Retrieval and Learning in Analogical Problem Solving

Retrieval and Learning in Analogical Problem Solving
Randolph M. Jones
Pat Langley
Artificial Intelligence Laboratory
Robotics Laboratory
University of Michigan
Computer Science Department
1101 Beal Avenue
Stanford University
Ann Arbor, MI 48109-2110
Stanford, CA 94305
rjones@eecs.umich.edu
langley@cs.stanford.edu
Abstract
Terminology
E
Before continuing, it is worth defining some terms to avoid
UREK A is a problem-solving system that operates through
a form of analogical reasoning. The system was designed to
future confusion. For analogical reasoning, a basic unit of
study how relatively low-level memory, reasoning, and learn-
knowledge is the analogical case, which is further decom-
ing mechanisms can account for high-level learning in human
posed into a set of concepts and relations between those con-
problem solvers. Thus, EUREKA’s design has focused on is-
cepts. For our purposes, every analogical case corresponds to
sues of memory representation and retrieval of analogies, at the
expense of complex problem-solving ability or sophisticated
a problem situation. A problem situation is a specific set of
analogical elaboration techniques. Two computational systems
relations describing a state of the world, together with a set
for analogical reasoning, ARCS/ACME and MAC/FAC, are
of goal relations that should be achievable by applying a se-
relatively powerful and well-known in the cognitive science lit-
quence of operators to that state. Note that cases in EUREKA
erature. However, they have not addressed issues of learning,
are a bit different from those in case-based reasoning, where
and they have not been implemented in the context of a per-
formance task that can dictate what makes an analogy “good”.
“case” typically denotes an entire problem solution. At any
Thus, it appears that these different research directions have
given time, EUREKA will have a current problem situation,
much to offer each other. We describe the EUREKA system
for which it must decide on an operator to apply. This is the
and compare its analogical retrieval mechanism with those in
target problem situation. The analogical reasoning process is
ARCS and MAC/FAC. We then discuss the issues involved in
generally divided into three stages. First, a retrieval mecha-
incorporating ARCS and MAC/FAC into a learning problem
solver such as E
nism identifies a number of candidate sources from the po-
UREK A.
tential analogies stored in memory. Next, the set of candidate
sources undergo further elaboration to fill out the potential
We are interested in the low-level memory, learning, and
mappings between each source and the target. Finally, evalu-
reasoning processes that give rise to improvement in problem-
ation of each candidate source determines how well each can-
solving behavior over time. EUREKA is the problem-solving
didate will serve as an analogical source for the target. Let us
architecture we are using to study these processes. An explicit
now turn to a description of E
assumption within E
UREKA in these terms.
UREKA’s design is that all processes are
aspects of analogical reasoning. In addition, we designed the
An overview of EUREKA
system so that the low-level retrieval and matching processes
would dominate its behavior. The system does not possess
Jones (1993) presents the computational details of EUREKA,
or learn the types of high-level control knowledge found in
but here we provide a general overview of the system. EU-
other problem-solving systems. Our intent is to investigate
REKA adopts a reasoning formulation called flexible means-
how much of human learning in problem solving can be mod-
ends analysis (Jones & VanLehn, 1994; Langley & Allen,
eled with such low-level mechanisms.
1991). As described above, each problem situation includes
This paper presents an overview of EUREKA’s architecture
a current world state and a set of goal conditions to which the
and some of the learning results it accounts for. We then turn
state should be transformed. Operator selection creates a goal
our attention to two well-known analogical retrieval mech-
to apply a particular operator to the current state of the prob-
anisms in the cognitive science literature. ARCS (Thagard,
lem situation. If the preconditions of the operator can all be
Holyoak, Nelson, & Gochfeld, 1990) and MAC/FAC (Gen-
matched to the current state, the operator executes, leading to
tner & Forbus, 1991) model psychological findings on ana-
a new problem situation with a different state but the same
logical retrieval and reasoning. However, neither has been
goals. Otherwise, the system sets up a new problem situation
examined in the context of a problem-solving system, or in a
with the same current state, but with the operator’s precondi-
system that learns with experience. The remainder of the pa-
tions as the new goals. EUREKA then treats this new problem
per focuses on the issues of analogical retrieval and learning,
situation in a recursive manner.
and discusses the possibilities of incorporating these alterna-
The difference between flexible means-ends analysis and
tive analogical retrieval mechanisms into a problem-solving
standard means-ends analysis (Ernst & Newell, 1969; Fikes
system.
