Original: A Brain-Inspired Hierarchical Reasoning Framework for Cognition-Augmented Prosthetic Grasping
Revised: A Brain-Inspired Hierarchical Reasoning Framework for Cognition-Augmented Prosthetic Grasping

Original: Abstract
Revised: Abstract

Original: This paper proposes a hierarchical framework that enables reasoning across multiple levels of abstraction, from perception to high level control.
Revised: This paper proposes a hierarchical framework that enables reasoning across multiple levels of abstraction, from perception to high-level control.

Original: The framework relies primar- ily in Hyperdimensional (HD) computing, a versatile braininspired computing paradigm that can produce composite dis- tributed representations for efficient classification, and can also facilitate symbolic reasoning.
Revised: The framework relies primar- ily on Hyperdimensional (HD) computing, a versatile braininspired computing paradigm that can produce composite dis- tributed representations for efficient classification and can also facilitate symbolic reasoning.

Original: We present this framework through an example use-case corresponding to cognitionaugmented prosthetic grasping, where user intent is prob- abilistically predictedin order to aid the user in successfully selecting and executing the most appropriate gripping mode.
Revised: We present this framework through an example use case corresponding to cognitionaugmented prosthetic grasping, where user intent is proba- bilistically predicted to aid the user in successfully selecting and executing the most appropriate gripping mode.

Original: Overall, this paper aims to illustrate how HD computing can constitute a mathematical formalism capable of integrating various levels of cognition under a common hierarchical framework.
Revised: Overall, this paper aims to illustrate how HD computing can constitute a mathematical formalism capable of integrating various levels of cognition under a common hierarchical framework.

Original: Introduction
Revised: Introduction

Original: Data-driven artificial intelligence (AI) and machine learning methods have demonstrated great success at perceptionbased benchmarks, sparking great interest in their adoption within various disciplines, including biomedical engineering, robotics, and edge computing (Park, Took, and Seong 2018;
Revised: Data-driven artificial intelligence (AI) and machine learning methods have demonstrated great success at perceptionbased benchmarks, sparking great interest in their adoption within various disciplines, including biomedical engineering, robotics and edge computing (Park, Took and Seong 2018;

Original: Zhou et al. 2019).
Revised: Zhou et al. 2019).

Original: Yet, the deployment of AI in practical scenarios may present heterogeneous requirements that range from the inclusion of uncertainty-capturing models of the world to the incorporation of expert knowledge and symbolic reasoning over it.
Revised: Yet, the deployment of AI in practical scenarios may present heterogeneous requirements that range from the inclusion of uncertainty-capturing models of the world to the incorporation of expert knowledge and symbolic reasoning over it.

Original: This calls for mathematical formalisms and frameworks that can integrate heterogeneous algorithms and representations.
Revised: This calls for mathematical formalisms and frameworks that can integrate heterogeneous algorithms and representations.

Original: Furthermore, human cognition is hierarchical by nature (Jeon 2014).
Revised: Furthermore, human cognition is hierarchical by nature (Jeon 2014).

Original: At the low level, cognition is established by the integration of massive sensory data from disparate sources.
Revised: At the low level, cognition is established by the integration of massive sensory data from disparate sources.

Original: In contrast, high level cognition involves abstract reasoning supported by pre-conceived notions of the world’s properties.
Revised: In contrast, high level cognition involves abstract reasoning supported by pre-conceived notions of the world’s properties.

Original: This hierarchical scheme ultimately drives human action , which, in turn, restarts the sense-reason-act closedloops thathuman brains are uniquely endowed with.
Revised: This hierarchical scheme ultimately drives human action , which, in turn, restarts the sense-reason-act closedloops that brains are uniquely endowed with.

Original: This paper posits that Hyperdimensional (HD) computing (Kanerva 2009), known also as Vector Symbolic Architectures (VSA) (Gayler 2003), constitutes a mathematical formalism capable of integrating the heterogeneous aspects of cognition described above.
Revised: This paper posits that Hyperdimensional (HD) computing (Kanerva 2009), known also as Vector Symbolic Architectures (VSA) (Gayler 2003), can constitute a mathematical formalism capable of integrating the heterogeneous aspects of cognition described above.

Original: HD computing builds upon a carefully designed algebra of vectors, that enables symbolic computations as well as the composite encoding of heterogeneous information.
Revised: HD computing builds upon a carefully designed algebra of vectors, that enables symbolic computations as well as the composite encoding of heterogeneous information.

Original: Moreover, due to its high-capacity and distributed nature, HD computing is robust against noise (Frady, Kleyko, and Sommer 2018) and can be memory efficient (Zhou, Muller, and Rabaey 2021; Kleyko, Frady, and
Revised: Moreover, due to its high-capacity and distributed nature, HD computing is robust against noise (Frady, Kleyko, and Sommer 2018) and can be memory efficient (Zhou, Muller, and Rabaey 2021; Kleyko, Frady, and

Original: Sommer 2021).
Revised: Sommer 2021).

Original: The posture above is motivated by the multiple successful implementations of HD computing available in literature; earlier on in its development focusing primarily on symbolic reasoning (Plate 1994b; Eliasmith et al. 2012;
Revised: The posture above is motivated by the multiple successful implementations of HD computing available in literature; earlier on in its development focusing primarily on symbolic reasoning (Plate 1994b; Eliasmith et al. 2012;

Original: Rachkovskij and Slipchenko 2012), and more recently demonstrating its usefulness for various perception tasks in the fields of robotics (Neubert, Schubert, and Protzel 2019; Mitrokhin et al. 2019) and bio-signal processing and classification (Rahimi et al. 2018).
Revised: Rachkovskij and Slipchenko 2012), and more recently demonstrating its usefulness for various perception tasks in the fields of robotics (Neubert, Schubert, and Protzel 2019; Mitrokhin et al. 2019) and bio-signal processing and classification (Rahimi et al. 2018).

Original: For perception tasks, in particular,
Revised: For perception tasks, in particular,

Original: HD computing enables the compact encoding of composite spatio-temporal and frequency distribution structures by exploiting the properties of high-dimensional binary and bipolar vector spaces (Kleyko et al. 2021a).
Revised: HD computing enables the compact encoding of composite spatio-temporal and frequency distribution structures by exploiting the properties of high-dimensional binary and bipolar vector spaces (Kleyko et al. 2021a).

Original: Classification often takes place by means of a simple nearest-centroid search, with the class-centroids also represented as highdimensional vectors and stored in an associative memory .
Revised: Classification often takes place by means of a simple nearest-centroid search, with the class-centroids also represented as highdimensional vectors and stored in an associative memory .

Original: Other approaches follow an iterative learning strategy that improves upon the often sub-optimal nearest-centroid search mentioned before (Diao et al. 2021).
Revised: Other approaches follow an iterative learning strategy that improves upon the often sub-optimal nearest-centroid search mentioned before (Diao et al. 2021).

