Towards dynamical system models of language-related brain potentials

Abstract

Event-related brain potentials (ERP) are important neural correlates of cognitive processes. In the domain of language processing, the N400 and P600 reflect lexical-semantic integration and syntactic processing problems, respectively. We suggest an interpretation of these markers in terms of dynamical system theory and present two nonlinear dynamical models for syntactic computations where different processing strategies correspond to functionally different regions in the system’s phase space.

Introduction

How are symbolic processing capabilities such as language realized by neural networks in the human brain? This is one of the most important problems in cognitive neurodynamics. Although methods such as event-related brain potentials (ERPs), event-related fields (ERFs), and functional magnetic resonance imaging (fMRI) have yielded considerable experimental evidence relating to the neural correlates of language processing, very little is known about the computational neurodynamical processes occurring at mesoscopic and microscopic scales that are responsible for macroscopically measurable effects.

Existing attempts to model language-related brain potentials are macroscopic-phenomenological, successfully relating particular ERP components to particular computational steps on the one hand, and to distinct cortical areas, on the other hand (Friederici 1995, 1998, 1999, 2002; Grodzinsky and Friederici 2006; Hagoort 2003, 2005; Bornkessel and Schlesewsky 2006; Frisch et al. 2008). However, these models are presently unable to relate ERPs to underlying neurodynamical systems.

In this paper we attempt to provide such an account for syntactic language processing. (A first step towards the modeling of the mismatch negativity ERP (MMN) for word recognition has recently been suggested by Wennekers et al. (2006) and Garagnani et al. (2007)). Using a particular ERP experiment on the processing of German subject–object ambiguities (Friederici et al. 1998, 2001; Bader and Meng 1999; beim Graben et al. 2000; Vos et al. 2001; Frisch et al. 2002, 2004; Schlesewsky and Bornkessel 2006), a locally ambiguous context-free grammar and two appropriate deterministic pushdown recognizers are constructed from a Government and Binding (Chomsky 1981; Haegeman 1994) representation of the stimulus material. Subsequently, we present two different implementations of these automata by nonlinear dynamical systems, resulting from universal tensor product representations (Dolan and Smolensky 1989; Mizraji 1989; Smolensky 1990; Mizraji 1992; Smolensky and Legendre 2006; Smolensky 2006). The first model is a parallel distributed processing (PDP) model (Rumelhart et al. 1986) in a high-dimensional activation vector space. Local ambiguity is processed in parallel (Lewis 1998) and model ERPs are obtained from principal components in the activation space. The second model generalizes previous work of beim Graben et al. (2004) on nonlinear dynamical automata (NDAs), where generalized shifts in symbolic dynamics (Moore 1990, 1991; Siegelmann 1996) are mapped onto a bifurcating two-dimensional dynamical system. In this model, local ambiguity is processed serially according to a diagnosis and repair account (Lewis 1998). Model ERPs are then described by the parsing entropy, obtained from a measurement partition.

Both models aim at a dynamical system interpretation of language-related ERPs (Başar 1980, 1998): The application of different processing strategies to ambiguous sentences is reflected by the exploration of different regions (probably with different volumes as well) in the processor’s phase space. These regions can be functionally related to differents ways of symbol manipulation (Fodor and Pylyshyn 1988; Smolensky 1991).

We begin with overview of language-relevant ERP components and how they could be interpreted in terms of dynamical system theory and other existing models of language processing. Then we describe a pilot study of a language processing ERP experiment and present two dynamical system models of the empirical data. This is followed by preliminary results of the ERP analysis to illustrate the dynamics of the two models in the light of their ERP correlates.

Event-related potentials in language studies

Event-related brain potentials (ERPs) are transient changes in the ongoing electroencephalogram (EEG) which are time-locked to the perception or processing of particular stimuli or cognitive events (Başar 1980, 1998; Regan 1989; Niedermeyer and da Silva 1999). ERPs differ with respect to their amplitude, polarity, latency (usually regarded as the time of maximal amplitude), duration, morphology, spatial topography, and, eventually, their putative neural generators. Some of these parameters vary depending on experimental manipulations that might be physical, such as pitch or volume for acoustic stimuli, size or color for visual stimuli, in the domain of psychophysics; or instructions or unexpectedness in the domain of cognitive neuroscience (Coles and Rugg 1995). As the ERP signal is about one order of magnitude smaller than the amplitude of the spontaneous EEG, signal analysis methods, such as ensemble averaging, source separation, or nonlinear techniques have to be employed (Regan 1989; Niedermeyer and da Silva 1999; Friederici et al. 2001; Makeig et al. 2002; Marwan and Meinke 2004; beim Graben et al. 2000, 2005; Allefeld et al. 2004; Schinkel et al. 2007). In the conventional averaging paradigm, the resulting waveforms are labeled by the polarities and latencies of peaks, such as N100, P300, N400, P600, denoting negativities around 100 and 400 ms, and positivities around 300 and 600 ms, respectively.

Since the pioneering work of Kutas and Hillyard (1980, 1984), ERPs have become increasingly important for the investigation of online language processing in the human brain. In this paper we focus on two important measures from the psycholinguistic perspective, the N400 and the P600.

The N400 component in language processing

Kutas and Hillyard (1980) reported a parietally distributed negativity around 400 ms after stimulus onset for semantically anomalous sentence continuations (shown in bold) such as (1), compared to semantically normal control sentences (2).

1

2

In subsequent work, Kutas and Hillyard (1984) reported that this N400 component is also sensitive to semantic priming, not only in sentence processing but also in lexical decision tasks (Osterhout and Holcomb (1995); see also Kutas and van Petten (1994) and Coles and Rugg (1995) for further reviews). According to Coles and Rugg (1995, p. 23), “the N400 appears to be a ‘default’ component, evoked by words whose meaning is unrelated to, or not predicted by, the prior context of the words” (Dambacher et al. 2006). This view is further supported by recent experimental findings on N400s evoked by purely semantic and thematic-syntactic manipulations (Frisch and Schlesewsky 2001; Bornkessel et al. 2004; Frisch and beim Graben 2005; beim Graben et al. 2005). For example, in the (extended) Argument Dependency Model (eADM) of Bornkessel and Schlesewsky (2006), the N400 reflects a mismatch with previous prominence information held in working memory (eADM is discussed in section “Introduction/Phenomenological models”).

The P600 component in language processing

The first syntax-related ERP component, observed by Neville et al. (1991), was the left anterior negativity (LAN). It is often more pronounced at left-hemispheric recording sites (hence the name). The LAN is related to phrase structure or word category violations 1 such as at the critical verb ‘of’ in2

3

The LAN was accompanied by a sustained parietal positivity around 600 ms, which turned out to be a very reliable indicator of syntactic processing problems. Osterhout and Holcomb (1992) found this P600 [or Syntactic Positivity Shift (SPS; Hagoort et al. (1993)] for phrase structure violations, such as (4), and for local garden path sentences (5).

4

5

When reading the sentence (5) from left to right, the preposition ‘to’ (in bold) renders the sentence temporarily ungrammatical because the comprehender expects the verb ‘persuaded’ to be followed by an object noun phrase. However, further downstream, when the phrase ‘was sent…’ is encountered, the sentence is recognized as grammatical: the verb ‘persuaded’ is a contraction of the full form ‘who was persuaded’, in other words, a reduced relative construction. Such local ambiguities thus lead the reader temporarily down a garden path; hence the name (Fodor and Ferreira 1998).

The P600 does not only reflect ungrammaticality or ambiguity. Its amplitude and latency varies with syntactic complexity and reanalysis costs (Osterhout et al. 1994; Kaan et al. 2000). Mecklinger et al. (1995); Friederici et al. (1998) and Frisch et al. (2004) reported an earlier positivity (P345) for sentences where syntactic reanalysis was easier to perform than for those evoking a P600. Late positivities which co-vary also with semantic and pragmatic contexts have been reported e.g. by Drenhaus et al. (2006). For this reason, Bornkessel and Schlesewsky (2006) propose distinguishing between the P600 and late positivites in their model.

The dynamical system approach

The brain is a complex dynamical system whose high-dimensional phase space is spanned by the activation space of its neural network. Measuring EEG, MEG or neuroimaging characteristics comprises a spatio-temporal coarse-graining that maps the phase space in an observable space spanned by multivariate time series (Amari 1974; Atmanspacher and beim Graben 2007; Freeman 2007). This observable space is explored by trajectories representing e.g. EEG time series, such as individual ERP epochs, that start from randomly scattered initial conditions.

Following Başar (1980, 1998), the experimental manipulations of an ERP experiment can be regarded as control parameters in the sense that individual ERP epochs recorded under different conditions will explore different regions of the observable space thereby revealing different topologies of the system’s flow. Changing the control parameters critically then causes bifurcations of the dynamics, that can be assessed by suitable order parameters, such as the averaged ERP, coherence, synchronization or information measures (Başar 1980, 1998; Allefeld et al. 2004; beim Graben et al. 2000).

