1. Introduction
In cognitive science, the mind is often described as working by constructing, manipulating, and exploiting models of the body and the environment. This idea was introduced by Craik (Reference Craik1943) and has become hugely influential.Footnote 1 But what exactly is meant by ‘models’ in this context? What distinguishes the claim that the mind models entities with which it interacts from the presumably weaker claim that the mind represents such entities? Recent philosophical work has linked the concept of a model to that of structural representation, often focusing on the hierarchical generative models of predictive processing (Gładziejewski Reference Gładziejewski2016; Gładziejewski and Miłkowski Reference Gładziejewski and Miłkowski2017; Kiefer and Hohwy Reference Kiefer and Hohwy2018; Williams and Colling Reference Williams and Colling2018). This connection has also been made in older work, such as by Cummins (Reference Cummins1989), Ryder (Reference Ryder2004), and Ramsey (Reference Ramsey2007). So one possible view is that modeling is the use of specifically structural representations, rather than representations with some other format. In this article my aim is to put forward an alternative view: I will argue that models should be distinguished from other representations by their content, not by their format.
More precisely, my aim is not to argue that philosophers and cognitive scientists never use the term ‘model’ to mean ‘structural representation’ or that they would always be wrong to do so. Instead, I want to draw attention to two points. First, the phenomena that philosophers and cognitive scientists call ‘models’ are typically structural representations, but they also have content of a certain kind, which distinguishes them from other representations. The generative models of predictive processing, for example, have both of these features. This raises questions about the relationship between the features—for example, whether the use of a structural format is particularly apt for representations with content of this kind—and also about the different ways in which they facilitate cognition, which may be obscured by talk of models that is insensitive to the difference between the two features. Second, there are examples of representations used in algorithms of quite central importance in cognitive science, which are naturally and universally referred to as models but which do not have a structural format. So having a structural format does not seem to be necessary for a representation to be a model.
I also certainly do not wish to deny that there is an important and interesting distinction between structural representations and those with other formats. Indeed, my argument relies on this distinction. My substantive claim is that there is a further important and interesting way to classify representations used in cognitive systems, which does not correspond exactly to the structural/nonstructural distinction but which captures part of what scientists intend when they speak of ‘models’, in a range of significant cases.
One complication is that talk of models contributes to modern computational cognitive science in two ways (Eliasmith Reference Eliasmith2007). As well as claiming that the mind models the body and the environment, cognitive scientists construct models of the systems they study. They engage in ‘model-based science’ (Godfrey-Smith Reference Godfrey-Smith2006b) by, for instance, developing computer learning algorithms and testing their performance on cognitive tasks. There is an extensive literature on the role and nature of the models used by scientists of all disciplines (e.g., Giere Reference Giere1988; Magnani and Nersessian Reference Magnani and Nersessian2002; Weisberg Reference Weisberg2013), and this includes debate on the means by which scientists’ models represent their targets, in which structural accounts have taken a prominent place (Suárez Reference Suárez2003). However, I must emphasise that my topic is not scientists’ models. My topic is the mental models that humans and animals are said to use for a wide variety of cognitive tasks.Footnote 2
This article has three main parts. In the next section, I define structural representation and give more detail on the claim that modeling is the use of representations of this kind. In section 3 I present the alternative, content-focused conception of the use of models in the accomplishment of cognitive tasks. Then in section 4 I argue against the structural representation view, using the example of a model-based reinforcement learning algorithm that may be implemented in simple computer programs.Footnote 3
2. The Format Conception
In this and the following section I describe two possible conceptions of a cognitive model, which I call the ‘format’ and ‘content’ conceptions. This section presents the format conception, in three stages: first, I clarify the notion of representational format; then, I explain what is meant by ‘structural representation’; and finally, I state the format conception and summarize the reasons why philosophers have adopted it.
2.1. Representational Format
In claiming that models are distinguished by their content rather than their format, I am claiming that one dimension of variation among representations is implicated in this distinction rather than another. However, there may be some uncertainty about what I mean by the ‘format’ dimension, and clearing this up will also help to make clear how I understand the notion of structural representation.
In my way of thinking about things, representations vary along at least three dimensions. The first is content, which is sufficiently familiar that I will not say anything more to explicate it. The second is format, and the third is what I will call ‘basis’. A representation’s format is the way in which its vehicle properties are used to perform its representational function. Maps, photographs, sentences, and musical scores are said to employ different formats, and in each of these cases vehicle properties are used for representation in different ways. The basis of a representation is the set of properties of the system within which it is embedded that make it the case that it has a representational function and that its vehicle properties are used to represent in a particular way. The basis of a representation might be, for example, a convention that governs its use or the biological function of a system that employs it.
