# Algorithmic ersatz for VSA: Macroscopic simulation of Vector Symbolic Architecture

Chloé Mercier, Thierry Viéville

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Chloé Mercier, Thierry Viéville. Algorithmic ersatz for VSA: Macroscopic simulation of Vector Symbolic Architecture. 2024. hal-04852953

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# Algorithmic ersatz for VSA: Macroscopic simulation of Vector Symbolic Architecture

Chloé Mercier<sup>a,*</sup> and Thierry Viéville<sup>a,**</sup>

<sup>a</sup> Mnemosyne Team, Inria Bordeaux, U. Bordeaux, LaBRI and IMN, France

E-mails: chloe.mercier@inria.fr, thierry.vieville@inria.fr

Submitted to Neurosymbolic Artificial Intelligence

## Abstract.

Spiking neuronal networks are biologically plausible implementations of brain circuit computations, meaning that they can provide a way to manipulate symbols embedded as numeric vectors that carry semantic information. More precisely, the Neural Engineering Framework (NEF), relying itself on a vector symbolic architecture (VSA) formalism, bridges the gap between tightly interleaved numerical and symbolic (including formal) computations. Determining how the brain can implement such processing is an important issue.

In the present work, following this track, we consider such a VSA-based formalism, and propose an implementation at a macroscopic level, thus a higher order of magnitude of scale than usual mesoscopic implementations. We also attempt to provide a better natural representation of usual human symbolic operations, considering an implementation of modal logic.

Beyond usual VSA data structures such associative memories we also introduce the notion of “relation maps” corresponding to relational memories, as observed in the brain.

An open-source implementation is provided with a benchmark and experimental observation of the method performance and limitation.

Keywords: Vector Symbolic Architecture, Semantic Pointer Architecture, Modal Logic, Neuro-symbolism

## 1. Introduction

### 1.1. Biologically plausible neurosymbolic representations

As a possible entry point to considering a biologically plausible implementation at a symbolic level, vector symbolic architectures (VSAs) were introduced as a way to manipulate symbolic information represented as numeric vectors (see, e.g., [302] for an introduction). VSAs have been proven to be helpful in modeling high-level cognition and accounting for multiple biological features [164, 189]. More specifically, the semantic pointer architecture (SPA) [164] instantiates so-called semantic pointers (i.e., vectors that carry semantic information) and makes it possible to manipulate them in networks of spiking neurons. This approach takes a significant step towards the unification of symbolic and sub-symbolic processing, in that it provides a way to translate the former into the latter. Consequently, complex knowledge representations in the form of compositional structures that are traditionally restricted to symbolic approaches can now be distilled in numerical and even neural<fnref n="1" /> systems [122].

<footnote>*Supported by https://team.inria.fr/mnemosyne/en/aide.E-mails: chloe.mercier@inria.fr, thierry.vieville@inria.fr.</footnote>

<footnote>**Corresponding author. E-mail: thierry.vieville@inria.fr.</footnote>

<footnote n="1">¹The term “neural refers to any type of nerve cell, whereas “neuronal is specifically related to neurons.”</footnote>
How can we represent a symbol in a neuronal assembly? A localist representation (one neuron or neuron group represented by a symbol) does not correspond to what is observed in the brain, and the basic idea is that a symbol corresponds to a pattern of activity distributed over the whole assembly. Let us consider a spiking neuron network and quantify its activity using, e.g., the neuron rates or higher-order statistics (see, e.g., [85] for a discussion). As developed by [165], this includes timing codes and population codes (i.e., relative timing codes between neurons). In the Neural Engineering Framework (NEF) [165], this high-dimensional set of bounded quantitative values can be collected and normalized, as a unitary stochastic vector in a high-dimensional space (with a few thousand dimensions for a biological neuronal map and often a few hundred dimensions at the simulation level), thus defining a SPA (building upon a particular case of VSA). The NEF provides a set of principles for implementing such an architecture through synaptic connections, including a time representation in spiking neuron systems (rather than, e.g., other representations based on synchrony within the neural assembly; (see [165] for technical details). This framework is a rather scalable alternative for a biologically plausible implementation of VSA, and it has already been implemented into a simulator called Nengo [38].

In the present study, we consider these developments as prerequisites and will simply consider that neural assem-
bly activity is represented by a high-dimensional unary stochastic vector. We also need to specify transformations
and define them at this abstract algebraic level. Mainly, following [335], we will consider the auto-association
mechanism, as developed in [472], and functional transformations, as detailed in [165].

1.2. What is this paper about?

We first revisit how to encode symbols within the VSA approach based on the framework introduced in [164],
targeting a macroscopic level of modeling. We describe how to generalize symbol encoding considering a related
degree of belief, beyond binary information, and following [335], we explain the semantic interest of such a gen-
eralization. We consider the Vector-Derived Transformation Binding (VTB) operator [203], for which we recall its
algebraic properties in Appendix C.

We then consider hierarchical knowledge structures in the sense of, e.g., [159], as a complement to associative
and sequential memorization, and we revisit how to implement such a memory structure using the VSA formalism.
To this end, we review VSA data structures and demonstrate that they are related to cognitive memory classification
according to [159]. To better understand their computational properties, we also illustrate how such data structures
compare to programming language containers, in Appendix B. We introduce a new data structure implementing
“relational maps” expressing semantic knowledge [396]. This data structure is important for symbolic derivations
and exemplifies the benefits of our macroscopic implementation.

We finally illustrate the proposed mechanism comparing to an existing simulation at the mesoscopic level, utiliz-
ing Nengo simulator [38], with a simulation at the macroscopic scale. We show that such computations may be, up
to a certain point, approximated without explicitly performing mesoscopic computations at the vector component
level; instead, an algorithmic ersatz can be used.

Notations and Layout. We write vectors and matrices in bold letters (bold capital letters for matrices), and scalars in italic. The dual quantity of a vector **x** is represented as its transpose **x**^T. The dot product between two vectors **x** ⋅ **y** can thus also be written **x**^T **y**. Components of vectors and matrices are represented here using subscripts.

We use the Kronecker notation $\delta_P \stackrel{\text{def}}{=} \begin{cases} 1 & \text{if } P \text{ is true} \\ 0 & \text{if } P \text{ is false} \end{cases}$

In this paper, we consider normal distributions $N(0, \sigma)$, i.e., Gaussian distribution with a null mean and a standard deviation $\sigma$, and we write the related random variable $v(\sigma)$.

Non straightforward derivations are done within a symbolic computing environment (our open-source program
is available on https://gitlab.inria.fr/line/aide-group/macrovsa) while trivial demonstrations of most statements are
given in footnotes.

## 2. Symbolic information encoding

Let us first revisit how VSA approaches implement symbolic computation, providing complementary details to
be used for the macroscopic simulation of mesoscopic mechanisms.
## 2.1. Symbol encoding

At the numerical level, each symbol is implemented as a randomly drawn fixed unit $d$-dimensional vector $\mathbf{x} \in \mathbb{R}^d$. Typically, $d \sim 100 \cdots 1000$, and we expect to manipulate $k \sim 100 \cdots 10000$ symbols at the simulation level. Here, our macroscopic implementation uses a dimension of 256, also typically found in related studies such as [335]. In a cortical or brain map, the order of magnitude is higher since the vector corresponds to the neuronal map activity (closer to $10^{5 \cdots 6}$) and the number of encoded symbols depends on which map is considered.

The vector components are drawn from a normal distribution $N(0, 1/\sqrt{d})$, in order to have an average magnitude<fnref n="2" /> of 1.

A similarity measure is now introduced in order to semantically compare two vectors. Classically, the cosine similarity (i.e., normalized dot product, denoted $\cdot$) is used to compute the semantic similarity between two unit vectors<fnref n="3" />:

$$ \mathbf{x} \cdot \mathbf{y} \stackrel{\text{def}}{=} \mathbf{x}^{\top} \mathbf{y} = \cos(\tilde{\mathbf{x}}, \tilde{\mathbf{y}}), $$

where $\mathbf{x}^\top$ denotes the transpose of $\mathbf{x}$. This measure also corresponds to the angular distance between the vectors.

The key property is that, provided that the space dimension $d$ is large enough, two randomly chosen different vectors will be approximately orthogonal. More precisely<fnref n="4" />,

$$ \mathbf{x}^\top \mathbf{y} \sim \delta_{\mathbf{x}=\mathbf{y}} + \nu(1/d), $$

i.e., it is almost 1 if equal and 0 otherwise, plus centered normal noise [443]. At the numerical level, using basic mean and standard-deviation calculi<fnref n="5" />, drawing vectors from such a distribution, and measuring the magnitude, orthogonality, and standard-deviation average values, we have verified for $d \sim 100 \cdots 1000$ that we generate unary vectors with a relative precision on the magnitude below 0.3%, while orthogonality is verified with a relative precision below 0.4%; the noise standard deviation prediction relative precision is below 0.3%.