Applying methods from information theory to ERP data requires a further coarse-graining of the continuous time series into a symbolic representation. Symbolic dynamics (Hao 1989; Lind and Marcus 1995) deals with dynamical systems of discrete phase space and time where trajectories are given by bi-infinite (for invertible maps) “dotted” symbol sequences

6

with symbols taken from a finite set, the alphabetA. In Eq. (6) the dot denotes the observation time t₀ such that the symbol right to the dot, displays the current state. The dynamics is given by the left shift

7

resulting to a new sequence with current state

In order to get a symbolic dynamics from a continuous system (X,Φ_t) with phase space X and flow one must first discretize time, such that Φ_t = Φ^t are the iterations of a map and secondly, one has to partition the observable space X into a finite number I of mutually disjoint sets: {A_i|i = 1, 2,…, I} which cover the whole phase space X, i.e. The index set A =  {1, 2,…, I} of the partition can then be interpreted as the alphabet of symbols a_i = i. Thus, the current state contained in cell A_i is mapped onto the dotted symbol “. a_i”. Accordingly, its successor x₁ = Φ (x₀) is mapped onto the symbol a_j if On the other hand, the left shift σ brings a_j onto the “surface” of the sequence s = … a_k . a_ia_j… by σ(… a_k . a_ia_j…) = … a_ka_i . a_j….

Partitioning the observable space of ERP time series, beim Graben et al. (2000, 2005), Frisch et al. (2004) and Drenhaus et al. (2006) were able to describe ERP components by suitable order parameters obtained from a symbolic dynamics of the time series. In this approach, each ERP epoch is represented by a string of length L of a few symbols {a_i|i = 1, 2,…, I} which form an epoch ensemble of N sequences for each experimental condition. A subset of these epochs that agree at a certain time t in a particular building block, a word of n = |w| symbols,

8

is called a cylinder set. Counting the members of cylinder sets (i.e. determing their probability measure) yields the word statistics from which event-related cylinder entropies

9

can be obtained.

Language processing models in psycholinguistics

In this section we provide a brief overview of language processing models discussed in the psycholinguistic literature (see e.g., Lewis (2000), Vasishth and Lewis (2006b), Lewis et al. (2006)). These can be classified into three broad types (some models cut across these broad categories, of course). The first type is the phenomenological model. Such a model pursues a top-down approach, accounting for macroscopic, phenomenological evidence from neuroimaging techniques, EEG, MEG, behavioral data and clinical studies; this type of model is usually neither algorithmically nor mathematically codified. The second type is the symbolic computational model; this class of model follows the classical cognitivistic account of formal language theory and automata theory (Newell and Simon 1976; Fodor and Pylyshyn 1988). The third type is the dynamical systems approach to cognition; these models attempt to bridge the gap between symbolic computation and continuous dynamics on the one hand, and between qualitative descriptions and quantitative predictions on the other.

These types of models can be classified along another dimension as well: they can be either serial or parallel (Lewis 1998). In a serial model, when a temporary ambiguity is encountered, only one possibility is pursued until a valid representation is computed, or the computation breaks down. In case of a breakdown, either the immediately preceding computational steps have to be retraced until a viable alternative can be found (backtracking), or the search space of possible continuations is locally modified such that the processing trajectory jumps into another admissible track (repair). By contrast, in a parallel model, the entire search space is globally explored by the processor. We study the implications of these two strategies by developing a parallel processing model (section “Methods/Tensor product top-down recognizer”) and a serial diagnosis and repair model (section “Methods/Nonlinear dynamical automaton”).

Phenomenological models

Based on the Wernicke-Geschwind model (Kandel et al. 1995, Chapt. 34), Friederici (1995, 1999, 2002) proposed her serial neurocognitive model of language processing (see also Grodzinsky and Friederici (2006)) that comprises three different phases: In a first phase around 200 ms, only word category information is taken into account to construct a preliminary syntactic representation of a sentence. Word category violations elicit the early left anterior negativity (ELAN) in the ERP (see section “Introduction/Event-related potentials in language studies”). In the second phase from 300 to 500 ms, semantic and thematic information such as the verb’s subcategorization frames are used to compute the argument dependencies and a first semantic interpretation. Violations of subcategorization information or semantic plausibility are related to the left anterior negativity (LAN) and the N400 component, respectively. Finally, in the third phase (between 500 and 800 ms), syntactic reanalysis of phrase structure violations and garden-path interpretations are accompanied by the P600 component.

This model became supplemented by a diagnosis and repair mechanism (Friederici 1998), which describes the recovery of the human language system from weak garden paths without backtracking. After diagnosing the need of reanalysis, the search space of possible grammatical continuations is modified by a local repair operation, into which the system moves directly afterwards (Lewis 1998). Friederici (1998) provides evidence that the diagnosis process is reflected by the onset of the “P600” which might appear as a P345 in constructions where only binding relations have to be recomputed (Friederici et al. 1998, 2001; Vos et al. 2001; Frisch et al. 2004). In contrast, amplitude and duration of the P600 reflect the amount of reanalysis costs. This view has been challenged by Hagoort (2003) who reported ERP results on weak and strong syntactic violations (11, 12) compared to a correct sentence (10) as in

10

11

12

where the weak violation (11) does not evoke an early positivity as it would have to be expected upon the diagnosis and repair model predicting that the weak violation would be easier to diagnose and to repair. However, this interpretation is at variance with the model because the weak violation (11) is morphosyntactic, which might be harder to diagnose than the strong word category violation (12) which in fact evoked an early starting, large and sustained P600.

Relying on Friederici’s (1995, 1999, 2002) neurocognitive model of language processing, Bornkessel and Schlesewsky (2006) have recently proposed their (extended) Argument Dependency Model, where only a syntactic core consisting of word category and argument structure information is maintained (van Valin 1993). The model accounts for cross-linguistic differences in ERP and neuroimaging patterns solely attributed to computational steps in the processing of semantic and thematic dependencies (Friederici’s phase 2). However, the third phase in the eADM comprises a generalized mapping and well-formedness checks that are reflected by late positivities in the ERP.

At the edge between the phenomenological and computational/dynamical models lies Hagoort’s Memory, Unification and Control (MUC) model (Hagoort 2003, 2005). It shares the property of a syntactic core (lexicalized structure frames) with the eADM and was computationally implemented by Vosse and Kempen (2000). The MUC attributes ERP and neuroimaging findings to a binding mechanism between lexical frames. The ELAN is evoked by a failure to bind lexical frames together, whereas the P600 is related to the time required to establishing such bindings. However, the MUC differs from the neurocognitive model and the eADM with respect to the priority of syntactic computation. While the former are serial/hierarchical, presupposing a syntax first mechanism, the latter is fully parallel where all kinds of information (word category, argument structure, semantics) are used as they become available.

Computational symbolic models

Classical cognitive science rests on the physical symbol system hypothesis that any cognitive process is essentially symbol manipulation that can be achieved by a Turing machine (or less powerful automata) (Newell and Simon 1976; Fodor and Pylyshyn 1988; beim Graben 2004). It is obvious that formal language theory provides the natural framework for treating such systems (Hopcroft and Ullman 1979; Aho and Ullman 1972).

Formal languages. A formal language is a subset of finite words over an alphabet A. Formal languages can be generated by formal grammars and recognized or translated by automata. Although natural languages are not context-free (Shieber 1985), context-free grammars (CFGs) and pushdown automata provide useful tools for the description of natural languages. A context-free grammar is a tuple G = (T, N, P, S) where T is the (terminal) alphabet of a language, N is an alphabet of auxiliary nonterminal symbols, is a set of rewriting rules and is a distinguished start symbol.

In the framework of Government and Binding Theory (GB) (Chomsky 1981; Haegeman 1994) the X-bar module supplies a context free grammar

where X and Y, respectively, denote one of the syntactic categories C (complementizer), I (inflection), D (determiner), N (noun), V (verb), A (adjective), or P (preposition). The start symbol of this grammar is usually CP or IP. The first rule determines that a maximal projection is formed from a specifier, SpecX, and an X′ phrase. The second rule allows for recursive adjunction and the third rule indicates that a head X⁰ takes a complement YP to form an X′ phrase. The head X⁰projects over its complements and adjuncts to the maximal projection. We use this framework in section “Methods/Processing models” to construct our particular language processing model.

Automata. Context-free languages can be parsed (i.e. recognized or translated) by serial pushdown automata where Q is a finite set of internal states, T is the input alphabet matching with the terminal alphabet of a context-free language, Γ is a finite stack alphabet, is a partial transition function, is the distinguished initial state, is the initial stack symbol, and is the set of final (accepting) states. An automaton with no final states: is said to accept by empty stack. Given a context-free grammar G, one can construct a pushdown automaton accepting the language generated by G by simulating rule expansions. Such an automaton is called a top-down recognizer. Formally, if G = (N, T, P, S) is a context-free grammar, then the pushdown automaton is a top-down recognizer for G, where δ is defined for all and all as follows:

13

14

where denotes the empty word of the language. For further discussion, see Aho and Ullman (1972); Hopcroft and Ullman (1979) and beim Graben et al. (2004).

For a deterministic top-down recognizer, the sets in Eq. 13 always contain only one element as the grammar G must be locally unambiguous. This can, however, be achieved by decomposing the grammar G into its locally unambiguous parts (beim Graben et al. (2004), see also Hale and Smolensky (2006) for a related approach). A deterministic top-down recognizer can be more conveniently characterized by its state descriptions where γ = γ₀γ₁…γ_k−1 is the content of the stack while w = w₁w₂…w_l is a finite word at the input tape. It follows from the definition of the transition function δ that the automaton has access only to the top of the stack γ₀ and to the first symbol of the input tape w₁ at each instance of time. Hence, we can define a map acting on pairs of the topmost symbols of stack and input tape respectively, by the transition function δ

15

where γ = γ₀…γ_k−1 if and if P contains a rule

Computational psycholinguistics does not only attempt to describe the grammar of natural languages, it also seeks to identify the mechanisms underlying human sentence processing (Lewis 2000). The existence of garden-paths is evidence that natural languages have local ambiguities; in addition, garden-paths have occasionally been taken to motivate serial processing theories (for contrasting parallel processing accounts, see section “Methods/Tensor product top-down recognizer” and Hale (2003, 2006), Boston et al. (in press)). One of the first serial processing models for human languages was Frazier and Fodor’s Sausage Machine (1978). It comprises two modules: the “sausage machine” generates phrase packages by looking a few symbols ahead into the input, while the “sentence structure supervisor” incrementally builds the phrase structure tree according to a context-free grammar. The model also accounts for particular processing strategies which minimize computational effort. Frazier and Fodor’s (1978) Minimal Attachment Principle favors phrase structure trees with a minimal number of nodes. Additionally, it has been shown that the model cannot be represented by a simple top-down recognizer (Fodor and Frazier 1980). However, there are proposals in the literature that the human parser may be a kind of a deterministic left-corner automaton with finite look-ahead (Marcus 1980; Staudacher 1990), as suggested by the sausage machine.