To consider an example in slightly more detail, one philosophical debate that is explicitly about representational format concerns whether there is a proprietary format for perceptual representation (Quilty-Dunn Reference Quilty-Dunn2020). Some contributors to this debate argue that perception has an iconic format, meaning that perceptual representations do not admit of canonical decompositions (Carey Reference Carey2009; Burge Reference Burge2010; Block Reference Block2014). That is, any part of a representation in iconic format is equally representational, as in a typical photograph. This contrasts with noniconic representations such as sentences, which have parts that do represent (such as words) and others that do not (such as part-words). Quilty-Dunn (Reference Quilty-Dunn2016) claims that at least some perceptual representations are not iconic, by arguing that objects are sometimes represented by discrete representational constituents, which behave more like words than like the parts of a photograph.
This debate fits my definition of format because it concerns different ways in which vehicle properties can be used to perform representational functions. If the iconic theory is correct, then in perception representational vehicles, instances of brain activity, are used in such a way that each of their parts plays an equally representational role. Perhaps any part of an instance of brain activity constituting a moment’s visual perception represents the distribution of colors over a region of the visual field. Alternatively, if Quilty-Dunn is right, then there are certain parts of the representational vehicles used in perception that represent objects, and they are such that their proper parts do not represent anything at all.
Representational basis should be distinguished from format because representations can be similar in format and different in basis or vice versa. Consider a weather map that represents landmasses by scale line drawings and forecast temperatures by colored regions. Such a map is an iconic representation, so if the iconic theory is right it is similar in format to perceptual representations. These differ in basis, however, because what makes the weather map a representation, and grounds its format, is the conventions governing its use. Perceptual representations, meanwhile, have their basis in either biological functions established by natural selection or something like Shea’s (Reference Shea2018) ‘task functions’. Conversely, if Quilty-Dunn (Reference Quilty-Dunn2020) is right that perception uses a plurality of representational formats, then it is possible for representations with the same basis to differ in format.
Two further points about format are important for my purposes. First, the concept of exploitable relations (Godfrey-Smith Reference Godfrey-Smith, MacDonald and Papineau2006a; Shea Reference Shea2018) is sometimes invoked in connection with structural representation, and this is helpful in one way and unhelpful in another. Shea (Reference Shea2018) argues that subpersonal mental representations can bear either of two exploitable relations to the phenomena they represent: they can either carry information about those phenomena or correspond to them structurally. Representations that differ in which of these relations are actually exploited by cognitive systems consequently differ in format because their vehicle properties are used for representational purposes in different ways. So this point is very helpful in distinguishing structural representations from others. But the idea can also be confusing because there is room for argument about whether all representations work by exploiting relations between vehicles and the phenomena represented. In particular, this is less clear in the case of representations with a conventional basis. The upshot is that if a representation exploits an exploitable relation, such as structural correspondence to its target, this has consequences for its format, but it is clearer that all representations have a format than that they all take advantage of exploitable relations.Footnote 4
Second, there are many ways to classify representational formats. Classifications can be more or less fine-grained, and different schemes may give overlapping classifications or perhaps even orthogonal ones. I take structural representation to be a broad class of formats, so some intuitively distinct formats (such as maps and photographs) may be brought together under this classification.
2.2. Structural Representation
Structural representations (SRs) are a species of representation defined by similarity in their formats. More specifically, SRs have the following two defining properties:
i) Relations over parts or states of the representational vehicle are used to represent relations over parts or states of the represented phenomenon.
ii) The relations over the vehicle are used in this way because there is a putative structural correspondence between them and the relations over the represented phenomenon.
This definition draws on discussions of structural representation by Swoyer (Reference Swoyer1991), O’Brien and Opie (Reference O’Brien, Opie, Clapin, Staines and Slezak2004), Ramsey (Reference Ramsey2007), Shagrir (Reference Shagrir2012), and Shea (Reference Shea2014, Reference Shea2018). Shea defines structural representation only in terms of the first of the two properties but makes clear that he also takes it to require structural correspondence. O’Brien and Opie do not use the expression ‘structural representation’ but theorize about a form of representation that relies on structural correspondence, defined in terms of structure-preserving mappings, which is how I define this concept below. Swoyer and Shagrir both define structural representation in terms of structural correspondence, and Ramsey describes SRs as relying on structural similarity, correspondence, or isomorphism. The point that the structural correspondence between the vehicle and the represented phenomenon is merely putative is rarely mentioned but necessary because of the possibility of inaccurate SRs, where the correspondence fails.
Following Shea and others, I define structural correspondence in terms of homomorphism, understood in the following way.
Let RA be a relation over parts or states of A, the representational vehicle, and let RB be a relation over parts or states of B, the thing represented. Then f, a function from parts or states of A to parts or states of B, is a homomorphism with respect to these relations if and only if for any pair a 1 and a 2,
.