This allows us to define a hypothesis to decide whether the $H_0$ hypothesis $\mathbf{x} \cdot \mathbf{y} = 0$ can be rejected, as detailed in Appendix A.

## 2.2. Modality encoding

### 2.2.1. The notion of belief

Most VSA approaches consider that two vectors $\mathbf{x}$ and $\mathbf{y}$ contain (not exclusively) equivalent information that is semantically equivalent, when the similarity $\tau$ equals 1, but there are different ways to interpret this result. Here, we enrich the notion of something being either false or true using a numeric representation of, e.g., partial knowledge, as illustrated in Fig. 1. The true value corresponds to 1 (fully possible and fully necessary), the false value to -1 (neither possible nor necessary, i.e., impossible), and the unknown value to 0, which corresponds to a fully possible but absolutely not necessary value.

Our representation is in one-to-one correspondence with the dual notions of necessity and possibility representation in the standard possibility theory. Information is always related to a certain degree of what is called “belief” in this formalism. While almost all partially known information is related to probability, Piaget proposed that the human “level of truth” is more subtle and related to possibility and necessity [460], as formalized in modal logic. These notions are further developed in the possibility theory discussed in [136] and [137].

<footnote n="2">²Given independent normal samples of zero mean and standard-deviation $\sigma$, the square of the magnitude is the sum of $d$ independent normal samples product, thus a chi-square distribution of standard-deviation $1/d$. Its sum is thus of standard-deviation 1 and so is the magnitude.</footnote>

<footnote n="3">³Let us consider two vectors $v_1 = \mathbf{u} + \mathbf{w}_1$ and $v_2 = \mathbf{u} + \mathbf{w}_2$, carrying the same semantic information encoded in $\mathbf{u}$, plus some additional information $\mathbf{w}_1$ and $\mathbf{w}_2$ independent from $\mathbf{u}$ and from each other. Since independent vectors are orthogonal, $v_1^T v_2 = u^T u = 1$: this is the meaning of semantic similarity.</footnote>

<footnote n="4">⁴The product of these two zero mean random variables of standard-deviation $1/\sqrt{d}$ is a random variable of standard-deviation $1/\sqrt{d^2}$. The product of these two normal random variables is not a normal variable but a linear combination of two independent Chi-square random variables, but we approximate them by a normal distribution, which is a conservative choice as detailed in Appendix A.</footnote>

<footnote n="5">⁵See https://raw.github.com/vthierry/onto2spa/main/figures/z_score.mpl for the open-source code used in this subsection.</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Representation of partial truth $\tau \in [-1, 1]$ in relation to necessity and possibility, as defined in the possibility theory. The interpretation is that something partially possible but not necessary is unlikely, whereas what is likely is entirely possible but only partially necessary. Such a formulation corresponds qualitatively to the human appreciation of the degree of belief as proposed by, e.g., [460].</figcaption></figure>

In other words, the possibility theory is devoted to the modeling of incomplete information, which is related to an observer's belief regarding a potential event and surprise after the event's occurrence. This is considered representative of what is modeled in educational science and philosophy [433]. Furthermore, in symbolic artificial intelligence, i.e., knowledge representation and logical inference, a link has been drawn between this necessity/-possibility dual representation and ontology [482]. This must be understood as a deterministic theory, in the sense that partial knowledge is not represented by randomness<fnref n="6" />. This modal notion of partial belief has several semantic interpretations depending on the context [176] (i.e. not only epistemic<fnref n="7" /> or doxastic<fnref n="8" /> but also deontic<fnref n="9" /> and so on). This representation has also been designed to be compatible with the ternary Kleene logic, in addition to being coherent with respect to the possibility theory, as discussed in detail in [502], where this deterministic representation of partial knowledge is generalized in order to include a probabilistic representation (using a 2D representation).

### 2.2.2. Implementing partial similarity knowledge

Let us now propose a design choice to apply this quantification to VSA symbols. A symbol representing a piece of information with a partial degree of belief $\tau \in [-1, 1]$ could be defined as:

$$ \hat{\mathbf{x}} \stackrel{\text{def}}{=} \tau \mathbf{x}, $$

where $\mathbf{x}$ corresponds to the numerical grounding of a symbol, and $\hat{\mathbf{x}}$ corresponds to the numerical grounding of a symbol, given its degree of belief $\tau$.

Interestingly enough, this representation is coherent with the semantic similarity in the following sense: Are two vectors containing similar information? Considering $\mathbf{x} \cdot \mathbf{y}$, if this value is close to 1, then it is considered true, and the modal representation and semantic similarity are coherent. If it is almost equal to 0, then the modal representation is *not true*. Since our design choice is to consider being in an open world in which all that is not true is not necessarily false, but that we simply can not claim is its true, say it is unknown. To take this a step further, if this value is negative (down to -1), the modal representation considers that it is false, i.e., that the contrary is true, which is coherent with the semantic similarity, although negative values are not explicitly used, to the best of our knowledge, in the literature quoted in this paper.

Given these atomic ingredients, let us now study how they can be stored and manipulated in different data structure.

<footnote n="6">⁶This deterministic representation of partial knowledge can be generalized in order to also include a representation of the randomness belief. In the vanilla possibility theory, the possibility can be seen as an upper probability: Any possibility distribution defines a set of admissible probability distributions, i.e., a consonant plausibility measure in the Dempster-Shafer theory of evidence [47]. In [494, 502], it is proposed to bound the approximate probability, reconsidering the original notion of necessity, in order to also consider a lower bound of probability. This could be an interesting extension of the present work.</footnote>

<footnote n="7">⁷Epistemic modal logic is concerned with reasoning about knowledge.</footnote>

<footnote n="8">⁸Doxastic logic is a type of logic concerned with reasoning about beliefs.</footnote>

<footnote n="9">⁹Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts.</footnote>
## 3. Knowledge structure encoding

### 3.0.1. Using bundling and binding to store information

As developed in detail in Appendix B and summarized in Table 1, the VSA formalism makes it possible to implement different data structures corresponding to different memory architectures, as defined by [159], and to different programming containers. Further details on a scalable biologically plausible knowledge representation can be found in [122]. A key point of the present work is to verify that these VSA mechanisms generalize to modal symbol encoding (see Appendix B for an exhaustive development, after [335] who introduce such a design choice).

All data structures rely only on the two following operations:

- The bundling operation of $N$ symbolic vectors $s_i$ corresponds to a simple addition $s \stackrel{\text{def}}{=} \sum_{k=1}^{N} s_i$ while the similarity operator $s \cdot s_j$ allows to detect if the symbol related to $s_j$ belongs to the bundling $s$.

- The binding operation of a symbol $s_1$ with another symbol $s_2$ writes $s \stackrel{\text{def}}{=} B_{s_1} s_2$ and enjoys the property that the corresponding unbinding operator $B_{s_1}^{-1}$ allows to retrieve $s_2$, i.e., $s_2 \simeq B_{s_1}^{-1} s$.

A reader not familiar with the related VSA formalism will find in Appendix B a didactic introduction, while the choice of the binding operator from among several available binding operators [443] is discussed in Appendix C. For this section to be self-contained, one may simply consider that binding makes it possible to create a key-value symbol pair, while the unbinding operation makes it possible to retrieve the value from the key, as discussed below.

<table><caption>Table 1
Biologically plausible data containers; see Appendix B for details.</caption><thead><tr><th>Container</th><th>VSA mechanism</th><th>Cognitive usage</th><th>Main available operations</th></tr></thead><tbody><tr><td>Set</td><td>Bundling or superposition</td><td>Self-associative memory</td><td>+ Element insertion/modification<br>+ Membership check<br>- No enumeration</td></tr><tr><td>Map or dictionary</td><td>Binding superposition</td><td>Auto-associative and hetero-associative memory</td><td>+ Element insertion/modification<br>+ Value·s check from key<br>+ Key·s check from value<br>+ Symbol check from approximate input<br>- No enumeration</td></tr><tr><td>Indexed and chained list</td><td>Ordinal binding superposition</td><td>Sequential memory</td><td>+ Element insertion/modification<br>+ Value enumeration</td></tr><tr><td>Relational map</td><td>(see next subsection)</td><td>Hierarchical memory</td><td>+ Element insertion/modification<br>- No enumeration</td></tr></tbody></table>

### 3.0.2. Relational maps

From early artificial intelligence knowledge representation to modern web semantic data structures, the basic idea of symbolic representation is to express knowledge through relationships, i.e., triple statements of the form ($\$subject, \$predicate, \$object$), as schematized in Fig. 2, [80, 300]. We consider this structure here to demonstrate that beyond existing data structures implemented with VSA, we can also easily define more complex structures at a macroscopic level.