Generalized shifts. Of special interest for our exposition in section “Methods/Nonlinear dynamical automaton” is that any automaton (even Turing machines and super Turing devices) can be reconstructed by a particular kind of symbolic dynamics, called generalized shifts (Moore 1990; 1991; Siegelmann 1996). These are given by sets of bi-infinite dotted strings Eq. 6 which evolve under the left shift Eq. 7 augmented with a word replacement mechanism. Let be a number of digits around the dot, i.e. d = 2, then the words w = a₋₁. a₀ of length d lie in a domain of dependence (DoD). The generalized shift is given by

16

17

18

where F(s) = l dictates a number of shifts to the right (l < 0), to the left (l > 0) or no shift at all (l = 0), G(s) is a word w′ of length e in the domain of effect (DoE) to be replacing the content w in the DoD of s, and denotes this replacement function.

A deterministic top-down recognizer as described by Eq. 15 can be easily represented by such a generalized shift, choosing

19

where the stack content γ had to be reverted to γ′ = γ_k−1… γ₀ in order to bring the topmost symbol γ₀ into the DoD of the generalized shift (beim Graben et al. 2004).

Although the repair procedure of Lewis (1998) goes beyond the domain of top-down parsing, it can be straightforwardly embedded into the framework of generalized shifts by a table of additional maps, such as

20

where γ_k could be a (reversed) string in the stack with arbitrary length |γ_k|. As the repair operation can only take place within the stack, the input string w remains unaffected. We exploit this possibility in our serial diagnosis and repair model in section “Methods/Nonlinear dynamical automaton”.

Cognitive architectures. To conclude this section we shall also consider computational models that are partly hybrid architectures of symbolic and dynamic systems (Vosse and Kempen 2000; Hale 2003, 2006; Hale and Smolensky 2006; van der Velde and de Kamps 2006). One particular type of these models is based on cognitive architectures such as ACT-R, which we discuss next as an illustration.

The cognitive theory ACT-R (Anderson et al. 2004) is implemented as a general computational model and incorporates constraints developed through considerable experimental research on human information processing. The ACT-R theory relevant for the present discussion and the parsing model are outlined next, and its empirical coverage is briefly discussed. For details of the architecture the reader should consult the article mentioned above.

In its essence, ACT-R consists of two distinct systems, declarative memory and procedural memory. Declarative memory consists of items (chunks) identified by a single symbol. Each chunk is a set of feature–value pairs; the value of a feature may be a primitive symbol or the identifier of another chunk, in which case the feature–value pair represents a relation. In addition to the memory systems, focused buffers hold single chunks. There is a fixed set of buffers, each of which holds a single chunk in a distinguished state that makes it available for processing. Items outside of the buffers must be retrieved to be processed. The three important cognitive buffers are: the goal buffer, the problem state buffer, and the retrieval buffer. The goal buffer serves to represent the current control state information, and the problem state buffer represents the current problem state. The retrieval buffer serves as the interface to declarative memory, holding the single chunk from the last retrieval.

The retrieval-buffer structure has much in common with conceptions of working memory and short-term memory that posit an extremely limited focus of attention of one to three items, with retrieval processes required to bring items into the focus for processing (McElree and Dosher 1993). All procedural knowledge is represented as production rules—asymmetric associations specifying conditions and actions. Conditions are patterns to match against buffer contents, and actions are taken on buffer contents. All behavior arises from production rule firing; the order of behavior is not fixed in advance but emerges in response to the dynamically changing contents of the buffers. The sentence processing model critically depends on the built-in constraints on activation fluctuation of chunks as a function of usage and delay. Chunks have numeric activation values that fluctuate over time; activation reflects usage history and time-based decay. The activation affects a chunk’s probability and latency of retrieval. ACT-R also assumes that associative retrieval is subject to interference. Chunks are retrieved by a content-addressable, associative retrieval process (McElree 2000). Similarity-based retrieval interference arises as a function of retrieval cue-overlap: the effectiveness of a cue is reduced as the number of items associated with the cue increases. Associative retrieval interference arises because the strength of association from a cue is reduced as a function of the number of items associated with the cue.

The sentence processing model embedded in ACT-R consists of a definition of lexical items in permanent memory defined in terms of feature–value pairs, and a set of production rules specifying a left-corner parser (Aho and Ullman 1972). The model has been applied to several different published reading experiments involving English (Lewis and Vasishth 2006), German (Vasishth et al. 2008), and Hindi (Vasishth and Lewis 2006a), using the self-paced reading and eyetracking methodologies. The simulations provide detailed accounts of the effects of length and structural interference on both unambiguous and garden path structures.

An important result in this model is that all fits were obtained with most of the quantitative parameters set to default values in the ACT-R literature. The remaining theoretical degrees of freedom in the model are the production rules that embody parsing skill, and these rules represent a straightforward realization of left-corner parsing conforming to one overriding principle: compute the parse as quickly as possible. This approach thereby considerably reduces theoretical degrees of freedom; both the specific nature of the strategic parsing skill and the mathematical details of the memory retrieval derive from existing theory, plus the assumption of fast, incremental parsing.

Dynamical system models

The dynamical systems approach to cognition claims that cognitive computation is essentially the transient behavior of nonlinear dynamical systems (Crutchfield 1994; Port and van Gelder 1995; van Gelder 1998; Beer 2000; beim Graben et al. 2004).

Connectionist models. Most attempts for such models of syntactic language processing (see Christiansen and Chater (1999); Lewis (2000) for surveys), were made using local connectionist representations. These are given by activation patterns in neural networks which are either extended over many processors (“neurons”), for distributed representations, or localized to single processors for local representations. All N network processors together span a finite vector space the activation space, in which the representational states of a connectionist network are situated. Formally, a representation

21

is local if exactly one processor is activated at a certain time, i.e. x_i = 1, x_j = 0, for all On the other hand, every other representation is distributed (Smolensky and Legendre 2006).

In his seminal work, Elman (1995) suggested a simple recurrent neural network (SRN) architecture that predicts word categories of an input string. This model was elaborated by Tabor et al. (1997) and Tabor and Tanenhaus (1999) using a gravitational clustering mechanism to account for reading time differences. However, local representations have been criticized by Fodor and Pylyshyn (1988) as being cognitively implausible because they lack the required compositionality and causality properties.

In response to these criticisms, Dolan and Smolensky (1989); Smolensky (1990) and independently Mizraji (1989, 1992) suggested a unifying framework for distributed representations of structured symbolic information such as lists or phrase structure trees. According to Smolensky (1990), this approach comprises three steps:

1. decomposing symbolic structures via fillers and roles,

2. representing conjunctions,

3. superimposing conjunctions formed by tensor products of filler/role bindings.

A distributed connectionist representation of symbolic structures s in a set S is a mapping ψ from S to activation vector space The set S contains strings, binary trees, or other symbol structures. Smolensky (1990) called the mapping ψ faithful if ψ is injective and no symbolic structure is mapped onto the zero vector.

In order to prescribe the desired representation, Smolensky (1990) defined the role decomposition F/R for the set of symbolic structures S as a cartesian product of sets F × R, where the elements of F are called fillers and those of R are dubbed roles. The role decomposition is then defined by the following mapping:

22

where each is associated to the set β(s) of filler/role bindings in (Smolensky 1990, p.169) .3 The mapping β is referred to as the filler/role representation of S which provides the first step of decomposing symbolic structures via fillers and roles.

Consider, e.g., a finite sequence (a word)

23

of n symbols a_i from a finite alphabet T. Following Smolensky, the symbols can be identified with the fillers, i.e. F =  T. On the other hand, the roles are given by the n string positions, R = {0, 1,…, n}. Then, we obtain from Eq. 22,

where the expression tells that filler is bound at role i_k in forming s.

A connectionist representation ψ comprises three different maps. First, all elements and are mapped onto different vectors of two finite-dimensional vector spaces respectively. Second, tensor product representations of the filler/role bindings are introduced by another function from β(s) to Therefore, Smolensky (1990, p. 174) defined the maps:

24

25

26

The Eqs. 24 and 25 map the fillers and roles onto the corresponding vector spaces. The filler/role binding is then formed by algebraic tensor products in Eq. 26.