For a vehicle to bear a structural correspondence to a represented phenomenon is just for a homomorphism to exist from the former to the latter. Homomorphisms are abundant, so to say that structural representation requires the existence of a homomorphism from the vehicle to the represented phenomenon is a very weak constraint. When combined with the first condition on structural representation, however, the constraint becomes significantly stronger.
Maps are often given as examples of SRs (Shea Reference Shea2014; Gładziejewski Reference Gładziejewski2016), and indeed they satisfy the definition. For example, consider the famous map of the London Underground network. This map has certain privileged parts, the marks symbolizing each station, which correspond in the obvious way to the stations themselves. A function taking each of these marks to the named station is a homomorphism with respect to some relations of particular salience on the map and some corresponding ones of practical importance in the world. It maps the relation being connected by a line of a single color over the marks for the stations to the relation being connected by a single London Underground line over the stations themselves, and it also preserves the orders in which stations are thus connected.
A crucial point for understanding the notion of structural representation is that for an entity to be an SR, a set of relations over its parts or states that structurally correspond to some relations over the parts or states of a further entity must be used to represent those relations. Many cases of structural correspondence are therefore not cases of representation at all. For example, there may be a very natural homomorphism between a set of 100 bricks arranged in offset rows in a wall and 100 football fans sitting in similarly offset rows in a stadium, with spatial relations between the bricks corresponding to similar spatial relations between the fans, but the theory of SR does not entail that the bricks represent the fans.
What is more, not all representations (or collections of representations) that are homomorphic to the things they represent are SRs because in many cases homomorphisms exist even though the relevant relations over the represented domain are not represented. Shea (Reference Shea2014, sec. 3) illustrates this point with the case of the honeybee’s waggle dance, in which the angle of the dance corresponds to the direction of the source of nectar, and the number of waggles corresponds to its distance. There is a homomorphism between dances and locations, and this homomorphism determines the representational content of any given dance. But this may not be a case of structural representation, because for that to be the case, relations between dances would have to condition the behavior of observer bees in ways that would indicate that they represent relations between the locations of nectar sources. For instance, bees might watch two dances and use the relation between them to fly first to one nectar source, then directly to another, without going back to the hive. But if bees do not do such things, merely using individual dances as guides to the location of single nectar sources, the relations between dances are not used in the way required for structural representation.
For a representation to be an SR, then, the details of the way in which it is used are crucial. The system that uses it must be capable of behaving in ways that are sensitive to relations over the represented domain, in virtue of the correspondence between these and certain relations over the representational vehicle. This sensitivity may, however, be subtle or indirect. If an organism uses some internal process as a dynamic model of the environment that predicts incoming sensory stimulation, this model may contribute directly only to perception. But by facilitating faster or more reliable responses to environmental contingencies, the model could affect the organism’s behavior, and these effects on behavior would constitute sensitivity to relations between features of the environment, such as the tendency for one kind of event to be followed by another.
The final aspect of structural representation that I want to discuss concerns lookup tables. Representations of this form play an important role in the argument to come, and there are some subtleties in how the definition just given applies to them. It will be helpful to have an example in mind, so consider table 1, into which many of the facts represented by the London Underground map could be transcribed. The table is in alphabetical order, and it lists each pair of stations connected by a single line, which line connects them, and how many stops are required and in which direction. Intuitively this is an example of a way to rerepresent much of the information in an SR in a nonstructural format. But in fact matters are slightly more complicated; I will note three points about how the definition of an SR applies to this case and to other lookup tables.
Table 1. London Underground
| Departure Station | Destination Station | Line | Stops | Direction |
|---|---|---|---|---|
| Acton Town | Aldgate East | District | 22 | East |
| Acton Town | Alperton | Piccadilly | 4 | West |
| … | … | … | … | … |
| Temple | Westminster | District | 2 | West |
| … | … | … | … | … |
First, the table certainly employs what might be called nonstructural elements. It identifies stations and lines by their names, and this way of representing does not make use of structural correspondence. Furthermore, it even represents some worldly relations other than by relations over the vehicle: both the number of stops between stations and the direction of travel between them are represented by symbols rather than relations. The use of nonstructural elements is common in artifactual SRs such as maps, especially to identify objects, and this should not prevent us from recognizing the crucial role that structural correspondence may still play. Unless we find a good reason to do otherwise, we should count all representations that satisfy the two conditions above as SRs, even if they also employ nonstructural elements.