Original: In this paper, we explore the capabilities of HD computing as a formalism to integrate classification, probabilistic inference, and symbolic reasoning within a single computational framework.
Revised: In this paper, we explore the capabilities of HD computing as a formalism to integrate classification, probabilistic inference, and symbolic reasoning within a single computational framework.

Original: We envision that this framework can enable reasoning and learning across different levels of abstraction by fully describing it in the HD space.
Revised: We envision that this framework can enable reasoning and, eventually, learning across different levels of abstraction by fully describing it in the HD space.

Original: The contributions of this paper can be summarized as follows:
Revised: The contributions of this paper can be summarized as follows:

Original: Proposal and illustration of a hierarchical reasoning framework .
Revised: Proposal and illustration of a hierarchical reasoning framework.

Original: Through a synthetic prosthetic grasping example, we illustrate how the proposed framework can hierarchically represent the different levels of abstraction of the application: at the lowest (perception) level it relies on biosignal classification; at the intermediate level it relies on a
Revised: Through a synthetic prosthetic grasping example, we illustrate how the proposed framework can hierarchically represent the different levels of abstraction of the application: at the lowest (perception) level it relies on biosignal classification; at the intermediate level it relies on a

Original: Dynamic Bayesian Network (DBN) to infer user-intent over time; and at the highest level it relies on symbolic reasoning for control and feedback.
Revised: Dynamic Bayesian Network (DBN) to infer user-intent over time; and at the highest level it relies on symbolic reasoning for control and feedback.

Original: Mapping of a probabilistic graphical model to HD space.
Revised: Mapping of a probabilistic graphical model to HD space.

Original: We demonstrate how to map the DBN mentioned above to the HD space.
Revised: We demonstrate how to map the DBN mentioned above to the HD space.

Original: We show that this mapping allows us to store the conditional probability tables (CPTs) of the model in superposition, with the potential of reducing memory footprint, and enabling an end-to-end HD representation for inference in the proposed framework.
Revised: We show that this mapping allows us to store the conditional probability tables (CPTs) of the model in superposition, with the potential of reducing memory footprint, and enabling an end-to-end HD representation for inference in the proposed framework.

Original: Discussion of the implications and extensions of the proposed framework We discuss the implications of the current framework and detail future work directions.
Revised: Discussion of the implications and extensions of the proposed framework. We discuss the implications of the current framework and detail future work directions.

Original: Notation
Revised: Notation

Original: Random variables are denoted with upper case letters X , and their instantiations with lower case letters x .
Revised: Random variables are denoted with upper case letters X , and their instantiated values with lower case letters x .

Revised: For long variable and value names, we capitalize the first letter to discern them (e.g. Task is a variable and task a value).

Original: Instantiations of multiple variables, as well as vectors, are denoted with bold lower case letters x .
Revised: Instantiations of multiple variables and vectors are denoted with bold lower case letters x .

Original: However, the high-dimensional vectors (or hypervectors) used as  primitives for HD computingare distinguished by denoting them with X .Furthermore, subscripts are used when expressing complex concepts, such as new or old, which would be written as X NEW and X OLD , respectively.
Revised: We distinguish the highdimensional vectors (or hypervectors, as introduced in the following section) that constitute the primitives of HD computing by denoting them with X .

Original: Hyperdimensional computing
Revised: Hyperdimensional computing

Original: Inspired by the understanding that the brain’s computations rely on massive circuits of neurons and synapses, HD computing operates on pseudo-random high-dimensional vectors through a well defined computational algebra of vectors, different from linear algebra (Kanerva 2009).
Revised: Inspired by the understanding that the brain’s computations rely on massive circuits of neurons and synapses, HD computing operates on pseudo-random high-dimensional vectors through a well defined computational algebra of vectors, different from linear algebra (Kanerva 2009).

Original: Pairing the aforementioned representation and algebra with an associative memory yields apowerful and versatile architecture that enables efficient learning and reasoning (Rachkovskij, Kussul, and Baidyk 2013).
Revised: Pairing such a representation and algebra with an associative memory yields a versatile architecture that enables efficient learning and reasoning (Rachkovskij, Kussul, and Baidyk 2013).

Original: Main properties and components
Revised: Main properties and components

Original: HD computing can materialize through different models, each with their own set of properties.
Revised: HD computing can materialize through different models, each with their own set of properties.

Original: See (Kleyko et al.
Revised: See (Kleyko et al.

Original: 2021b)
Revised: 2021b)

Original: for an extensive list.
Revised: for an extensive list.

Original: In general, all HD computing instances rely on the following components and properties (Neubert, Schubert, and Protzel 2019):
Revised: In general, all HD computing instances rely on the following components and properties (Neubert, Schubert, and Protzel 2019):

Original: High-dimensional spaces
Revised: High-dimensional spaces.

Original: The primitives of HD computing are vectors with n dimensions commonly larger than 10 2 , denoted as X = x ∈ R n .
Revised: The primitives of HD computing are vectors with n dimensions commonly larger than 10 2 , denoted as X = x ∈ R n .

Original: We refer to these high-dimensional vectors as hypervectors .
Revised: For the rest of the paper, we refer to these high-dimensional vectors as hypervectors .

Original: Depending on the model,these hypervectors can be real-numbered tensors (Smolensky 1990), real and complex valued vectors (Plate 1994a), binary dense
Revised: Depending on the model, hypervectors can be real-numbered tensors (Smolensky 1990), real and complex valued vectors (Plate 1994a), binary dense

Original: (Kanerva et al.
Revised: (Kanerva et al.

Original: 1997) or sparse (Rachkovskij 2001) vectors or bipolar vectors (Gayler 1998).
Revised: 1997) or sparse (Rachkovskij 2001) vectors or bipolar vectors (Gayler 1998).

Original: In this paper, we use the Multiply Add Permute (MAP) model (Gayler 1998), which relies on the latter bipolar representation.
Revised: In this paper, we use the Multiply Add Permute (MAP) model (Gayler 1998), which relies on the latter bipolar representation.

Original: The choice of high-dimensional spaces is motivated by several of their properties, as explained throughout the following paragraphs, such as the near orthogonality of random hypervectors, their robustness to noise and their huge capacity, even when represented sparsely (Frady, Kleyko, and Sommer 2018).
Revised: The choice of highdimensional spaces is motivated by several of their properties, as explained throughout the following paragraphs, such as the near orthogonality of random hypervectors, their robustness to noise and their huge capacity, even when represented sparsely (Frady, Kleyko, and Sommer 2018).

Original: Orthogonality and similarity metrics HD computing exploits the properties of high-dimensional spaces.
Revised: Orthogonality and similarity metrics. HD computing exploits the properties of high-dimensional spaces.

Original: In particular, the hypervectors mentioned above are created randomly by uniformly and independently sampling each dimension from the underlying HD space (Neubert, Schubert, and Protzel 2019).
Revised: In particular, the hypervectors mentioned above are created randomly by uniformly and independently sampling each dimension from the underlying HD space (Neubert, Schubert, and Protzel 2019).