The second step, representing conjunctions, is achieved by pattern superposition, which is basically vector addition. If two vectors are each represented in one connectionist system by an activation pattern, then the representation of the conjunction of these patterns is the pattern resulting from superimposing the individual patterns. Smolensky (1990, p. 171) defined that a connectionist representation ψ employs the superpositional representation of conjunction if and only if:

27

where are the filler/role bindings for s conceived as propositions: “filler f binds to role r”. Thus, the connectionist representation of a conjunction of filler/role bindings is the sum of the representations of the individual bindings.4 The connectionist representation ψ restricted to β(s) is defined in Eq. 26 as ψ_b, such that we arrive eventually at

28

The string from example (23) is thus represented by the sum of tensor products

29

Depending on the dimensionality of the filler and role vector spaces, the representation Eq. 28 is faithful, if the filler and role vectors are both linearly independent. Contrarily, if this is not the case, the tensor product representation Eq. 28 describes graceful saturation of activation (Smolensky 1990; Smolensky and Legendre 2006). Note that a faithful tensor product representation allows the projection of activation patterns back into the respective filler and role spaces, thus enabling the unbinding of constituent structures. We shall use such a faithful representation for phrase structures trees in section “Methods/Tensor product top-down recognizer” for modeling our massively parallel top-down recognizer. Such models are important as they possess a compositional architecture whose representations by patterns of distributed activation are in fact causally efficacious for cognitive computations, thereby disproving Fodor and Pylyshyn (1988).

On the other hand, tensor product representations that are not faithful are interesting for other reasons. One example for such representations are fractal encodings, proposed by Tabor (1998, 2000) for constructing dynamical automata in case of (Smolensky and Legendre 2006; Smolensky 2006). Also Gödel encodings (Moore 1990, 1991; Siegelmann 1996; beim Graben et al. 2004) are special, namely, one-dimensional tensor product representations, for integer fillers and real-valued roles: where n is the number of fillers.

Dynamical recognizers. Pollack (1991) introduced dynamical recognizers and implemented them through cascaded neural networks in order to describe classification tasks for formal languages. A dynamical recognizer is a quadruple where is a “phase space”, T is a finite input alphabet of symbols a_i, is an iterated function system, parameterized by the symbols from and is a “decision function”. The recognizer accepts a finite string as belonging to a certain formal language, when

for a particular initial condition Note that this dynamical system is completely non-autonomous as there is no intrinsic dynamics prescribed generating trajectories from initial conditions as itineraries These systems have been further developed by Moore (1998); Moore and Crutchfield (2000), and especially by Tabor (1998, 2000) towards “quantum automata” and dynamical automata, respectively. We exploit these ideas in section “Methods/Nonlinear dynamical automaton” to present a “quantum representation theory for dynamical automata”.

Nonlinear dynamical automata. The simplest variant of the tensor product representation is the Gödel encoding, where a one-sided string (Eq. 23) is mapped onto a real number in the unit interval [0, 1] by a b_T-adic expansion

30

Here, is an arbitrary encoding of the b_T symbols in T by the integers from 0 to b_T − 1, and denotes its extension to sequences from

The Gödel encoding Eq. 30 has the advantage that also dotted bi-infinite strings (Eq. 6) can be described by numeric values, simply by splitting the string at the dot and representing the left-hand-side from the dot by one number and the right-hand-side from the dot by another number such that

31

32

Thus, a bi-infinite string is mapped onto by a point (x, y) in the unit square [0, 1]². This “symbologram” provides a phase space representation of a symbolic dynamics (Cvitanović et al. 1988; Kennel and Buhl 2003).

Next, consider a generalized shift with DoD of length d as in section “Introduction/Computational symbolic models”. Then, the word contained in the DoD at the left-hand-side of the dot, partitions the x-axis of the symbologram, while the word contained in the DoD at the right-hand-side of the dot partitions its y-axis. In our case, for d = 2, the DoD is formed by the words w = a₋₁. a₀, such that the symbol a₋₁ produces a partition of the x-axis and the symbol a₀ of the y-axis. Taken together, the DoDs partition the unit square into rectangles which are the corresponding domains of the symbologram dynamics. Moore (1990, 1991) has proven that this dynamics is given by piecewise affine linear (yet, globally nonlinear) maps. In contrast to the non-autonomous, forced dynamical recognizers and automata (Pollack 1991; Tabor 1998, 2000), the nonlinear symbologram dynamics, resulting from generalized shifts is autonomous and intrinsic. Therefore, we shall refer to these systems as to nonlinear dynamical automata (NDA). It is obvious that rectangles in the NDA’s phase space, which correspond to cylinder sets of the underlying generalized shift, are symbolic representations and that these are causally efficacious as they are contained in the cells of the partition defining the NDA’s dynamics (beim Graben 2004). Less obvious is yet, that these rectangles are also constituents obeying compositionality demands (Fodor and Pylyshyn 1988). However, this becomes clear by considering the symbolic dynamics of the NDA: constituents form admissible words of generalized shifts (Atmanspacher and beim Graben 2007).

In sum, we realize that faithful high-dimensional tensor product representations as well as two-dimensional NDAs fulfill Fodor and Pylyshin’s (1988) constraints on dynamical system architectures for cognitive computation, where different regions in phase space represent different symbolic contents with different causal effects.

Methods

In this section we present two different models of syntactic language processing both comprised by Smolensky’s universal tensor product representations. The first model makes explicit use of a high-dimensional and faithful filler/role binding architecture that allows the description of syntactic parsing as a massively parallel process in the phase space of a nonlinear dynamica l system. The second model, on the other hand, generalizes the nonlinear dynamical automata approach of Gödel encoded generalized shifts. It is a serial diagnosis and repair model, augmented with a non-autonomous forcing to describe incoming sentence material. Both models will be developed on the basis of a context-free grammar that describes the processing difficulties in a pilot experiment on language-related brain potentials. The models are able to describe, at least qualitatively, the obtained ERP results by trajectories that explore different regions in phase space in pursuing different processing steps.

ERP experiment

In an ERP experiment on the processing of locally ambiguous German sentences, 14 subjects were asked to read 120 sentences each, belonging to two different condition classes:

33

34

Sentences were visually presented phrase-by-phrase with 500 ms (plus 100 ms blank screen resulting in 600 ms interstimulus interval) at the screen of a computer monitor. The sentences were presented in pseudo-randomized order and subjects had to answer probe questions after each sentence to control for their attendance.

The sentences (33) and (34) began with a feminine noun phrase (NP) die Rednerin (‘the speaker’), ambiguous with respect to its grammatical role (either subject or direct object). The disambiguating information was provided by the determiner of the second noun phrase: Berater (‘the advisor’) is explicitly marked as accusative case, therefore assigned to the direct object role in sentence (33). By contrast, Berater bears nominative case, hence indicating that this NP must be interpreted as the subject of the sentence. Correspondingly, die Rednerin has to be disambiguated towards the direct object role of the sentence (34).

According to psycholinguistic principles such as the minimal attachment principle, discussed in section “Introduction/Computational symbolic models”, a subject interpretation of the ambiguous NP is preferred against the alternative direct object interpretation. This preference, the subject preference strategy, leads to processing problems for object-first sentences such as (34). These difficulties are reflected by changes in the event-related brain potential.

In order to measure ERPs, subjects were attached to an EEG recorder by means of 25 Ag/AgCl electrodes mounted according to the 10–20 system (Sharbrough et al. 1995). EEG was recorded with a sampling rate of 250 Hz (impedances < 5kΩ) and was referenced to the left mastoid (re-referenced to linked mastoids offline). Additionally, EOG was monitored to control for eye-movement artifacts.

For the ERP analysis only artifact-free trials (determined by visual inspection) with correct answers to the probe questions were selected. Epochs covering the whole sentence from −1,400 to 2,500 ms relative to the presentation of the critical second NP were baseline corrected by subtracting the time-average of a pre-sentence interval of 200 ms. Mean ERPs were computed for each subject and condition and subsequently averaged into the grand averages.

Moreover, a symbolic dynamics analysis (section “Introduction/The dynamical system approach”) was performed. As ERPs are strongly nonstationary over a whole sentence, the binary half-wave encoding (beim Graben 2001; beim Graben and Frisch 2004; Frisch et al. 2004) was employed with parameters: width of secant slope window, T₁ = 70 ms; width of moving average window, T₂ = 10 ms; dynamic encoding lag, l = 8 ms. This symbolization technique detects local maxima and minima of the EEG time series, mapping time intervals between their successive inflection points onto words consisting of only “0”s and “1”s, respectively. Note that the half-wave encoding resembles the first sifting step of the empirical mode decomposition (Huang et al. 1998; Sweeney-Reed and Nasuto 2007). Due to its local operations, the half-wave encoding technique is able to reduce baseline problems and nonstationary drifts. From the symbolically encoded ERP epochs, event-related cylinder entropies Eq. 9 were computed.

As the ERP results have only illustrative purpose in this study, we do not perform statistical analyses here. A follow-up of this provisional pilot experiment is currently scheduled; and comprehensive results will be published later.

Processing models

In order to construct a formal representation of our ERP language stimuli, we present the phrase structure trees of the sentences (33) and (34) according to Government and Binding Theory (GB) (Chomsky 1981; Haegeman, 1994) in Fig. 1. Figure 1a shows the tree of (33) and 1b that for (34).

Fig. 1.

GB phrase structure trees. (a) For the sentence (33). (b) For the sentence (34)

These trees are derived on the basis of the X-bar module of GB (section “Introduction/Computational symbolic models”). Their binding relations result from movement operations that are represented by co-indexed nodes (hence the nodes DP_k and DP_i are essentially the same, they only differ in their binding partners, the tracest_k, t_i). Note that binding and movement cannot be captured within the framework of context-free grammars; one possibility to do so are minimalist grammars (Stabler 1997; Michaelis 2001; Stabler and Keenan 2003; Hale 2003; Gerth 2006).

