Representations in Distributed Cognitive Tasks

In this article we propose a theoretical framework of distributed representations and a methodology of representational analysis for the study of distributed cognitive tasks—tasks that require the processing of information distributed across the internal mind and the external environment. The basic principle of distributed representations Is that the representational system of a distributed cognitive task is a set of internal and external representations, which together represent the abstract structure of the task. The basic strategy of representational analysis is to decompose the representation of a hierarchical task into its component levels so that the representational properties at each level can be independently examined. The theoretical framework and the methodology are used to analyze the hierarchical structure of the Tower of Hanoi problem. Based on this analysis, four experiments are designed to examine the representational properties of the Tower of Hanoi. Finally, the nature of external representations is discussed.     

Distributed representation as a principle for the analysis of cockpit information displays

This article examines the representational properties of cockpit information displays from the perspective of distributed representations (Zhang & Norman, 1994). The basic idea is that the information needed for many tasks in a cockpit is distributed across the external information displays in the cockpit and the internal minds of the pilots. It is proposed that the relative distribution of internal and external information is the major factor of a display's representational efficiency. Several functionally equivalent but representationally different navigation displays are selected to illustrate how the principle of distributed representations is applied to the analysis of the representational efficiencies of cockpit information displays.     

A Cognitive Theory of Graphical and Linguistic Reasoning: Logic and Implementation

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 differences that affect facility of inference. Our theory proposes that graphical representation limit abstraction and thereby aid “processibility”. We discuss the ideas of specificity and abstraction, and their cognitive relevance. Empirical support both comes from tasks which involve the manipulation of external graphics and tasks that do not. For the former, we take Euler's (1772) circles, provide a novel computational reconstruction, show how it captures abstractions, and contrast it with earlier construals and with Johnson-Laird's (1983) mental models representations. We demonstrate equivalence of the graphical Euler system, and the nongraphical mental models system. For tasks not involving manipulation of external graphics, we discuss text comprehension, and the mental performance of syllogisms. By positing an internal system with the same specificity 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 specific representations are relatively easy to store.     

Representing task context: proposals based on a connectionist model of action

Representations of task context play a crucial role in shaping human behavior. While the nature of these representations remains poorly understood, existing theories share a number of basic assumptions. One of these is that task representations are discrete, independent, and non-overlapping. We present here an alternative view, according to which task representations are instead viewed as graded, distributed patterns occupying a shared, continuous representational space. In recent work, we have implemented this view in a computational model of routine sequential action. In the present article, we focus specifically on this model's implications for understanding task representation, considering the implications of the account for two influential concepts: (1) cognitive underspecification, the idea that task representations may be imprecise or vague, especially in contexts where errors occur, and (2) information-sharing, the idea that closely related operations rely on common sets of internal representations.     

The reliability of and the relation between non-symbolic numerical distance effects in comparison, same-different judgments and priming

The development of number processing is generally studied by examining the performance on basic number tasks (comparison task, same-different judgment, and priming task). Using these tasks, so-called numerical distance effects are obtained. All these effects are generally explained by assuming a magnitude representation related to a mental number line: magnitudes are represented from left to right with partially overlapping representations for nearby numbers. In this study, we compared the performance of adults on these different tasks using non-symbolic stimuli. First, we investigated whether the effects obtained in these behavioral tasks are reliable. Second, we examined the relation between the three different effects. The results showed that the observed effects in the case of the comparison task and the same-different task proved to be reliable. The numerical distance effect obtained in the priming task, however, was not reliable. In addition, a correlation was found between the distance effects in the comparison task and the same-different task. The priming distance effect did not correlate with the other two effects. These results suggest important differences between distance effects obtained under automatic and intentional task instructions regarding the use of them as indices of mathematical ability.     

The role of representations in age differences in analogical reasoning

Age-related declines in the efficiency of a number of cognitive tasks have been postulated to be attributable to decreases with age in the quality of internal representations used to mediate performance on those tasks. This proposal was investigated in a geometric analogies task by manipulating variables (i.e., the number of elements per term and the temporal delay between presentation of pairs of terms) assumed to affect the quality or stability of internal representations. As expected, the performance of older adults was impaired more than that of young adults by these manipulations. Further analyses revealed that these representational deficits may be due to a reduction of approximately 40% in the quantity of some type of processing resource between, approximately, 20 and 70 years of age.     

