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A Change for the Better? Assessing the Computational Cost of Re-representation

Todd Wareham (harold@mun.ca) and Robert Robere (robere.it@gmail.com)

Department of Computer Science, Memorial University of Newfoundland

St. John’s, NL Canada A1B 3X5

Iris van Rooij (i.vanrooij@donders.ru.nl)

Donders Institute for Brain, Cognition, and Behaviour

Radboud University Nijmegen, Nijmegen, The Netherlands

Abstract

The ability to re-represent information—i.e., to see things in

new ways—is essential for human reasoning, creativity, and

learning. It forms the foundation of insight problem solving

and scientiﬁc explanation, and is hypothesized to play a pivotal

role in concept development in children. Re-representation is

useful because it allows a cognizer to make sense of things

in ways that were previously impossible. Yet, invoking this

operation can quickly become computationally intractable in

light of the combinatorial explosion of re-representation op-

tions to consider. Although this intractability may explain

why discovering useful ways of re-representing information

can be cognitively challenging at times (as in insight puz-

zles and creativity), it seems difﬁcult to reconcile with au-

tomatic and apparently effortless forms of re-representation

(as in everyday analogizing and children’s development of

concepts). To get more insight into the conditions that can

make re-representation tractable, we performed computational

complexity analyses of a formal model of re-representation

as invoked in analogy derivation. We will discuss how

our complexity results can help explain when and why re-

representation can be invoked effectively and efﬁciently.

Keywords: Re-representation; Analogy; Structure Mapping;

Computational Complexity; Fixed-Parameter Tractability

Introduction

Many theories of cognitive abilities operate on mental repre-

sentations of information, each of which assumes a particular

encoding of relevant situations and concepts. As there are

typically many possible encodings, one’s initial representa-

tion may in fact be inappropriate for the task at hand, slowing

down or even stopping the ability from functioning. In such

cases, it is hypothesized that humans change encodings and

re-represent stored information.

Re-representation has been invoked in many cognitive the-

ories. It allows natural analogies that rely on semantic rather

than syntactic matching (e.g., “Bob running into the cave”

is like “Alice walking into a room” ⇒“Bob moves into the

cave” is like “Alice moves into the room” (Gentner & Kurtz,

2006)). The powerful abstractions generated by such re-

representation in turn have been hypothesized to underlie cer-

tain types of concept development in children (e.g., the emer-

gence of abstract relations and attributes: “X is hotter than Y”

⇒“temperature(X) is greater than temperature(Y)” (Gentner,

Rattermann, Markman, & Kotovosky, 1995)). More radi-

cal types of re-representation can in turn lead to totally new

ways of envisioning particular situations and concepts, and

thus have been invoked in theories of insight problem solving

(Ohlsson, 1992), scientiﬁc discovery (Gentner et al., 1997),

and creativity (Welling, 2007).

Investigating re-representation experimentally is difﬁcult,

but there is increasing evidence for its psychological reality

(Gentner & Kurtz, 2006; Kurtz, 2006). This has motivated the

development of computational theories of re-representation

(e.g., Ohlsson (1992); Yan, Forbus, and Gentner (2003);

Krumnack, Gust, K¨

uhnberger, and Schwering (2008)), which

has raised the following conundrum: The combinatorial ex-

plosion of re-representation options that must be considered

in such theories seems to be computational intractable. This

intractability may explain why certain cognitive activities in-

voking re-representation like insight problem solving and cre-

ativity are challenging, but is at odds with how other forms of

re-representation invoked in everyday reasoning or cognitive

development seem so effortless and automatic.

In this paper, we assess the computational difﬁculty of a ba-

sic type of re-representation, namely individual predicate re-

representation within Gentner’s Structure Mapping Theory of

analogy derivation (SMT) (Gentner, 1983; Yan et al., 2003).

We give the ﬁrst proof that such re-representation is compu-

tationally intractable, even when invoked in the context of

incremental rather than general analogy derivation. This ﬁnd-

ing indicates that constraints on both the re-representation

process and its inputs must be exploited to yield tractability.