From the trees in Fig. 1, a context-free grammar is readily obtained. Though, for the sake of simplicity, we first discard the lexical material regarding the next to last nodes in the trees as the terminal symbols T = {DP₁, C⁰, t, DP₂, DP₃, V⁰, I⁰} where:

Additionally, only binary branches of the trees are taken into account such that the expansion of I⁰ is also abandoned. The corresponding nonterminal alphabet is then given by N =  {CP, C′, I₁, I₂, VP, IP, V′} where we have introduced two categories I₁, I₂ for the I′ nodes in Fig. 1b. The grammar G = (T, N, P, CP) comprises the rewriting rules

35

36

37

38

39

40

41

42

43

44

The grammar G is locally ambiguous for there is more than one rule to expand the nonterminals IP, I₁, VP, respectively. Following beim Graben et al. (2004), the grammar can be decomposed into two locally unambiguous grammars G₁, G₂, resulting in the production systems

45

for generating sentence (33), and

46

for generating sentence (34), (this decomposition is analogue to the harmonic normal form decomposition suggested by Hale and Smolensky (2006)).

Each of the languages produced by the disambiguated grammars G₁, G₂, can be deterministically parsed with a suitable top-down recognizer as explained in section “Introduction/Computational symbolic models”. In the following two subsections, we use Smolensky’s tensor product representations in order to construct two different top-down recognizers. The first, a parallel, high-dimensional dynamical system using faithful filler/role bindings (Dolan and Smolensky 1989; Smolensky 1990, 2006; Smolensky and Legendre 2006); the second, a serial diagnosis and repair NDA based on a Gödel encoding of generalized shifts (Moore 1990, 1991; Siegelmann 1996) combined with a non-autonomous update dynamics (Pollack 1991; Moore 1998; Tabor 2000; Moore and Crutchfield 2000).

Tensor product top-down recognizer

Following Dolan and Smolensky (1989), Smolensky (1990, 2006) and Smolensky and Legendre (2006), hierarchical structures can be represented in activation space of neural networks by filler/role bindings through tensor products. A labeled binary tree supplies three role positions: Parent, LeftChild, and RightChild which can be identified with basis vectors of a three-dimensional role vector space,

47

To each of these roles, either a simple or a complex filler can be attached. Simple fillers are terminal and nonterminal symbols of the grammar G, which in turn are also identified with basis vectors of another, 14-dimensional space:

48

The binding of a category to a tree position is expressed by a tensor product A simple tree is then given by the sum over all filler/role bindings Since tree positions can be occupied by subtrees, a recursive application of tensor products, as e.g. in instantaneously increases the dimensionality of the respective subspaces. Therefore, the sum has to be replaced by a direct sum over tensor product spaces leading to

49

where is the infinite-dimensional Fock space generated by the finite-dimensional filler and role vector spaces (Smolensky and Legendre 2006, p. 186, footnote 11). In fact, a faithful representation of infinitely recursive phrase structures would demand the complete Fock space (beim Graben et al. 2007).

However, the grammar G is fortunately not recursive thus making Smolensky’s approach tractable with finite means. In the following, we shall devise a tensor product top-down recognizer. Its state description at time t comprises two Fock space elements and corresponding to the stack and the input of the automaton defined in Eq. 15, respectively. denotes the tensor product tree at processing time t, while the tensor is of the form i.e. all fillers, which correspond only to terminals, are assigned to the root position. Thereby, encodes the input being processed. Because the direct sum in these expressions is commutative and associative, we additionally require two pointers, to the next, not yet expanded, nonterminal filler in the stack and to the topmost position in the input. Therefore, a tensor product top-down recognizer τ_i for processing grammar G_i (i = 1, 2) is characterized by a semi-symbolic state description

50

The parser recursively expands the nonterminal filler beneath into an elementary tree when there is a rule corresponding to in the grammar. Thereafter, it looks into the expanded tree, whether or agrees with the terminal symbol in pointed to by If this is the case, the particular filler will be removed only from the input, readjusting to the next input symbol.

Since filler/role bindings can be established recursively, we introduce four additional roles indicating the positions in the state description list above. Binding to to etc., yields the representation of a state description of τ_i

51

As there are two sentences, (33) and (34), which can be processed by two deterministic tensor product top-down recognizers τ₁ and τ₂, there are four different parses, to be examined. Parsing trajectories are presented in the Appendix.

For numerical implementation of the model, we chose the arithmetic vector spaces and and identify the fillers and roles ) with their canonical basis vectors respectively. Thus, e.g. the role for the parent position in vector space is This leads to a local and faithful representation of these symbols.

In this representation, the tensor products are then given by Kronecker products (Mizraji 1989, 1992) of filler and role vectors, where

Generally, the calculation of iterated tensor products leads to a vast number of subspaces of different dimensions. Consider, e.g., the tree representation attached to for state 2 of the parse,

saying that CP is attached to Parent, DP₁ to LeftChild and C′ to RightChild. Since the filler is beneath the “stack pointer” in rule (36), corresponding to the state transition

applies next. This results into state 3 of the trajectory with

in position.

Obviously, the first two direct summands and in this term belong to another, lower dimensional, subspace of the Fock space than the remaining ones. Therefore, usage of direct sums is crucially required for proper tensor product representations. However, as our model grammars are not recursive at all, we know that the representation of the last phrase structure tree for well-formed parses is an element of a finite tensor product space with maximal dimension M. Therefore, we take as an embedding space, interpreting the direct sums as those of finite subspaces in Technically, we achieve this by multiplying all vectors from lower-dimensional subspaces with tensor powers of the root role from the right, where

For the example above, we hence obtain the correct expression

where the usual vector addition is now admissible.

Since all representations are now activation vectors from the same high-dimensional vector space we can easily construct the desired parallel parser from the four trajectories obtained by respectively, simply by superimposing the state descriptions of and for each processing step separately. This entails the parallel parse of the sentence (33). On the other hand, superimposing the instantaneous state descriptions of and yields the parallel parse of the sentence (34). Note that the trajectories of the parses have to be filled up with zero vectors to adjust their different durations.

At the end, our construction yields a time-discrete dynamical system with phase space and flow , representing a parallel tensor product top-down recognizer. The iterates generate the parsing trajectory for t = 0, 1, 2… starting with initial condition

The results of this model are presented in section “Results/Tensor product top-down recognizer”. Here, we have standardized the numerical data by the z-transform in order to make the different parses comparable. Additionally, we employ a principal component analysis (PCA) for reducing the dimensions of the activation vector space in order to facilitate visualization. In short, the PCA is an orthogonal linear transformation that rotates the data into a new coordinate system such that the greatest variance of the data has the direction of the first principal component, PC#1, the second greatest variance of the second principal component and so on.

Nonlinear dynamical automaton

In this section, we construct an NDA top-down recognizer for the stimulus material of the ERP experiment along the lines of beim Graben et al. (2004). Given the two, locally unambiguous grammars G₁, G₂ with productions Eqs. 45, 46, we first construct two generalized shifts τ₁,τ₂ for recognizing the context-free languages generated by them.

Pushdown automata generally process finite words , whereas generalized shifts are operating on bi-infinite strings Therefore, we first have to describe finite words through infinite means. This can be achieved by forming equivalence classes of bi-infinite sequences that agree in a particular building block around the separating dot. Yet, such classes of equivalent bi-infinite sequences are exactly the cylinder sets introduced in section “Introduction/The dynamical system approach”. Thus, every state description γ′ . w of a generalized shift emulating a top-down recognizer, with (γ′ denotes γ in reversed order, again) and corresponds to a cylinder set

52

in For the sake of subsequent modeling and numerical implementation, we regard the finite strings γ in the stack and w in the input as one-sided infinite strings, continued by random symbols and

53

Next, we address the Gödel encoding to construct the NDAs. Because the stack alphabet and the input alphabet T of the pushdown automata differ in size, two separate Gödel encodings, one for the x- and another one for the y-axis of the symbologram are recommended. Using the random continuations Eq. 53, we obtain from Eq. 30

54

55

where b_T is the number of terminal symbols in T, b_Γ = b_T + b_N is the number of stack symbols in is the number of nonterminal symbols in N,), and g is the arbitrary Gödel encoding of these symbols.

Fixing the prefixes γ₀…γ_k−1 of the stack and w₁w₂…w_l of the input, results to a cloud of points randomly scattered across a rectangle in the unit square of the symbologram. These rectangles are compatible with the symbol processing dynamics of the NDA, while individual points do not have an immediate symbolic interpretation. We shall refer to arbitrary rectangles as to macrostates, distinguishing them from the microstates of the underlying dynamical system.

In order to achieve our modeling task, we first determine the Gödel codes of the context-free grammar derived in section “Methods/Processing models” by arbitrarily introducing the integer numbers

56

where we have used “hexadecimal” notation for numbers 10 to 13.

Disregarding the numerical values of these codes for a moment, we can prescribe the two generalized shifts τ₁,τ₂ by their transition functions. For the shift that emulates the top-down recognizer τ₁, processing the productions P₁ (Eq. 45), this function is

57

Here, w₁ always stands for the topmost symbol in the input. The last transition describes any attachment of a successfully predicted terminal. All other transitions describe the prediction by expanding a rule in P₁. Table 1 presents the dynamics of this generalized shift processing the sentence (33).

Table 1.

Sequence of state transitions of the generalized shift τ₁, processing the well-formed string 0361645

Time	State	Operation
0	7 · 0361645
1	80 · 0361645
2	8 · 361645
3	93 · 361645
4	9 · 61645
5	A6 · 61645
6	A · 1645
7	5C · 1645
8	5D1 · 1645
9	5D · 645
10	546 · 645
11	54 · 45
12	5 · 5
13

The operations are indicated as follows: means prediction according to rule (X) in the productions Eq.  45 of a context-free grammar; means cancelation of successfully predicted terminals both from stack and input; and means acceptance of the string as being well-formed

Accordingly, the generalized shift τ₂ for recognizing the well-formedness of the string 0326645, which encodes sentence (34), with respect to the productions P₂ is constructed. Table 2 displays the resulting symbolic dynamics.