Original: These seed hypervectors are often saved in an item memory (I.M.), implemented as a code-book or look-up table.
Revised: These seed hypervectors are often saved in an item memory (I.M.), implemented as a code-book or look-up table.

Original: Due to their high dimension, two randomly generated seed hypervectors are very likely orthogonal.
Revised: Due to their high dimension, two randomly generated seed hypervectors are very likely orthogonal.

Original: This results in significant robustness against noise when trying to recognize them or recover them from the I.M. through appropriate distance metrics, such as Hamming distancefor binary hypervectors, orangular distances (e.g. cosine similarity) for real-valued hypervectors.
Revised: This results in significant robustness against noise when trying to recognize them or recover them from the I.M. through appropriate distance metrics, such as Hamming distance or cosine similarity (Kleyko et al. 2021b).

Original: We refer to the similarity between two hypervectors as sim( · , · ) .
Revised: We refer to the similarity between two hypervectors as sim( · , · ) .

Original: Depending on the metric of choice, similarity is directly (e.g. cosine similarity) or inversely (e.g. Hamming distance) proportional to it.
Revised: Depending on the metric of choice, similarity is directly (e.g. cosine similarity) or inversely (e.g. Hamming distance) proportional to it.

Original: Moreover, due to their approximate orthogonality, seed hypervectors stored in the I.M. can be interpreted and deployed as discrete symbols.
Revised: Moreover, due to their approximate orthogonality, seed hypervectors stored in the I.M. can be interpreted and deployed as discrete symbols.

Original: This elementary property of high-dimensional spaces constitutes the key of HD computing, and is exploited for the construction of data structures through specialized operations, and for recovering information from noisy inputs as well as for symbolic and subsymbolic queries through a simple nearest neighbor search, as detailed in the following paragraphs.
Revised: This elementary property of high-dimensional spaces constitutes the key of HD computing, and is exploited for the construction of data structures through specialized operations, and for recovering information from noisy inputs as well as for symbolic and subsymbolic queries through a simple nearest neighbor search, as detailed in the following paragraphs.

Original: HD computing operations As mentioned before, HD computing relies on three main operations: 1) Binding ◦ : The binding operation combines two input vectors into an output vector dissimilar to the input.The
Revised: HD computing operations. HD computing relies on three main operations: 1) Binding ◦ : The binding operation combines two input hypervectors into an output hypervector dissimilar to the input.

Original:  MAP modelused in this paper implementsthe bindingoper- ation as an component-wise product of the input hypervec
Revised: The MAP model implements binding as an componentwise product of the input hypervectors.

Original: - tors. One of the most common uses of the binding operation is to form associations between two concepts or variables in a role-filler or key-value fashion, such as SENSOR =
Revised: One of the most common uses of the binding operation is to form associations between two concepts or variables in a role-filler or key-value fashion, such as a record describing sensor type

Original:  TYPE ◦ EMG .
Revised: SENSOR = TYPE ◦ EMG .

Original: Moreover, each of the original vectors can be recovered by unbinding it from the original association.
Revised: Moreover, each of the original vectors can be recovered by unbinding it from the original association.

Original: In the MAP model, each vector is its own multiplicative inverse, such that binding a vector to itself results in a hypervector of ones (i.e. X ◦ X
Revised: In the MAP model, each vector is its own multiplicative inverse, such that binding a vector to itself results in a hypervector of ones (i.e. X ◦ X

Original: = 1 , with X ∈ R n and 1 = (1 , 1 , ..., 1) ∈ R n ).
Revised: = 1 , with X ∈ R n and 1 = (1 , 1 , ..., 1) ∈ R n ).

Original: This self-inverse property means that the unbinding operation is the same as the binding one (component-wise multiplication) allowing to easily recover the elements of associations.
Revised: This self-inverse property means that the unbinding operation is the same as binding, allowing to easily recover the elements of associations.

Original: For example, the type of sensorin the previous example can be recovered from the SENSOR association by unbinding TYPE from it (i.e. 
Revised: For example, the type of sensor can be recovered from the SENSOR association by unbinding TYPE from it (i.e. TYPE ◦ SENSOR =

Original: [Equation]
Revised: TYPE ◦ TYPE ◦ EMG = 1 ◦ EMG = EMG ).

Original: Note also that binding is associative and commutative and distributes over the bundling operation,which is explained below.
Revised: Binding is associative and commutative and distributes over the bundling operation, explained below.

Original: 2) Bundling + : The bundling operation typically implemented as an component-wise sum of hypervectors, combines or superposes two input vectors into an output that is similar to the two inputs.
Revised: 2) Bundling + : The bundling operation, typically implemented as an component-wise sum of hypervectors, combines or superposes two inputs into an output that is similar to the two inputs.

Original: This property results from the near orthogonality of random hypervectors.
Revised: This property results from the near orthogonality of random hypervectors.

Original: For example, suppose we have a database D
Revised: For example, suppose we have a database D

Original: = { A , B , C } of randomly generated hypervectors
Revised: = { A , B , C } of random hypervectors

Original: and we generate the bundle X =
Revised: and we create the bundle X =

Original: A + B .
Revised: A + B .

Original: Then, sim( A , X ) >> sim( C , X ) and sim( B , X ) >> sim( C , X ). This is because A and B act symmetrically as noise for the recognition of the otherfrom X , and their noisy versions are more similar to their original versions than to any other randomly generated vector.
Revised: Then, sim( A , X ) >> sim( C , X ) and sim( B , X ) >> sim( C , X ) because A and B act symmetrically as noise for the recognition of the other and their noisy versions are more similar to their original versions than to any other random hypervector.

Original: 3) Permutation (also called protect) ρ : Permuting a vector produces a dissimilar vector to the original one.
Revised: 3) Permutation (also called protect) ρ : Permuting a vector produces a dissimilar vector to the original one.

Original: This operation is often used to “protect” ordering that could be lost due to the associative and distributive rules of binding.
Revised: This operation is often used to “protect” ordering that could be lost due to the associative and distributive rules of binding.

Original: Note that permutations distribute over the previous two operations.
Revised: Note that permutations distribute over the previous two operations.

Original: Data structures
Revised: Data structures

Original: Through the components described above, HD computing offers the possibility of encoding data structures into compound hypervectors. Additional computations on these data structures can be performed through the operations described above (Kleyko et al. 2021a). Examples of common data structures are:
Revised: Through the components described above, HD computing offers the possibility of encoding data structures into compound hypervectors (Kleyko et al. 2021a), for example:

Original: 1) Key-value pairs : Binding can be used to represent associations between elements in a key-valuepair or variable-value fashion.
Revised: 1) Key-value pairs : Binding can be used to represent associations between elements as key-value or variable-value pairs.

Original: We already saw an example of this data structure through the SENSOR association, where we could consider that TYPE is the key and EMG the value.
Revised: For example, in the SENSOR association TYPE is the key and EMG the value.

