The present study offers an explicit process model that is implemented in computational form to explain the mapping from online to offline responses in a single-analyzer architecture. The model is based on the proposal by Lewis & Phillips (2015) that the mapping from online to offline responses involves extended re-processing of the sentence in memory to minimize the signal-to-noise ratio.

Since the Lewis & Phillips (2015) proposal has not been implemented before, some architectural assumptions must be clarified. For explicitness, the proposal will be framed as a process of sequential memory sampling in the cue-based memory retrieval framework (e.g., McElree 1993; 2000; McElree, Foraker & Dyer 2003; Lewis & Vasishth 2005; Lewis, Vasishth & Van Dyke 2006), in which a stimulus response is based on accumulation of evidence over time. In the cue-based memory framework, incorrect memory retrieval (i.e., retrieval of a non-target or “grammatically irrelevant” item) can trigger a “backtracking” process to reanalyze the sentence using sequential memory sampling (i.e., repeated retrieval attempts) (McElree 1993; McElree et al. 2003; Martin & McElree 2018). In the technical use of the term, backtracking refers to the process of returning to a choice point in the parse for reanalysis, and is often evoked to explain how the parser recovers from garden path effects (see Lewis 1998, for discussion). For present purposes, the notion of backtracking can be extended to memory retrieval processes, whereby retrieval mechanisms perform the same retrieval process multiple times over the same representation using the same set of cues used in the initial retrieval attempt, and aggregating the outcomes to minimize the signal-to-noise ratio, leading to more accurate representation of the current parser state (McElree 1993). This account is also inspired by “analysis-by-synthesis” models of perception, in which pattern recognition, symbolic generative processes, and hypothesis confirmations are performed by comparing a predicted pattern to the actual input, computing the error, and iterating the process until the error is minimized (see Bever & Poeppel 2010, for a review). Crucially, if linguistic dependency formation relies on cue-based retrieval, as previously claimed (Lewis 1996; McElree 2000; McElree et al. 2003; Lewis & Vasishth 2005; Lewis et al. 2006; Van Dyke & McElree 2006; 2007; 2011; Van Dyke 2007; Jonides et al. 2008; Martin & McElree 2008; 2009; 2011; Vasishth et al. 2008), then it is reasonable to assume that backtracking would apply uniformly to retrieval for linguistic dependencies, such as subject-verb agreement.

To provide a brief sketch of how this process plays out, consider again the sentence in (1). Here, incorrect retrieval of the attractor during online processing fails to satisfy the grammatical constraints on subject-verb agreement, e.g., it is not the subject of the verb, triggering a backtracking process to recover the target subject. Since backtracking takes time to complete, different outcomes are predicted at different points in time: initially, the wrong item can be retrieved, giving rise to agreement attraction in time-restricted online measures, but this retrieval error can be rectified via backtracking operations triggered by the grammar, eventually leading to the correct analysis reflected in offline judgments.

Three experiments were designed to test Lewis and Phillips’ (2015) proposal that the mapping from online to offline responses reflects extended re-processing of sentences in memory. Experiments 1 and 2 used an agreement attraction paradigm to verify that online and offline measures yield contrasting profiles with respect to illusory licensing. The results of those experiments served as the basis for the computational implementation of the proposed process model in Experiment 3. To preview, the model generates a good fit to the data from Experiments 1 and 2, providing proof-of-concept for the single analyzer account.

Results from Experiment 1 revealed n effect of agreement attraction in a time-restricted acceptability task, which appear as increased acceptability for ungrammatical sentences with an attractor that matched the number of the verb, relative to ungrammatical sentences that lacked a number-matching attractor. These results replicate those reported in previous studies that have used speeded acceptability judgments to elicit agreement attraction (e.g., Wagers et al. 2009; see also Parker & Phillips 2016), and provide a clear measure of time-restricted responses that will be directly compared to the untimed acceptability judgments in Experiment 2.

Results from Experiment 2 showed that participants are sensitive to the number match between the subject head noun and the verb but are not misled by a number matching attractor when they are given ample time to make their judgment. These results replicate previous studies showing that attraction effects are reduced in untimed tasks (see Table 1). Crucially, Experiments 1 and 2 tested the same item sets and held constant the mode of presentation (RSVP) and the requirement for a forced-choice judgment, but showed contrasting profiles that hinged on whether or not judgments were elicited with a time restriction. This contrast is illustrated in Figure 6, which shows how much the presence of the plural attractor boosts (or fails to boost) acceptance rates in the ungrammatical conditions for timed and untimed judgments. This figure highlights that attraction is significantly reduced in untimed judgments. A statistical analysis supporting this contrast is presented in Table 7. In addition, average response times for Experiment 2 were also considerably longer than those from the speeded judgment task in Experiment 1, which is consistent with proposal that additional time for re-sampling reduces susceptibility to attraction. This proposal will be explored in-depth in the modeling experiment in the next section.