Table 2.

Sequence of state transitions of the generalized shift τ₂, processing the well-formed string 0326645

Time	State	Operation
0	7 · 0326645
1	80 · 0326645
2	8 · 326645
3	93 · 326645
4	9 · 26645
5	A2 · 26645
6	A · 6645
7	B6 · 6645
8	B · 645
9	5C · 645
10	546 · 645
11	54 · 45
12	5 · 5
13

The operations are indicated as above

In order to describe the processing problem arising from parsing the string 0326645 by τ₁, we present this dynamics in Table 3.

Table 3.

Sequence of state transitions of the generalized shift τ₁, processing the unpreferred string 0326645

Time	State	Operation
0	7 · 0326645
1	80 · 0326645
2	8 · 326645
3	93 · 326645
4	9 · 26645
5	A6 · 26645
6	A6 · 26645

The operations are indicated as above, yet refers to the processing problem that a predicted VP is not present in the input

A possible solution for describing diagnosis and repair processes (Lewis 1998; Friederici 1998) in generalized shifts was proposed by beim Graben et al. (2004). Comparing step 6 in Table 3 with step 5 in Table 2, suggests the construction of a third generalized shift τ₃ with only one nontrivial transition

58

This repair shift replaces the stack content A6 by another content A2 which now lies at the admissible trajectory of τ₂ shown in Table 2, steps 5 to 13.

Now, we brought all necessary ingredients together to construct the NDA. The DoD of the generalized shifts τ₁, τ₂ has width d = 2 such that both symbols next to the dot partition the x- and the y-axis of the unit square symbologram through the Gödel encoding, Eqs. 54, 55. Note that the DoD of the repair shift has width d = 3, as the two most significant symbols in the stack have to be replaced. Because there are b_Γ = 14 stack symbols and b_T = 7 input symbols, the x-axis is divided into 14 intervals whereas the y-axis is covered by 7 intervals. The resulting partitioning of the phase space is depicted in Fig. 7 below.

Fig. 7.

Measurement partition of the NDA phase space. Grid density is shown logarithmically

The top-down recognizers that are emulated by the generalized shifts τ_p (p = 1, 2) can either categories according to a uniquely given rule in their production systems P_p if the topmost symbol γ₀ in the stack is a nonterminal which is subsequently expanded into the right-hand-side of the production since the grammars are locally unambiguous by construction. Correspondingly, the NDA dynamics Φ_p maps a point to its image if x belongs to one of the intervals labeled by 7 to D in Fig. 7. These points are translated parallel to the x-axis. Moreover, a point with is subjected to the numerical counterpart of Summarizing these effects, all points contained within one rectangle of the partition are equivalently transformed by the piecewise linear map Φ_p acting as

59

where (x_t, y_t)^T is a state at time t, (x_t+1, y_t+1)^T is its successor at time t + 1, and (i, j) indicates the ith rectangle along the x-axis and the jth rectangle along the y-axis, given by the Gödel numbers i = g(γ₀), j = g(w₁) of the topmost symbols of stack and input tape. The coefficients and a^{p, (i,j)}_y of the flow Φ_p (p = 1, 2) describe a parallel translation of a state whereas the matrix elements λ^p,(i,j)_x and λ^p,(i,j)_y mediate the stretching (λ > 1) or squeezing (λ < 1) of a rectangular macrostate.

Thus, the Gödel encoding provides three different functions Φ_p (p = 1, 2, 3) assigned to the generalized shifts τ_p. It is therefore obvious to regard the index p as a control parameter of only one dynamical system living at the unit square. Doing so, relates diagnosis and repair processes in language processing to bifurcations in dynamical systems, namely qualitative changes in their global behavior.

Nevertheless, the NDA model constructed so far exhibits serious disadvantages concerning psycholinguistic plausibility. Since it is a deterministic, autonomous dynamical system, the complete future of the system’s evolution is already encoded in its initial conditions, which is clearly at variance with psycholinguistic evidence. First, psycholinguistic experiments are generally conducted either acoustically or visually in a word-by-word (or phrase-by-phrase) presentation paradigm. In both cases, the human language processor is an open, non-autonomous system that is continuously perturbed by new information supplied from the environment (beim Graben 2006). Therefore, an interactive computational account appears to be more appropriate (Wegner 1998). Second, predictions of the human parsing module are often incorrect. Garden path interpretations (Osterhout and Holcomb 1992; Hagoort et al. 1993; Mecklinger et al. 1995; Osterhout et al. 1994; Fodor and Ferreira 1998; Kaan et al. 2000) or the failure of particular processing strategies, as exemplified in section “ERP experiment”, clearly demonstrate the non-determinacy of the human language processor.

In order to remedy these shortcomings, we suggest the following solution: First, restrict the automaton’s input to a working memory, or look-ahead, of finite length (Frazier and Fodor 1978), say, l = 2, such that w = w₁w₂. Second, after each attachment (i.e. when w = w₂), the next symbol is scanned from the environment, that is regarded as an information source (Shannon and Weaver 1949; beim Graben 2006). Third, define a map such that w′ = w₂a_i. I.e., inserts a_i into the second-most significant position of the working memory. A generalized shift described in this way, is not longer an autonomous system. Now, it is interacting with its environment which non-deterministically perturbs its state descriptions (Wegner 1998). Table 4 illustrates the parsing of the string 0361645 by the generalized shift τ₁ augmented with the operation.

Table 4.

Sequence of state transitions of the generalized shift τ₁ supplemented by the operation, processing the well-formed string 0361645

Time	State	Operation
0	7 · 03
1	80 · 03
2	8 · 3
3	8 · 36
4	93 · 36
5	9 · 6
6	9 · 61
7	A6 · 61
8	A · 1
9	A · 16
10	5C · 16
11	5D1 · 16
12	5D · 6
13	5D · 64
14	546 · 64
15	54 · 4
16	54 · 45
17	5 · 5
18	ϵ · ϵ

The operations are indicated as above; additionally denotes the non-autonomous action of the scanned new input symbol a_i upon the state description of the shift. In contrast to Table 1 the initial state contains only the symbols 03 in the working memory

How does the operation affect the corresponding NDAs? A possible solution to this problem is offered by the dynamical recognizers, discussed in section “Introduction/Dynamical system models” (Pollack 1991; Moore 1998; Tabor 1998, 2000; Moore and Crutchfield 2000), where functions acting on the phase space X are assigned to symbols However, the assumption of an arbitrarily parameterized iterated function system is apparently inconvenient in the present context, where the Gödel encoding Eqs. 31, 32 already provides a suitable interface between the symbolic and the numeric levels of description. Fortunately, the mathematical discipline of algebraic representation theory (van der Waerden 2003), and its physical counterpart, algebraic quantum theory (Haag 1992), supply adequate concepts for formulating a proper solution.

Taking the word semigroup (T*, ·) as our starting point5 we can “represent” the individual symbols by “quantum operators” acting on the phase space through

60

where w′ = w₂ ·a_i. That is, the scanned input a_i only acts at the y-coordinate of the state in the symbologram according to

61

This mapping is indeed a semigroup homomorphism and hence a representation as is proven in the appendix. Numerically, Eqs. 60, 61 lead to the functions

62

where denotes the Gaussian integer function. Thereby, a dynamical recognizer with iterated function system is implemented upon the phase space of a nonlinear dynamical automaton (X, Φ_p). Supplementing the generalized shifts τ₁,τ₂ with the operation and representing it by Eqs. 60, 61, 62 in the NDA’s phase space, yields a non-autonomous, interactive dynamical system. Each operation is now reflected by a vertical squeezing of rectangular macrostates in phase space.

Finally, we have to address the measurement of the NDA’s states at macroscopic scales. Can we find a suitable “order parameter” accounting for the occurrence of processing problems? In our ERP analysis (sections “Introduction/The dynamical system approach”, “Methods/ERP experiment”) we used information-theoretic entropies as indicators of disorder–order transitions in language-related brain dynamics. The key concept for this analysis is the cylinder entropy Eq. 9. Obviously, this quantity is also appropriate for assessing the dynamics of generalized shifts and NDAs, since the symbolically meaningful states of the generalized shifts are essentially cylinder sets and these are given by rectangular macrostates in the phase space X = [0, 1]² of the NDA.

In order to measure the cylinder entropy of the processing states, we equip the phase space with a measurement partition whose mesh width determines the probability

63

that a rectangle R is covered by a cell B_i of the partition. Here, the geometrical function area(R) = (x₂−x₁) · (y₂−y₁) determines the area of the rectangle R =  [x₁, x₂] × [y₁, y₂] (beim Graben et al. 2004).

Results

In this section, we present the results of our dynamical system simulations on syntactic language processing and the qualitative findings of the illustrative ERP experiment for comparison.

ERP data

Figure 2 displays the voltage grand averages of the sentence processing ERPs (a) and the running cylinder entropy resulting from the half-wave encoding (b) at parietal electrode site Pz. The blue waveforms were obtained for the control condition (33). The critical (34) condition exhibits a P600 ERP (Fig. 2a), evoked by the crucial determiner der versus den in the second NP. The P600 is reflected by a large drop in the cylinder entropies (Fig. 2b). Additionally, Fig. 2b shows the N400 deflections elicited by each integration of incoming lexical material. The comparison of Fig. 2a and b also reveals a strong nonstationary drift in the averaged ERPs that became completely suppressed by the local algorithm of the half-wave encoding.