Original: The hypervectors forming these associations are all randomly generated and stored in the I.M. 2) Sets and histograms : Sets can be represented by binding the hypervectors corresponding to the elements in the set.
Revised: 2) Sets and histograms : Sets can be represented through hypervector bundling.

Original: For example, the set S =
Revised: For example, the set S =

Original: { a, b, c } can be represented through a bundleof hypervectors corresponding to each element: S =
Revised: { a, b, c } can be represented as the bundle S =

Original: A + B + C , where the elements of S are random seed hypervectors stored in I
Revised: A + B + C .

Original: M. For multi-sets, such as H = { a, a, b, c, c, c } , a frequency distribution can be repre- sented through a bundleof hypervectors H =
Revised: For multi-sets, such as H = { a, a, b, c, c, c } , a frequency distribution can be represented through the bundle H =

Original: 2A + B + 3C .
Revised: 2A + B + 3C .

Original: The frequency of a specific hypervector, can be approximately recovered from a bundle through their dot product, which we denote as dot( · , · ) .
Revised: The frequency of a specific hypervector, can be approximately recovered from a bundle through their dot product, which we denote as dot( · , · ) .

Original: Thus, the frequency of, for example, A in H is dot( H , A ) /n ∼ 2 , where n denotes the hypervector’s dimension.
Revised: For example, the frequency of A in H can be calculated as dot( H , A ) /n ≈ 2 , where n denotes the hypervectors’ dimension.

Original: 3) Sequences : The ordering of the components in a sequence can be preserved through the permutation operation ρ .
Revised: Sequences : The ordering of the components in a sequence can be preserved through the permutation operation ρ .

Original: As such, the sequence Z = ( a, b, c ) can be represented in HD space as Z = ρ 2 ( A )+ ρ 1 ( B )+ ρ 0 ( C ) , where ρ m ( V ) denotes that hypervector V is permuted m times.
Revised: As such, the sequence Z = ( a, b, c ) can be represented in HD space as Z = ρ 2 ( A )+ ρ 1 ( B )+ ρ 0 ( C ) , where ρ m ( V ) denotes that hypervector V is permuted m times.

Original: Parsing representations and queries
Revised: Parsing representations and queries

Original: The data structures above can be combined to create com- pound representations (Kleyko et al. 2021a).
Revised: The data structures above can be combined to create com- pound representations (Kleyko et al. 2021a).

Original: For example,suppose theSENSOR association can be used within a record listing the characteristics of a specific prosthetic device  in terms of the sensorsit includes, where the user is an adult, and the location (or context) where it is normally used.
Revised: For example, the record listing the characteristics of a specific prosthetic device can be specified in terms of its sensors and the location (or context) where it is normally used.

Original: For example, DEV = SENSOR + CONTEXT , where SENSOR = TYPE SENS ◦ EMG and CONTEXT =
Revised: For exam- ple, DEV = SENSOR + CONTEXT , where SENSOR = TYPE SENS ◦ EMG and CONTEXT =

Original: LOC ◦ HOME .
Revised: LOC ◦ HOME .

Original: Thanks to the properties described in the previous section, we can, for example, probe the record to de- termine the location where the device is used by unbinding LOC from DEV : 
Revised: Thanks to the properties described above, we can, for example, probe the record to determine the location where the device is used by unbinding LOC from DEV : DEV

Original: [Equation]
Revised: [Equation]

Original:  Sincethe term TYPE SENS ◦ EMG ◦ LOC is dissimilar to any of the seed hypervectors in I.M., it can be regarded as noise.
Revised: + LOC ◦ [Equation] [Equation] Since TYPE SENS ◦ EMG ◦ LOC is dissimilar to any of the seed hypervectors in I.M., it can be regarded as noise.

Original: Theseed hypervector corresponding to HOME can thenbe recovered by choosing the vector in I.M. with the highest similarity to HOME + noise .
Revised: The HOME hypervector is then recovered by selecting the neighbor nearest to HOME + noise in the I.M.

Original: Thus, this probing operation “cleans up” the noise of the compound representation.
Revised: Thus, the probing query “cleans up” the noise of the compound representation.

Original: Proposed hierarchical framework
Revised: Proposed hierarchical framework

Original: The proposed hierarchical framework is illustrated in Figure 1 and corresponds to a cognition-augmented prosthetic grasping use-case.
Revised: The proposed hierarchical framework is illustrated in Figure 1 and corresponds to a cognition-augmented prosthetic grasping use case.

Original: Use-case description
Revised: Use case description

Original: This synthetic use-case is intended to augment the EMGbased gesture recognition application in (Moin et al. 2021) and (Zhou, Muller, and Rabaey 2021).
Revised: This synthetic use case is intended to augment the EMGbased gesture recognition application in (Moin et al. 2021) and (Zhou, Muller, and Rabaey 2021).

Original: The goal of these works is to classify EMG (and accelerometer in the latter) data into a set of gestures, which can be used by prosthesis users to select the appropriate gripmode (Figure 3.b shows an example of gesture to grip mappings).
Revised: The goal of these works was to classify EMG (and accelerometer in the latter) data into hand-gestures, which can be used by prosthesis users to select the appropriate grip (Figure 3.b shows an In practice, permutation commonly consists of cyclic shifts.

Original:  Our proposed use-case is motivated by the observation thatmany prosthesis users  struggle to remember and selectthe appropriate gesture-to-grip mapping while trying to engage in their daily activities (Espinosa and Nathan-Roberts 2019).
Revised: example of these mappings). Our proposed use case is motivated by the observation that prosthesis users often struggle to remember and select appropriate gesture-to-grip mappings (Espinosa and Nathan-Roberts 2019).

Original: Therefore, we propose to augment this applicationbeyond gesture classification by incorporating user-intent prediction, which can ultimately be used in aiding prosthesis users throughout the grip selection and execution process.
Revised: We propose to augment this application by incorporating user-intent prediction, which can ultimately aid prosthesis users through grip selection and deployment.

Original: Moreover, as described later on in the paper, the higher level specification of user intent can be used for analogical reasoning towardsidentification of, and adaptation to new environments.
Revised: Moreover, as described later on, the higher level specification of user intent can be used for analogical reasoning towards adaptation to new contexts.

Original: Hierarchical reasoning framework
Revised: Hierarchical reasoning framework

Original: The proposed framework comprises three layers:1)
Revised: The proposed framework comprises three layers:

Original:  Perception layer The role
Revised: 1) Perception layer.

Original: of the perception layer, high lighted in blue, is to map heterogeneous sensory inputs to higher level descriptive concepts, such as hand-gesture and arm-position.
Revised: The perception layer, highlighted in blue, maps heterogeneous sensory inputs to higher level descriptive concepts, such as hand-gesture and arm-position.