Second, this lookup table does have one of the two defining features of SRs, because the vehicle relation being on the same row of the table is used to represent the worldly relation being connected by a single London Underground line. Something similar will also be true of many other lookup tables. However, there is no structural correspondence between these two relations, because the function from station names on the table to stations themselves is not one to one, but only one token of each name on the table stands in the relevant vehicle relation to each other name. Going more slowly, let us apply the definition of a homomorphism given above to the current case in the following way:
| Let: | a 1, … , an be token names on the table |
| b 1, … , bm be stations on the London Underground network | |
| RA be the relation being on the same row of the table | |
| RB be the relation being on the same London Underground line. |
Then the function mapping station names on the table to the stations that they name is not a homomorphism with respect to these two relations because it is not true that for any pair a 1 and a 2,
. Consider the first token of the name ‘Acton Town’ and the token of the name ‘Westminster’ on table 1. If these are a 1 and a 2 respectively, then f(a 1)RBf(a 2) is true because Acton Town and Westminster are both on the District Line. But a 1RAa 2 is not true because these two tokens do not appear on the same line of the table. So the lookup table as a whole is not an SR. Its structure does not correspond to the structure of the system it represents, because relations over that system are represented piecemeal. While in the system represented each object stands in many relations to other objects, in the representational vehicle there are no parts that stand in the corresponding set of relations.
This pattern is likely to be common when information is stored in lookup tables. Part of what defines a lookup table is that the rows and columns have representational significance, but there are few structures of relations that can be represented in a lookup table other than in the piecemeal way just described.
Third, despite this, each row of the table we are considering is an SR, according to the definition. The function that maps each station name on the first row of the table to the corresponding stations is a homomorphism with respect to the two relations being on the same row and being on the same line. This shows that a composite representation made up of parts that are SRs will not always be an SR itself. It also shows that not all SRs make any very interesting use of structural correspondence. If each row on the table we are considering were replaced by a sentence of English, the same information would be represented with very little change in efficiency or accessibility. Yet English sentences are not SRs; for example, in the sentence ‘Acton Town is on the same line as Aldgate East’ the relation of being on the same line is represented by a part of the vehicle, not a relation over parts.Footnote 5 There is a stark contrast between the rows of our table and SRs that do make significant use of structural correspondence, such as the London Underground map itself. It is thanks to its use of structural correspondence that this map stores and presents information in a highly efficient and accessible way.
2.3. Arguments for the Format Conception
Now that we have the notion of SR in hand, we are ready for a statement of the format conception. The format conception is the claim that cognitive models are SRs.
Format Conception: A cognitive model is a structural representation used in performing a cognitive task.
Since we know what SRs are, it remains only to briefly consider arguments for the format conception.
The format conception of cognitive models is motivated by at least three lines of thought. First, many artifacts that we refer to as ‘models’ are used as SRs. For instance, Ryder (Reference Ryder2004) mentions dynamic models of the solar system, which can be used to answer questions about possible spatial relations between the planets. Swoyer (Reference Swoyer1991) mentions a model airplane used for wind-tunnel testing. This point certainly makes it natural to use the term ‘model’ to describe at least some SRs in cognitive systems. Second, it is sometimes argued that causal and informational theories of mental representation (also called ‘indicator’ or ‘detector’ theories) suffer from insuperable difficulties, which can be avoided if we explain representation in terms of exploitable structural correspondence (Ramsey Reference Ramsey2007; Williams and Colling Reference Williams and Colling2018). So the thought is that by showing that cognitive models are SRs it is possible to defend their status as representations. And third, it has been argued that cognitive models of some specific kinds, especially the generative models implicated in the predictive processing theory of cognition, are in fact SRs (Gładziejewski Reference Gładziejewski2016; Kiefer and Hohwy Reference Kiefer and Hohwy2018; for the predictive processing theory, see Clark [Reference Clark2013, Reference Clark2016] and Hohwy [Reference Hohwy2013]). Gładziejewski’s argument for this claim works by comparing these models to cartographic maps, which are taken to be prototypical nonmental SRs.
These arguments are persuasive, but they are not conclusive. The first and third arguments strongly suggest that models and SRs are connected, but they do not show that an alternative conception of cognitive models could not be more illuminating, and the second argument relies on a highly contentious claim about the prospects of indicator theories. So I now turn to the content conception of cognitive models.
3. The Content Conception
The content conception builds on the work of Lake and colleagues (Reference Lake, Ullman, Tenenbaum and Gershman2017), who seem to rely on differences in representational content, rather than format, in drawing a distinction between what they call ‘pattern recognition’ and ‘model-building’ algorithms for cognitive tasks. Lake et al. argue that in order to construct human-like artificial intelligence we must develop algorithms of the latter kind (see also Tenenbaum et al. Reference Tenenbaum, Kemp, Griffiths and Goodman2011; Garnelo, Arulkumaran, and Shanahan Reference Garnelo, Arulkumaran and Shanahan2016). They suggest that only model-building cognizers can perform tasks with understanding and apply their existing knowledge of the world effectively in learning to perform new tasks or adapting to changing environments. If they are correct, then the content conception of a model may contribute to distinguishing between truly intelligent artificial systems and programs that perform impressively on cognitive tasks only through brute force or clever tricks. In essence, Lake et al.’s distinction is between one group of algorithms that learn to proceed in a single step from input to output (the pattern recognition type) and another that constructs and employs structures that represent underlying features of the domain with which the system interacts (the model builders). This is intended to capture common features of two distinctions: between model-free and model-based reinforcement learning, and between discriminative and generative classifiers.