Fig. 2.

Event-related brain potentials for the (blue) and (red) sentences ((33 and 34), respectively) at the parieto-central electrode Pz. (a) ERP voltage averages. (b) Event-related cylinder entropies obtained from a half-wave symbolic encoding with parameters T₁ = 70ms, T₂ = 10ms, and l = 8ms. Waveforms are digitally low-pass filtered with a cut-off frequency of 10Hz for better visibility. The disambiguating words der versus den appeared at t₀ = 0s. Language-related ERPs, N400 and P600, are indicated by arrows

Parsing dynamics

Tensor product top-down recognizer

First, we present the successive increase of subspace dimensions of the tensor product top-down recognizer, constructed in section “Methods/Tensor product top-down recognizer”. Table 5 displays these dimensions for the parsing trajectories per time step. Starting with an initially 224-dimensional space, the final embedding space has dimension

Table 5.

Dimensions of the subspaces for parsing the and the sentences, by a tensor product top-down recognizer, respectively

Iteration
1	224	224
2	224	224
3	224	224
4	896	896
5	896	896
6	3,584	3,584
7	3,584	3,584
8	14,336	14,336
9	57,344	14,336
10	57,344	57,344
11	57,344	229,376
12	229,376	229,376
13	229,376	229,376
14	229,376	229,376

Applying the PCA, entailed a remarkable result: The very high-dimensional dynamics shown in Table 5, turned out to be effectively one-dimensional. Only the first principal component, PC#1, of the standardized trajectories exhibits considerable variance. Figure 3 depicts a return plot of x_t = PC#1 for a given iteration at the x-axis plotted against its next iteration, x_t+1 at the y-axis, for both parallel parses. Blue lines represent the sentence (33), while red lines stand for the sentence (34). Additionally, the green line shows the identity function x_t+1 = x_t.

Fig. 3.

Return plot of the first principal component, PC#1, of the tensor product top-down recognizer processing the (blue) and (red) sentences (33 and 34), respectively

As Fig. 3 reveals, both parses start with different initial conditions in phase space, reflecting the initially different inputs to be processed: for the sentence (blue), it is DP₁C⁰t DP₂t V⁰ I⁰ and for the sentence (red), DP₁ C⁰ DP₃ttV⁰ I⁰. The map Φ, whose action is indicated by the lines connecting the points, is obviously nonlinear in phase space. For the first iterations, the trajectories converge. However, they are significantly diverging after the fifth iteration. Eventually, both trajectories settle down in a stable fixed point attractor (0, 0)^T approaching the accepting states of the automaton.

The surprising finding that the dynamics is effectively one-dimensional, can be easily explained by looking at the core transformation of the tensor product top-down recognizer. Consider again the transition from state

to its successor

where we provisionally assume that the last vector belongs to the embedding space. Because all fillers and roles are represented through canonical basis vectors in their respective vector spaces, their Kronecker tensor products are rather sparse vectors, consisting mostly of zeros and only a few ones. The parser’s action expanding a grammatical rule is therefore described by inserting more and more ones into the state vector as the processing evolves. In the limit such dynamics would reach the—though not well-formed—vector (1, 1,…,1)^T which is given by the identity line in the high-dimensional embedding space. Accordingly, also approaching the accepting states of the tensor product top-down recognizer appears as convergence towards such limit ray. Its direction is obtained by the PCA.

Finally, we present the first principal component PC#1 depending on time as the parser’s “time series” in Fig. 4.

Fig. 4.

Time series of the first principal component of the tensor product top-down recognizer in phase space processing the (blue) and (red) sentences (33 and 34), respectively. The model “P600” is indicated by the arrow

Again, we see that both parses start from different initial conditions, subsequently converging for a number of iterations. At step 6, the time series diverge significantly from each other. At this point one path of the processes maintained in parallel, breaks down due to the garden path and becomes abandoned. Afterwards, only successful paths are continued, thereby correctly parsing the subject–object sentence with grammar G₁ and the object–subject sentence with its appropriate grammar G₂. The behavior of the first principal component exhibits considerable resemblance with the reanalysis P600 observed in the ERP data in section “Results/ERP data”. Therefore, we propose this observable as model EEG and consider its behavior as a “model P600”.

Nonlinear dynamical automaton

The NDA model was constructed as a dynamical system (X, Φ_p). The phase space X = [0, 1]² is partitioned into rectangles given by the cartesian products of 14 intervals at the x-axis and 7 intervals at the y-axis labeled by the syntactic categories of the grammar as discussed in section “Methods/Processing models”. The control parameter p distinguishes two syntactic processing strategies, subject preference by default (p = 1) and object-first (p = 2). A third value (p = 3) is assigned to a repair map, connecting those strategies with each other. The intrinsic dynamics is augmented with a non-autonomous counterpart, a representation ρ of the word semigroup by phase space operators establishing the operation. Incoming linguistic material thereby interactively perturbs the autonomous evolution of the NDA.

Figure 5 presents the macroscopic evolution of the NDA processing the string 0361645 according to subject preference strategy p = 1. Each horizontal layer corresponds to one processing step in Table 4. Initial conditions are 700 points randomly scattered over the blue rectangle indicated by the star “*” which corresponds to row 0 in Table 3. After each attachment, the NDA’s state covers exactly one domain of the piecewise linear maps Eq. 59 given by the partition. Then, the next word is scanned from the external information source, acting as a quantum operator at X. These transitions are indicated by “°”s. Obviously, only the extend of the rectangles in the y-axis is influenced by a squeezing operation described through Eq. 62. The parsing process ends when the states are spread across the whole unit square. Then, the stack and the working memory both contain the empty word which corresponds to the 0-cylinder [].

Fig. 5.

Phase space dynamics of input-driven nonlinear automaton processing the sentence (33). Each layer displays one iteration of the NDA top-down recognizer. The initial condition is indicated by “*”; non-autonomous operations are indicated by “°”

Correspondingly, Fig. 6 displays the NDA dynamics starting from 700 randomly distributed initial conditions comprising the first macrostate in Table 3. Again, the non-autonomous operations are indicated by “°”s. Now the “*” depicts the failure of the subject preference strategy applied to the sequence 0326645. State 5 is invariant under the action of Φ₁ which is diagnosed during the transition to state 6. At this time, an external control system intervenes into the dynamics by tuning the control parameter to p = 3 for one iteration (steps 6 to 7), thus destabilizing the unwanted invariant state. Here, the repair map replaces the stack content A6 by A2. After repair, the control system intervenes again by setting p = 2 such that the NDA now emulates the object-first strategy. The processing is completed when the whole phase space is occupied by the cloud of microstates at time 20.

Fig. 6.

Phase space dynamics of input-driven nonlinear automaton processing the sentence (34). Here, only one non-autonomous operation is indicated by “°”. The diagnosis and repair transitions are indicated by “*”. (a) Global view. (b) Enlargement

In order to assess the cylinder entropy as the NDA’s order parameter, the measurement partition Fig. 7 is used. The partition is constructed in such a way, that fine-grained meshes lead to high cylinder entropy, while coarse-grained meshes give rise to low entropy values. Note the relatively coarse partitioning of cell 6 × 2 accounting for the garden path in Fig. 6.

Figure 8 eventually, presents the cylinder entropies for both sentences (blue) and (red). In order to synchronize the respective operations, some idling cycles have been introduced. Therefore, processing time is now measured in arbitrary units instead of iterations.

Fig. 8.

Time series of the cylinder entropy of the NDA top-down recognizer processing the (blue) and (red) sentences (33 and 34), respectively. The model “N400’s and the “P600” are indicated by arrows

Figure 8 shows that the unwanted invariant state processing according to subject preference has low entropy similar to the P600 in the ERP data (Fig. 2b). This is due to the measurement partition Fig. 7 whose grid in the cell 6 × 2 is too coarse to capture the invariant state. Moreover, Fig. 8 reveals that each operation is reflected by a drop in cylinder entropy as well, because the volumes of the macrostates are reduced by the vertical squeezing. Hence the probabilities to cover these states by the cells of the measurement partition are lowered as well. Thus, we suggest to consider entropy drops in the NDA dynamics as “model ERPs” as in section “Results/Tensor product top-down recognizer”. Our “model P600” is now reflected by a drop in entropy caused by arriving at an unwanted invariant macrostate in phase space, a region that has no functional significance for the processing of the input. Likewise, our “model N400s” are elicited by the integration of incoming new material into working memory, which reduces the available phase space volume each time.

Discussion

In this paper, we presented two psycholinguistically plausible nonlinear dynamical system models for language-related brain potentials. In a paradigmatic ERP experiment on the processing of local object–subject ambiguities in German, we found a strong P600 at the disambiguating clausal region for dispreferred object-first sentences in comparison with subject-first sentences which are more easily understood due to the subject preference strategy for the interpretation of ambiguous noun phrases (Bader and Meng 1999; Friederici et al. 2001; Frisch et al. 2004). Consistently, a symbolization analysis of the ERP data revealed a large drop in event-related cylinder entropy in the same time window as the P600. Smaller drops in entropy reflected the N400 component for integrating incoming new lexical material into working memory (Coles and Rugg 1995; Dambacher et al. 2006).