In the second part of this paper we illustrate a methodology

suitable for identifying such constraints. We also discuss how

our results can help explain when and why re-representation

can be invoked effectively and efﬁciently.

Computational-level Models

Analogies are deﬁned over concepts, which are repre-

sented in SMT by predicate-structures composed of enti-

ties (e.g.,SUN ,PL AN ET) and predicates relating those en-

tities (as well as other predicates) (e.g., ATT RACT S(SU N,

PL AN ET)). Predicate-structures are naturally represented as

vertex-labelled directed acyclic graphs in which entities are

leaves, predicates are internal vertices, and predicates are

linked to their arguments by arcs (see part (a) of Figure 1).

An analogy “Tis (like) a B”, where Band Tare predicate-

structures, is a mapping from a portion of Bto a portion of T

that satisﬁes the following three conditions:

1. The mapping is structurally consistent,i.e., match-

ing relations must have matching arguments and any

element in one predicate-structure matches at most

one element in the other.

111

Attracts Revolve

Mass

Greater Attracts Revolve

nucleus electron

ChargeCharge

sun

Mass

Cause

Greater Attracts Revolve

And

Cause

Mass Mass

sun planet

Cause

Opposite−Sign Greater Attracts RevolveGravity

nucleus electron

ChargeCharge

Greater

planet

a)

b)

Figure 1: Analogy Derivation in Structure-Mapping Theory.

(a) Two graph representations of predicate structures encod-

ing descriptions of the solar system (left) and the Rutherford

model of the atom (right). (b) An optimal analogy between

the structures in (a), where dotted arrows indicate the map-

pings between corresponding pairs of predicates and objects.

2. Relational focus: The mapping must involve com-

mon predicates but need not involve common objects,

i.e., matched predicates must have the same type, ar-

gument, number and order but matched objects need

not have the same name.

3. Systematicity: The mapping tends to match inter-

connected, deeply-nested predicate-substructures.

Let val(A)be the systematicity of an analogy A. The most

systematic analogy between a pair of predicate-structures is

an optimal analogy (see part (b) of Figure 1).

Under SMT, re-representation of predicates is only invoked

to better the analogical match between two given predicate-

structures. As such, it relaxes identical-only predicate-type

matches (e.g., ATTRACTS →ATTRACTS) to allow selected

non-identical matches (e.g., WAL K →MOVE). There are two

classes of mechanisms for performing re-representations:

1. Rule-guided (part (a) of Figure 2): A predicate of type

xcan be re-represented as a predicate of type yif there

is a rule x→y. Collections of rules can be encoded

as predicate-type similarity tables (represented explicitly

(Holyoak & Thagard, 1989) or generated implicitly by

predicate decomposition (Gentner et al., 1995)) or gen-

eralization lattices (in which the most speciﬁc predicate-

types are at the bottom of the lattice and the most abstract

predicate-types are at the top) (Winston, 1980).

2. Context-guided (part (b) of Figure 2): A predicate pof

type xcan be re-represented as a predicate of type yif p

appears in a structural context immediately “outside” an

(a)

(b)

Figure 2: Re-Representation Mechanisms in Structure-

Mapping Theory. (a) Rule-guided. (b) Context-guided.

Analogically-matched regions are enclosed by dashed boxes.

existing analogy between two predicate-structures which,

if p’s type is changed, will allow an incremental addition

to that analogy which increases its systematicity. The most

basic type of context is a “hole”, in which a pair of pred-

icates in Band Thave different types but both their argu-

ments and parents have the same types and can be matched.

Analogy derivation alternates with such re-representation un-

til a satisfactory analogy is reached. In any one round of re-

representations, it is assumed that each predicate in the given

predicate-structures can change at most once. Though we

focus here on single-predicate re-representation, more com-

plex re-representations involving larger changes in structure

are also possible (Yan et al., 2003).