One surprising effect concerning the finishing times for Experiments 1 and 2 is that participants consistently took longer to respond in the grammatical condition with a singular attractor. A similar, albeit smaller effect is observed in the parallel ungrammatical conditions with a singular attractor. In these conditions, both the target and subject overlap in features with the retrieval cues (e.g., both are singular nouns). A likely possibility is that the increased time in these conditions reflects a “fan” effect at the stage of retrieval (Anderson 1974; Anderson & Reder 1999), which can lead to increased processing times when multiple items match the retrieval cues (Badecker & Straub 2002; Autry & Levine 2014; but cf. Chow, Lewis & Phillips 2014). Alternatively, it could reflect an effect of feature-overwriting at the stage of encoding (Nairne 1990; Vasishth, Jäger & Nicenboim 2017), where the overlap in features degrades the quality of the target representation, making recovery of the target more difficult at the stage of retrieval.

Taken together, Experiments 1 and 2 confirm that online/offline mismatches involving agreement attraction reflect the time sensitivity of the task (Lewis & Phillips 2015). These results will form the empirical basis of the single-analyzer process model developed and tested in Experiment 3.

To derive quantitative predictions for timed and untimed responses, the current study used a variant of the ACT-R model of sentence processing described in Lewis & Vasishth (2005), which implements a cue-based retrieval mechanism for syntactic dependency formation [using code originally developed by Badecker & Lewis (2007)]. ACT-R (Adaptive Control of Thought—Rational; Anderson et al. 2004) is a general cognitive architecture based on independently motivated principles of memory and cognition, and has been applied to investigate a wide range of cognitive behavior involving memory access, attention, executive control, and learning. The ACT-R model of sentence processing applies the cognitive principles embodied in the general ACT-R framework to the task of sentence processing.

In the model, the words and phrases of a sentence are encoded as “chunks” (Miller 1956) in content-addressable memory (Kohonen 1980), and hierarchical sentence structure is represented using pointers that index the local relations between chunks. Chunks are encoded as bundles of feature-value pairs, which are inspired by the attribute-value matrices described in head-driven phrase structure grammars (Pollard & Sag 1994). Features are specified for lexical content (e.g., morpho-syntactic and semantic features), syntactic information (e.g., category, case), and local hierarchical relations (e.g., parent, daughter, sister). Values for features include symbols (e.g., ±singular, ±animate) or pointers to other chunks (e.g., NP1, VP2).

Linguistic dependencies, such as subject-verb agreement, are constructed using a domain-general cue-guided retrieval mechanism. This mechanism probes all previously encoded chunks in memory to recover the left part of the dependency (i.e., the target/licensor) using a set of retrieval cues that are compiled into a retrieval probe. Retrieval cues are derived from the current word, the linguistic context, and grammatical constraints, and correspond to a subset of the features of the target (Lewis et al. 2006).

The current model falls under the class of “activation-based” models of memory access, as chunks are differentially activated based on their match to the retrieval cues (see Jonides et al. 2008, for a review). In this class of models, the probability of retrieving a chunk is proportional to the chunk’s overall activation at the time of retrieval, modulated by decay and similarity-based interference from other items that match the retrieval cues. The activation of an item Ai is defined in Equation 1, which makes explicit four principles that are known to impact memory access: (i) an item’s baseline activation B_i, (ii) the match between the item and each of the j retrieval cues in the retrieval probe S_ji, (iii) the penalty for partial matches PM between the cues of the retrieval probe and the item’s feature values, and (iv) stochastic noise. 2

Baseline activation B_i is calculated according to Equation 2, which describes the usage history of chunk i as the summation of n successful retrievals of i, where t_j reflects the time since the jth successful retrieval of i to the power of the negated decay parameter d. The output is passed through a logarithmic transformation to approximate the log odds that the chunk will be needed at the time of retrieval, based on its usage history. After a chunk has been retrieved, the chunk receives an activation boost, followed by decay.

The degree of match between chunk i and the retrieval cues reflects the weight W associated with each retrieval cue j, which defaults to the total amount of goal activation G available divided by the number of cues (G/j). Weights are assumed to be equal across all cues. The degree of match between chunk i and the retrieval cues is the sum of the weighted associative boosts for each retrieval cue S_j that matches a feature value of chunk i. The associative boost that a cue contributes to a matching chunk is reduced as a function of the “fan” of that cue, i.e., the number of competitor items in memory that also match the cue (Anderson 1974; Anderson & Reder 1999), according to Equation 3.

Partial matching makes it possible to retrieve a chunk that matches only some of the cues (Anderson & Matessa 1997; Anderson et al. 2004), creating the opportunity for retrieval interference of the sort that leads to agreement attraction errors (Wagers et al. 2009). Partial matching is calculated as the matching summation over the k feature values of the retrieval cues. P is a match scale, and M_ki reflects the similarity between the retrieval cue value k and the value of the corresponding feature of chunk i, expressed by maximum similarity and maximum difference.

