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Rather than working memory capacity acting largely as a distinct subordinate function of fluid intelligence, there is an emerging consensus that the WM-Gf link (e.g., Ackerman et al., 2005; Engle, Tuholiski, et al., 1999; Kyllonen & Christal, 1990; Unsworth & Engle, 2005) can be understood as the outcome of common functions dictated by the strength and flexibility of relational bindings between integrated representations (Chuderski, 2014; Oberauer et al., 2007; Shipstead et al., 2016). In the current study, we manipulated a single task (the Arithmetic Chain Task; Oberauer et al., 2001) in order to differentially demand retention, access, and binding. The layering of manipulations allowed us to pinpoint the contributing functions while reducing task artefacts typically present in comparing multiple task formats. Our manipulations began on a similar premise to Oberauer et al. (2001), in distinguishing retention from access by comparing how stored contents (the ABC mappings) were assessed during arithmetic performance: either passively (recalled at the end of the task only) or accessed during active processing (incorporating the mappings as part of the

arithmetic). We extend on Oberauer et al.’s (2001) research by comparing fixed-access to random-access which prevents systematic chunking (Halford et al., 1998). Our findings partially replicated Oberauer et al. by finding that access does incur a larger cost to arithmetic processing over passive retention – but we have demonstrated this only occurs if the stored variable mappings must be accessed in a random order. This indicates that Oberauer et al. would not have found their results (that access incurred a greater load than retention) if they had made the seemingly minor change of not randomly ordering the XYZ=ABC mappings. This is demonstrated most clearly in Figure 4.5, where breaking down the access cost (Figure 4.5B) into its constituent access conditions (Figures 4.5D and 4.5E) produced markedly different outcomes: only random-access shows a significant cost over retention (the

access specifically (i.e., layered over an otherwise identical task format in fixed-access) is uniquely influenced by what is common to WM and Gf – which we have argued is the demands of binding (Chuderski, 2014; Oberauer et al., 2008).

Figure 4.6 adapts Oberauer’s (2009) concentric model, including the source of demands specific to each version of the ACT presented in the current study. Retention costs were defined as the demand imposed by encoding and maintaining additional task-irrelevant mappings for later recall. These mappings are theorized to be stored in long-term memory (outside the region of direct access) for the duration of the task. Although we identified a retention cost that Oberauer et al. (2001) did not, this cost was largely driven through a failure to recall these task-irrelevant mappings, as opposed to a load influencing the arithmetic itself (see Figure 4.3). In contrast, access costs incorporated task-critical

mappings, requiring establishing and maintaining bindings within the direct access region of

WM throughout the task. We introduced an additional cost associated with ensuring multiple,

flexible bindings in the direct access region by restricting systematicity through random rather than fixed mappings. Our proposition was that the systematicity that facilitates a single strong schema set where mappings are in a fixed serial order cannot be exploited when mappings are random, necessitating maintaining access to multiple independent bindings (Chuderski, 2014). The breakdown in systematicity results in unstable bindings that must be flexibly bound and unbound in light of the updated ordering only indicated during the executive processing of the primary arithmetic.

Figure 4.6. Diagram of Oberauer’s (2009) concentric model of WM adapted to specify task

manipulations in the featured Arithmetic Chain Task. Each small circle depicts a

representation within memory, which can either be active (filled circle) or inactive (unfilled circle). The larger oval represents the region of direct access, a capacity-limited store where representations are active above threshold and available for immediate binding and further processing. Representations in the direct-access region can be connected into a common schema set by binding them into a related context. In the ACT, retention costs involve passively storing ABC mappings (e.g., A = 6) outside the direct-access region during the arithmetic processing. Access costs are incurred by ABC mappings which must be kept active in the direct-access region for use in the arithmetic processing (called upon in cases like X = A). Binding costs are incurred by ABC mappings which must be flexibly unbound and rebound into an updated order during the arithmetic processing (cases such as X = B).

Based on work such as Oberauer et al. (2008) and Chuderski (2014), we defined a

relational binding (RB) factor as what is common to WM and Gf (defined by the SSPAN and

APM, respectively). While it is unusual to run a factor analysis on just two variables, we argued that this was more appropriate than including the SSPAN and APM separately in a regression analysis, where their respective regression coefficients would reflect unique contributions, and the common features would be obscured as shared variance without direct assessment. Thus, the EFA was used to a create a simple RB indicator from prototypical measures to approximate what is common to WM and Gf. Performance on the RB factor

indicates the extent to which participants performed well on what is common to WM and Gf – theorized to be the capacity for flexible binding.