Acting on all available re-representation opportunities can

both be computationally expensive and potentially result in

analogies that are meaningless or misleading, e.g., “ana-

logical hallucinations” (Gentner & Kurtz, 2006, Page 616).

There are many possible strategies for selecting which re-

112

representations to perform. Two general principles underlie

all such strategies (Yan et al., 2003, Page 1269):

1. Systematicity: All else being equal, re-representation

suggestions that lead to increases in the systematicity

of the derived analogy will be preferred.

2. High Selectivity: The selection process should be

tightly controlled, so that very few of the possible op-

portunities are selected for consideration.

The above considerations yield the following

computational-level models of representation under SMT.

All three models assume that re-representation is done to

improve on a given optimal analogy. The ﬁrst of these

models is general, in that it does not require that the created

analogy be an extension of the given analogy.

ANAL OGY DERIVATION WITH RE-REP RE SEN TATIO N

(ADR)

Input: Predicate-structures Band T, an optimal analogy

A(B,T), a rule-set R, and integers kand c.

Output: Predicate-structures B0and T0and an analogy

A0(B0

,T0)such that (i) B0and T0are derivable from B

and Tby at most kapplications of rules from Rand (ii)

val(A0)−val(A)≥c.

The second and third models are restrictions of the ﬁrst, as

they are required to extend the given analogy.

ANAL OGY IMPROVEMENT WITH RULE-GUIDED

RE-RE PR ESE NTATI ON (AIR[R])

Input: Predicate-structures Band T, an optimal analogy

A(B,T), a rule-set R, and integers kand c.

Output: Predicate-structures B0and T0and an analogy

A0(B0

,T0)such that (i) A⊂A0, (ii) B0and T0are derivable

from Band Tby at most kapplications of rules from R,

and (iii) val(A0)−val(A)≥c.

ANAL OGY IMPROVEMENT WITH CONTE XT-

GUIDED RE-RE PR ESE NTATI ON (AIR[C])

Input: Predicate-structures Band T, an optimal analogy

A(B,T), and integers kand c.

Output: Predicate-structures B0and T0and an anal-

ogy A0(B0

,T0)such that (i) A⊂A0, (ii) B0and T0are

derivable from Band Tby at most kcontext-guided re-

representations, and (iii) val(A0)−val(A)≥c.

For simplicity, we will assume that all context-guided re-

representations in the third model are of the basic “hole” type

shown in part (b) of Figure 2.

It is possible that the act of analogy derivation rather than

re-representation may artiﬁcially boost the difﬁculty of the

models described above. To this end, we will also ana-

lyze a fourth model of re-representation, whose goal is to

re-represent a given predicate-structure in order to satisfy a

polynomial-time computable function Prop that returns ei-

ther True or False,e.g., does the re-represented Tcontain a

particular type of easily-recognizable structure?

GEN ER AL DERIVATION WITH RE-RE PR ESE NTATI ON

(GDR)

Input: Predicate-structure Tsuch that Pro p(T) =

False, rule-set R, and integer k.

Output: Predicate-structure T0such that (i) T0deriv-

able by at most kapplications of rules from Rand (ii)

Prop(T0) = T rue.

The four models above are those that will be considered

below. However, as will be explained later in the paper, re-

sults derived relative to these models have implications for a

broad range of cognitive theories invoking re-representation.

Re-representation is Intractable

To investigate the computational (in)tractability of the mod-

els of re-representation given in the previous section, we have

adopted standard complexity-theoretic proof techniques from

Computer Science (Garey & Johnson, 1979). Using these

techniques, we have proven the following (see the supple-

mentary materials for proofs1):

Result 1 ADR, AIR[R], AIR[C], and GDR are NP-hard.

These results imply that there do not exist any algorithms for

performing basic re-representation in the sense of the models

considered here in polynomial time for all inputs (i.e., time

upper-bounded by some function ncwhere nis a measure of

input size and cis some constant).2In other words, all algo-

rithms for these models will run in exponential time or worse

(i.e., time upper-bounded at best by some function cnfor c

and nas above). As exponential-time algorithms have unre-

alistically long runtimes for all but very small inputs, they

are generally considered to be computationally intractable

(Garey & Johnson, 1979).

