A cognitive theory of graphical and linguistic reasoning : logic and implementation

A cognitive theory of graphical and linguistic reasoning:
logic and implementation
Keith
Stenning
Jon
Ob erlander
Human Communication Research Centre
University of Edinburgh
2 Buccleuch Place
Edinburgh eh8 9lw Scotland
Abstract
We discuss external and internal graphical and linguistic representational systems. We
argue that a cognitive theory of peoples' reasoning performance must account for a the
logical equivalence of inferences expressed in graphical and linguistic form; and b the
implementational di erences that a ect facility of inference. Our theory proposes that
graphical representations limit abstraction and thereby aid processibility. We discuss the
ideas of speci city and abstraction, and their cognitive relevance. Empiricalsupport comes
from tasks i involving and ii not involving the manipulation of external graphics. For
i, we take Euler's Circles, provide a novel computational reconstruction, show how it
captures abstractions, and contrast it with earlier construals, and with Mental Models'
representations. We demonstrate equivalence of the graphical Euler system, and the non-
graphical Mental Models system. For ii, we discuss text comprehension, and the mental
performance of syllogisms. By positing an internal system with the same speci city as
Euler's Circles we cover the Mental Models data, and generate new empirical predictions.
Finally, we consider how the architecture of working memory explains why such speci c
representations are relatively easy to store.
Acknowledgements The support of the Economic and Social Research Council for the Hu-
man Communication Research Centre is gratefully acknowledged. Our initial thinking
on the topics discussed here was greatly in uenced by the work of John Etchemendy
and Jon Barwise. Our research progressed through helpful discussions with colleagues
in the Inference Working Group of the hcrc, and it has bene tted from the suggestions
of audiences at meetings in Kinloch Rannoch, Edinburgh, and Stanford, and from the
constructive advice of our reviewers. Continuation of the work has been supported by
three grants: `signal', Special Project Grant G9018050, from the Joint Councils Initia-
tive in Cognitive Science and hci; Collaborative Research Grant 910954, from nato;
and `grace', Basic Research Action P6296, from the cec esprit Programme.
1

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1 Introduction
Humans can use a variety of external representational systems to perform the same task. The
same reasoning task can be performed with linguistic representations, such as logical formulae,
or with graphical representations, such as diagrams. Di erent representational systems can
give rise to di erent performance characteristics. Humans also use internal representations
which may intuitively be di erentiated as linguistic or imagistic, and which exhibit di erent
processing characteristics. There is a long history of controversy about how these internal
representations can be distinguished Galton 1883, Pylyshyn 1973, Kosslyn et al. 1979, or
indeed whether this is possible even in principle see Anderson 1978.
In this paper, we argue that a cognitive theory of peoples' reasoning performance is required
which can account for two things. First, the fundamental equivalence of inferences expressed
in graphical and linguistic form; and secondly, the di erences in facility of inference in the
two modes and in heterogeneous combinations. We thus contrast the logic of a task with its
implementation.1
This general argument will be advanced with respect to a particular theory of cognitive
implementation. The kernel of the theory is that graphical representations such as diagrams
limit abstraction and thereby aid processibility. We term this property of graphical systems
of representation speci city|the demand by a system of representation that information
in some class be speci ed in any interpretable representation. We thus identify speci city
as the feature distinguishing graphical and linguistic representations, rather than low level
visual properties of graphics. We take speci city to be a general, logically-characterisable
property of representational systems, which has direct rami cations for processing e ciency.
Our account thus has two virtues. It allows computational speci cation of the processing
di erences between di ering systems. But also, by detaching the distinctions from low level
di erences to do with media, it reveals features of natural language discourse which resemble
graphical limitations on abstraction. These features will play a similarly important part in
maintaining the processibility of natural language discourse.
The paper is structured as follows. In Section 2, we sketch our main working hypotheses. In
Section 3, we introduce some of the ideas which underpin our account. The trade-o in pro-
cessing, between between expressiveness and e ciency, applies to any computational system,
human or arti cial. We therefore go on to consider two domains which provide suitable empi-
rical tests for the theory. The rst is examined in Section 4, where we reconstruct a traditional
system for externally supported graphical reasoning, Euler's Circles ecs, and compare it
with another notation for solving syllogisms, Johnson-Laird's 1983 `Mental Models' system
mms. In Section 5, we turn to the second domain, and consider the relation between exter-
nal graphical representations, and internal cognitive structures. There, we discuss how our
theory bears on text comprehension, and on the mental performance of syllogism tasks.
1
This parallels Larkin and Simon 1987, who emphasise the distinction between informational equivalence
and computational equivalence. Like them, we depart somewhat from Marr's 1982 terminology. For Marr,
the computational level characterises a process in terms of abstract mathematical functions; implementation is
a matter of the hardware level. However, relative to a logic, computing with the logic is an implementational
issue. We understand computational issues to be less, rather than more, abstract than logical issues, and
therefore adopt the latter way of speaking, in which to compute with a logic, we must implement it.