I will discuss these two examples in turn and argue that, in both cases, the model-building algorithms employ representations with content that is apt for use in explaining the inputs they receive and for justifying outputs. In the two examples, the representations in question have more in common than just this: they also represent relations that make it possible to calculate the probabilities of states of the environment and of forms of input given these states. But for reasons I explain further below, I think that specifying the content conception in terms of aptness for explanation and justification is more likely to give a fully general characterization than any more direct account of the content involved. For the avoidance of doubt, my claim is not that the use of a cognitive model requires the capacity to give explanations or justifications but only that it requires the use of representations with content that could contribute to explanations or justifications. So on this account what makes a representation a model will indeed be its content, not the cognitive functions for which the system in question can use it.
In the rest of this section, I first discuss the two examples and then present the content conception and draw out an important implication.
3.1. Two Examples
In model-based reinforcement learning (RL), agents learn to select rewarding actions by coming to represent two kinds of information about their environment. They learn about the values of possible outcomes and about relations between actions and outcomes. To choose the most rewarding action in a given situation, agents using this system may use these representations to generate a decision tree—showing the outcomes accessible from their current situation by a single action, the outcomes accessible from each of those, and so on—and then to evaluate each possible course of action on the basis of the value of the sequence of outcomes it will bring about. In contrast, in model-free RL agents learn only about the levels of reward brought about by actions in given situations, without learning which outcomes result from these actions. In effect, they learn the values of actions, rather than of outcomes, and they do not learn about relations between actions and outcomes. Model-free algorithms learn directly what output should be produced (the action) in response to each input to the system (the situation); model-based algorithms learn facts about the environment that allow them to reason about which action to perform.
Model-free RL algorithms such as temporal difference learning can learn the long-run values of actions (Sutton and Barto Reference Sutton and Barto2018), thus avoiding the computational costs associated with generating and evaluating decision trees. But they are inflexible compared to model-based algorithms in the following sense: model-based algorithms can quickly adapt to small changes in outcome values or action-outcome contingencies, by updating the corresponding representations, but model-free algorithms must relearn policies from scratch to accommodate such changes. As Lake and colleagues (Reference Lake, Ullman, Tenenbaum and Gershman2017) explain, a human video game player could immediately perform competently in a new version of a game with a different objective or pattern of rewards and punishments, but Mnih et al.’s (Reference Mnih2015) deep neural network that achieved human-like performance on a range of Atari games requires a great deal of retraining to cope with such changes (Rusu et al. Reference Rusu, Rabinowitz, Desjardins, Soyer, Kirkpatrick, Kavukcuoglu, Pascanu and Hadsell2016) because their system relies solely on model-free RL.
This difference in flexibility and the ability to apply knowledge in new conditions is illustrated by the standard experimental methods for distinguishing model-based from model-free action selection. In one method, outcome devaluation (Balleine and Dickinson Reference Balleine and Dickinson1998), animals learn to perform an action, such as pressing a lever, for a reward, which is typically an unfamiliar food. The reward is then devalued, away from the setting in which the action has been learned. For example, the food may be paired with an injection of a substance that causes gastric illness. The animals are then tested to see whether they resume performing the action in the original setting, without the reward being delivered. Continuing to perform the action is considered to be evidence of model-free RL because this indicates that the action itself is represented as rewarding. Reduced performance is evidence of model-based RL because it indicates knowledge that the action leads to the now-devalued reward.
A second method, the two-step task (Gläscher et al. Reference Gläscher, Daw, Dayan and O’Doherty2010; Lee, Shimojo, and O’Doherty Reference Lee, Shimojo and O’Doherty2014), involves participants making two choices in sequence. Each choice leads to a subsequent state with a certain probability, and reward depends only on which state is reached after the second step. There are different ways to use tasks of this form to test for model-based control, but a simple one is to allow participants to explore the state space in the absence of reward first, then inform them about which states are rewarding and introduce the rewards. Participants who perform above chance when rewards are first available must be using model-based RL because no actions have previously been rewarded, so a model-free system would have learned nothing (Gläscher et al. Reference Gläscher, Daw, Dayan and O’Doherty2010).