Given that it is NP-hard to derive analogies of a speciﬁed

systematicity (van Rooij, Evans, M¨

uller, Gedge, & Wareham,

2008; Veale & Keane, 1997), the NP-hardness of ADR is

not unexpected. The N P-hardness of AIR[R] and AIR[C] is

surprising, as deriving analogies that must be built on and in-

clude given analogies (Forbus, Ferguson, & Gentner, 1994)

was not previously thought to be intractable. This suggests

that the act of re-representation all by itself is intractable,

which is conﬁrmed by the NP-hardness of GDR. That all

of these results hold in the most basic case as well – that

is, re-representation of individual predicates — has addi-

tional power, as this means that these results may actually

under-estimate the complexity of more complex types of re-

representation invoking larger scale structural changes such

as those proposed in Yan et al. (2003).

All this being said, the above does not say that re-

representation is impossible – rather, it suggests that re-

representation in practice may require one or more additional

constraints on the inputs and/or the re-representation process

1http://www.cs.mun.ca/∼harold/Papers/ICCM12supp.pdf

2This assumes that the conjecture P6=NP is true, which is widely

believed within the Computer Science community on both theoreti-

cal and empirical grounds (Fortnow, 2009).

113

not considered so far in order to be computationally practical.

In the next section, we describe a methodology that can be

used to both model such speciﬁc constraints and investigate

their computational effects.

A Method for Identifying

Tractability Conditions

A computational problem that is intractable for unrestricted

inputs may yet be tractable for non-trivial restrictions on the

input. This insight is based on the observation that some

NP-hard problems can be solved by algorithms whose run-

ning time is polynomial in the overall input size and non-

polynomial only in some aspects of the input called parame-

ters. In other words, the main part of the input contributes to

the overall complexity in a “good” way, whereas only the pa-

rameters contribute to the overall complexity in a “bad” way.

In such cases, the problem Πis said to be ﬁxed-parameter

tractable (Downey & Fellows, 1999) for that set of parame-

ters. The following deﬁnition states this idea more formally.

Deﬁnition 1 Let Πbe a problem with parameters k1,k2,

. . .. Then Πis said to be ﬁxed-parameter (fp-) tractable

for parameter-set K ={k1,k2,...,kc}if there exists at least

one algorithm that solves Πfor any input of size n in time

f(k1,k2,...,kc)nc, where f (·)is an arbitrary function and c

is a constant. If no such algorithm exists then Πis said to be

ﬁxed-parameter (fp-) intractable for parameter-set K.

In other words, a problem Πis fp-tractable for a parameter-set

Kif all superpolynomial-time complexity in solving Πcan be

conﬁned to the parameters in K. In this sense the unbounded

nature of the parameters in Kcan be seen as a reason for

the intractability of the unconstrained version of Π. For any

given ﬁxed-parameter (in)tractability result, other results may

be implied courtesy of the following lemmas:

Lemma 1 If Πis fp-tractable for K then Πis fp-tractable

for any K0such that K ⊂K0.

Lemma 2 If Πis fp-intractable for K then Πis fp-

intractable for any K0such that K0⊂K.

It follows from the deﬁnition of fp-tractability that if an in-

tractable problem Πis fp-tractable for parameter-set K, then

Πcan be efﬁciently solved even for large inputs, provided

only that all the parameters in Kare relatively small. In the

next section we report on our investigation of whether or not

parameters may be used in this way to render the models

ADR, AIR[R], AIR[C], and GDR tractable.

What Does (and Doesn’t)

Make Re-representation Tractable?

Table 1 lists the parameters that we will consider in our ﬁxed-

parameter analyses of re-representation. Each of these pa-

rameters is of interest for different reasons. Parameters oand

pare already known to individually render analogy deriva-

tion fp-tractable (van Rooij et al., 2008; Wareham, Evans,

Table 1: Overview of Parameters Considered.