Such information can be implemented through a distributed representation using bundling and binding operations, i.e., associative maps. Following and generalizing [335], we propose to start with an architecture of combined associative memories as represented in Fig. 3. Each associative map stands for a predicate, as proposed and developed by, e.g., [469]. It integrates a demultiplexor which is nothing but another associative map, allowing to index the previous subject → object associative maps. At the algebraic level, this writes:
<figure><img src="image_4.png" /><figcaption>Fig. 2. Atomic representation of knowledge: To express some knowledge regarding a symbol, the subject, we define a feature with a predicate that has an object as an attribute (i.e., a quantitative data value or a qualitative symbol).</figcaption></figure>

$$t_{\text{pso}} \stackrel{\text{def}}{=} \sum_{i} \mathbf{B}_{p_i} t_{p_i}, \text{ with } t_{p_i} \stackrel{\text{def}}{=} \sum_{j, p_j = p_i} \mathbf{B}_{s_j} o_j,$$

where:

- $s_i, p_i, o_i$ are vectors encoding the subject, predicate and object symbols;

- each $\mathbf{B}_y \mathbf{x}$ is a binding which corresponds to a key-value pair (the key is $y$ and the value is $x$);

- combining these with bundling (i.e., a simple sum) in $t_{p_i}$ allows to define an associative map, which unbinding operation allows to retrieve the object of a given subject and predicate:

$$\mathbf{B}_{s_i} \sim t_{p_j} \approx o_{ij} + \text{unknown};$$

where “unknown” stands for a vector which is almost orthogonal, thus not similar, to any other vectors, as made explicit in Appendix B.

- selecting the associative map, given a predicate, is also done through the $t_{\text{pso}}$, implementing the demultiplexor:

$$\mathbf{B}_{p_i} \sim t_{\text{pso}} = t_{p_i} + \text{unknown}.$$

To obtain such properties the choice of a non-commutative binding operator from among several available binding operators [443] is important, in order not mix predicates, subjects and objects.

<figure><img src="image_5.png" /><figcaption>Fig. 3. A relational map as a row of associative memory. For each predicate, an associative memory stores the hetero-associations between subject and object. An input associative memory acting as demultiplexor allowing to choose for a given predicate which associative memory to choose. At the algebraic level this corresponds to a simple combination of bundling and binding operations.</figcaption></figure>

Given a triple $(s_0, p_0, o_0)$, it is straightforward to verify to what extent it is stored in the relational map through unbinding:

$$(B_{s_0} \sim B_{p_0} \sim t_{\text{pso}} \cdot o_0),$$

and this obviously generalizes to a triple multiplied by a modal $\tau$ value.
We can also further obtain all objects of a given subject for a given property,

$$ \mathbf{t}_{\mathbf{p}_j, s_j} \stackrel{\text{def}}{=} \sum_{\mathbf{p}_j = \mathbf{p}_i, s_j = s_i} \mathbf{o}_i \approx B_{s_j} \tilde{B}_{\mathbf{p}_j} \mathbf{t}_{\text{pso}} + \text{unknown}, $$

using the notation of Appendix C. We can also easily define:

$$ \mathbf{t}_{\mathbf{p}_j, \mathbf{o}_j} \stackrel{\text{def}}{=} \sum_{\mathbf{p}_j = \mathbf{p}_i, \mathbf{o}_j = \mathbf{o}_i} \mathbf{s}_i \approx B_{\mathbf{s}_j} \tilde{B}_{\mathbf{p}_j} \mathbf{t}_{\text{pso}} + \text{unknown}. $$

However such a construction, up to the best of our knowledge, cannot allows to retrieve, for instance each predicate of a given subject. More precisely, there is no operation to recover $\mathbf{t}_{s_j}$ or $\mathbf{t}_{s_j, \mathbf{o}_j}$ from $\mathbf{t}_{\text{pso}}$, and no operation to recover $\mathbf{t}_{\mathbf{p}_j}$ or $\mathbf{t}_{\mathbf{p}_j, \mathbf{o}_j}$ from $\mathbf{t}_{\text{spo}}$. nevertheless, to this end, a dual construction, $\mathbf{t}_{\text{spo}} \stackrel{\text{def}}{=} \sum_i \mathbf{B}_{\mathbf{s}_i} \mathbf{B}_{\mathbf{p}_i} \mathbf{o}_i$, with similar decoding formulae makes it possible to further access the properties of a given subject $\mathbf{t}_{s_j} \stackrel{\text{def}}{=} \sum_{i, s_j = s_i} \mathbf{B}_{\mathbf{p}_i} \mathbf{o}_i$ or the properties of a given subject-object couple $\mathbf{t}_{s_j, \mathbf{o}_j} \stackrel{\text{def}}{=} \sum_{i, s_j = s_i, \mathbf{o}_j = \mathbf{o}_i} \mathbf{p}_i$ using similar formulae. We thus shave now two relation maps $\mathbf{t}_{\text{pso}}$ and $\mathbf{t}_{\text{spo}}$. This is an important constraint, and it would be interesting to verify if such a constraint is observed at the level of the brain's semantic memory.

Again, in order to enumerate the different elements of these maps $\mathbf{t}_{\bullet}$, we need the corresponding indexing mechanisms discussed previously. If the basic operation is to enumerate all triples, with order constraints, then the choice of the storage architecture is not crucial; this is going to be the case later in this paper.

To take this a step further, we can also consider an additional symbol “something,” and each time a triplet $(\mathbf{s}_i, \mathbf{p}_i, \mathbf{o}_i.)$ is added, we can also add $(\sigma, \mathbf{p}_i, \mathbf{o}_i.)$, $(\mathbf{s}_i, \sigma, \mathbf{o}_i.)$, and $(\mathbf{s}_i, \mathbf{p}_i, \sigma.)$. This makes it possible to retrieve the fact that there is a link between the predicate and object, subject and object, and subject and predicate, without requiring the enumeration of the different elements<fnref n="10" />.

At the cognitive level, this corresponds to cognitive maps interacting with each other and is a proposal to formalize the notion of hierarchical memory organization, as discussed in, e.g., [159].

At the computer programming level, this corresponds to a “triple store” used in ontology reasoners and is in fact a distributed representation of an oriented graph, in the form of an adjacency set for $\mathbf{t}_{\text{spo}}$ construction and a hierarchical edge set for $\mathbf{t}_{\text{pso}}$ construction.

However, comparing to existing development in the literature, defining and manipulating such relational maps, especially when considering rather large data ensembles, might becomes intractable to simulate at the microscopic level, or even at the mesoscopic level, and it might be useful for large scale application to study to what extent this could also be implemented at a macroscopic level, as developed now.

# 4. Implementation at the macroscopic scale

The VSA, when implemented using the NEF, allows a microscopic simulation of the neuronal processes, at the spiking neuronal network level, for memorization and processing operations. At a higher scale, when the VSA is implemented as described in Appendix C using linear algebra and permutation operations, we are at a mesoscopic scale, allowing us to perform the same operations without explicitizing the neuronal state value and evolution, but only random vectors linear algebra. This is one major advantage of this class of approaches.

To take this a step further, at a higher macroscopic scale, we could directly consider the previous operations, predicting the results of the different algebraic operations without explicitly working at the vector component level. Let us describe how this approach can be designed and implemented using what could be called an “algorithmic ersatz.”

## 4.1. Symbol indexing

In the VSA, each symbol of the vocabulary is associated with a $d$-dimensional random vector. At the macroscopic scale, we only need to register each unary vector $\mathbf{u}_k$ using an integer number $k$, incremented for each new symbol. At

<footnote n="10">¹⁰This has been proposed by Gabriel Doriath Döhler (unpublished research report).</footnote>
the input/output level, the human-readable string ($s_k$) representation of the symbol is utilized, but it is not considered further here.

Weighted symbols of index $i$, correspond to a unary vector number $k_i$, with also a “belief” value $\tau_i \in [-1, 1]$, as discussed in subsection 2.2, that is equal to 1 by default. They are also estimated up to a certain normal centered additive noise $v(\sigma)$ of standard deviation $\sigma_i \ge 0$, which is equal to 0 by default when no approximate operation has been applied to the symbol.

Two symbols may thus have the same unary vector number but different belief levels or different noise levels. The associative table of symbols is thus a simple associative array data structure of ($k_i$, $\tau_i$, $\sigma_i$), corresponding to

$$ \mathbf{x}_{k_i} = \tau_i \mathbf{u}_{k_i} + v(\sigma_i), $$

as illustrated in a more programmatic format in Fig. 4.