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2 A general cognitive theory of graphical representations
We can sketch the main points which characterise our working theory in the following way:
1. Graphical representations are one sort of representation which exhibit `speci city'|
they compel speci cation of classes of information, in contrast to systems that allow
arbitrary abstractions.
2. Actual graphical systems permit the expression of some, but not all, abstractions.
3. Together, this means that such representations are relatively easy to process.
4. This speci city helps explain why graphical techniques, such as Euler's Circles, for
teaching abstract reasoning are so widespread, and presumably e ective.
5. The internal working memory representations we use in some reasoning tasks share with
graphical representations this property of speci city.
6. Natural language discourse conventions stay closer to graphics in respect of speci city
than do fully abstractive logical languages, in order to preserve processibility.
It is worth observing that our theory is intended to avoid emphasis on the particularly visual
properties of graphics. We instead emphasise some general logical properties of representa-
tions, which have computational rami cations. It is easy to imagine a blind reasoner using
embossed Euler's Circles to solve syllogisms. In this paper, we do not discuss point 6, the
2
role of natural language discourse conventions, in any detail; some preliminary remarks are
made in Stenning and Oberlander 1991:613 615. Point 4, which relates to processing with
external graphical representations, is dealt with in Section 4, and point 5, which relates to
processing with internal graphical representations, is dealt with in Section 5. Section 3 lays
out some groundwork, by making more precise the idea of speci city and the related notions
which underpin points 1 to 3.
3 Speci city and limited abstraction
The rst part of our working theory raises a number of questions. First: what does speci city
in a representational system actually mean? Secondly, what does it mean to be able to express
some, but not all, abstractions? Thirdly, how does this limited expressiveness purchase ease
of processing? In answering these questions, we attempt to de ne speci city more precisely,
and therefore make use of some further new terms. In particular, we introduce three types
of representational systems, organised by their increasing expressiveness. These are mimi-
mal abstraction, limited abstraction, and unlimited abstraction representational systems. We
illustrate them with two simple cases, and then in Section 3.4 indicate their computatio-
nal signi cance by comparing them with Levesque's 1988 vivid systems. Such a tripartite
hierarchy obviously evokes the Chomsky language hierarchy cf. Aho and Ullman 1972; we
address this parallel, and the cognitive relevance of our proposal, in Section 3.5.
2
Tactile Venn diagrams have been used with good e ect in teaching blind students elementary logic Gold-
stein and Moore personal communication.

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3.1 Minimal abstraction representational systems
The simplest characterisation of speci city can be given in semantic terms. Imagine a repre-
senting world and a represented world. The former re ects at least some aspects of the latter.
To characterise a representational system, we must state i the represented world; ii the
representing world; iii what aspects of i are being modelled; iv what aspects of ii are
doing the modelling; and v the correspondences between the two worlds Palmer 1978:262.
Let us require a characterisation ideally to provide an extra component: vi a key: that part
of the mapping from representation to world which has to be made explicit to users of the
representation because they do not carry it as part of their general knowledge. A system
will then have a set of possible representations, constructible out of basic elements, each of
which represents some world as being some way. Rearranging the elements in a particular
representation may cause it to correspond to a di erent possible world.
Now, when a system of representation is a language, either natural or logical, it is relati-
vely straightforward to give a model-theoretic semantics for the system, and for its possible
representations. An interpretation function will map representational elements into model
elements; di ering choices of domain for the model would lead to di ering interpretation
functions. For example, we could choose to model temporal expressions in natural language
using a timeline with a domain of integers, or reals, or whatever. Now, suppose we x both
the domain and the interpretation function; then there is particular question we may ask:
how many models correspond to a representation? Under the intended interpretation for the
language, how many ways are there of making a sentence true?
By contrast, consider a system of representation which is|at least super cially|not like a
natural language. Take a graphical system in which a well-formed representation is a xed
arrangement of squares containing a set of solid black circles. The intended interpretation
for this system tells us three things. The squares denote the set of o ces in my building; the
black circles denote researchers; and the relation of spatial containment denotes the relation
of working in an o ce. Just as with a language, we can ask: how many models correspond
to a particular representation in this system? Under the intended interpretation, how many
ways are there of making a graphical representation true?
The basic semantic point here is just this: a minimal abstraction representational system
mars is one in which there is exactly one model for each representation in the system,
under the intended interpretation. We can put this another way via the notion of a relation
dimension cf. Palmer 1978:268. A dimension is a set of mutually exclusive relations, only
one of which holds for each object or set for which the relation is de ned. For example, colour
is a unary dimension whose values are properties such as redness; interobject distance is a
binary dimension whose values are distances. The current point is thus: take a represented
world, choose which relation dimensions the representing world is to capture; a mars is then
one which, for every chosen dimension, must have a single value for every object in the domain.
This semantic characterisation has a syntactic re ex; a representational system which is
minimally abstract will embody certain restrictions on its possible representations, which
ensure that each representation corresponds to exactly one intended model. The particular
manifestation of this syntactic re ex will depend a good deal on the overall form of the
representational system. To illustrate this, consider in turn two trivial marss: a graphical