Name Description

oMaximum number of objects over Band T

pMaximum number of predicates over Band T

kAmount of allowed re-representation

|R|Rule-set size

aTotal number of re-representation

opportunities in Band T

& van Rooij, 2011) and may in turn make analogy deriva-

tion with re-representation fp-tractable. Parameters kand |R|

explicitly and implicitly, respectively, encode the High Se-

lectivity principle for re-representation selection strategies,

and should thus be small in practice. Finally, in addition

to considering parameters that separately characterize the in-

puts (o,p) and the re-representation process (k,|R|), we will

investigate parameter a, which in a sense encodes the de-

gree of interaction between the given predicate-structures and

the re-representation mechanisms (either rules in Ror hole-

contexts) in terms of the number of opportunities that these

inputs provide for the application of these mechanisms.

The results of our analyses relative to these parameters are

given below (see the supplementary materials for proofs). As

we are still in the early stages of our investigation, these re-

sults in tandem with Lemmas 1 and 2 do not yet fully charac-

terize the parameterized complexity of our models relative to

all possible combinations of the considered parameters. How-

ever, even at this initial stage, we can still draw some inter-

esting conclusions and conjectures.

Let us start with the fp-intractability results:

Result 2 ADR and GDR are fp-intractable for parameter-

sets {o,k,a}and {k,|R|}.

Result 3 AIR[R] is fp-intractable for parameter-set {o,k,a}.

Result 4 AIR[C] is fp-intractable for parameter-set {o,k}.

Though there are still some open questions (in particular, the

parameterized status of AIR[R] relative to {|R|}, AIR[C] rel-

ative to {a}, and GDR relative to {p}), these results in tan-

dem with Lemma 1 establish that almost none of the param-

eters considered here can, if individually restricted to small

values, render any of our models computationally feasible.

The same also holds for any combinations of the parameters

within the parameter-sets mentioned in these results. Of par-

ticular note is the fact that neither of the four models consid-

ered here can be made feasible by restricting kalone. This

suggests that other principles in addition to High Selectiv-

ity must underlie re-representation selection strategies if re-

representation is to be made feasible. These principles may

have to be model-speciﬁc; for example, the current scarcity

of fp-intractability results for AIR[R] and AIR[C] suggests

that requiring derived analogies to build on given analogies

may provide model-speciﬁc opportunities for restrictions that

yield fp-tractability.

114

Consider now the fp-tractability results:

Result 5 ADR, AIR[R], and AIR[C] are fp-tractable for

parameter-set {p}.

Result 6 GDR is fp-tractable for parameter-set {p,|R|}.

Result 7 ADR and AIR[R] are fp-tractable for parameter-set

{o,|R|,a}.

Result 8 AIR[C] is fp-tractable for parameter-set {o,a}.

Result 9 GDR is fp-tractable for parameter-set {|R|,a}.

Each of these results implies that if all parameters in that re-

sult’s parameter-set have small value, then the model men-

tioned in that result can be computationally feasible on in-

puts of arbitrary size. For example, Result 8 says that if o

and aare simultaneously of small value, then AIR[C] may be

computationally feasible. Results 7, 8, and 9 are of partic-

ular interest. The constraint on predicate-structure size im-

posed by ois not overly onerous, as many kinds of predicate-

structures are based on a relatively small number of objects

(Schlimm, 2008); moreover, it seems reasonable to conjec-

ture that for certain applications (e.g., those involving large-

scale re-representation rules), aand |R|may be suitably small.

Generality of Results

All of the intractability results reported in this paper, though

deﬁned relative to a speciﬁc theory of analogy derivation,

have broad applicability. This is because the models exam-

ined here are restricted versions of models for other cognitive

theories that invoke re-representation, e.g.,

•The re-representation modes encoded in our models are

used in many cognitive theories (e.g., GDR’s single-

structure re-representation parallels re-representation in

insight problem solving (Ohlsson, 1992)).

•The predicate-structures on which our models are based are

a powerful but basic form of representation, and it seems

reasonable to conjecture that these other theories can be

phrased in terms of predicate-structures.