If and only if comparison with mesoscopic computation is required, the explicitization of the unary vector value with $||\mathbf{u}_{k_i}|| = 1$ is added to the data structure.

```c
Symbol : {
    unsigned int key;          // index
    double       tau;           // belief level
    double       sigma;          // noise level
    enum          type;          // symbol type
    string        name;          // human readable name
    double       mesoscopic_value[dimension]; // optional vector
}
```

Fig. 4. Programmatic implementation of a symbol at the macroscopic scale, adding an optional mesoscopic numeric vectorial value, to compare macroscopic and mesoscopic calculations.

The symbol type can be *atomic*, i.e. scalar, such a symbol is only defined by its name, without including binding or bundling, or either *bundling* or *binding* with a complementary data structure, as developed below.

With this representation

- Two symbols **x** and **x'** are approximately colinear if and only if they have the same unary vector index.

- Two colinear symbols **x** and **x'** are indistinguishable if and only if:

$$ 0 \approx \|\mathbf{x} - \mathbf{x}'\| \approx (\tau - \tau') \mathbf{u}_k + v(\sigma + \sigma'), $$

up to the first order, with $|\tau - \tau'| < \sigma + \sigma'$ as developed in Appendix 11.

- The similarity between two symbols **x** and **x'** writes:

$$ \mathbf{x}^T \mathbf{x}' \approx \tau \tau' \delta_{k=k'} + v(\sigma + \sigma'), $$

up to the first order.

At this stage we thus have specified atomic symbol, or the dual of such a symbol.

## 4.2. Symbolic derivation of compounded symbols

Given atomic symbols that are randomly drawn, using the enumerating operations given in Appendix C, we have to compute compounded symbols composed through bundling and binding (while unbinding is a simple binding with the dual symbol, thus taken into account with binding), while in order to compute entailment rules, we need to be able to compute the similarity between any of these compounded symbols.

At the macroscopic level, it is straightforward to define an “oracle” that can calculate the result of all operations as follows:

**Bundling canonical representation.** A symbol corresponding to the bundling of other symbols is fully defined by the symbol set, as illustrated in Fig. 5, and an operation over a bundling (binding or similarity) mainly results from applying the operation to each element. The key point is to have chosen an implementation which allows to maintain a normalized representation of the bundling elements.
Bundling : {
map< symbol-id, Symbol> symbols;

}

Fig. 5. Programmatic implementation of a bundling:

- When a new symbol is added, if already present in the bundling, the $\tau$ and $\sigma$ values are updated, otherwise a new map entry is created.

- If the added symbol is itself a bundling, its component are directly expanded, making use of associativity.

- Deleting a symbol corresponds to adding it with zero $\tau$ value.

This allows to maintain a canonical representation of a bundling set of symbols.

The representation is canonical in the sense where, if equality is well-defined on their components, two bundling are semantically equal, if and only if they are syntactically equal, i.e., if the have the same symbol map, and each corresponding symbols are indistinguishable.

**Binding canonical representation.** A symbol corresponding to the binding of one symbol onto another is fully defined by the pair of symbols and yields either a reduction, if it is the corresponding unbinding operation, or a binding combination.

Then, to reduce such an expression we thus simply have to recursively¹¹:

+ Expands $\mathbf{B}_y \mathbf{x}$ binding on the $y$ argument if it is a bundling.

+ Expands $\mathbf{B}_y \mathbf{x}$ binding on the $x$ argument if it is a bundling.

+ Reduces dual $\mathbf{B}_y \mathbf{B}_y \sim \mathbf{x}$ or $\mathbf{B}_y \sim \mathbf{B}_y \mathbf{x}$ binding / unbinding operation, on elementary or compounded symbols.

Let us call “atomic binding case”, when for every $\mathbf{B}_y, \mathbf{y}$ is an atomic symbol. When considering all binding operation regarding used data structures, as discussed in details in Appendix B, or literature quoted in this paper, we are in this atomic binding case. In such a case, the iterative application of these three operations allows to obtain all expression in a canonical form, written:

$$ \sum_k \prod_{l=1}^{l_k} \mathbf{B}_{ykl} \mathbf{x}_k $$

where $\mathbf{x}_k$ and $\mathbf{y}_{kl}$ are atomic symbols and $\mathbf{y}_{kl} \neq \mathbf{y}_{k(l+1)}$, while we may have $l_k = 0$ thus omitting the binding operation.

At the programmatic level, binding and unbinding are simply defined by their two operators, and a flag to specify binding or unbinding.

For the sake of generality, let us also discuss canonical forms beyond the “atomic binding case”. Interesting enough, there are no more reduction in terms of a flatter or reduced expression in this general case¹². In other words,

¹¹More formally, we consider the following rules:

$$ e_y \quad \mathbf{B} \sum_k y_k \mathbf{x} \to \sum_k \mathbf{B}_{yk} \mathbf{x} $$

$$ e_x \quad \mathbf{B}_y \sum_k \mathbf{x}_k \to \sum_k \mathbf{B}_y \mathbf{x}_k $$

$$ r_1 \quad \mathbf{B}_y \mathbf{B}_y \sim \mathbf{x} \to \mathbf{x} $$

$$ r_2 \quad \mathbf{B}_y \sim \mathbf{B}_y \mathbf{x} \to \mathbf{x} $$

where

- $e_y$ and $e_x$ corresponds to bundling expansion over binding,

- $r_1$ and $r_2$ corresponds to the binding / unbinding reductions.

Applying recursively $e_y$ and $e_x$ obviously yields to an expression of the form

$$ \sum_k \prod_l \mathbf{B}_{ykl} \mathbf{x}_k $$

where $\mathbf{x}_k$ are scalar symbols; otherwise one of the three expansion can be applied to reduce the expression.

Applying recursively $r_1$ and $r_2$ on the resulting expression allows to guaranty that no binding / unbinding couple of operation remains.

¹²Let us consider the idempotent mirroring matrix $\mathbf{B}_{\leftrightarrow}$, and the related dual operator $\sim$, also well-defined and detailed in appendix C.:

$$ \mathbf{B}_{\leftrightarrow} \mathbf{x} = \mathbf{x} \sim \text{ and } \mathbf{B}_{\leftrightarrow} \mathbf{B}_y \mathbf{x} = \mathbf{B}_y \mathbf{y} $$

We now have to consider expressions where the binding left parameter is also a binding, i.e., of the form $\mathbf{B}_{B_y z} \mathbf{x}$. Thanks to the dual operator, it expands as:

$$ e_b \quad \mathbf{B}_{B_y z} \mathbf{x} \to \mathbf{B}_{\leftrightarrow} \mathbf{B}_x \mathbf{B}_y \mathbf{z} = [\mathbf{B}_x \mathbf{B}_y \mathbf{z}]^{\sim} $$
we still derive a canonical form of an expression with binding.

These results enlighten the symbolic manipulations performed at the macroscopic level.

As a consequence, the notion of approximate colinearity or equality (i.e., indistinguishably) between two binding symbols is easily defined, as it is simply induced by the same notion of the two left and right symbols.

*Limit of the implementation.* This symbolic derivation assumes that no symbol with two different names have a hidden similarity, i.e.:

$$ \forall \mathbf{x}, \mathbf{y}, \mathbf{x} \cdot \mathbf{y} \approx 0 \text{ and } \mathbf{x} \cdot \mathbf{y} \sim 0. $$

These symbolic derivation rules are in fact nothing but a subset of usual algebraic normalization rules, where the bundling stands for the sum, what it is here, and the bundling a non-commutative product, with a left-inversion mechanism. We refer to symbolic computation textbook for further details.

### 4.3. Symbol noise derivation

At the mesoscopic scale, calculations are made up to the floating-point machine precision, which is not taken into account here. The operations rely on the fact that we consider random vectors in a high-dimensional space, and thus they are approximately orthogonal up to, up to the first order, a normal centered additive noise. The main operations are the dot product used to calculate the similarity, as detailed in subsection 2.1, and the approximation of the matrix inverse using its transpose for unbinding, as detailed in Appendix C.

We must thus consider a level of noise for each symbol and update this level of noise after each calculation; this noise, up to the first order, can still be represented by a centered normal distribution. This cannot be neglected, because we also introduce a level of belief value that can be small and thus is not negligible with respect to the noise level. We denote by $\sigma_{\bullet} \stackrel{\text{def}}{=} O(1/d)$ the order of magnitude added by an approximate operation, as discussed previously in this paper.