& Nilsson, 1971) is that the flexible form does not require se-

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lected operators to apply directly to the current goal conditions
Table 1: EUREKA’s algorithm for spreading activation.
(i.e., it is not necessary that the selected operator obviously
“reduce any differences”). Rather than using this heuristic
Let ACTIVATION_THRESHOLD be 0.01;
to limit search, EUREKA relies on its retrieval and learning
Let DAMPING_FACTOR be 0.4;
mechanisms to control which operators are suggested to ap-
Let INITIAL_ACTIVATION be 1.0;
ply to any particular problem situation. Because operator se-
lection depends on the entire problem situation (and not just
SPREAD_INIT(Source)
the goals), EUREKA can blend goal-driven and opportunistic
SPREAD(Source,INITIAL_ACTIVATION,NIL)
behavior when appropriate.
Every time EUREKA generates a new problem situation, it
SPREAD(Source,Value,Path)
stores a representation of the situation (as well as the opera-
If (Value < ACTIVATION_THRESHOLD) or
tor the led to this situation) into its long-term semantic net-
Source is in Path
work. Each object and relation in a problem situation becomes
Then EXIT
a node in the semantic network. In addition, the network
Else Increase Source.Activation
stores nodes representing instances of architecturally defined
by Value;
concepts, such as problem situations and operators. Items
For each link X from Source
are never deleted from long-term memory, and memories are
Let Target be the node
never stored in an abstract form. Rather, the semantic mem-
connected to Source by X;
ory stores all the specific problem situations that it encoun-
Let Newvalue be
ters. Situations become linked together in memory when they
SPREAD_VALUE(Source,X,Value)
share objects, relations, or object types. If a particular con-
* DAMPING_FACTOR;
cept from a problem situation already exists in memory, EU-
PUSH Source onto Path;
REKA increases the trace strengths of the links from the con-
SPREAD(Target,Newvalue,Path)
cept, rather than adding a new copy of the concept.
When EUREKA is working on a particular problem situa-
SPREAD_VALUE(Source,Link,Value)
tion, it must select an operator to apply to the problem. To
Let Total be 0;
this end, EUREKA retrieves a subset of the stored problem
For each link X from Source
situations from long-term memory. This small set of candi-
If X has the same type as Link
date sources is further elaborated and evaluated, to see which
Then Increase Total
would provide the best candidate analogy for the current prob-
by X.trace_strength;
lem situation. EUREKA chooses one candidate stochastically,
Return Value *
based on the evaluation score, and identifies the operator as-
(Link.trace_strength / Total)
sociated with that source analogy. Finally, the system creates
a goal to apply to the newly mapped operator to the current
state.
first retrieves a set of candidate source problem situations
EUREKA proceeds in this manner until it solves the prob-
from memory. The second involves a relatively expensive
lem or the current solution path fails (by exceeding a time limit
computation to elaborate the mapping between each candidate
or detecting a cycle in the solution path). Upon failure, EU-
source and the target problem situation. Because the elabora-
REKA does not have the luxury of backtracking, which would
tion process is so expensive, it is important that the cheaper
allow the system to search the problem space systematically
retrieval process return a relatively small set of candidates.
and possibly exhaustively. Rather, EUREKA begins the prob-
However, the system must also do what it can to make sure
lem anew from the initial problem situation. The inability to
it does not miss good candidates in memory. In EUREKA, we
backtrack systematically greatly hinders the system’s ability
have focused on the retrieval phase, to analyze how changes in
to solve problems, but we feel that this is a psychologically
retrieval patterns can lead to higher-level changes in problem
plausible limitation. The limitation also places further impor-
solving.
tance on effective learning.