Original: Therefore, for this layer, we reuse the representations and models proposed in (Zhou, Muller, and Rabaey 2021).
Revised: Therefore, for this layer, we reuse the representations and models proposed in (Zhou, Muller, and Rabaey 2021).

Original: In this work, EMG signals are windowed over 50 millisecond  sections, and are mapped to spatiotemporal hypervector representations.
Revised: EMG signals are windowed over 50 millisecond overlapping sections, and are mapped to spatio-temporal hypervector representations

Original: The input “query” hypervectors are then mapped to specific gestures by means of a nearest centroid classifier.
Revised: The input “query” hypervectors are then mapped to specific gestures through a nearest centroid classifier.

Original: The centroids in the model arealso specified in HD space by means of class-prototype hypervectors, and classification takes place by selecting the class that maximizes the  similarity(measured by cosine similarity) between its prototype and the query vector.
Revised: The centroids in the model are specified as class-prototype hypervectors, and classification takes place by selecting the class that maximizes the (cosine) similarity between its prototype and the query vector.

Original: Accelerometer data is used to classify the user’s arm position through a Support Vector Machine (SVM).
Revised: Accelerometer data is used to classify arm-position through a Support Vector Machine (SVM).

Original: In (Zhou, Muller, and Rabaey 2021),this position information is used to improve gesture classification performance in a dual-stage classification fashion.
Revised: In (Zhou, Muller, and Rabaey 2021), position information is used to improve gesture classification performance through dual-stage classification.

Original: For our framework, we feed the classified arm position as well as the classified gesture to the following layer.
Revised: In this work, the classified positions and gestures are both fed to the following layer.

Original: 2) Probabilistic reasoning layer The second layer of abstraction, highlighted in green, builds a high-level proba- bilistic world model of the application.
Revised: 2) Probabilistic reasoning layer. The second layer, highlighted in green, builds a high-level probabilistic world model of the application.

Original: In our use case, we realize this model with a DBN that encodes the probabilistic relations among the tasks performed by the user and the observed hand-gestures and arm-positions, written as ges and pos , respectively.
Revised: In this work, we realize this model with a DBN that encodes the probabilistic relations among the tasks performed by the user and the observed handgestures and arm-positions, written as ges and pos , respectively.

Original: This DBN makes a first order Markov assumption for the temporal relations among tasks, gestures and positions.
Revised: This DBN makes a first order Markov assumption for the temporal relations among tasks, gestures and positions.

Original: The probabilistic reasoning layer in our example framework performs filtering on this DBN.
Revised: The probabilistic reasoning layer in our example framework performs filtering on this DBN.

Original: That is, it computes the posterior distribution over the task at time t , given all evidence over ges and pos to date.
Revised: That is, it computes the posterior distribution over the task at time t , given all evidence over ges and pos to date.

Original: That is, we compute Pr( task t | ges 0:

Original: t , pos 0: t ) . In practice, the inference algorithm maintains a belief over the task at time t and updates it at time t + 1 given the evidence over t + 1 and t in a recursive fashion (Russell and Norvig 2002).
Revised: In practice, the inference algorithm maintains a belief over the task at time t and updates it at time t + 1 given the evidence over t + 1 and t in a recursive fashion (Russell and Norvig 2002).

Original: Specifically, for all values of task at time t + 1 :
Revised: Specifically, for all values of Task at time t + 1 :

Original: [Equation]
Revised: [Equation]

Original: where the variable task t is marginalized as it remains unobserved; Pr( task t | ges 0: t , pos 0: t ) is calculated in the previous iteration with the evidence over ges and posavailable at time t and t − 1 ; and α is a normalization constant that ensures that the posteriors sum to 1.
Revised: where the variable Task t is marginalized as it remains unobserved; Pr( Task t | ges 0: t , pos 0: t ) is calculated in the previous iteration with the evidence over ges and pos at time t and t − 1 ; and α is a normalization constant that ensures that the posteriors sum to 1.

Original: We map the DBN to HD space 2 by using the data structures discussed in the Hyperdimensional computing Section, as shown in Figure 2.
Revised: We map the DBN to HD space 2 by using the data structures discussed in the Hyperdimensional computing Section, as shown in Figure 2.

Original: First, we generate random hypervector seeds for each value of the task , ges and pos variables, and store them in the Value-I.M.
Revised: First, we generate random hypervector seeds for each value of the Task and Ges variables, and store them in the Value-I.M.

Original: Then,as indicated in Figure 2, we randomly project the probability of each value to HD space by multiplying it times a new randomly generated hypervector.
Revised: Then, we randomly project the probabilities to HD space through scalar multiplication with new random hypervectors.

Original: Note that we scale the probabil- ity values ( by e.g. 10 3 ) such that we only  store integers.
Revised: Note that we scale the probabilities (by e.g. 10 3 ) such that we only have to store integers.

Original: Then, we bind this real-valued vector with the hypervector corresponding to the observed value (for example, GES1 in the first row of the CPT).
Revised: Then, we bind this real-valued vector with the hypervector corresponding to the observed value (for example, GES in the first row of the CPT).

Original: As such, we can create bundles of real-valued hypervectors corre- sponding to each of the conditioned variables (e.g. we create a realvalued bundle for the ges t variable conditioned on task t = task 1 and ges t − 1 = ges 1 , labeled COND1 in the figure).
Revised: As such, we can create bundles of real-valued hypervectors corresponding to each of the conditioned variables (e.g. we create a real-valued bundle for the Ges t variable conditioned on Task t = task 1 and Ges t − 1 = ges 1 , labeled COND1 in the figure).

Original: We can further superimpose these bundles by binding them to their corresponding conditioning values, as indicated by the labels Conditional bundle .
Revised: We can further superimpose these bundles by binding them to their corresponding conditioning values, as indicated by the labels Conditional bundle .

Original: Note that, in this example, we apply the permute operation to the values of ges t − 1in order to  indicate that this variable was observed attime t − 1 .
Revised: Note that, in this example, we apply the permute operation to the values of Ges t − 1 to explicitly indicate they were observed at t − 1 .

Original: Finally, we superimpose (again through bundling) all the conditional probability vectors.
Revised: Finally, we superimpose (through bundling) all these conditional probability vectors.

Original: This process of recursive bind- ing and bundlingultimately allows us to store a single hypervector per CPT in the model.
Revised: This process of recursive binding and bundling allows us to store a single hypervector per CPT in the model.

Original: At inference time,the probabilities are recovered from thesuper- imposed CPT hypervectors by unbinding them from their condition (which will normally be the observed or marginalized variables), taking the  dot product between the HD vector corresponding to the value of interest, and dividing by the dimension of the HD space (as discussed for histograms in the data structure section).
Revised: At inference time, probabilities are recovered from the CPT hypervectors by unbinding them from their condition (normally observed or marginalized variables) and taking the normalized dot product with the hypervector corresponding to the value of interest (see histograms in the data structure section).