On the one hand, considering the similarity operation between two symbols, we obtain¹³ for the dot product

$$ (\tau_i \mathbf{u}_{k_i} + \nu(\sigma_i)) \cdot (\tau_j \mathbf{u}_{k_j} + \nu(\sigma_j)) = \tau_i \tau_j \delta_{k_i=k_j} + \nu(\sigma_{ij}), \sigma_{ij} < \sigma_i + \sigma_j + \sigma_{\bullet}. $$

up to the first order, considering that the noise is independent of the vector values up to the first order. We can thus perform this operation without explicitly computing the dot product.

However, as discussed in Appendix C, beyond idempotence, the dual operator ~ neither expands nor simplifies over binding operations. It only expands over bundling. This means that we can not further reduce or normalize binding expressions for which the left argument, named y here, is not atomic.

In other words, the three form on the left-hand and right-hand size of $e_b$ are simply equivalent and do not lead to further derivation, except trivial useless ones (e.g., adding a zero). Therefore, up to our best understanding, no other expression with x, y, and z, can be semantically equal to one of the three equivalent expressions.

It means that two expressions for which the right parameter is in canonical form as discussed previously and the left parameter is a binding itself in canonical form, are semantically equal if and only if they are syntactically equal.

¹³The derivation is written as follows:

$$ ((\tau_i \mathbf{u}_{k_i} + \nu(\sigma_i)) \cdot (\tau_j \mathbf{u}_{k_j} + \nu(\sigma_j))) = \tau_i \tau_j \mathbf{u}_{k_i} \cdot \mathbf{u}_{k_j} + \tau_i \mathbf{u}_{k_i} \cdot \nu(\sigma_j) + \tau_j \mathbf{u}_{k_j} \cdot \nu(\sigma_j) + \nu(\sigma_i) \cdot \nu(\sigma_j). $$

If $k_i \neq k_j$, then $\mathbf{u}_{k_i} \cdot \mathbf{u}_{k_j} = \nu(\sigma_{\bullet})$ since these random vectors are approximately orthogonal up to normal noise with a standard deviation with an order of magnitude of $\sigma_{\bullet}$ [505], whereas if $k_i = k_j$, then $\mathbf{u}_{k_i} \cdot \mathbf{u}_{k_j} = 1$ since these are unary vectors.

Then, $\mathbf{u}_{k_i} \cdot \nu(\sigma_j)$ is the dot product, and it is a random variable of mean $\mathbb{E}[\mathbf{u}_{k_i} \cdot \nu(\sigma_j)] = 0$, since vectors are assumed to be independent of other sources of noise up to the first order, and variance $\mathbb{E}[\mathbf{u}_{k_i}^T \nu(\sigma_j) \nu(\sigma_j)^T \mathbf{u}_{k_i}] = \sigma_j^2$ since the covariance $\nu(\sigma_j) \cdot \nu(\sigma_j)^T = \sigma_j^2 \mathbf{I}$ because the noise is isotropic; meanwhile, $\mathbf{u}_{k_i}^T \mathbf{u}_{k_i} = 1$ since it is a unary vector. Note that here, $\mathbf{u}_{k_i}$ is not a random variable; it stands for the mean of the random vector drawn.

The derivation for $\mathbf{u}_{k_j} \cdot \nu(\sigma_i)$ is identical.

Assuming that $\nu(\sigma_i)$ and $\nu(\sigma_j)$ are independent and of zero mean, as hypothesized, its related variance is known to be $\sigma_i^2 \sigma_j^2$. The product of these two normal distributions is not a normal distribution; instead, it is a linear combination of chi-square distributions in the general case. However, here, as it is a second-order term, with respect to the expected small values of $\sigma_i$ and $\sigma_j$, it is negligible. Collecting these results, we obtain that up to the first order,
Here, we show a conservative choice by proposing an upper bound for the noise, while the exact value, up to the first order, of $\sigma_{ij}$ can also easily be used, and is implemented. This design choice is also conservative with respect to a mesoscopic implementation, because it increases the noise at each operation, whereas at the mesoscopic level, each numerical random vector is drawn once; thus, depending on the combination of operations, the noise may not increase. We consider here that noise must be added at each step, and we wonder if this is not more realistic than a frozen noise value. However, it would have been possible (but quite computationally heavy) to consider freezing noise values and caching them in some table.

On the other hand, considering a symbol of index $j$ bined by a symbol of index $i$ and unbined by a symbol of index $i'$ so that $k = k_i = k_{i'}$, in order to ensure a valid binding/unbinding operation, we obtain, up to the first order¹⁴,

$$ \mathbf{B}_{(\tau_i \mathbf{u}_i + v(\sigma_i))} (\tau_j \mathbf{u}_j + v(\sigma_j)) = \tau_i \tau_j \bar{\mathbf{u}}_{ij} + v(\sigma'_{ij}), \sigma'_{ij} \le (1 + \sigma_i + \sigma_j) \sigma_{\bullet}^{\frac{1}{2}}, $$

where $\bar{\mathbf{u}}_{ij} \stackrel{\text{def}}{=} \mathbb{E}[\mathbf{B}_{\mathbf{u}_i} \mathbf{u}_j]$ is a new vector orthogonal to $\mathbf{u}_i$ and $\mathbf{u}_j$, while the noise related to the binding operation is integrated into $\sigma'_{ij}$.

Since an unbinding operation is simply a binding operation with a dual vector, the noise calculation is the same.

Finally, since bundling is a simple summation, additive noise standard-deviations simply add. The tricky point is the approximation of $\tau$, as summarized here¹⁵:

$$ \sum_{i=1}^{I} \mathbf{x}_i = \tau_{\bullet} \mathbf{u}_{\bullet} + v(\sum_{i} \sigma_i), \tau_{\bullet} \stackrel{\text{def}}{=} \sqrt{\sum_{k} (\sum_{i, k_i=k} \tau_i)^2}, $$

where $\mathbf{u}_{\bullet}$ is a new unary vector approximately orthogonal to the others, while $\tau_{\bullet}$ is the related magnitude. We thus will use $\tau_{\bullet}$ as the approximate value of a bundling belief, which is rather arbitrary at this stage, but could easily be improved.

### 4.4. Other possible features

Similar considerations would easily allow us to implement the same approach at a macroscopic for the dual operator ~, related to the commutator operator $\mathbf{B}_{\leftrightarrow}$ and the composition operator ⊗ given in Appendix C. However,

$$ (\tau_i \mathbf{u}_{k_i} + v(\sigma_i)) \cdot (\tau_j \mathbf{u}_{k_j} + v(\sigma_j)) = \tau_i \tau_j \delta_{k_i k_j} + v \left( \underbrace{|\tau_j| \sigma_i + |\tau_i| \sigma_j + |\tau_i \tau_j| \sigma_{\bullet}}_{\stackrel{\text{def}}{=} \sigma_{ij}} \right), $$

yielding the expected result.

¹⁴As made explicit in Appendix C, the binding of two independent vectors $\mathbf{y}$ and $\mathbf{x}$ is a random vector and we can write

$$ \begin{align*} \mathbf{B}_{(\tau_i \mathbf{u}_i + v(\sigma_i))} (\tau_j \mathbf{u}_j + v(\sigma_j)) &= \\ \tau_i \tau_j \mathbf{B}_{\mathbf{u}_i} \mathbf{u}_j + \tau_i \mathbf{B}_{\mathbf{u}_i} v(\sigma_j) + \tau_j \mathbf{B}_{v(\sigma_i)} \mathbf{u}_j + \mathbf{B}_{v(\sigma_i)} v(\sigma_j) &= \\ \tau_i \tau_j \mathbb{E}[\mathbf{B}_{\mathbf{u}_i} \mathbf{u}_j] + \tau_i \tau_j v(1/d^{1/4}) + \tau_i \sigma_j v(1/d^{1/4}) + \tau_j \sigma_i v(1/d^{1/4}) + \sigma_j \sigma_i v(1/d^{1/4}) &\approx \\ \tau_i \tau_j \bar{\mathbf{u}}_{ij} + (\tau_i \tau_j + \tau_j \sigma_i + \sigma_j \sigma_i) v(1/d^{1/4}) &\approx \end{align*} $$

$$ \tau_i \tau_j \bar{\mathbf{u}}_{ij} + v \left( \underbrace{(|\tau_i \tau_j| + |\tau_i| \sigma_j + |\tau_j| \sigma_i) \sigma_{\bullet}^{1/4}}_{\sigma'_{ij}} \right), $$

up to the first order, since $v(\sigma)$ is a random vector of magnitude $\sigma$, while

$$ \mathbb{E}[\mathbf{B}_{\mathbf{u}_i} v(\sigma_j)] = \mathbb{E}[\mathbf{B}_{v(\sigma_i)} \mathbf{u}_j] = \mathbb{E}[\mathbf{B}_{v(\sigma_i)} v(\sigma_j)] = 0, $$

because these random vectors correspond to centered random vectors.

$$ \begin{align*} \sum_i \mathbf{x}_i &= \sum_i \tau_i \mathbf{u}_{k_i} + v(\sigma_i) \\ &= \sum_k \tau_k \mathbf{u}_k + v(\sum_i \sigma_i) \tau_k \stackrel{\text{def}}{=} \sum_{i, k_i=k} \tau_i \\ &= \tau_{\bullet} \mathbf{u}_{\bullet} + v(\sum_i \sigma_i) \end{align*} $$

with:

$$ \tau_{\bullet} \approx \sqrt{\sum_k \tau_k^2}, \|\mathbf{u}_{\bullet}\| = 1, $$

since $\mathbf{u}_k$ are approximately orthogonal.
as explored in the previous section, unless explicitly used to defined expressions, they a not useful in our case. This
is also the case for the **i** identity element.