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P Q R S T
a 1 0 0 1 0
b 1 1 0 0 1
c 0 1 1 1 0
d 0 0 1 0 1
Figure 1: A graphical tabular representation of a world W
Pa Qa Ra Sa Ta
^
:
^
:
^
^
:
Pb Qb Rb Sb Tb
^
^
^
:
^
:
^
Pc Qc Rc Sc Tc
^
:
^
^
^
^
:
Pd Qd Rd Sd Td
^
:
^
:
^
^
:
^
Figure 2: A comprehensive sentence of L representing world W
0
system of two-dimensional tables, and a linguistic system of restricted predicate calculus.
Two dimensional tables. Consider a system of tabular representation. In the represented
world W, there are four objects and ve unary property dimensions; each dimension has
just two values, which means that an object either has the property, or it doesn't. For the
tabular representational system to be minimally abstract, the representing world must always
represent each of the objects and dimensions, and must assign each object exactly one value on
each dimension. Figure 1 provides a representation which can be interpreted as an element
of a minimally abstract tabular system. So: where is the syntactic re ex of the semantic
constraint? Our representation contains symbols for objects, properties, 1s and 0s; what is
speci c about it? The answer is that a well-formed tabular representation has no cells which
are not occupied by a 1 or a 0. There are no empty cells occupied by Blanks, and there are
no crowded cells occupied by more than one symbol.
Restricted predicate calculus. A related but rather di erent syntactic re ex arises when
we consider a predicate calculus representation of the same world W. To represent W,
we can stipulate that we have a representational language with the following properties.
We take a rst order predicate logic with identity, but without quanti ers and with only
negation and conjunction as connectives. We make the unique names assumption, and insist
that only one constant denote each element in the domain. Call this language L ; here, it
0
contains four constants and ve predicates. We can say that a sentence of L in conjunctive
0
normal form is comprehensive when it contains the minimum number of clauses|here, 20|
required to exhaust the combinatorial possibilities of predicate and constant symbols. Figure 2
provides a representation which can be interpreted as an element of a minimally abstract
linguistic system based on L . In L , every sentence corresponds to a single interpretation
0
0