Retention costs did produce a significant impact on performance over the control

condition when recall was considered as part of the scoring, yet these costs were not associated with the RB factor. In the retention condition, passive storage demands were incurred by encoding a set of task-external mappings at the beginning of the trial and recalling them at the end of the trial. In this way, the unrelated storage could be relegated to long-term memory outside the direct-access region. The current results indicate that this passive retention is not associated with the RB factor. This manipulation of retention is different from traditional CSPANs, where repeated unrelated, trial-specific processing

temporally overlaps with storage in which the running sequence of list items must be updated regularly. Since our retention involved encoding at the beginning and recall at the end, this storage was more passive than that required by the within-trial updating of CSPAN where the direct-access region is frequently probed with intermittent processing. Despite this, the CSPAN included in the current experiment correlated substantially better with retention than any of the other ACT conditions, while also providing a version of the ACT similar enough to the access conditions where the specific effect of access could be isolated.

In contrast to retention, access costs represent ABC mappings which must be kept active in the direct-access region during the arithmetic processing. The present results suggest that the direct-access region may be a source of capacity limits, but it is one that can be circumvented with systematic chunking of consistently-ordered bindings. We speculated that exploiting the fixed-order of mappings could systematically reduce the number of bindings from three to one. To account for this, we contrasted two access conditions: fixed and random. The conceptual difference between fixed and random is the flexibility of the

demands on the direct-access region, only the random-access requires maintaining three independent bindings. It is possible a systematic rearrangement of mappings could occur before the arithmetic processing has begun (but after the random order is revealed), but this still requires rapid binding and unbinding – a clearly isolated function above and beyond the otherwise identical fixed-access condition. There is a higher chance of losing the bindings during this rearrangement, and we observed small statistically significant differences in recall performance between the access conditions. While loss is a contributing factor, crucially, and consistent with Oberauer et al. (2007), Wilhelm, Hildebrandt, and Oberauer (2013), and our own task analyses incorporating systematicity (Halford et al., 1998), binding costs were significantly related to the RB factor, and this is by way of the random-access condition. Given that the fixed and random manipulations use an otherwise identical task format, this provides supporting evidence that what is common to WM and Gf is a capacity for flexible binding. It is worth reiterating the insights provided by this result. The binding costs inferred through the random-access condition already account for all other aspects of the ACT format. That is, the mental arithmetic involved in the core task and the additional burden of encoding, accessing, and recalling the ABC mappings through the task have already been accounted for. The exclusive component of random-access that remains after this incremental cost-analysis is the restriction that multiple bindings cannot be systematically reduced by way of fixed ordering. This restriction necessitates multiple bindings, and our results indicate that this is associated with the common factor between WM and Gf. As predicted by Oberauer et al.’s (2008) hypothesis, performance on WM tasks appears to be dictated by the binding capacity of the direct-access region. Here, we further demonstrate that the ability to rapidly establish and dissolve flexible bindings uniquely explains what is common to WM and Gf. In the current analyses, we labelled this commonality RB to represent our theoretical position. It is of course possible this commonality could be interpreted differently (e.g., controlled

attention; Kane et al., 2001), though these interpretations would also need to provide a theoretical account of the difference between fixed-access and random-access, as we have done using systematicity (Halford et al., 1998).

4.4.1. Conclusion

In conclusion, our data suggest that it is not mere passive retention, nor systematic access, that defines the common WM-Gf link but rather, the ability to establish and maintain flexible bindings. In this way, CSPAN is a useful tool not simply because it taps storage capacity, but because the interim processing frequently interrupts the strength and stability of bindings of to-be-remembered elements. Passive retention of the to-be-remembered elements does not appear important, but providing direct access to durable, flexible bindings is. A version of the ACT which incorporates the temporal overlap of processing and storage (seen in CSPANs) between ABC mappings and the arithmetic may provide further insight into this notion, as may an RB factor defined through additional tasks. For now, our results provide preliminary but insightful evidence of the importance of a binding flexibility function in WM.