For our models, we first described the stimulus material of the experiment by phrase structure trees according to Government and Binding theory (Chomsky 1981; Haegeman 1994), and derived a locally ambiguous context-free grammar from these trees. This grammar was subsequently decomposed into its unambiguous parts, following a technique suggested by beim Graben et al. (2004). These unambiguous grammars represented the two alternative processing strategies, namely subject preference against object preference. Processing both stimulus sentences, either subject–object or object–subject by deterministic top-down recognizers (Aho and Ullman 1972; Hopcroft and Ullman 1979) for both grammars, respectively, lead to four different parses for subsequent modeling.

Both dynamical system processing models were essentially grounded in Smolensky’s and Mizraji’s tensor product representations of symbolic content in neural activation spaces (Dolan and Smolensky 1989; Mizraji 1989, 1992; Smolensky 1990). The first model represented the syntactic categories of the disambiguated grammars as linearly independent filler vectors and positions in a labeled binary tree as a basis of three-dimensional space. Hence, our tensor product top-down recognizer was obtained as a nonlinear function from the high-dimensional activation space onto itself. The four different parses corresponded to itineraries of this nonlinear map. In order to build a parallel processor (Lewis 1998), the two parses for the subject–object sentence and the other two for the object–subject sentence were linearly superimposed in activation space. In each of these superpositions one process leading into a garden path extinguished.

For this parallel processing model we found that the dynamics was essentially one-dimensional, as revealed by a principal component analysis. The trajectories of the parser, starting from different initial conditions which represented the subject–object and object–subject order, respectively, settled down into one stable fixed point in PCA space. During its transient computation, the trajectories diverged exactly when the garden path was encountered. Regarding the first principal component as our model EEG, its time series showed a remarkable resemblance with the P600 effect in averaged ERPs.

Our second model deployed a two-dimensional, linearly dependent, representation of the tensor product scheme through a Gödel encoding. The two deterministic top-down recognizers corresponding to subject preference and object preference strategy were first translated to generalized shifts (Moore 1990, 1991; Siegelmann 1996). In order to cope with the garden path, a third repair shift was constructed. Using the symbologram method (Cvitanović et al. 1988; Kennel and Buhl 2003), a nonlinear dynamical automaton, i.e. a time-discrete dynamical system at the unit square depending on a control parameter was obtained. This model implemented a serial diagnosis and repair processor (Lewis 1998; Friederici 1998) where the garden path corresponds to an unwanted invariant set which became destabilized by a bifurcation into the repair mode.

The intrinsic, autonomous parsing dynamics was supplemented by a non-autonomous counterpart, inspired by dynamical recognizers (Pollack 1991; Moore 1998; Tabor 1998, 2000; Moore and Crutchfield 2000), where the environment interactively perturbs the states of the dynamics by incoming new words (Wegner 1998; beim Graben 2006). We implemented these interactions using algebraic representation theory of quantum operators (Haag 1992; van der Waerden 2003), or in other words, by representations of the “phrase space” on the phase space. Since symbolic content is represented by a partition of the model’s phase space into rectangular macrostates, symbolic dynamics provides appropriate measures for this system in terms of information theory (Shannon and Weaver 1949; beim Graben et al. 2000). Therefore, we considered the cylinder entropy with respect to a particular measurement partition as our model ERP. Making the measurement partition rather course-grained in the phase space region of the garden path thus entailed an entropy drop corresponding to the P600 component. On the other hand, integrating new words into the model’s working memory shrinks the volume of the macrostates thus leading to lower entropy as well. Therefore our model is also able to reflect the N400 component, at least partially.

Both dynamical systems models have their respective advantages and disadvantages. First of all, they are both consistent with the dynamical system interpretation of event-related brain potentials (Başar 1980, 1998; beim Graben et al. 2000). According to this view, the brain is a high-dimensional, complex dynamical system whose trajectories transiently explore different regions in phase space when performing different computational tasks. On the other hand, these regions are functionally significant and therefore causally distinguished (Fodor and Pylyshyn 1988). Neurophysiological measurements, such as EEG/ERP, MEG, fMRI, or others map this high-dimensional phase space to a less dimensional observable space by spatio-temporal coarse-graining, i.e. by taking signal averages over space, time and trial dimensions (Amari 1974; Freeman 2007; Atmanspacher and beim Graben 2007). However, robust, general features of the underlying dynamics might be preserved in this observable representation. This is especially the case using symbolic dynamics (Hao 1989; Lind and Marcus 1995; beim Graben et al. 2000) for modeling and analysing experimental data. Atmanspacher and beim Graben (2007) have shown that spatio-temporal coarse-grainings generally lead to a coarse-graining, namely a partitioning of the system’s phase space into equivalence classes as well.

The same holds actually for our models. The parallel tensor product top-down recognizer lives in a 229,376-dimensional embedding space. However, as “meaningful” symbols are only represented by sparse vectors of zeros and ones (i.e. by vertices in a 229,376-dimensional hypercube), most of the available space is a meaningless vacuum. After choosing the PCA as coarse-graining method, the one-dimensional observable space spanned by PC#1 turned out to be a viable description. We saw that the effectively one-dimensional dynamics is due to the representation of grammatical rules where more and more ones are introduced in the state vector.

By contrast, the non-autonomously driven nonlinear automaton lives in a two-dimensional partitioned phase space by construction. Also by construction, these rectangular cells of the partition are the domains of the intrinsic piecewise affine linear map that represents the symbol processing of the NDA. Therefore, these regions are in fact functionally significant and causally distinguished. On the other hand, we had to introduce a further coarse-graining by means of a measurement partition. Two microstates belonging to the same cell of this partition are epistemically indistinguishable with respect to a particular observable, which could hence be regarded as the model EEG (Atmanspacher and beim Graben 2007).

In sum, we presented two nonlinear dynamical language processors where ERP effects are reflected by functionally and causally different regions in phase space. Nevertheless, we are acutely aware that these models are only a first step towards a proper understanding of the neurophysiological bases of linguistic computation. We shall conclude this discussion by addressing some open problems and an outlook to ongoing and future research.

Concerning the tensor product top-down recognizer, we were in the lucky situation that our context-free model grammar was not recursive. Yet, recursion is an important property of natural languages allowing, e.g., for center embedding of clauses. In such cases, using linearly independent filler and role vectors would lead to an explosion of the dimension of embedding space. One possible resort for this problem has recently been suggested by beim Graben et al. (2007) making use of the full-fledged Fock space of tensor product representations instead of finite-dimensional embeddings thereof. This line of research might lead to a field-theoretical account to computational psycholinguistics (Amari 1977; Jirsa and Haken 1996; Thelen et al. 2001; Erlhagen and Schöner 2002; Wright et al. 2004). By contrast, Smolensky (2006) and Smolensky and Legendre (2006) argue in favor of linearly dependent representations in order to account for processing problems of center embeddings by graceful saturation in activation space.

Another issue is that natural languages are definitely not context-free (Shieber 1985) and that language processing is certainly not simply top-down (Fodor and Frazier 1980; Marcus 1980; Staudacher 1990). However, since our model grammar is based on Government and Binding theory (Chomsky 1981; Haegeman 1994) and includes at least the necessary traces resulting from the non-context-free movement operations, it was suited in its restricted domain describing only the two stimulus sentences of our ERP experiment. More appropriate grammars are e.g. minimalist grammars (Stabler 1997; Michaelis 2001; Stabler and Keenan 2003; Hale 2003). A minimalist tensor product parser has meanwhile been constructed by Gerth (2006).

Although Moore (1990, 1991) proved that any Turing machine can be implemented by an NDA, this two-dimensional representation might actually not be sufficient for such minimalist grammars. The last proposal of Stabler and Keenan (2003) indicates that a multiprocessor system where each processor would be one NDA could be better for this aim. Therefore, neural network or neural field implementations suggest themselves for NDA and tensor product automata as well.

This leads to our final issue. In the present paper, we have formally constructed time-discrete nonlinear systems in phase space without explicitly devising neural networks. In this sense, we presented training patterns for connectionist models of language processing. The remaining step of implementing these models would be important to provide continuous-time dynamical models (Vosse and Kempen 2000; van der Velde and de Kamps 2006; Wennekers et al. 2006; Garagnani et al. 2007). Such models are also required for predicting latencies of ERP components and to bridge the gap to other dependent measures such as reading times and eye movements (Lewis and Vasishth 2006; Vasishth and Lewis 2006a; Vasishth et al. 2008; Boston et al. in press).

Appendix

Trajectories of the tensor product top-down recognizer

The processing trajectory for the sentence (33) according to grammar G₁ () comes out to be

On the other hand, processing the sentence (34) with grammar leads into the garden path. Its trajectory is then

Here, the parser expands the first two rules of grammar G₁ without any difficulty, since they are the same as the two first rules of the appropriate grammar G₂. After processing rule (37), IP → t I₁, the first symbol on the input stack is the filler which represents the category DP₃. However the upcoming filler to be handled is At this point the parser breaks and does not reach the accepting final state by its trajectory. Similarly, another garden path occurs for processing the sentence (33) according to grammar while accepts the sentence (33) as being well-formed regarding G₂ in the end.

Proof of the word semigroup representation theorem

What finally remains is to prove the assertion stated in section “Nonlinear dynamical automaton” that the quantum operators provide a semigroup homomorphism and hence a representation of the word semigroup in the phase space X of an NDA (X, Φ). To this end, we have to generalize the definition Eqs. 60, 61 to proper words.

Let therefore be words over the terminal alphabet T of the NDA (X, Φ) with lengths As defined in Eq. 55, the Gödel code of the input word w of (X, Φ) with respect to the terminal number base b_T is given as

Call ρ′(u) the operator ρ (u) (Eq. 60) projected onto the y-component of the phase space. Its impact is then given by Eq. 61

which can be written as

after introducing the function

For the proof we calculate

QED.