Learning and development of embodied numerosity

Recent empirical evidence indicates that seemingly abstract numerical cognitions are rooted in sensory and bodily experiences. In particular in finger counting finger-based representations reflect a specific case of embodied cognition, we termed embodied numerosity. Furthermore, we suggest that finger-based representations should be considered a distinct representation of number (magnitude) and argue that this representation is activated automatically whenever we encounter a number. We discuss in what way such a theoretical framework can account for the associations of fingers and numbers observed so far. In the final part, we evaluate whether the concept of embodied numerosity should be generalized beyond finger-based representations with particular focus on whether bodily-sensory experiences (such as moving the whole body along the mental number line) may corroborate numerical capabilities. In a series of intervention studies, we consistently observed more pronounced training effects for our embodied numerosity trainings for different age groups, different digital media, different number ranges, and different control conditions. Taken together, we conclude that embodied representations of number (magnitude) exist, are not limited to finger-based representations, and influence number processing in a systematic and functional way that can be used to foster the efficiency of numerical trainings.     

Directions in Connectionist Research: Tractable Computations Without Syntactically Structured Representations

Should connectionists abandon the quest for tractably computable cognitive transition functions while retaining syntactically structured mental representations? We argue, in opposition to Horgan, that it should not. We argue that the case against tractably computable functions, based upon the claimed isotropic and Quinean character of cognition, fails since cognition is not as isotropic and Quinean as Fodor and Horgan contend. Moreover, we illustrate how current research in both connectionism and cognitive neuroscience suggests that tractability can be preserved through division of overall computations into modular sub-processes, each of which is tractable. As to syntactically structured representations, we argue that they are unneeded for most cognitive tasks organisms confront, and that when they are needed, they may be provided by external representational media such as natural language. Moreover, we note that increasingly cognitive linguistics has become the ally of connectionism and that the research program of cognitive linguistics suggests that abilities to use natural languages may be developed without requiring syntactically structured mental representations to exist prior to natural language.     

Mental multiplication: Nothing but the facts?

Current models of adult arithmetic performance assume that representation includes only facts and procedures. However, other kinds of representations such as an analog scale or sets of number multiples might be useful in a variety of multiplication-related tasks. Introducing the practice transfer paradigm, we demonstrate that associations between distinct representational structures can be detected via cross-task transfer, provided that initial performance is retrieval based. Results support the predictions of the integrated-structures model of multiplication knowledge. Implications for well-established item differences such as the problem-size effect are addressed, and the question of how integration occurs is considered.     

What Is the Model in Model-Based Planning?

Flexibility is one of the hallmarks of human problem-solving. In everyday life, people adapt to changes in common tasks with little to no additional training. Much of the existing work on flexibility in human problem-solving has focused on how people adapt to tasks in new domains by drawing on solutions from previously learned domains. In real-world tasks, however, humans must generalize across a wide range of within-domain variation. In this work we argue that representational abstraction plays an important role in such within-domain generalization. We then explore the nature of this representational abstraction in realistically complex tasks like video games by demonstrating how the same model-based planning framework produces distinct generalization behaviors under different classes of task representation. Finally, we compare the behavior of agents with these task representations to humans in a series of novel grid-based video game tasks. Our results provide evidence for the claim that within-domain flexibility in humans derives from task representations composed of propositional rules written in terms of objects and relational categories.     

Mental evolution and development: evidence for secondary representation in children, great ages, and other animals

Recent interest in the development and evolution of theory of mind has provided a wealth of information about representational skills in both children and animals. According to J. Perner (1991), children begin to entertain secondary representations in the 2nd year of life. This advance manifests in their passing hidden displacement tasks, engaging in pretense and means-ends reasoning, interpreting external representations, displaying mirror self-recognition and empathic behavior, and showing an early understanding of "mind" and imitation. New data show a cluster of mental accomplishments in great apes that is very similar to that observed in 2-year-old humans. It is suggested that it is most parsimonious to assume that this cognitive profile is of homologous origin and that great apes possess secondary representational capacity. Evidence from animals other than apes is scant. This analysis leads to a number of predictions for future research.     