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Stenning and Oberlander
6
of its constants and predicates; one and only one sentence is true in any interpretation. The
system is minimally abstract because each sentence of the language corresponds to exactly one
model. The syntactic re ex here is that well-formed representations have to be comprehensive
sentences of L . They must have exactle 20 conjuncts, and each combination of predicate
0
and constant symbol must appear once.
We can see that the restriction of a system to minimal abstraction is quite a radical one. For
the restricted predicate language, we used only a nite vocabulary in xed length sentences;
we did not use quanti ers and variables, and we did not disjoin comprehensive sentences.
Usual logical languages obviously o er these facilities; less obviously, actual uses of tables
are rarely as restricted as that exempli ed in Figure 1. Let us now therefore turn to a less
restrictive class of representational systems, which are related to marss.
3.2 Limited abstraction representational systems
Each representation in a mars under interpretation could represent only one model, only one
way for the world to be. Yet real graphical systems surely do not labour under this constraint.
Consider again the o ce-allocation diagram mentioned above. Suppose I wanted to represent
the fact that all the o ces have two persons in them, apart from one, which has either two
or three persons in it. There are two general strategies for enriching diagrams that could be
applied here. First I could create multiple diagrams: that is, I could produce two alternate
3
diagrams representing the alternatives and place them side-by-side in a complex diagram.
Notice that each of the representations gives exactly one value for each object o ce on
the relevant dimension number of occupants. But a representational system which allows
multiple diagrams has enriched its expressive power, albeit in a rather simple way. For now,
we would say that the complex diagram actually represents two ways the world could be; the
single complex diagram represents two models.
We will say that such a system is one type of limited abstraction representational system
lars. This particular type of lars is such that a complex diagram abstracts over several
models; the number of its multiple subdiagrams corresponds to the number of models; each
subdiagram corresponds to one model. For each type of system, there will be a syntactic
re ex for this semantic property. With our tabular system, the re ex will be that we allow
juxtaposition of multiple tables, one for each element of the disjunction. With the predicate
system, the re ex will be that we allow well-formed formulae to be those which consist of one
or more disjuncts, each of which is a comprehensive sentence of the old system.
But we need not adopt multiple diagrams to solve the o ce-representation problem. A second
strategy would be to augment diagrams with new symbols. We could introduce a new type of
white circle into the squares-and-black-circles representation; one which stands for a worker
who might or might not be there. With the new symbol, we can collapse the two diagrams
of the multiple method into one. This would contain a set of squares, all but one of which
contain only two black circles; the nal square containing two black and one white circle.
A system which introduces this type of symbol is another type of lars. Here, a single
diagram corresponds to several models, the number depending on the precise interpretation
3
There is actually a third strategy, which we turn to when we discuss unlimited abstraction, below.

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Stenning and Oberlander
7
Pa Qa Ra Sa Ta
^
3
^
:
^
:
^
^
:
Pb Qb Rb Sb Tb
P Q R S T
^
^
^
:
^
:
^
Pc Qc Rc Sc Tc
a
0 0 1 0
^
:
^
^
^
^
:
Pd Qd Rd Sd Td
b 1 1 0 0 1
^
:
^
:
^
^
:
^
c 0 1 1 1 0
d 0 0 1 0 1
Figure 3: Modi ed predicate and tabular representations of multiple worlds
of the new symbol. In terms of relation dimensions, a lars of either kind is a system which
permits some object to take more than one value on some dimension. The semantic move is,
of course, re ected in the syntax of the lars. With our tabular system, consider introducing
a Blank, de ned as Either 1 or 0. Now, we can abstract over worlds. Each blank appearing
in the diagram doubles the number of cases. In our predicate system, we can introduce a ,
3
so that, for instance Pa Pa Pa. permits disjuncts of this form to be conjuncts
3

_
:
3
in the old comprehensive sentences of our system. Equivalently, we could permit `partial'
sentences, which simply omit such internal disjunctive clauses. Figure 3 illustrates both
options. Symbols like these do not permit the expression of dependencies between values
in cells of a table or polarities of clauses in a sentence. Semantically, abstraction is only
permitted over models which di er with regard to one object's value on exactly one dimension.
So abstraction really is limited, in that little exibility is allowed in picking out regions of the
space of possible models. Using a new symbol to capture abstractions, the number of models
abstracted over is exponential in the number of occurrences of the symbol.
Thus, the semantic power introduced by this type of new symbol falls short of that a or-
ded by genuinely `linguistic' symbols, in the following sense. Only the latter, occurring in
a representation, permit the expression of arbitrary dependencies between entities in the re-
presented world. Introducing expressions for arbitrary dependencies corresponds to the third
general strategy mentioned in Footnote 3. In the tabular case, for example, we could express
the idea that one object's value on a dimension depends on another object's value on ano-
ther dimension by inserting an equational expression into the appropriate cell of the table.
Alternatively, we could place more complex information in the key which is part of the re-
presentational system. Our new symbols would be de ned here, in terms of the dimensional
values to which they correspond. And so too could arbitrary dependencies; these would di er
from other parts of the key because they would refer to speci c parts of the representation to
which they were adjoined. Compare a key entry for a table which stated Blank anywhere: 1
or 0 in that location" with another entry which said Blank in column P row b: 1 if Qc = Sd,
0 otherwise". Let's call statements of the former type key terminology, and of the latter
type key assertions, loosely following the distinction introduced between terminological and
assertional knowledge cf. Brachman, Fikes and Levesque 1983.