There are two abstract features of model-based RL in virtue of which it may be said to use models, while model-free RL does not. First, model-based RL represents relations that model-free RL does not and that crucially go beyond the relations between inputs and suitable outputs that are most directly relevant to the task of action selection. Model-based RL represents how the environment will change over time, contingent on possible actions, and makes use of these representations to select actions. Model-free systems do represent current states of affairs and the range of actions that are currently available, but they do not represent possible transitions between states of the environment, which are relations between such states. Model-free systems represent only the expected levels of reward of actions in each possible state, which is to say that they represent only the suitability of available outputs for each possible input.
Second, these relations allow the formation of expectations about environmental contingencies and are apt to facilitate hypothetical reasoning and to ground explanations of inputs and justifications of outputs. Both model-based and model-free RL algorithms are good for returning representations of actions that are likely to be rewarding as outputs when fed representations of states of affairs as inputs. But model-based RL also has the resources to perform further related tasks. It employs representations that have the potential to be used to predict what will follow from either the current state of affairs or some hypothetical alternative, given each of a range of possible actions. This information can be used to give nontrivial justifications of selected actions—model-free RL says only that the chosen action is the best available, whereas model-based RL says what is good about it.
Turning now to the distinction between discriminative and generative classifiers (Jebara Reference Jebara2004; Lake et al. Reference Lake, Ullman, Tenenbaum and Gershman2017), a typical task in machine learning is to classify inputs, such as handwritten characters. In this context inputs are sometimes called ‘data’, and outputs are ‘labels’. Discriminative classifiers are those that work by learning and applying a representation of the probability distribution of labels given data. These representations allow them to perform classification tasks directly; given a particular handwritten numerical character, the system will read off from the distribution the probabilities that the correct label for this shape is 0, 1, 2, … , or 9 and can simply pick the label with the highest probability. Generative classifiers learn the distribution of data given labels and the prior probabilities of labels, which enables them to perform the task thanks to Bayes’s rule. Compared to discriminative algorithms, they work the ‘other way around’; they attempt to match the data to representations of likely shapes corresponding to each character. In a sense, a generative algorithm for classifying handwritten characters relies on knowledge about what each character is like, rather than knowledge about which shapes constitute which characters.
As I mentioned above, generative models are central to the predictive processing theory of cognition (Clark Reference Clark2013, Reference Clark2016; Hohwy Reference Hohwy2013). As it applies to perception, this theory claims that we perceive the world through an ongoing process of hypothesis formation and testing. The way that I represent the world as being at time t causes my perceptual system to represent a probability distribution over ways the world might be at time
, from which a hypothesis is formed. This hypothesis predicts that I will undergo certain sensory stimulation at
, and this prediction can be used to test and revise the hypothesis, until one is found that minimizes prediction error. In order to perform these steps, the perceptual system must employ a generative model. It requires representations of the probabilities of forms of sensory stimulation (the input to the system) given hypotheses about how things are (the output) and also of the causal or informational relations between states of the world at successive times or in neighboring locations.
Generative classification algorithms are therefore distinguished from discriminative algorithms by the same two abstract features that distinguish model-based from model-free RL, although it is the second that defines the generative-discriminative distinction. First, in some important cases generative algorithms represent relations that are not represented by discriminative algorithms. As just described, a prediction error minimization algorithm for perception will use a representation of relations between successive states to generate hypotheses concerning incoming stimuli; without this knowledge far too many hypotheses would have to be tested for the system to perform the task effectively. In terms of data and labels, this representation of relations provides the prior probabilities of labels.
Second, unlike discriminative algorithms, which proceed directly from input to output, generative models represent features of the task domain that explain the data. These features include both relations between hidden variables and the typical sensible qualities of the kinds that the system aims to identify. If human perceptual systems use generative models, these represent relations that have the potential to explain why we receive the sensory stimulation we do; for example, I may represent that a goal has just been scored, that goal scoring tends to cause cheering, and that cheering sounds a certain way, and these facts will together explain the sounds I am currently hearing. The same aspects of the representational content of generative models allow the formation of expectations about stimuli and are apt to make hypothetical reasoning possible.
3.2. Formulating the Content Conception
Model-based RL algorithms and generative classifiers have in common that, unlike their respective alternatives, they represent features of task domains that are capable of explaining their inputs or justifying their outputs. Model-based RL algorithms represent causal relationships between states, actions, and subsequent states, and these have the potential to contribute to both explaining why agents find themselves in the states they do and justifying their chosen actions. Generative classifiers represent the probabilities of data given labels, and these representations can contribute to explaining why the data they receive take the forms they do. Generative models in predictive processing, moreover, represent causal or informational relations between successive states, which can contribute further to explaining sensory stimulation.
These features distinguish model building from pattern recognition. So the content conception of a cognitive model is as follows:
Content Conception: A cognitive model is a representation used in performing a cognitive task, which represents features of a task domain that are apt to explain inputs to the cognitive system or justify its outputs.