The combination of E
As mentioned above, EUREKA stores in its semantic net-
UREKA’s learning mechanisms and
its stochastic selection process encourage the system to ex-
work an episodic memory of every problem situation it en-
plore alternative solution paths on subseqent attempts to solve
counters. Retrieval is implemented as a spreading-activation
a problem. However, there is no guarantee that a previous
process similar to that found in ACT (e.g., Anderson, 1976).
Each node in the representation of the target problem situation
search will not be duplicated. If the system fails to find a solu-
tion after a preset number of attempts (50 in our experiments),
becomes a source of activation, which then spreads to other
it abandons the problem completely.
nodes according to the strengths of the links to those nodes.
The activation algorithm appears in Table 1.
Analogical retrieval in EUREKA
After the spread of activation terminates, EUREKA checks
EUREKA’s analogical reasoner incorporates two stages. The
the “top-level” node for each problem situation stored in

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memory. This node contains a unique name for the problem
Table 2: Robust learning behaviors identified in human prob-
situation and has links to all the nodes representing relations
lem solvers (VanLehn, 1989).
in the problem situation. The problem situation whose top-
level node has the highest level of activation becomes a candi-
1. Subjects reduce their verbalizations
date source. In addition, any other problem situation becomes
of task rules as they become more
a candidate source if its top-level node has at least one percent
experienced with practice.
of the level of activation of the strongest source.
2. Improvement occurs quickly in
knowledge-lean domains.
Learning in EUREKA
3. There is a power-law relationship
As Table 1 indicates, the activation that spreads from a source
between the speed of performance on
node, , to another node, , depends on the number of nodes
perceptual-motor skills (and
i
j
to which is linked (because activation is divided among links
possibly problem-solving skills) and
i
of the same type), as well as the strength of the link between
the number of practice trials.
i
and . This highlights an important aspect of the retrieval pro-
4. Problem isomorphs do not become more
j
cess within EUREKA. That is, the spread of activation (and,
difficult simply by changing surface
therefore, patterns of retrieval) can change for primarily two
features of the problems.
reasons: new nodes and links being added to memory, and
5. Other representation changes can
changes in link strengths. It follows that these are the two
make problem isomorphs substantially
ways that learning can change behavior in EUREKA.
more difficult.
Thus, one way in which EUREKA learns is simply by
6. There is asymmetric transfer between
adding new experiences by rote into memory.
Because
tasks when one task subsumes
spreading activation is a competitive process, introducing
another.
more competitors into memory can change what gets retrieved
7. Negative transfer is rare.
in the future. However, simply adding experiences to memory
8. "Set" effects (or Einstellung) can
will not necessarily improve problem-solving behavior, which
lead to negative transfer.
is what we really want. Therefore, EUREKA also changes its
9. Spontaneous noticing of a potential
behavior by updating link trace strengths. When the system
analogy is rare.
solves a particular target problem situation, it checks which
10. Spontaneous noticing is based on
source analog was used to help solve the problem. The system
superficial features.
then strengthens the links between the target problem situation
and the successfully applied analogical situation. Note that
a problem presented to EUREKA generally involves a set of
details here so we can discuss other issues. EUREKA’s re-
problem situations, so EUREKA can learn about solved prob-
trieval and learning methods directly account for most of the
lem situations, even if the attempt to solve the global prob-
behaviors identified by VanLehn. Behaviors 1 and 3 require
lem fails. In the long run, stored problem situations that help
a bit of extra interpretation, and are not modeled as well as
solve new problem situations become strongly connected to
the others. In general, the results indicate that these types of
the problem situations that they help solve. Thus, they “soak
learning can indeed arise from rather low-level processes.
up” more activation from future problem situations, and be-
There are other models of analogical problem solving (e.g.,
come more easily retrieved.