•The basic single-predicate-change rules and hole-contexts

used in our models are special cases of the more complex

re-representation invoked in these other theories.

Results for models of other theories that satisfy the above then

follow from the well-known observation that intractability re-

sults for a problem Πalso hold for any problem Π0that has

Πas a special case and can hence solve Π(suppose Πis in-

tractable; if Π0is tractable, then it can be used to solve Π

efﬁciently, which contradicts the intractability of Π– hence,

Π0must also be intractable).

Our fp-tractability results are more fragile, as innocuous

changes in the form of the inputs or the re-representation rules

and contexts may in fact violate assumptions critical to the

operation of the algorithms underlying these results. For now,

we can say that as the parameters mentioned in Results 7, 8,

and 9 encode only the combinatorics of re-representation pos-

sibilities (via |R|and/or a) and require only that the structures

generated by each such possible set of re-representations can

be evaluated to determine if they comprise a viable solution in

a reasonable amount of time, these results apply to all models

whose input-types and re-representation mechanisms satisfy

these conditions.

Discussion

Our research was motivated by the question of how the com-

putational difﬁculty of re-representation in general can be

reconciled with the ease of many instances of everyday re-

representation. To address this question, we ﬁrst set out

to assess using formal methods whether re-representation as

proposed in one such instance, namely analogy derivation,

was computationally tractable. We found that this is not the

case. In contrast, even these models of analogy derivation

that only allow the simplest forms of re-representation can

be proven NP-hard (Result 1). This means that no practi-

cal (read: polynomial time) algorithm can exist that perform

such re-representation for all representations. To our knowl-

edge, this is the ﬁrst formal proof of the intractability of re-

representation in the context of analogy derivation.

As this left the questions of how and under what condi-

tions re-representation can become tractable, we performed

complexity analyses to identify parameters that when re-

stricted to small values render re-representation tractable (see

Table 1 and its associated section for results). We believe that

the following two of our ﬁndings are of particular interest:

1. Limiting the amount of re-representation (i.e., small k)

does not by itself (nor when combined with many other pa-

rameters) make re-representation tractable (Results 2–4).

2. What does make re-representation tractable in the case of

analogy (and, as noted above, many other more complex

models) is when all of the parameters in the sets {p}(Re-

sult 5) or {o,|R|,a}(Results 7 and 8) are simultaneously

restricted to take small values.

The latter set in (2) may be applicable to re-representation

in everyday analogy derivation (especially those cases apply-

ing large-scale re-representations) and the former set may be

reasonable for re-representation in concept development, as

it is strongly hypothesized that children’s representations are

object- and attribute-rich and relationally poor (i.e., small p)

(Gentner et al., 1995). The question now is whether these

properties actually hold in these and other observedly fast

forms of re-representation. If empirical evidence of these

properties can be found, then our tractability results provide a

psychologically plausible explanation of how the modelled

forms of re-representation can be tractable despite the in-

tractability of re-representation in general.

To summarize, in this paper we have given the ﬁrst for-

mal proofs not only that re-representation is computation-

ally difﬁcult even by itself, but that there are restrictions that

115

may allow it to operate quickly in practice. Promising direc-

tions for future work include extending parameterized anal-

yses of the models deﬁned here to other parameters (in par-

ticular, parameters like athat describe interactions between

the given input and the re-representation process), developing

good ﬁxed-parameter algorithms for re-representation within

analogy derivation for implementation in large-scale AI sys-

tems like the Companions architecture (Forbus & Hinrichs,

2006), and investigating in detail the extent to which results

and conclusions presented here apply to other models of re-

representation-assisted analogy such as AMBR (Kokinov &

Petrov, 2000) and HDTP (Krumnack et al., 2008) as well as

models of insight problem solving and creativity.

Acknowledgments

The authors would like to thank three anonymous reviewers

for comments that improved the presentation of this paper.

TW was supported by NSERC Discovery Grant 228104.

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