Domain-specific and domain-general changes in children's development of number comparison

The numerical distance effect (inverse relationship between numerical distance and reaction time in relative number comparison tasks) has frequently been used to characterize the mental representation of number. The size of the distance effect decreases over developmental time. However, it is unclear whether this reduction simply reflects developmental changes in domain-general speed of processing and whether it is specific to numerical compared with non-numerical magnitude. To examine these open questions, we conducted a cross-sectional study with 6-, 7-, and 8-year-old children as well as adult college students. Participants performed comparisons on Arabic numerals, arrays of squares, squares of varying luminance and bars of varying height. To control for general age-related changes in reaction time, a measure of speed of processing was used as a covariate in the analysis. A significant developmental decrease in the distance effect was found across numerical and non-numerical comparison tasks over and above general changes in processing speed. However, this change was not found to differ as a function of format. These data suggest that developmental changes in the distance effect are reflective of changes in a domain-general comparison process, rather than domain-specific developmental changes in number representations. However, analysis of overall reaction times revealed significantly greater developmental changes for numerical relative to non-numerical comparison tasks. These findings highlight the importance of taking multiple measures into account when characterizing developmental changes in numerical magnitude processing. Implications for theories of numerical cognition and its development are discussed.     

Working memory as internal attention: Toward an integrative account of internal and external selection processes

Working memory (WM) and attention have been studied as separate cognitive constructs, although it has long been acknowledged that attention plays an important role in controlling the activation, maintenance, and manipulation of representations in WM. WM has, conversely, been thought of as a means of maintaining representations to voluntarily guide perceptual selective attention. It has more recently been observed, however, that the contents of WM can capture visual attention, even when such internally maintained representations are irrelevant, and often disruptive, to the immediate external task. Thus, the precise relationship between WM and attention remains unclear, but it appears that they may bidirectionally impact one another, whether or not internal representations are consistent with the external perceptual goals. This reciprocal relationship seems, further, to be constrained by limited cognitive resources to handle demands in either maintenance or selection. We propose here that the close relationship between WM and attention may be best described as a give-and-take interdependence between attention directed toward either actively maintained internal representations (traditionally considered WM) or external perceptual stimuli (traditionally considered selective attention), underpinned by their shared reliance on a common cognitive resource. Put simply, we argue that WM and attention should no longer be considered as separate systems or concepts, but as competing and influencing one another because they rely on the same limited resource. This framework can offer an explanation for the capture of visual attention by irrelevant WM contents, as well as a straightforward account of the underspecified relationship between WM and attention.     

Learning representations in a gated prefrontal cortex model of dynamic task switching

The prefrontal cortex is widely believed to play an important role in facilitating people's ability to switch performance between different tasks. We present a biologically‐based computational model of prefrontal cortex (PFC) that explains its role in task switching in terms of the greater flexibility conferred by activation‐based working memory representations in PFC, as compared with more slowly adapting weight‐based memory mechanisms. Specifically we show that PFC representations can be rapidly updated when a task switches via a dynamic gating mechanism based on a temporal‐differences reward‐prediction learning mechanism. Unlike prior models of this type, the present model develops all of its internal representations via learning mechanisms as shaped by the demands of continuous periodic task switching. This advance opens up a new domain of research into the interactions between working memory task demands and the representations that develop to meet them. Results on a version of the Wisconsin card sorting task are presented for the full model and a number of comparison networks that test the importance of various model features. Furthermore, we show that a lesioned model produces perseverative errors like those seen in frontal patients.     

Representations in dynamical embodied agents: re-analyzing a minimally cognitive model agent

Understanding the role of 'representations' in cognitive science is a fundamental problem facing the emerging framework of embodied, situated, dynamical cognition. To make progress, I follow the approach proposed by an influential representational skeptic, Randall Beer: building artificial agents capable of minimally cognitive behaviors and assessing whether their internal states can be considered to involve representations. Hence, I operationalize the concept of representing as 'standing in,' and I look for representations in embodied agents involved in simple categorization tasks. In a first experiment, no representation can be found, but the relevance of the task is undermined by the fact that agents with no internal states can reach high performance. A simple modification makes the task more "representationally hungry," and in this case, agents' internal states are found to qualify as representations. I conclude by discussing the benefits of reconciling the embodied-dynamical approach with the notion of representation.     

Decade breaks in the mental number line? Putting the tens and units back in different bins.