Hammond, 1986; Veloso & Carbonell, 1993), which rely on
It is also worthwhile to note that EUREKA requires its learn-
indexing methods for analogical retrieval, and generally fo-
ing mechanism to be noise tolerant. Because operator selec-
cus on learning and reasoning at a higher architectural level
tion is based on the current structure of memory and the sys-
than EUREKA. In contrast, there are other analogical mecha-
tem cannot systematically backtrack, a search path that leads
nisms that share EUREKA’s spirit in modeling analogical rea-
to failure now may turn into a successful path later. EUREKA
soning as a relatively low-level memory process. The follow-
might fail simply because it does not “remember” or retrieve
ing section discusses two of the more well-known models of
the appropriate operator in a particular situation. As the sys-
this type.
tem gathers experience, it may learn to retrieve such opera-
tors, turning bad search paths into good ones.
Learning to solve problems with ARCS and
Qualitative behaviors exhibited by EUREKA
MAC/FAC
Table 2 presents VanLehn’s (1989) list of a number of robust
ARCS (Thagard et al., 1990) and MAC/FAC (Gentner & For-
qualitative results that have been observed in humans learning
bus, 1991) are the analogical retrieval algorithms associated
to solve problems. EUREKA addresses these issues to vary-
with two relatively well-known and sophisticated systems for
ing degrees. Jones and Langley (1994; Jones, 1989) present
analogical elaboration and evaluation: ACME (Holyoak &
a number of detailed experiments with EUREKA that address
Thagard, 1989) and SME (Falkenhainer, Forbus, & Gentner,
these results. Due to a lack of space, we will not present the
1989). It is attractive to consider incorporating these sys-

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tems into a learning, problem-solving architecture such as
space of such concepts ahead of time. MAC then similarly
EUREKA. We feel such an attempt could benefit research on
computes a content vector for the target of the analogy. The
both sides. On the one hand, EUREKA’s analogical mecha-
retrieval score for each potential source is computed as the dot
nisms have been used to model human learning in problem
product of the source’s content vector with the target’s content
solving, but it is questionable whether they can model some
vector. This gives an estimate of the degree to which relations
of the psychological findings on analogical retrieval and eval-
are shared between the target and each source, and it is very
uation (e.g., Gick & Holyoak, 1983). On the other hand,
quick to compute. The source with the largest dot product is
ARCS/ACME and MAC/FAC have both been demonstrated
marked as a retrieved candidate. In addition, any other source
on the retrieval and evaluation results, but they have so far
with a dot-product value of at least 10% of the highest value
been used to model analogical retrieval in relative isolation
is retrieved.
from other tasks. Although both systems have built-in no-
Changes in retrieval as knowledge increases
tions of what makes a good analogy, it is sometimes difficult
to judge why other potential analogs or mappings might not
Having a feel for how ARCS and MAC/FAC work, let us turn
be better in particular situations. A problem solver provides
our attention to how their behavior might adapt with experi-
a context by which to judge the quality of analogies more ob-
ence. We have stressed that our primary focus in EUREKA
jectively: A good analogy is one that helps solve a problem.
is on how learning can change retrieval patterns, leading to
In addition, our work with EUREKA has focused on how ana-
larger changes in problem-solving behavior. Thus, it is most
logical reasoning can adapt with experience, but the two other
important for us to examine the types of events that allow EU-
systems have so far not incorporated mechanisms for chang-
REKA’s retrieval mechanism to learn, and how they would ap-
ing their behavior over time. The remainder of this paper dis-
ply to MAC/FAC or ARCS. Let us first consider how the mere
cusses the issues we foresee in incorporating the ARCS and
storage of new experiences can influence retrieval. There are
MAC/FAC retrieval mechanisms into a problem-solving sys-
two aspects of performance to examine when the knowledge
tem that learns. First, let us provide a quick overview of the
base increases in size. First, new knowledge may change the
ARCS and MAC/FAC retrieval methods.
time it takes for the retrieval algorithm to execute. Second,
the resulting set of candidate sources may change as new po-
Retrieval with ARCS
tential sources are added to memory.