Interesting enough, deriving a macroscopic ersatz of complex instead of real VSA specification, seems straight-
forward, since algebraic relations easily generalizes to these case, as reviewed in Appendix C.

Furthermore, we consider binding/unbinding operations using the VTB algebra, but the same reasoning could
also be performed for other binding/unbinding operations, including and especially for heavily calculated operators,
including if a commutative binding operation is considered, yielding obvious additional reduction rules.

This is another argument to consider macroscopic algorithmic ersatz: while VTB-algebra or almost one order
of magnitude heavier than, for instance, convolution operators ($O(d^{3/2})$ instead of $O(d \log(d))$), and this is also
the case for other non-commutative binding operations (see Appendix C), the macroscopic simulation of both have
similar computational costs.

4.5. Available implementation

We are thus in a position to propose an algorithmic ersatz of the usual VSA mesoscopic linear algebra calculations
involving high-dimensional random vectors. This has been implemented and made available as public documented
open-source code<fnref n="16" />. For the generation of a symbol, at a given level of belief $\tau$ and for a given level of first-order
random normal noise with a standard deviation $\sigma$, the usual similarity, bundling, and binding operations are made
available, and the data-structures made explicit here also<fnref n="17" />.

5. Experimental results

5.1. Benchmarking macroscopic simulation

Symbolic expression syntax. In order to input or output expression we use the usual JSON<fnref n="18" /> syntax in a weak form<fnref n="19" />, namely:

- Bundlings are represented by lists:

[symbol_1 ...].

- Bindings are represented by the construct:

{b y: symbol x: symbol },

where *b* stands for binding and is replaced by *u* for unbinding.

- Atomic symbols are represented by the construct:

{ name: symbol-name tau: tau-value sigma: tau-value },

tau and sigma being optional, while atomic symbols with tau=1 and sigma=0 are also represented by strings.
At the implementation level, this requires no more than a few lines of code.

Symbolic derivation examples. In Table 2, four illustrative examples of symbolic reduction are given:

- The 1st one illustrates the canonical representation of bundling where, in this case [*a*, *b*] and [*b*, *a*] have the same internal representation, while expansion of binding over bundling is performed for the reduced form.

- The 2nd one illustrates that bundling of bundling are flatten while empty bundling or singleton bundling are reduced, it also shows an example of tau and sigma computation.

- The 3rd and 4th ones illustrate binding/unbinding reduction and related belief computation including in complex expressions.

<footnote n="16">¹⁶https://line.gitlabpages.inria.fr/aide-group/macrovsa.html.</footnote>

<footnote n="17">¹⁷Additional mechanisms of rule derivations are also present, beyond the present work.</footnote>

<footnote n="18">¹⁸See https://www.json.org.</footnote>

<footnote n="19">¹⁹See https://line.gitlabpages.inria.fr/aide-group/wjson.</footnote>
Numerical noise estimation. In Table 3, the comparison between mesoscopic similarities measures of elementary expressions, and the related macroscopic prediction is reported. Noise of order of magnitude $O(1/d)$ for similarities and $O(1/d^{1/4})$ for binding has been considered. We observe that with respect to our simple 1st order and conservative derivations, noise is overestimated at the macroscopic level, and this overestimation increases with the dimension.

This overestimation increases almost logarithmically with the space dimension, and may partially be compensated by a rather simple rule of thumb, which could easily be improved in the future. This is especially important since macroscopic simulation is mainly useful when the space dimension is to be increased.

After numerical adjustment, it appears that an experimental rule of thumb of:

$$ \sigma_{\bullet} = O(1/d) \approx \underbrace{\frac{1}{1024 d}}_{\sigma_{\bullet}^{0}} \approx \underbrace{\frac{1}{1024 (6 \log_{10}(d) - 11) d}}_{\sigma_{\bullet}^{1}} $$

leads to reasonable results: using $\sigma_{\bullet}^{0}$ allows one to maintain the standard-deviation over-estimation above almost 1 and below 5, for $d < 10^{5}$, while using rule of the thumb defined by $\sigma_{\bullet}^{1}$ allows one to maintain the standard-deviation over-estimation 0.75 and below 2, for $d < 10^{5}$, which corresponds to the result in Fig. 3.

This allows to conclude from these numbers that our rather simple, 1st order, estimation of mesoscopic noise, allowing to predict it at the macroscopic level, being conservative, we observe an expected overestimation of the noise, which increases with the space dimension. Obtaining such an overestimation is a conservative choice, in the sense that some deductions may be missed, whereas false deductions will be avoided, as discussed in Appendix A.

A step further, we have investigated to what extents the usual approximation, here and in the literature, of chi-square distribution by a normal distribution, regarding the dot-product similarity operation is appropriate. This is shown in Fig. 6. The main result is that this difference is below 1 bit (i.e., we miss less than 1 bit for information, about half a bit, to consider it as a normal distribution instead of a combination of chi-square ones), and it might decrease with the dimension, although the result is not obvious.

<figure><img src="image_6.png" /><figcaption>Fig. 6. Average divergence in bits (i.e., using $log_2$ in the formula) between the observed mesoscopic noise distribution and a normal distribution with the same standard-deviation, as a function of the space dimension.</figcaption></figure>

Another interesting aspect is that our macroscopic model is consistent with that of [443], which obtains the following (from Fig. 4 of that paper) for the VTB representation:

$$ d \gtrsim 32 (s + 0.575) $$

represents the minimal dimension $d$ needed to obtain a 99% accuracy with a bundling of size $s$, using a similarity calculation to extract vectors from the bundling. Our model does not take the negligible bias 0.575 into account but allows us to calibrate the level of noise to $\sigma \approx \frac{0.016}{d}$, in order to perform a simple z-score test under the normal hypothesis
$$ \tau > 2\sigma $$

to decide if the related $\tau$ value of the similarity is distinguishable from the noise. On the other hand, we do not consider two vectors to be similar if the similarity is below the standard deviation of the noise, preferring a more conservative threshold.

To take this a step further, we also implemented the $(1/d^{1/4})$ noise dependency for unbinding, with the same calibration.

Studying, at the mesoscopic level, the numerical precision of unbinding on associative maps is to the best of our knowledge, this was not studied in the papers quoted in this one.

**Binding magnitude verification** In Table 4, we have verified another aspect of our formal developments: The VTB binding magnitudes. This is important, contrary to previous studies, because we have introduced the notion of belief via the $\tau$ magnitude parameter. As formally derived $||\mathbf{B}_y \mathbf{x}|| \simeq 1$ and $||\mathbf{B}_y \mathbf{y}|| \simeq \sqrt{2}$, while other magnitudes have not been derived a-priory only observed numerically. If simulation is run at the mesoscopic level, these values matters, and computations must be re-normalized. If simulation is run at the macroscopic level, the normalization assumption is always considered.

**Binding computation duration.** In Fig. 7, we observe the VTB binding mechanism computation time, which is expected to evolve with $O(d^{3/2})$. In fact, at the implementation level, we observe that due to compiler and on-line processor optimization, the average computation time is an order of magnitude lower, because the VTB binding formula in eq. (1) is easy to optimize, at both the compilation and multi-core processor levels. Fitting the results we obtain, for one binding computation average time, something like:

$$ T_{\text{milli-seconds}} = 0.048 + \frac{0.28 d^{1.35}}{10000}, $$

on a standard Intel® Core™ i5-8265U CPU @ 1.60GHz × 8 processor, while the obtained result is not very stable around $O(d^{1.3})$. This instability obviously depends on the processor multi-core actual state, when running the simulation.