As Lake et al. put it, “cognition is about using … models to understand the world, to explain what we see, to imagine what could have happened that didn’t, or what could be true but isn’t, and then planning actions to make it so” (Reference Lake, Ullman, Tenenbaum and Gershman2017, 2).
A potential objection to this account is that it offers quite an indirect characterization of the kind of content that distinguishes cognitive models. Instead of saying which features of the task domain models represent, it says only that they represent features that are apt for further tasks. Before I give my response to this objection, it will be helpful to note an important implication of the account.
The implication is that cognitive models do not have their status as such outright but only in relation to particular tasks for which they are used; many representations are models relative to some task but not to others. One example of this comes from the theory of forward models used in motor control (Wolpert et al. Reference Wolpert, Ghahramani and Jordan1995; Grush Reference Grush2004). Grush describes subsystems of the motor control system, which he calls ‘emulators’, which take efference copies of motor commands as input and produce representations of likely sensory feedback as outputs (I will use Grush’s term to refer to these subsystems and the term ‘forward model’ for the representations they employ). He explains that one function of emulators is to allow motor commands to be corrected more rapidly than if real sensory feedback were required. He further explains that emulators might use either associatively learned lookup tables representing motor command-sensory feedback pairs or more sophisticated ‘articulated models’, with parts that correspond to at least some of the parts of the musculoskeletal system itself, and generate representations of likely feedback by simulating the interaction of these parts. We can focus on the unarticulated case and consider whether a forward model that consists of command-feedback pairs is a model at all, by the standards of the content conception.
If we take the task to be motor control, the answer is yes. The inputs to the motor control system specify goal behaviors, and the task of the system is to produce sequences of motor commands that will generate behaviors matching these specifications. Relative to this task, the unarticulated forward model is a model because the production of a particular motor command may be justified by the fact that the previous command is likely to cause a certain form of sensory feedback, in combination with facts about the current goal behavior. If an agent is trying to post a card through a slot, and the forward model entails that the card will move toward a position a little to the left of the slot, this justifies a new motor command to move the card to the right. However, relative to the emulator’s task, the unarticulated forward model is, despite its name, not a model at all. The emulator’s task is to take motor commands as input and produce representations of likely sensory feedback as output, and the representation in question links these inputs and outputs in a single step. Relative to this task, the forward model is just like the action value representations in model-free RL, or the representations of the distribution of labels given data in discriminative classifiers. In contrast, the articulated forward model that Grush suggests would count as a model relative to both tasks.
I now return to the objection that the content conception characterizes models indirectly. The points just raised about task relativity illustrate the difficulty of giving an alternative, more direct account of the kind of content that characterizes models. To start with, consider the proposal that what characterizes models is that they represent probabilistic relations between world states; this is an obvious common feature of the models in both model-based RL and predictive processing. This characterization is problematic because, as in the case of motor control, the systems for which representations of this form are models may have subsystems relative to which they are not models. For example, model-based RL systems include subsystems with the function of taking representations of state-action pairs as inputs and providing representations of likely outcomes as outputs. Relative to this function, some representations of relations between world states may not be models. Note that this is an implication not just of the content conception as stated but also of the first-pass account of pattern recognition/model-free algorithms, which described them as those that link inputs and outputs in a single step.
A different possible alternative would be to characterize models as representations that link features other than the inputs and outputs of the systems concerned. This account has difficulty with model-free RL, however, because algorithms of this type do typically represent the levels of reward associated with each state-action pair, as opposed to merely maintaining representations of which action should be performed in each state. The content conception as stated does a better job of accommodating this example because actions cannot be justified by saying that they are rewarding to a particular degree or more rewarding than alternatives. In this context, to say that a chosen action was rewarding gives no new information to justify the choice. So I suggest that the content conception as stated offers the most promising way to capture Lake et al.’s insight.
4. An Argument against the Format Conception
In this section I argue against the format conception of cognitive models by reference to Dyna, a simple model-based RL algorithm for ordinary computers (Sutton Reference Sutton1991). I claim that Dyna uses a model but does not use structural representation. So Dyna constitutes a counterexample to the format conception: it shows that there are models that are not SRs.
Before I get into the argument, a comment on methodology. Earlier I distinguished cognitive models, which are my topic in this essay, from scientists’ models of the phenomena they study. A possible concern about my argument is that Dyna and similar programs are scientists’ models of cognitive systems, as opposed to cognitive systems themselves. Perhaps a mere scientists’ model of model-based RL need not use a model itself. This objection is ill founded, however, because there is no reason to deny that Dyna actually undergoes RL. My method is therefore not unlike some scientists’ use of models; I use a particularly simple example of the phenomenon I am interested in, in order to see what is essential to it.