Lastly, stochastic noise contributes to the activation level of chunk i. Noise is generated from logistic distribution with a mean of 0, controlled by the noise parameter s, which is related to the variance of the distribution, according to Equations 4 and 5. Noise is recomputed at each retrieval attempt. Activation noise plays a critical role in the current analysis. Activation creates the opportunity for memory errors (Anderson & Matessa 1997), such as agreement attraction in real-time comprehension. The notion of noise in this framework is based on the hypothesis that memory trace activation fluctuates over time both randomly and as a function of usage (see Lewis & Vasishth 2005, for discussion).

Ultimately, activation A_i determines the probability of retrieving a chunk according to Equation 6. The probability of retrieving chunk i is a logistic function of its activation with gain 1/s and threshold τ. Chunks with a higher activation are more likely to be retrieved.

Typically, the target item will have the highest probability of retrieval, because it has the highest degree of activation at retrieval due to its match to the retrieval cues. However, non-target items, such as attractors, can be activated based on a partial match to the retrieval cues (see the third term of Equation 1) and subsequently retrieved if their activation is higher than that of the target due to noise, giving rise to attraction effects.

Once an item is accessed in memory as described in Equations 1–6, it is checked by the grammar (the sole structure-building system) to determine whether it meets the grammatical requirements for dependency formation. If the item satisfies these requirements, it will be integrated into the current context by combining the cues and contents of the item to form a new memory trace (Eich 1982; Murdock 1983; Dosher & Rosedale 1989) with a feature reflecting its downstream dependency (Parker, Shvartsman & Van Dyke 2017). However, if the item does not satisfy grammatical requirements, e.g., because it is not in the required structural position, then the grammar will trigger a subsequent retrieval process that engages in iterative sampling over the same representation using the same cues that were used in the initial retrieval attempt to recover the target. This process sequentially aggregates the outcomes from each retrieval iteration to minimize the signal-to-noise ratio such that retrieval of the target becomes the dominant outcome over time.

In an attraction configuration such as (1), if a non-target item has a subset of the required features, such as a plural feature for plural subject-verb agreement, the process of checking the plural feature can temporarily boost acceptability, giving rise to attraction effects, but resampling will still occur because the attractor does not satisfy the grammatical constraints on subject-verb agreement, i.e., it is not the subject of the verb. Crucially, sequential sampling will decrease the probability of retrieval error over time, eventually leading to the grammatically correct analysis revealed in later offline judgments. This process will take time to complete, predicting different outcomes depending on the amount of time that comprehenders have to process the sentence, e.g., initial time-sensitive vs. untimed responses. Importantly, during reprocessing, the model relies on the same rules and representations used in the initial retrieval attempt, consistent with the single-analyzer account of sentence comprehension proposed by Lewis & Phillips (2015). That is, sequential sampling in the model does not resort to different rules, build a different set of representations, or invoke different mechanisms for timed and untimed tasks.

The goal of the present study was to sharpen the issues concerning the debate over the cognitive architecture of language by testing the hypothesis that online and offline responses for sentence comprehension are the product of a single structure-building system embedded in a noisy cognitive architecture, and that mismatches between timed and untimed judgments about a sentence reflect extended re-processing to minimize the signal-to-noise ratio in grammatical processing over time (Lewis & Phillips 2015). To test this hypothesis, the current study focused on a specific type of online/offline mismatch involving agreement attraction.

Experiments 1 and 2 verified the online and offline generalizations reported in the literature using a single set of items across experimental methods: comprehenders treat ill-formed agreement dependencies with a feature-matching attractor as acceptable in time-restricted measures, but judge those same sentences as less acceptable in untimed measures. Experiment 3 then offered an explicit process model based on the single-analyzer account of the linguistic cognitive architecture (Phillips 2004; 2013; Phillips et al. 2011; Lewis & Phillips 2015). The model captured the mapping between online and offline responses as a process of error-driven sequential sampling in the cue-based memory retrieval framework (Lewis & Vasishth 2005; Lewis et al. 2006). The key prediction of the model is that different outcomes are expected at different points in time, which can be tracked by timed and untimed measures. Modeling results were closely aligned with the behavioral data, showing attraction in initial timed judgments, and a rapid reduction and eventual nullification of attraction in offline tasks as a function of sequential sampling over time.

