Halford’s Relational Complexity Framework

The final theory to consider is Halford’s Relational Complexity (RC) model (Halford, Baker, McCredden, & Bain, 2005; Halford et al., 1998). This model differs from previous models in that it was not developed as a model of WM per se but rather, a framework for assessing processing aspects of WM. Despite this, it has been a strong influence in future models of WM (Cowan, 2001; Oberauer, 2009a). Where Baddeley’s model is insufficient in explaining the central executive, Halford designed RC as a metric for quantifying processing capacity. As it turns out, the pervasiveness of chunking in quantifying WM capacity means that RC may be a more appropriate method for measuring WM capacity in general, not just the processing features of WM.

In this framework, processing capacity is defined as the number of arguments that must be simultaneously represented to instantiate a relation between the arguments. Like similar theorizing (Hummel & Holyoak, 2001; Oberauer, 2009a), arguments are bound dyads (elements bound to roles, slots to fillers, contents to contexts). A relation of binary

complexity would consist of two arguments. For instance, comprehending a relation of size between a rat and a mouse (RC=2, because there are two arguments) involves instantiating4 a relation between rat and mouse, such as rat-larger, mouse-smaller allowing us to

comprehend that the rat is larger than the mouse. Each additional level of RC involves an additional argument: ternary relations involve three arguments, quaternary relations involve four, and so on. Unary relations consisting of a single argument are also possible, but they only allow the isolated comprehension of an element’s category (e.g., rat-rodent) or attribute (e.g., firetruck-red) which do not act as an operator to the relation.

Transitive inference problems are a clear way of showing how RC level can increase systematically. For instance, given the premises “John is taller than Mary” and “Mary is taller than Anne”, we can construct relations using John, Mary, and Anne as elements and their heights as roles. Comparing John and Mary or Mary and Anne involves instantiating a binary relation because we need only consider one of the premises (two arguments): John is taller than Mary or Mary is taller than Anne (each are given in the premises) Conversely,

comparing John and Anne would involve instantiating a ternary relation because both premises (all three arguments) must be simultaneously considered to deduce that John is taller than Anne: John is taller than Mary and Mary is taller than Anne, therefore John must also be taller than Anne. Once this ternary relation has been comprehended and we know that John is taller than Anne, we can compress the relationship between John and Anne into a simpler binary relation, with the proviso that this new binary relation cannot (on its own) offer information on how Mary fits into this equation.

4 _{Instantiating a relation, constructing a relation, and generating a relation can be considered largely} synonymous though there are subtle differences which dictates their use in this thesis. Halford prefers instantiation as a verb as it is similar to representing the relation in memory (you instantiate a relation as you would represent an element within WM), while Oberauer prefers the more processual term constructing which demonstrates that the relation is built through a set of bindings. Thus, this thesis will use ‘instantiate’ for the use of representing a relation in WM, and ‘construction’ for the more general building of a relation, while

Halford et al. (2005) present evidence that quaternary (RC=4) relations represent the typical upper limit of complexity that humans can process (uncoincidentally similar to Cowan’s (2001) chunk limit of four). An important caveat to the RC framework is that the content of the elements or roles are distinct from the complexity of the relations, as is the format of the task. We could, for instance, make individual arguments more difficult to comprehend (e.g., by blurring the rat so it may be confused with the mouse, or by describing John’s appearance rather than simply naming him) but this increase in difficulty would be on a separate scale to the difficulty being represented by RC. In this way, a task requiring binary integration could be more difficult (as in the likelihood of answering correctly) than one requiring quaternary integration, because there are factors independent on RC influencing the difficulty. Systematically increasing RC means keeping these other sources of difficulty constant across levels of complexity, else we introduce noise in the metric. In general, we cannot directly compare relations of equal complexity across tasks because there are many other factors that play into the difficulty of a task. For instance, solving the simple arithmetic problem 3+5=? involves instantiating a ternary relation5 between three arguments: 3-addend,

5-addend, ?-sum, through which the simple incrementing of 3 to 5 results in 8, which can be

retrieved and bound to the sum position for the solution. Conversely, a task like Raven matrices (J. Raven, 1989) also involves ternary relations but is unquestionably more difficult than one-digit arithmetic because, although the RC remains at three throughout the test, items range dramatically in difficulty due to the range of unknown rules (Carpenter et al., 1990) and the embedding of complexity in superimposed (3x3) sets of ternary relations (Birney,

5 _{I acknowledge that one-digit addition likely involves immediate retrieval of solutions in adults due to over-} learnt associations between single integers, but mental arithmetic is theoretically a ternary process (Halford et al., 1998). If this scenario seems unrealistic, consider the same example but with two-digit numbers.

2002). Thus, it is not advised to compare complexity across tasks but, with all else equal, increases in RC should equate to increases in binding capacity required.