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Stenning and Oberlander
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3.3 Unlimited abstraction representational systems
Finally, let us say that a system is an unlimited abstraction representational system uars if it
expresses dependencies either inside a representation, with equations or whatever, or outside
the representation, via key assertions. In itself, this choice of terms is purely stipulative;
however, as should emerge below, the processing di erences between lars and uars are
likely to be accounted for in terms of the expressivness of representation and key combined,
rather than in terms of representation alone. Hence we choose to say that a lars is a system
which keeps its representations simple, and keeps assertions out of its keys.
The kinds of lars that are of interest to us, then, are ones which achieve abstraction by
using multiple diagrams and key terminology new symbols. What is limited about multiple
diagrams is that we need n diagrams to represent n models. What is limited about diagrams
with new symbols is that, for m occurrences of symbol in a single diagram, we cannot help
but represent a number of models exponential in m. We contend that that normal graphical
systems are larss; much of their usefulness, and their limitations, arise from this property.
3.4 The computational signi cance of limited abstraction
Our working cognitive theory of graphical representations distinguishes marss, larss and
uarss on semantic and hence syntactic grounds. But the actual reasons for picking out
these classes of representational systems lie in the computational properties which ow from
the semantic properties. We would predict that a lars would be more computationally
e ective than a uars, and that this e ectiveness would be of use both to human and arti cial
information processors. To investigate the validity of such a prediction, consider Levesque's
1988 ndings. Levesque approached the problem of inferential tractability from a rather
di erent direction. He observed that various well-known metalogical results prevent even rst
order predicate logic from providing a computationally tractable reasoning system. He then
asked: what modi cations to or deviations from classical logic will be necessary to ensure the
tractability of reasoning"? His claim is that these deviations are exactly the same deviations
as are necessary to make logic more psychologically realistic.
Levesque's basic suggestion is that if reasoning tasks are arranged so as to minimise the
number of cases to be considered, they can be kept tractable. Requiring a kb to be vivid
is one way to help minimise cases. A kb is vivid if it is in a certain syntactic form. For
sentences of rst-order predicate calculus, the kb can only contain i ground, function-
free atomic sentences; ii inequalities between all distinct constants assumption of unique
names; iii universally quanti ed sentences over the domain, which for each predicate and
constant express the closed world assumptions; and iv the axioms of equality. A vivid kb is
consistent and complete; and more importantly, it is tractable, via this theorem of Levesque:
Theorem 1: Suppose kb is vivid and uses m constants. Let Q ;:::;Q be quan-
1
n
ti ers, let be quanti er-free. Then determining if kb = Q
Q has an
j






1
n
Om
 algorithm.
n+1
j
j
The worst cases will be exponential in n, but where n is much less than the length of ,
and is much less than the size of kb, things will be much better; and these are plausible

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Stenning and Oberlander
9
assumptions. Whether or not inference with a vivid kb is tractable will still depend on the
algorithm used, but in theory large kbs of the order of 10 sentences remain tractable.
9
Like a mars, a vivid kb cannot represent universals or disjunctions. However, Levesque
suggests various extensions to vivid-form kbs which retain its computational characteristics,
while increasing expressiveness. Universals are represented by the addition of function-free
Horn clauses; disjunctions by a switch to semi-Horn form
4
kbs. The latter encode taxono-
mies which allow some disjunctions to be re-expressed non-disjunctively, using subsuming
predicates. Vividness can also be improved via the use of observer-centered visually salient
properties", which will irresistibly be applied in cases such as Berkeley's triangle.
We agree wholeheartedly with the claim that a primary reason for the appeal of visual in-
formation lies in what it cannot leave unsaid about the observed situation compared to
unrestricted linguistic information" p387 . Of course, on neither our view nor Levesque's is
this property con ned to visual representation. His major point is that tractability is best
maintained by minimising the number of cases that must be computed over. His preferred
method of case minimisation involves syntactic constraints on representational systems. Our
major point so far has been that graphical systems are syntactically constrained; and it turns
out that these constraints are very similar to those suggested by Levesque. In itself, this is
not surprising, since our claim about limited abstraction is e ectively a claim that i larss
can help minimise cases; and ii their power to do so is somewhat limited. But the conse-
quence of arriving at a type of representational system which resembles Levesque's is that it
too should have computationally desirable properties.
Let us explore the correspondences. An element of a mars will be of a certain syntactic form,
the precise restrictions depending on the particular mars. In the case of the restricted nite
predicate language L discussed earlier, a comprehensive sentence can be regarded as a kb
0
of a special type, actually less expressive than a vivid kb. Inference with respect to this kb
will indeed correspond to tractable table-lookup. Of course, all such an inference e ectively
tells us is the polarity of a given conjunct. larss have slightly more complex properties. A
system based on L permitting disjunctions of comprehensive sentences will require an upper
0
bound of n look-ups for every query, where n is the maximal number of disjuncts required to
cover a set of models. If each look-up gives the same answer, that answer will be returned;
otherwise, the lack of an answer will be returned. In principle n for L could be very large,
0
but in practice, the multiple representation technique would not be used when n is large.
A system based on L permitting `partial sentences', or the new de ned symbol , will not
3
0
always allow the polarity of every conjunct to be found on table look-up, since the answer
will not be in the table to be found. However, the lack of an answer can be found on look-up,
and the interpretation of any new symbol found by look-up can be determined by consulting
the key terminology statements, again by look-up. This type of system will have the same
general properties as Levesque's semi-Horn form kbs, since the key terminology is equivalent
to the de nition of subsuming predicates in a taxonomic component.
When key assertions must be consulted, as when we are dealing with a uars, the complexity
of inference will depend largely upon the syntactic complexity permitted in the assertions.
If an L -based lars were supplemented with expressions of unrestricted quanti ed predicate
0
4
Of the form:


 with
0 and atomic.
8x
:
:
:
8x
p
^
:
:
:
p
!
p
n;
k

p
1
n
1
k
k +1
i

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Stenning and Oberlander
10
calculus, inferential complexity would degrade accordingly. However, one can envisage uars
which permit only some syntactically limited key assertions, and thereby maintain a desirable
level of tractability. For example, we could require key assertions to contain only universal
quanti ers; inference in such a setting should then be tractable.
3.5 The cognitive signi cance of limited abstraction
There is a sense, then, in which mars, lars and uars form of hierarchy, in which expres-
siveness and tractability are inversely related. This naturally recalls Chomsky's hierarchy of
languages, ranging from type 3 languages  nite-state, through types 2 and 1 context-free
and context-sensitive, respectively to type 0 languages recursively enumerable sets. Thus,
it is natural to raise two further issues, concerning the relation between the proposed hierarchy
and Chomsky's; and the cognitive relevance of such hierarchies.
On the rst issue, we have little to say. Chomsky was concerned with systems containing
linear sequences of symbols, and we have cast our net somewhat more widely. It is thus
not obvious what kinds of correspondences hold; what might constitute a context-free lars?
On the second issue, we would concede that it's obvious that most actual graphical systems
function as larss. Now, Chomsky's hierarchy has perhaps proved to be of limited use to
cognitive science. Most natural languages, after all, are at least type 1, and thus we do
not easily locate interesting constraints on processing. By contrast, we would maintain that
placing graphical systems at the la point in the rs hierarchy has signi cant rami cations for
processing. For instance, we indicate in Section 4.2.4 that the lars version of a particular
graphical system is superior to the earlier mars version, which had been justly criticised on
the grounds of its combinatorial ine ciency.
We acknowledge that the computational constraints discussed above are|in a sense|
relatively weak, for two reasons. First, such characterisations tend to dwell on worst-cases,
which may not be a concern for computational agents which exist in forgiving environments.
Secondly, actual performance pro les are only partially determined by the complexity con-
traints. Even if a representation system has a certain complexity, the choice of a particular
representation for a given problem has a considerable impact on its solubility.
Nonetheless, if it is accepted that humans have a set of special purpose reasoning mechanisms
rather than a single general purpose mechanism, then we can show that at least one of these
mechanisms performs e ciently precisely in virtue of the limited abstraction permitted by
the representations it manipulates. To substantiate our schematic theory, we must develop
detailed analyses of actual graphical systems and their cognitive impact. There are two broad
lines of enquiry which could provide empirical evidence for the development of the theory.
Study of externally implemented graphical systems can reveal whether their expressive power
is that of a lars. But to show that the logical distinctions between mars, lars and uars
have cognitive consequences requires study of human performance and cognitive structure.
Evidence may come from tasks involving the manipulation of external graphics, but para-
doxically, most of the existing work which addresses our evidential needs actually studies
internal cognitive structures which arise during the performance of tasks involving no exter-
nal graphics. We will consider both types of evidence here, in Sections 4 and 5 respectively.
Our method is to start from studies of external graphical systems and then to ask how such

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