Dyna is an algorithm for learning to exploit sources of reward in relatively simple virtual environments. These environments generate finite Markov decision processes (MDPs), which the algorithms must solve (Sutton and Barto Reference Sutton and Barto2018). In an MDP time proceeds in discrete steps, and at each time step the agent finds itself in an identifiable state and must select an action. At the next time step, perhaps partly as a consequence of its action, the agent will find itself in a further state and will receive a numerical reward. Then it must choose a further action. Transitions between states may be probabilistic, but the probabilities depend only on the previous state and the action selected. In finite MDPs, there are only finitely many possible states, actions, and levels of reward.
In small finite environments, RL researchers have used lookup tables to represent what is learned in both model-free and model-based algorithms. This point is made clear in works such as Boyan and Moore (Reference Boyan, Moore, Leen, Tesauro and Touretzky1995) and Kuvayev and Sutton (Reference Kuvayev and Sutton1996), which discuss the problem of extending algorithms for finite MDPs to ones in which there is continuous variability in states and actions. This is a crucial point because, as I have argued, lookup tables are not usually SRs.
In particular, the Dyna architecture maintains three lookup tables. One represents the long-run values associated with states, which depend on the values of likely subsequent states. A second represents the next states and rewards that follow from each state-action pair; this is referred to as the ‘world model’ (Sutton Reference Sutton1991). And the third represents a policy—a set of rules concerning what to do in each state, which is used to select actions. The algorithm causes the world model to be updated from the agent’s experience of the environment, and the policy and record of long-run state values are updated both by ‘real’ experience and ‘simulated’ experiences generated by the world model. The model is therefore used for planning, understood as updating the policy in advance of action.
To see that the world model in Dyna is not an SR, we can apply the argument used in section 2.2 to show that lookup tables do not usually bear a structural correspondence to the phenomena they represent. The key point is that because each possible state will be represented by multiple entries in the world model, there is no homomorphism between this representation and the causal structure of the virtual environment.
For example, consider the fragment of a possible world model shown in table 2. The presence of a given state symbol in the right-hand column represents that the corresponding state is caused by the performance of action A1 in the state represented by the symbols on the same row in the left-hand column. It may be suggested, then, that the relation over parts of the table being to the right on the same row is used to represent the relation causation via A1 in the virtual environment.
Table 2. Possible World Model
| Initial State | Resulting State under A1 |
|---|---|
| S1 | S2 |
| S2 | S3 |
| S3 | S4 |
| S4 | S2 |
If we define a function that takes symbols in the table to the corresponding states of the virtual environment, however, we can see that this will not be a homomorphism with respect to these two relations. For the function to be a homomorphism, it would have to be the case that a given state symbol is to the right on the same row as another if and only if their corresponding states are linked by causation via A1 in the virtual environment. This is not the case because the S1 on the first row and the S2 on the last row are such that their corresponding states are linked by causation via A1, but they themselves are not linked by the relation being to the right on the same row. Similar points will hold for world models for almost every possible environment.
As we also saw in section 2.2, if each row of a lookup table is considered to be a separate representation, these do satisfy the definition of an SR. But this is hardly grounds on which to defend the format conception because the point remains that the lookup table—the means by which the causal structure of the environment is represented—does not exploit any correspondence with this structure. Its parts correspond in structure to parts of the environment, but only when such small parts are considered that this is near trivial.
The upshot of these considerations is that Dyna does not use structural representation. With this point in hand, all we need to show that the format conception of cognitive models is mistaken is that Dyna uses such a model. I take the account of the content conception in section 3 to support this claim because it shows how the distinction between model-based and model-free RL can be assimilated to that between generative and discriminative classifiers; this indicates that the term ‘model’ is not used idiosyncratically by RL researchers. So I conclude that having a structural format is not necessary for a representation to be a cognitive model.
5. Conclusion
Philosophers have often connected the concept of a cognitive model to that of structural representation, suggesting that what it is for a representation used in a cognitive process to be a model is for it to be an SR. I have presented an alternative conception, according to which models are distinguished by their representational content, not their format. Following Lake et al. (Reference Lake, Ullman, Tenenbaum and Gershman2017), this conception emphasizes connections between model building, expectation, explanation, and justification. I have also argued that having a structural format is not necessary for a representation to be a model.
However, this does not mean that there are no important connections between the phenomena of structural representation and the use of models of task domains by cognitive systems. It could be that the use of structural representation is sufficient for modeling—although one reason to doubt this is that discriminative classifiers may use iconic (and hence structural) formats in representing their inputs. This point deserves further investigation. Even if structural representation is neither necessary nor sufficient for modeling, it could still be the case that there is a great deal of overlap between these categories, for deep reasons, perhaps to do with surrogative reasoning (Swoyer Reference Swoyer1991). Nonetheless, theorists should pay more attention to content, and perhaps less to format, in seeking to understand cognitive models.