Original: For example, starting fromthe CPT illustrated in Figure 2, the value of Pr( ges t = ges 1 | task t = task 2 , ges t − 1 = ges 1 ) can be recovered through dot( CPT ges t ◦ TASK2 ◦ ρ ( GES1 ) , GES1 ) / (1000 · n ) , where  1000 accounts for the scaling factor used during the mapping (whose purpose is to facilitate integer storage).
Revised: For example, starting from CPT ges t in Figure 2, the value of Pr( Ges t = ges 1 | Task t = task 2 , Ges t − 1 = ges 1 ) can be recovered through dot( CPT ges t ◦ TASK2 ◦ ρ ( GES1 ) , GES1 ) / (1000 · n ) , where n is hypervector dimension and 1000 accounts for the scaling factor that facilitated integer storage during mapping.

Original: Inference can  beagain performed with Equation 1.
Revised: Inference can then be performed with Equation 1.

Original: As illustrated in Figure 2,the process of unbinding thevalues of the conditional variables pro- duces a noisy version of COND2 .
Revised: As illustrated in Figure 2, unbinding the conditional variables hypervectors from the CPT hypervector produces a noisy version of COND2 .

Original: Similarly, there is information loss associated with recovering the final value of the probability of interest by means of the dot product.
Revised: Similarly, there can be information loss when recovering the final probabilities through the dot product.

Original: Thus, an important future work direction, as discussed towards the end of this paper, consists on quantifying the error induced bythis strategy of parameter superposition and storage.
Revised: Thus, an important future work direction consists on deriving bounds on the error induced by parameter superposition and storage.

Original: Despite the induced information loss discussed above, the moti- vation behind mappingthis probabilistic model to the HD space is two-fold: 1)
Revised: Despite the induced information loss, the motivation behind mapping probabilistic models to hypervectors is twofold:

Original: The potential improvements in memory efficiency for large models, since the space complexity of each CPT remains constant and depends only on the capacity 3 of the hypervectors (Frady, 
Revised: The potential improvements in memory efficiency for large models, since the space complexity of the CPTs remains constant and depends only on the capacity 3 of the hypervectors (Frady, Kleyko, and Sommer

Original: Kleyko, and Sommer 2018; Kleyko et al.
Revised: Kleyko et al.

Original: 2018), whereas the space complexity of the regular tabular representation can grow proportional to the number of involved variables and their cardinality.
Revised: ), unlike traditional tabular CPTs, which can have exponential space complexity. 4 We leave the precise memory Capacity

Original: We leave the analysis of HD vector storage capacity as well as a precise quantification of the memory efficiency  for future work, but prior work has found that storing multiple models in superpo- sition (Cheung et al. 2019) can lead to memory savings.
Revised: recover- efficiency analysis for future work, but prior work has found that storing multiple models in superposition (Cheung et al. 2019) can lead to memory savings.

Original: For example, (Zhou, Muller, and Rabaey 2021) demonstrated parameter memory savings of one order of magnitude when implementing dual-stage classification using high-dimensional model superposi- tion.
Revised: For example, (Zhou, Muller, and Rabaey 2021) demonstrated parameter memory savings of one order of magnitude when implementing dual-stage classification using high-dimensional model superposition.

Original: The work in (Zeman, Osipov, and Bosnic 2021) also inte- grates superposition to achieve neural network compression rates of up to 100x in multi-task settings, compared to individual neural network deployment.
Revised: The work in (Zeman, Osipov, and Bosnic 2021) also exploits superposition to achieve neural network compression rates of up to 100x in multi-task settings, compared to individual neural network deployment.

Original: Additionally, specialized HD computing hardwareimplementations that generate the seed hypervectors dynamically at run-time (Kleyko, Frady, and Sommer 2021)–instead of storing them all in I.M.– have shown potential energy savings of between 5x and 10x (Menon et al. 2021) compared to a tradi- tional machine learning implementation.
Revised: Additionally, specialized HD computing hardware that generates the seed hypervectors dynamically at run-time (Kleyko, Frady, and Sommer 2021) have shown potential energy savings of between 5x and 10x (Menon et al. 2021) compared to a traditional machine learning implementations.

Original: The second motivation for this mapping is that encoding the entire framework with the same primitives can be advantageous, as it can potentially enable reasoning and, eventually, learning across all levels of abstraction including the Control Layer described next.
Revised: The second motivation for this mapping is that encoding the entire framework with the same primitives can be advantageous, as it can potentially enable reasoning and, eventually, learning across all levels of abstraction including the Control Layer described next.

Original: Moreover, a common representation can enable hardware re-use and integration across different types of machine learning workloads, a trend with growing interest in the hardware-algorithm co-optimization community (Verhelst and Murmann 2020; Shafique et al. 2021).
Revised: Moreover, a common representation can enable hardware re-use and integration across different types of machine learning workloads, a trend with growing interest in the hardware-algorithm co-optimization community (Galindez Olascoaga et al. 2019; Shafique et al. 2021).

Original: 3) Control layer The last layer serves as an interface between the proposed framework and other external components of the sys- tem.
Revised: 3) Control layer. The last layer serves as an interface between the proposed framework and other external components of the system.

Original: There are several ways in which this layer can be realized, depending on the needs of the system and the application.
Revised: There are several ways in which this layer can be realized, depending on the needs of the system and the application.

Original: This ver- satility is further facilitated by the used hypervector representation, as discussed through the two following examples:
Revised: This versatility is further facilitated by the used hypervector representation, as discussed through the two following examples:

Original:  Symbolic reasoning: for example, to transfer knowledge to unknown contexts and environment through analogical reasoning.
Revised: - Symbolic reasoning: for example, to transfer knowledge to unknown contexts and environments through analogical reasoning.

Original: Section Context Knowledge Transfer Using Symbolic Reasoningwithin the experimental section discusses this idea in detail.
Revised: Section Context Knowledge Transfer Using Symbolic Reasoning discusses this idea in detail.

Original:  Reasoning to actuation (input-output) mapping: the predicted user intent (i.e. belief in the set of tasks at time t given ev- idence) can, for example, be used to fine tune the output grip selection.
Revised: - Reasoning to actuation (input-output) mapping: the predicted user intent (i.e. belief in the set of tasks at time t given evidence) can, for example, be used to fine tune the output grip selection.

Original: One way of doing that in HD space is by augmenting the functionality of HD computing implementations of state-automatabased control (Osipov, Kleyko, and Legalov 2017) or recall of reac- tive behaviour (Neubert, Schubert, and Protzel 2017).
Revised: One way of doing that in HD space is by augmenting the functionality of HD computing implementations of state-automata-based control (Osipov, Kleyko, and Legalov 2017) or recall of reactive behaviour (Neubert, Schubert, and Protzel 2017).

Original: In the recall technique, illustrated in Figure 1, a composite hypervector repre- sentation encodes sensor-action pairs, such that behavior can be resembled at run-time by extracting (through unbinding) the action corresponding to the current sensor value.
Revised: In the recall technique, illustrated in Figure 1, a composite hypervector representation encodes sensor-action pairs, such that behavior can be resembled at run-time by extracting (through unbinding) the action corresponding to the current sensor value.