Most models of number recognition agree that among other number representations there is a central semantic magnitude representation which may be conceptualized as a logarithmically compressed mental number line. Whether or not this number line is decomposed into different representations for tens and units is, however, controversial. We investigated this issue in German participants in a magnitude comparison (selection) task in which the larger of two visually presented Arabic two-digit numbers had to be selected. Most importantly, we varied unit-decade-compatibility: a number pair was defined as compatible if the decade magnitude comparison and the unit magnitude comparison of the two numbers would lead to the same response (e.g. 52 and 67) and as incompatible if this was not the case (e.g. 47 and 62). While controlling for overall numerical distance, size and other variables, we consistently found compatibility effects. A control experiment showed that this compatibility effect was not due to perceptual presentation characteristics. We conclude that the idea of one single number line representation that does not additionally assume different magnitude representations for tens and units is not sufficient to account for the data. Finally, we discuss why decade effects were not found in other experimental settings.     

How Reliably Do Eye Parameters Indicate Internal Versus External Attentional Focus?

Eye behavior is increasingly used as an indicator of internal versus external focus of attention both in research and application. However, available findings are partly inconsistent, which might be attributed to the different nature of the employed types of internal and external cognition tasks. The present study, therefore, investigated how consistently different eye parameters respond to internal versus external attentional focus across three task modalities: numerical, verbal, and visuo-spatial. Three eye parameters robustly differentiated between internal and external attentional focus across all tasks. Blinks, pupil diameter variance, and fixation disparity variance were consistently increased during internally directed attention. We also observed substantial attentional focus effects on other parameters (pupil diameter, fixation disparity, saccades, and microsaccades), but they were moderated by task type. Single-trial analysis of our data using machine learning techniques further confirmed our results: Classifying the focus of attention by means of eye tracking works well across participants, but generalizing across tasks proves to be challenging. Based on the effects of task type on eye parameters, we discuss what eye parameters are best suited as indicators of internal versus external attentional focus in different settings.     

Knowledge on the line: Manipulating beliefs about the magnitudes of symbolic numbers affects the linearity of line estimation tasks

It has been suggested that differences in performance on number-line estimation tasks are indicative of fundamental differences in people’s underlying representations of numerical magnitude. However, we were able to induce logarithmic-looking performance in adults for magnitude ranges over which they can typically perform linearly by manipulating their familiarity with the symbolic number formats that we used for the stimuli. This serves as an existence proof that individuals’ performances on number-line estimation tasks do not necessarily reflect the functional form of their underlying numerical magnitude representations. Rather, performance differences may result from symbolic difficulties (i.e., number-to-symbol mappings), independently of the underlying functional form. We demonstrated that number-line estimates that are well fit by logarithmic functions need not be produced by logarithmic functions. These findings led us to question the validity of considering logarithmic-looking performance on number-line estimation tasks as being indicative that magnitudes are being represented logarithmically, particularly when symbolic understanding is in question.     

Stochastic approaches to understanding dissociations in inflectional morphology

Computer modelling research has undermined the view that double dissociations in behaviour are sufficient to infer separability in the cognitive mechanisms underlying those behaviours. However, all these models employ multi-modal representational schemes, where functional specialisation of processing emerges from the training process. Targeted lesioning of different regions of functional specialisation leads to varied but predictable deficits in model performance. We argue that multi-modal representational schemes are not a necessary condition for the observation of double dissociations in an information processing system that shares resources across multiple tasks. Using a uni-modal representational system, we demonstrate that double dissociations may also result from stochastic processes. Lesioning experiments on a single-route, uni-modal connectionist model of regular and irregular noun and verb morphology confirm and extend earlier work demonstrating that selective impairment across tasks can result from damage to a distributed information processing system. A systematic investigation of the degree to which performance deteriorates across different inflectional classes reveals that simple and double dissociations can occur in this single-route, uni-modal model. An important prediction of the model is that double dissociations between regular and irregular inflection, resulting from stochastic processes should be extremely rare. However, they are particularly likely to occur when the researcher uses test batteries consisting of a small number of items. Given that cognitive neuropsychologists rarely provide details about the distribution of performance in a disordered population, it is concluded that a stochastic interpretation of double dissociations may have wider applicability than is normally supposed.     

The development of numerical estimation: evidence against a representational shift

How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number‐line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in children’s estimation performance are identified and discussed. This work demonstrates that proportion‐judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number.