ARCS divides the retrieval process into two stages, beginning
Taking the first issue, EUREKA’s spreading-activation
with a table look-up for each concept in the target. This table
mechanism performs a limited search through memory, and
provides a list of all the concepts in memory that are immedi-
many portions of memory (those distant from the target con-
ately related to a concept (e.g., by subordinate, superordinate,
cepts) will be completely ignored by the retrieval process.
or part-of relationships). ARCS then considers retrieving any
Jones (1989, 1993) has demonstrated empirically and analyti-
source that includes at least one of the collection of concepts
cally that EUREKA’s form of spreading activation takes a con-
related to the target. This happens by creating a constraint net-
stant amount of time relative to the size of memory, even when
work of possible concept matches, linked together by excita-
implemented as a serial algorithm. On the other hand, the
tory and inhibitory links. ARCS only sets up match hypothe-
specific representation or “shape” of memory can sometimes
ses between semantically similar concepts (from the look-up
have a significant impact on retrieval time. This is an impor-
table). More complete match hypotheses are saved for the
tant issue, related to the utility problem (Minton, 1988) in ma-
more expensive ACME matcher. In addition, special nodes
chine learning. It would not be desirable for a system to slow
are created to link concepts that have been marked as impor-
down merely because its memory is growing. However, both
tant in various ways. Finally, activation spreads throughout
ARCS and MAC/FAC are guaranteed to take longer as new
the network until the network settles. Each candidate source
analogical sources are added to memory, because they both
receives a retrieval score, computed from the activation of
examine every potential source as part of the retrieval pro-
the concepts in the source. It is not clear, however, that the
cess. This means that processing is at least linearly related to
ARCS system makes a distinction between candidate sources
the number of potential source analogs in memory. ARCS in-
that “are retrieved” vs. those that are not.
cludes an even more expensive construction of the constraint
network, which depends on the number of sources that include
Retrieval with MAC/FAC
concepts similar to those in the target. Thagard et al. (1990)
MAC/FAC1 takes quite a different approach to retrieval.
propose, however, that much of their algorithm can execute
MAC computes a content vector for each source stored in
in parallel, so time will be constant if we assume an arbitrary
memory. The content vector ignores concepts that represent
number of processors (one per potential source stored in mem-
simple objects, and records the number of occurrences of each
ory). Presumably the same is true of MAC/FAC.
concept that can take other concepts as arguments (e.g., rela-
Let us next consider how the mere addition of cases to the
tions and functions). This requires MAC to know the entire
knowledge base can change retrieval patterns. New problem
situations in EUREKA’s memory imply new competitors for
1
Note that MAC corresponds to the analogical retrieval mecha-
nism, while FAC is the more expensive elaboration and evaluation
activation. Thus, if a newly stored situation shares concepts
algorithm.
with other problem situations, activation levels necessarily

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change. In contrast, MAC/FAC independently associates a
an attractive option, because it would allow a more pragmatic
content vector and retrieval score with each stored situation.
view of similarity that adapts to problem-solving experience,
The retrieval score has absolutely nothing to do with other
rather than requiring a fixed table of similarities to be provided
stored cases. Thus, adding new cases will have a limited im-
to the system. In addition, ARCS gives extra strength to con-
pact on the set of retrieved candidate sources. Because the
cepts that are marked as “important” and mappings that are
retrieval threshold is based on a percentage of the highest-
marked as “presumed”. These marks are specified and fixed
valued retrieval item, a newly stored problem situation can
before retrieval begins. One of the benefits of a performance
only significantly change retrieval patterns in the cases where
task, such as problem solving, is that it is possible to induce
it receives the highest retrieval score.
important concepts over time. Perhaps such an induction al-
ARCS uses an activation process that is inherently com-
gorithm could use experience to adapt the measure of “impor-
petitive like Eureka’s. Thus, the final retrieval values for
tance” of concepts over time.
each candidate source can certainly change as memory grows.