Original: We posit that the posterior distribution from the layer below could further be used to weight this recall representation, an interesting direction that we leave for future work.
Revised: We posit that the posterior distribution from the layer below could further be used to weight this recall representation, an interesting direction that we leave for future work.

Original: Experiments Experimental setup and dataset
Revised: Experiments Experimental setup and dataset

Original: We validated our approach with the dataset in (Zhou, Muller, and Rabaey 2021), which includes measurements from a 64-channel EMG electrode array with an on-board tri-axial accelerometer, shown in Figure 3.a.
Revised: We validated our approach with the dataset in (Zhou, Muller, and Rabaey 2021), which includes measurements from a 64-channel EMG electrode array with an on-board tri-axial accelerometer, shown in Figure 3.a.

Original: We consider 3 hand-gestures and 4 arm positions, as illustrated in 3.b and 3.c.
Revised: We consider 3 handgestures and 4 arm positions, as illustrated in 3.b and 3.c.

Original: In addition, we define 3pos- sible tasks based on common activities of daily living and based on the available gestures/grips and postures from the aforemen- tioned dataset: brushing hair, tying shoelace and open jar.
Revised: In addition, we define 3 tasks based on common activities of daily living and based on the available gestures/grips and postures from the aforementioned dataset: brushing hair, tying shoelace and opening jar.

Original: Sequential data over the different tasks is not available in (Zhou, Muller, and Rabaey 2021), therefore, we generate a 10000 sample sequential dataset by sampling an ideal DBN modeled with expert knowl- edge and we use a random 70%, 30% train-test split over 10 trials to evaluate the performance of the two lowest layers of the proposed framework.
Revised: Sequential data over the different tasks is not available in (Zhou, Muller, and Rabaey 2021), therefore, we generate a 10000 sample sequential dataset by sampling an ideal DBN modeled with expert knowledge and we use a random 70%, 30% train-test split over 10 trials to evaluate the performance of the two lowest layers of the proposed framework.

Original: Task description
Revised: Task description

Original: The task we evaluated included predicting the most likely task at time t , given evidence over ges and pos over time.
Revised: The task we evaluated consisted on predicting the most likely Task at time t , given evidence over Ges and Pos over time.

Original: That is, we eval- uate the accuracy of predicting the most likely task t after evaluat-

Original: ing the filtering algorithm described in Equation 1.
Revised: This is done through the filtering algorithm described in Equation 1.

Original: Note that the evidence variables in the DBN, ges and pos , are the output of the classification of EMG and accelerometer signals in the perception layer.
Revised: Note that the evidence variables in the DBN, Ges and Pos , are the output of the classification of EMG and accelerometer signals in the perception layer.

Original: In particular, EMG signals are classified as hand-gestures through an HD computing implementation of a nearest centroid classifier, and accelerometer signals are classified as arm-positions through an SVM classifier.
Revised: In particular, EMG signals are classified as hand-gestures through an HD computing implementation of a nearest centroid classifier, and accelerometer signals are classified as arm-positions through an SVM classifier.

Original: Moreover, we evaluatethe performance of two DBNs with dif- ferent structures: the complex model has the same structure as the model depicted in Figure 1, and the simple model ignores the time relationships among gestures and positions, that is, at any timeslice t of the DBN, ges t and pos t only depend on task t .
Revised: Moreover, we evaluate two DBNs with different structures: the complex model has the same structure as the model depicted in Figure 1, and the simple model ignores the time relationships among gestures and positions. That is, at any time-slice t of the DBN, ges t and pos t only depend on task t .

Original: The goal of evaluating two model structuresis to compare how their hypervector  implementation degrades in comparison to the traditional tabular CPT implementation, as described below.
Revised: We evaluate these two model structures to compare how their hypervector CPT implementation degrades in comparison to the traditional tabular CPT implementation.

Original: Results
Revised: Results

Original: Figure 4 shows an example of the posterior distributions evalu- ated witheach of the aforementioned DBNs (simple in blue and complex in red) when using a standard tabular CPT representa- tion (cross markers) and when mapping CPTs to hypervectors (circle markers).
Revised: Figure 4 shows an example of the posterior distributions evaluated with the aforementioned DBNs (simple in blue and complex in red) when using a standard tabular CPT representation (cross markers) and when mapping CPTs to hypervectors (circle markers).

Original: The real distribution (from the DBN we sampled to generate the datasets) is shown with a black dotted line.
Revised: The real distribution (from the DBN we sampled to generate the datasets) is shown with a black dotted line.

Original: It is clear that the posterior calculated by the complex model is closer to the real distribution.
Revised: It is clear that the posterior calculated by the complex model is closer to the real distribution.

Original: However, note that the hypervector CPT mapping of the complex model degrades more thanwith the simple model.
Revised: However, note that the hypervector CPT mapping of the com- [Figure] plex model degrades more than the simple model’s.

Original: This is expected due to the limited superposition capacity of the hypervector representation, which is strained more by the complex model due to the additional variable-wise dependencies among ges and pos .
Revised: This is expected due to the limited superposition capacity of the hypervector representation, which is strained more by the complex model due to the additional variable-wise dependencies among Ges and Pos .

Original: Figure 5 comparesthe task prediction accuracy of all models.
Revised: Figure 5 compares prediction accuracy for all models.

Original: Overall, filtering accuracy is preserved throughout the 10 time slices shown, and the complex model shows superior performance than the simple one.
Revised: Overall, filtering accuracy is preserved throughout thetime slices, and the complex model shows superior performance than the simple one.

Original: Yet, once again, the hypervectorCPT representation of the complex model loses nearly 3% accuracy, whereas the hypervectorCPT representation  does not lead to significant accuracy degradation.
Revised: Yet, once again, the hypervector representation of the complex model loses nearly 3% accuracy, whereas the hypervector representation of the simple one does not lead to significant accuracy degradation.

Original: Overall, there is a clear trade-off between hypervector capacity (driven by its representation type and dimension), model complexity, and memory efficiency, which we leave for future work.An
Revised: Overall, there is a clear trade-off between hypervector capacity (driven by its representation type and dimension), model complexity, and memory efficiency, which we leave for future work.

Original: [Figure] interesting direction would be to systematically quantify the super- position capacity of hypervectors when used to represent graphical probabilistic models such as the DBN studied in this work.
Revised: An interesting direction would be to systematically quantify the superposition capacity of hypervectors when used to represent graphical probabilistic models such as the DBN studied in this work.

Original: Context Knowledge Transfer Using Analogical Reasoning
Revised: Context Knowledge Transfer Using Analogical Reasoning

Original: The Control Layerof the proposed framework can allow us to de- duce generalizable outcomes that can be carried from one context to another.
Revised: The Control Layer can allow us to deduce generalizable outcomes that can be carried from one context to another.