The current results have several implications for our understanding of the source of agreement attraction effects and the cognitive architecture of language. First, the behavioral experiments from the current study (Experiments 1–2) sharpened the empirical issue concerning contrast between online and offline tasks involving agreement attraction by isolating the effect of timing in a way that previous studies on agreement attraction had not. Holding constant the mode of presentation, Experiments 1 and 2 provided empirical support for the claim that previously observed contrasts between online and offline data are distinguished by the time sensitivity of the response. Second, and more importantly, the results of the current study provide proof-of-concept that one type of online/offline mismatches involving agreement attraction can be captured in the single-analyzer framework (illustrated in Figure 2), without positing separate analyzers for online and offline tasks. Specifically, the current study drew on a widely-used model (ACT-R) and showed that by extending the model to perform iterative memory sampling, we are able to capture the contrast between online and offline data without recourse to a special class of extra-grammatical strategies or heuristics. In this way, the notion of resampling provides an explicit proposal for what constitutes “reflection” in linguistic judgment tasks, namely that it might involve repeated re-sampling of an activation-based memory to better distinguish between grammatical and ungrammatical strings. More broadly, the current results lend further support to the claims that reanalysis entails additional processing time (Martin & McElree 2018), and that multiple retrieval attempts can account for reanalysis effects without recourse to a specialized reanalysis mechanism (Van Dyke & Lewis 2003; Martin & McElree 2018).

A concern with the current study is that the proposed model does not predict acceptability judgments per se. In the current study, it was simply assumed that memory activations and the output of retrieval processes feed judgments in a monotonic fashion. However, there are alternative ways in which differences in activation for the target vs. attractor could impact judgments, and an important task for future research is to test this assumption more rigorously. For instance, activation values may have a non-monotonic, probabilistic relation with judgments that incorporates uncertainty at various levels of representations, starting at the level of the input and ending with motor command for the button press. What is needed is an “end-to-end” model that maps directly from input to the button press for the judgment. A modest next step would be to integrate the current model with recent modeling efforts that simulate judgment distributions (e.g., Dillon et al. 2015), which would draw directly from the activation distributions observed in the current study.

The current results do not disconfirm the dual-analyzers account. But they do provide the necessary proof-of-concept that at least one piece of evidence taken to support the dual- analyzers account can be captured in a single- analyzer architecture by drawing on independently motivated principles of general cognition. The current single-system account offers several advantages over the dual-analyzers account. First, the current account offers a plausible explanation for why grammatically accurate judgments are often slow or delayed. According to the dual- analyzers account, slow but accurate judgments are taken to reflect a grammatical analyzer that is distinct from the fast acting parser (Townsend & Bever 2001). However, slow responses do not necessarily entail a separate linguistic analyzer. Under the current single-analyzer account, the reason we sometimes see delayed accuracy is because comprehension relies on complex, multiple-step computations (constraint application, cue-generation, memory access, retrieval, integration, interpretation, etc.) that take time to complete, even for a relatively straightforward dependency like subject-verb agreement. If online and offline measures can access the internal stages of those computations, then it should be unsurprising to find different responses at different points in time.

Second, the current proposal offers a detailed linking hypothesis that relates the underlying cognitive architecture with observable linguistic behavior. If the grammar operates independently of observable online parsing behavior, as assumed under a dual-analyzers account, then grammatical computations will be difficult to pinpoint in time, making it impossible to develop or test linking hypotheses for linguistic knowledge and behavior (see Phillips 2004, for discussion). However, if online and offline phenomena are treated as different reflections of the same system, then the mental operations of the grammar become easier to pinpoint in time.

The current process model captured the mapping from online to offline responses for agreement attraction effects. Importantly, the model is not simply a “one off” model built to explain a narrow range of effects for subject-verb agreement. As noted in the Introduction, attraction effects are observed for a wide range of dependencies involving anaphora, ellipsis, case licensing, and negative polarity items (Drenhaus et al. 2005; Vasishth et al. 2008; Xiang et al. 2009; Martin et al. 2012; Sloggett 2013; Xiang et al. 2013; Parker et al. 2015; Parker & Phillips 2016; 2017). The current model can be applied similarly to capture attraction effects for each of these dependencies. However, recent work suggests that there are subtle, qualitative differences in attraction effects across dependencies (Dillon et al. 2013; Parker & Phillips 2016; 2017), and an important task for future research is to test whether those nuances are captured in the current model.

Lastly, it is also worth noting that the proposed process model is compatible with the broader conclusions drawn in the perceptual and cognitive domains. For instance, Keren & Schul (2009) argued that in the visual system, conflicting responses at different points in time, such as those involving visual illusions, reflect a single representational system that relies on two different types of criteria to evaluate the system’s output, resulting in contrasting percepts, rather than the output of multiple visual systems. Under the current single-analyzer view, the conflicting responses observed for agreement attraction also reflect different evaluation criteria, such as the initial feature match at retrieval for online measures, and the aggregate response based on sequential sampling for offline measures (when there is an initial retrieval error), resulting in contrasting percepts. A potentially fruitful line of future research would be to examine the extent to which the evaluation criteria are structured similarly across cognitive domains.