses, and
d e t e r m i n i n g Irrelevant variables. Like its predecessor,
BACON.3 is a general discovery system. One piece of
evidence for this claim is that BACON.3 solved five
d i f f e r e n t tasks using the same small set of heuristics.
A second point in favor of BACON.3's generality was
its ability to resolve tasks when the data were
g a t h e r e d in different orders.
In conclusion, the
p r o g r e s s to date has been encouraging, and suggests
that more interesting discoveries lie ahead.
[ 1 ] Langley, P.
BACON. 1: A general discovery
system. In Proc. CSCSI, 1977, 173-180.
of BACON.3? The
were used in each
tasks in which each
used. Most of the
[ 2 ] Forgy, C. and McDermott, J.
0PS2 Manual.
P i t t s b u r g h , Pa.: Carnegie-Mellon University,
Department of Computer Science, 1977.
507