<figure><img src="image_7.png" /><figcaption>Fig. 7. Binding average computation time as function of the space dimension. Although not very stable, a better fit is always obtain for a $O(d^p)$, $1 < p < \frac{3}{2}$ interpolation.</figcaption></figure>

This result is not directly useful for our purpose, but is in favor of using the VTB algebra, including at the mesoscopic level, despite a less performing computation time.
## 5.2. A preliminary experiment

To illustrate the use of macroscopic mechanism beyond basic formulae, we reconsider the example proposed in [335] which has been simulated at the mesoscopic level and is based on a minimal ontology in Fig. 8.

<figure><img src="image_8.png" /><figcaption>Fig. 8. An example of a simple ontology with three individuals. The black arrows correspond to factual statements input into the database and the green arrows correspond to inferred statements. Rectangular boxes stand for individuals, round boxes stand for classes, and properties are used to label arrows. Here, from the fact that a subject eats an object, we deduce that this subject is a Person, and the object is Food. This is illustrated by a red arrow. From the fact that the object is a Margherita pizza, which is a Pizza, which is a Food, according to the class hierarchy, we deduce that the object is a Pizza, and re-deduce that it is a Food. Furthermore, because Luigi (among other activities, since it is an open world) eats a pizza, we deduce that Luigi is a Person. Because of property heritage, meaning that here a Topping is an Ingredient, we also deduce from the fact that this pizza has mozzarella as a topping that it also has mozzarella as an ingredient. In the macroscopic implementation, this property is deduced from the fact that Margherita pizza always has mozzarella as a topping, allowing us to generate compounded inferences. From [335].</figcaption></figure>

We have considered a symbol encoding dimension of $d = 256$ to be consistent with previous mesoscopic experiments, such as those in [335]. This is tested with an associative map and relational map, as described in the previous
section, and we have implemented the tiny Pizza experiment<fnref n="20" />, obtaining, in the simplest case, the expected closure, as given in Fig. 9.

## 5.3. Macroscopic implementation of the VSA system

<table><caption>Fig. 9. The expected inferences using the proposed RDFS subset of entailment rules obtained by the macroscopic algorithmic ersatz of the VSA implementation.</caption><thead><tr><th>Input triples</th><th>Inferred triples</th></tr></thead><tbody><tr><td>(Luigi eats thisPizza)</td><td>(Luigi rdf:type Person)</td></tr><tr><td>(thisPizza rdf:type MargheritaPizza)</td><td>(thisPizza rdf:type Pizza)</td></tr><tr><td>(MargheritaPizza rdfs:subClassOf Pizza)</td><td>(thisPizza rdf:type Food)</td></tr><tr><td>(Pizza rdfs:subClassOf Food)</td><td>(MargheritaPizza<br>rdfs:subClassOf Food)</td></tr></tbody></table>

More interesting is what happens when modality is considered, e.g.,

(Luigi 0.5 eats thisPizza).

In other words, it is possible but not completely necessary that Luigi eats the given pizza. In that case,

- it is still possible but no longer entirely true that Luigi is a person;

- it is still entirely true that this pizza is some food, even if Luigi did not eat it, because it is true that it is a pizza, which is food.

This is what is obtained by the implementation, as shown by the open-source tiny experiment output<fnref n="20" />.

Although it is far from being complete, this macroscopic implementation of an algorithmic ersatz of VSA mesoscopic operations seems sound and it is consistent with previous results. It has a final non-negligible advantage: It is quite “simple” in the sense that it does not require very complicated or twisted mechanisms. It requires a bit more than 500 lines of formatted C++ code, including formal symbolic operations on the algebraic operators.

# 6. Discussion and conclusion

## 6.1. Contributions

In this paper, we have been able to propose, up to the implementation level, a reformulation of the powerful VSA approach with a few additions:

- We explicitized a degree of belief for each knowledge item that is linked to the possibility theory related to modal logic, and we revisited the main proposed abstraction of biologically plausible data structures to verify their compatibility with this generalization while comparing them with usual programming data structures and discussing how to efficiently scan (i.e., enumerate) such data structures.

- We proposed an implementation of hierarchical or relational semantic data structures within the VSA formalism in relation to hierarchical cognitive memory, allowing us to introduce symbolic derivations.

- We introduced the idea of simulating such a mechanism at a macroscopic, more symbolic level in order to obtain computations independent of the VSA dimension space, thus making it possible to scale up such mechanisms. This idea has been applied to VTB algebra but is also obviously reusable with other VSA algebras, through the macroscopic implementation is even more interesting for binding operations requiring more operations.

<footnote n="20">²⁰ The source code is available at https://gitlab.inria.fr/line/aide-group/macrovsa/-/blob/master/src/pizza_experiments.cpp and it is noticeable that the C/C++ implementation of such rules is straightforward to write, as documented in the source code. This piece of code output is available at https://gitlab.inria.fr/line/aide-group/macrovsa/-/raw/master/src/pizza_experiments.out.txt, in accordance with Fig. 9, and it also shows the intermediate inference steps.</footnote>
## 6.2. On biological plausibility and numerical versus semantic grounding

We thus propose an anchoring, i.e., a numerical grounding of semantic information. This, indeed, does not mean that the brain performs such operations as it, but this anchoring is biologically plausible in the sense that algorithms can be implemented in a VSA, as developed in this paper, which itself is a model of spiking neuron assembly activity, as developed by [164] with the NEF approach reviewed in this paper.

Numerical grounding, or anchoring, fundamentally differs from the semantic symbol grounding problem, as reviewed and discussed in [479], where symbols are linked to their meanings and anchored in sensorimotor features, which involves the capacity to pick referents of concepts and a notion of consciousness. In a nutshell, this is still an open problem that we are not going to address here. However, proposals of methods to link abstract symbols to neuronal reality enrich the issue of how mental states can be meaningful. Furthermore, the fact that our abstract representation is anchored in sensorimotor features means that it is also a link between symbols and their potential referents. To take this a step further, when we represent concepts, the chosen design choice associates prototypes, allowing us to anchor an abstract element to a concrete example.

Another aspect not targeted by the present study is the emergence of symbols, i.e., the fact that a symbolic representation emerges from a biological or any physical system in interaction with its environment. This issue corresponds to the ungrounding of concrete signs<fnref n="21" />, as discussed in, e.g., [400], in relation to the emergence of symbolic thinking (see, e.g., [498] for a detailed discussion). At the computational neuroscience level, the issue is addressed in [428] for a toy experiment; that paper emphasizes that to address such an issue, we must avoid explicitly embedding any symbol anywhere in the model, a priory or a posteriory. Here, we do not address the emergence issue, but in a sense, we do address a feasibility issue: To what extent can sophisticated symbolic processing be anchored in numerical processing, not just rudimentary operators? We also address an interpretation issue, i.e., we consider to what extent sub-symbolic sensorimotor anchored processing corresponds to symbolic processing, as discussed later in this paper.

## 6.3. Approach limitations and perspectives

The macroscopic simulator is operational but still limited to basic VSA operations of bundling and binding, and their applications to set, associative maps, and other data structures detailed in Appendix B. The symbolic mechanisms are implemented directly at a procedural level, because relatively simple. More complex symbolic algebra could require specific software such as Maple<fnref n="22" /> or equivalent ones, but at the cost of relying on close software or middle-ware and at the cost of a decrease of performances, because using very general mechanisms instead of tuned dedicated coding. Interesting enough, the present implementation is easily generalize to other binding operators or other VSA mechanism, and also to VSA in with complex numbers, as detailed in appendix C. As a next step, the present macroscopic package will be used to study biologically plausible inference mechanisms implemented with VSA.

Following the usual VSA approaches, the symbolic information is embedded in a compact Riemannian manifold with a very simple topology, a hyper-sphere, and we have made explicit the fact that finally, the number of encodable symbols is rather limited. Other geometries may offer better performances, and the particular hyperbolic embedding of hierarchical representations benefits from the fact that due to the hyperbolic negative curvature of the space, even an exponentially growing data structure can be parsimoniously represented [363] because of the expanding geometry (to make a long story short). The idea to embed the data representation in non-Euclidean spaces and especially hyperbolic spaces has already been explored in detail, for instance, in [132], showing that the satisfiability and algorithmic complexity can be drastically different<fnref n="23" />. This might be an interesting extension of typical VSA approaches that makes it possible to consider the symbol's numerical embedding in such a Riemannian differential manifold. This could be a fruitful perspective of such work.