MAC/FAC is another story entirely, because it does not
Law, Forbus, and Gentner (1994) showed that increasing the
share the notion of competition between candidate analogi-
number of cases can be quite detrimental to ARCS, but did
cal sources. As we have mentioned above, retrieval compu-
not adversely affect MAC/FAC’s behavior. Presumably, sim-
tations are independent for each target-source pair. However,
ilar detrimental effects would be seen in EUREKA’s retrieval
the one thing that is common across candidate sources is the
mechanism as cases begin to compete with each other. On the
algorithm for computing the content vector. When construct-
other hand, EUREKA (and most likely ARCS) can also ben-
ing the vector, MAC/FAC counts each occurrence of every
efit from the addition of appropriate knowledge. MAC/FAC
feature. These counts could instead be weighted by param-
can only benefit in the sense that it may have a new case to
eters for each feature, which would be tuned with experience.
retrieve, but it cannot benefit in any competitive sense. Thus,
There is a potentially more interesting alternative for learn-
we interpret Law, Forbus, and Gentner’s result not as a con-
ing. It almost seems that there is a built-in assumption to
demnation of competitive retrieval algorithms, but as further
MAC/FAC: changes in analogical retrieval should only arise
evidence of the importance of associating a learning method
through a reformulation of the representation of the sources
with retrieval.
and targets. It is not clear whether the creators of MAC/FAC
intend this to be a fixed, architectural constraint, but it is cer-
Tuning retrieval with experience
tainly interesting to view it that way. In this case, the only
The second aspect of learning has to do with tuning the
way for MAC/FAC to tune its retrieval patterns would be
retrieval process to improve and focus itself with experi-
for it to change the representation of its stored cases. These
ence. Again, EUREKA achieves this by increasing link trace
changes would be based on knowledge of how the retrieval
strengths associated with stored problem situations when they
scoring mechanism works, and would be designed to award
aid in the solution of new problem situations. With appro-
useful candidate sources higher scores in future similar situ-
priate experiences, the system will eventually retrieve fewer
ations. It is not clear to us what the details of such a mecha-
items, but they will have higher estimated quality. None of
nism would be. However, it would provide a pragmatic ap-
the presentations of ARCS or MAC/FAC have addressed the
proach to changing representation within a cognitive agent.
issue of learning. Thus, rather than comparing learning mech-
The agent would change its representations in response to
anisms, we are free to hypothesize the types of learning mech-
problem-solving success (or failure), and would change them
anisms that might be amenable to ARCS and MAC/FAC.
in such a way as to improve future behavior.
ARCS has a competitive activation-based retrieval mecha-
nism, so it is tempting to assume that it would benefit from a
Summary
learning mechanism similar to EUREKA’s. However, it is im-
EUREKA provides a model of analogical retrieval in problem
portant to note that activation spreads through a very different
solving. In addition, it incorporates a learning mechanism that
type of network in each case. EUREKA’s semantic network
focuses on tuning the retrieval of candidate analogical sources
is a long-term structure encoding the representation of prob-
from memory. These relatively low-level mechanisms give
lem situations and the relations and objects they share. In con-
rise to larger qualitative changes in problem-solving behav-
trast, the constraint networks in ARCS are constructed anew
ior. We have used the EUREKA model to explain the primary
each time a target is presented, and represent potential ways to
learning effects that have been identified in human problem
match the concepts in each source to the concepts in the tar-
solvers. Because the system includes a memory-based mech-
get. Despite these differences, there still seems to be some po-
anism for the retrieval of analogies, it is natural to compare
tential to altering link strengths in ARCS. Some link strengths
this mechanism to ARCS and MAC/FAC, two well-known
are fixed measures of the degree of similarity between differ-
retrieval mechanisms in the cognitive science literature. We
ently related concepts (e.g., synonyms have a similarity value
have used the lessons learned from building EUREKA to guide
of 0.6, superordinates a value of 0.3, and subordinates a value
our analysis of how ARCS and MAC/FAC would fit into the
of 0.2). There does not seem to be any reason in principle
context of a performance task (problem solving) where learn-
that these similarity measures could not be learned for specific
ing can and should take place. EUREKA demonstrates that
concept pairs, rather than fixed by these abstract types. This is
low-level mechanisms can have a significant impact on high-

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level behavior. It will be interesting to see what qualitative
Minton, S. (1988). Learning effective search control knowl-
differences arise in learning problem solvers that incorporate
edge: An explanation-based approach. Boston: Kluwer
the ARCS or MAC/FAC algorithms.
Academic.
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