<footnote n="21">²¹In the semiotic hierarchical meaning of an "icon" built only from sensorimotor features, structures at an "index" level built by concrete relationships between given objects give rise to a "symbol" in the semiotic sense, which corresponds to abstract general relationships between concrete concepts or sensorimotor features.</footnote>

<footnote n="22">²²See https://www.maplesoft.com.</footnote>

<footnote n="23">²³A version of these elements intended for a wider audience is available in a science popularization journal: ..</footnote>
**Acknowledgments** Terrence C. Stewart is gratefully acknowledged for his inspiring advice that helped us with some aspects of this work. Gabriel Doriath Döhler is thanked for his work on clarifying the use of the VTB algebra and introducing interesting ideas during his undergraduate internship. Frédéric Alexandre and Hugo Chateau-Laurent are gratefully acknowledged for their valuable advice and their contributions to previous works on this subject. We are especially thankful for the reviewing work on this article, which deeply helped us to make the draft valuable. This work is supported by the https://team.inria.fr/mnemosyne/en/aide exploratory action.

*Conflict of interest:* This work is not subject to any conflict of interest.

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## Appendix A. Hypothesis testing regarding symbol similarity

The design choice of a symbol implementation as a random normal unary vector, as reviewed in subsection 2.1 allows us to define a hypothesis to decide whether the $H_0$ hypothesis $x \cdot y = 0$ can be rejected.

In our case, we approximate the chi-square distribution average of $x \cdot y$ by a normal distribution of the same standard-deviation, which is a conservative choice as shown in Fig. 10.

<figure><img src="image_9.png" /><figcaption>Fig. 10. Comparison between a chi-square distribution in black (numerically drawn from $10^5$ samples) and the normal distribution of same mean and standard-deviation, in red: The choice is conservative because much more samples have values closed to zero in the former case. More precisely the kurtosis is of about 6 (i.e. the sharpness estimation with respect to a normal distribution using 4-th order momenta).</figcaption></figure>

We can consider a two-tailed “z-test” with the alternative hypothesis $H_0$, which states that $x \cdot y \neq 0$. Here, the z-score$^{24}$, with $d$ samples and a known standard deviation with an order of magnitude $O(1/d)$, is the following:

$$z \equiv \sqrt{d}(x \cdot y).$$

It follows an almost distribution, which can be easily verified numerically, as shown in Fig. 11 (left column). For two vectors that are not independent but have an angular dependency, we can numerically observe, in Fig. 11 (right column), the similarity dependency as a function of the vector’s relative orientation. This obvious fact is quite important, allowing us to develop a macroscopic simulation of VSA operations.

This makes it possible, on the one hand, to consider, for instance, a threshold:

$$\theta \stackrel{\text{def}}{=} \pm 2\sigma,$$

along with considering this z-score to have a confidence interval better than 99%, and to relate the similarity estimation to an angular dependence between two vectors, as detailed in Fig. 11. To the best of our knowledge, this obvious implementation has not yet been made explicit, and it is used in subsection 5.3, as detailed in section 4, allowing us to propose to simulate the different operations defined later in this paper at a macroscopic scale.

$$z \stackrel{\text{def}}{=} \frac{\bar{x} - \mu}{\sigma / \sqrt{d}},$$

<footnote>$^{24}$Given a distribution, the z-score for $d$ samples is defined as</footnote>

<footnote>where the expected mean is $\mu = 0$, the a priori standard deviation is $\sigma = O(1/d)$, and the experimental mean $\bar{x} = (x \cdot y)$ is obtained from the dot product.</footnote>
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

Fig. 11. Numerical observations of the similarity defined by the dot product of two random vectors for *d* = 100 in the upper row and *d* = 1000 in the lower row. The left column shows the histogram of the z-score ($\sqrt{d} \cdot (\mathbf{x} \cdot \mathbf{y})$) for two normal vectors, in comparison with a normal distribution. These experimental distributions have a kurtosis of about 10; this is lower than the kurtosis of a normal distribution, which is expected to have a kurtosis of 3. The right column shows the z-score as a function of the angle $a \stackrel{\text{def}}{=} \bar{\mathbf{x}} \cdot \bar{\mathbf{y}} = \arccos(\mathbf{x} \cdot \mathbf{y})$, making it possible to visualize the dispersion with respect to the expected cosine profile.

## Appendix B. On VSA data structures

This section revisits the literature, emphasizing the properties of the data structures; it discusses in more detail their computational properties and limitations and links them to usual programming language data structures.
## B.1. Unordered set or bundling

We first consider an unordered set $\mathcal{S}$ of $N$ symbols grounded to values $\{\mathbf{s}_1, \cdots, \mathbf{s}_i, \cdots, \mathbf{s}_N\}$, and we would like to be able to store them in such a way that we can check if a given symbol is in the set. Very simply, we ground $\mathcal{S}$ to the vector $\mathbf{s}$:

$$ \mathbf{s} \stackrel{\text{def}}{=} \sum_i \mathbf{s}_i, $$

which provides a solution, because given a symbol $\mathbf{s}_\bullet$, we obviously observe that $\mathbf{s}_\bullet \cdot \mathbf{s} \simeq 1$ if it corresponds to a certain symbol $\mathbf{s}_i$, and it is almost 0 otherwise, because random vectors are almost orthogonal, as previously explained. This is called bundling [443] or superposition.

Furthermore, the representation intrinsically includes a notion of transitivity: If a set includes another subset, by construction, it includes the subset elements. More precisely,

$$ \mathbf{s} \stackrel{\text{def}}{=} \sum_i \mathbf{s}_i \text{ and } \mathbf{s}_i \stackrel{\text{def}}{=} \sum_j \mathbf{s}_{ij} \Rightarrow \mathbf{s} \stackrel{\text{def}}{=} \sum_{ij} \mathbf{s}_{ij}, $$

and thus, $\mathbf{s}_{ij} \cdot \mathbf{s} > 0$ for all subset elements.

This obviously generalizes to weighted symbols $\hat{s}_i$, i.e., symbols with modality weighting. In that case, $\mathbf{s}_\bullet \cdot \mathbf{s} \simeq \tau$ makes it possible to retrieve the belief weight. This is equivalent to inputting a symbol $\mathbf{s}_\bullet$ that is approximately similar to a given symbol $\mathbf{s}_i$, thus indicating an approximate similarity; however, it neither allows us to retrieve the exact value of $\mathbf{s}_i$ nor indicates if a positive value below 1 corresponds to a weighted symbol that has been exactly retrieved or to a symbol approximation.

This has an interesting biological interpretation: $\mathcal{S}$ has features in common with a Hopfield network or other related attractor networks, where information has been stored in a distributed way while activating the map with an input makes it possible to determine whether the symbol is stored or not. This is also called self-associative. The main difference with auto-associative, or clean-up memory, is that in Hopfield network attractor networks converge to the exact stored value, providing a mechanism of associative memory, which is now developed while introducing superposition, allowing us to better understand the need for a more sophisticated mechanism.

Let us provide an analogy with programming data structures, explicitizing the similarities and differences between what is proposed here and what is available in common programming languages<fnref n="25" />. This unordered set representation corresponds to a “set” container (e.g., a `std::unordered_set` in C++ or a `set()` in Python) that has only an insertion method and a membership test function, without the capability to intrinsically enumerate the elements, as formerly discussed.

### B.1.1. Symbol enumeration

At this stage, this structure does not allow us to directly enumerate all symbols $\mathbf{s}_i$, because from $\mathbf{s}$, it is not possible to decode the superposed vectors. In [122], for instance, where data structures are defined using superposition, the intrinsic memory enumeration of the stored information is not addressed. We thus need an external mechanism to select all elements and perform an operation on each one. However, at the implementation level, in NENGO [164], an explicit list of the defined vocabulary $\{\cdots \mathbf{s}_i \cdots\}$ is maintained, and the way to select the elements is to test $(\mathbf{s}^T \mathbf{s}_i)$ for each element of the vocabulary. This select operator has a complexity of $O(K)$, where $K$ is the size of the vocabulary. Later in this paper, we will also propose a biologically plausible indexing mechanism, in order, for instance, to manipulate sequences.

## B.2. Associative map

We now consider an unordered associative memory, or “map,” of $N$ correspondences $\{\mathbf{s}_1 \to \mathbf{o}_1, \cdots, \mathbf{s}_i \to \mathbf{o}_i \cdots \mathbf{s}_N \to \mathbf{o}_N\}$ between subjects and objects. To this end, we use the binding operation $B_{\mathbf{s}_i}$, defined in Appendix C, with a pseudo-inverse, i.e., an unbinding operator, $B_{\mathbf{s}_i}^\sim$:

$$ \mathbf{m} \stackrel{\text{def}}{=} \sum_i B_{\mathbf{s}_i} \mathbf{o}_i, $$

so that

<footnote n="25">²⁵We will do the same for other cognitive structures because we think that it illustrates the computing capability of the cognitive object.</footnote>