Original: This is especially important because learning is typi- cally performed based on sporadic samples that are attributed to a limited number of contexts.
Revised: This is especially important because learning is typically performed based on sporadic samples that are attributed to a limited number of contexts.

Original: This section presents some examples of how the symbolic and, more specifically, analogical reasoning capabilities of HD computing can be exploited for the transfer of knowledge across context in the prosthetic use-case.

Original: In most practical scenarios, learning is performed based on lim- ited samples generated from a fixed set of sensors under a specific context.

Original: Some samples may exhibit common features with respect to learnt tasks among different contexts, whereas other samples lead to different task definitions depending on the context.
Revised: Some samples may exhibit common features with respect to learnt tasks among different contexts, whereas other samples lead to different task definitions depending on the context.

Original: The question is: how can we optimize our learning method such that it can systematically transfer task definitions between contexts so that our system can adapt to unknown settings in other contexts?
Revised: The question is: how can we optimize our learning method such that it can systematically transfer task definitions between contexts so that our system can adapt to unknown settings in new contexts?

Revised: This section presents some examples of how the symbolic and, more specifically, analogical reasoning capabilities of HD computing can be exploited for the transfer of knowledge across context in the prosthetic use case.

Original: For example, suppose that the DBN in Figure 1 was trained for the interactions of a prosthetic arm with objects in a kitchen, and that the DBN’s current task prediction (i.e. task t ) is openinga jar.
Revised: For example, suppose that the DBN in Figure 1 was trained through the interactions of a prosthetic arm with objects in a kitchen, and that the DBN’s current task prediction (i.e. Task t ) is opening jar.

Original: Thus, we could create a composite hypervector that describes the current situation, defined in terms of the current context (the loca- tion in this example), and the current task:
Revised: We can create a composite hypervector that describes this situation, defined in terms of the current context (location), and task (activity):

Original: [Equation]
Revised: [Equation]

Original: where LOC ◦ KIT denotes that the location in “situation 1” is the kitchen and ACT ◦ OJ denotes that the currenttask or activity is openinga jar.
Revised: where LOC ◦ KIT denotes that the location in “situation 1” is the kitchen and ACT ◦ OJ denotes that the current activity is opening jar.

Original: Because our DBN was trained solely with data from a kitchen context, it may be unable to generalize towards the correct pre- diction of activities in a new context, such as a “front-door” location.
Revised: Because our DBN was trained solely with data from a kitchen context, it may be unable to generalize towards the correct prediction of activities in a new context, such as a “front-door” location.

Original: Instead of seeking to gather more training data and retrain the DBN, we can explicitly encode our expert knowledge about all the different scenarios in the Control Layer, such that the framework can automatically exploit this information at run-time.
Revised: Instead of seeking to gather more training data and re-train the DBN, we can explicitly encode our expert knowledge about all the different scenarios in the Control Layer, such that the framework can automatically exploit this information at run-time.

Original: For example, we know that opening a door in the front-door con- text requires a twisting motion similar to the one used to open a jar in the kitchen context.
Revised: For example, we know that opening a door in the “frontdoor” context requires a twisting motion similar to the one used to open a jar in the kitchen context.

Original: Thus, we can encode the “front-door” twist- ing situation as the hypervector:
Revised: Thus, we can encode the “front-door” twisting situation as the hypervector:

Original: [Equation]
Revised: [Equation]

Original: where LOC ◦ FDR denotes that this second situation takes place at a front-door location and ACT ◦ OD denotes that the twisting activity in this situation is opening the door.
Revised: where LOC ◦ FDR denotes that “situation 2” takes place at a front-door location and ACT ◦ OD denotes that the twisting activity in this situation is opening the door.

Original: Then, we can instruct our framework to follow a similar analog- ical reasoning strategy as the example query  “What is the Dollar of Mexico?”
Revised: Then, we can instruct our framework to reason analogically through substitution about the two situations, with a query similar to “What is the Dollar of Mexico?”

Original: (Kanerva 2010).
Revised: (Kanerva 2010).

Original: In our case, we wish the Control Layerin our framework to answer the question, “what is the equivalent activity to opening a jar  in a front-door environment?”.
Revised: Specifically, in our case, we wish the Control Layer to answer the question, “what is the analogous of opening a jar when the user is in a front-door environment?”.

Original: This can be done with analogical reasoning through substitution to answer this question as follows.

Original: First, hypervectors corresponding to the two situations must be bound, such that the values occupying the same keys are bound (i.e. kitchen isbound to front-door and open-jar  to open-door):
Revised: First, hypervectors corresponding to the two situations must be bound, such that the values occupying the same keys are bound (i.e. kitchen is to front-door as open-jar is to open- door):

Original: [Equation]
Revised: [Equation]

Original: Then, the  queryabove –“what is the equivalent task to open jar in a front door environment?” – can simply be answered by probing theabove bound structure  with the open-jar ( OJ ) hypervector:
Revised: Then, the desired analogical query can simply be answered by probing the bound structure above with the open-jar ( OJ ) hypervector:

Original: [Equation]
Revised: [Equation]

Original: As such, our framework is enabled to conclude that the analogous to opening jar in a new “front door” environment is opening  door.
Revised: Our framework is thus capable of concluding that the analogous to opening jar in a new “front door” environment is opening a door.

Revised: Note that no training data from the new front-door environment nor re-tuning of the DBN were required to reach to this conclusion.

Original:  These symbolic reasoning mechanisms could further be used to provide feedback to the probabilistic model for continual learning and adaptation. Moreover, the analogical reasoning example ex- plained here could be exploited when training data collection is too costly and expert knowledge is available.
Revised: This is useful when expert knowledge is available and training data is costly. These symbolic reasoning mechanisms could further be used to provide feedback to the probabilistic model for continual learning and adaptation, an interesting direction for future work.

Original: Related work
Revised: Related work

Original: This work presents a hybrid framework comprising different types of reasoning and computational paradigms.
Revised: This work presents a hybrid framework comprising different types of reasoning and computational paradigms.

Original: As such, the body of related work is vast and spans multiple fields.
Revised: As such, the body of related work is vast and spans multiple fields.

Original: For brevity, we focus mostly on related neurosymbolic computing strategies and existing hybrid HD computing/VSAs solutions, but we are aware of many other relevant works, which we would like to further discuss in future and extended versions of this paper.
Revised: For brevity, we focus mostly on related neurosymbolic computing strategies and existing hybrid HD computing/VSA solutions, but we are aware of many other relevant works, which we would like to further discuss in future and extended versions of this paper.

Original: Neurosymbolic approaches Several state-of-the-art approaches in complex event processing have attempted to leverage expert knowledge by diving the problem into low-level perception and high-level reasoning, and enabling neural networks to learn the former from data and emulate the latter from user defined rules (Xing et al. 2020).
Revised: Neurosymbolic approaches. Several state-of-the-art approaches in complex event processing have attempted to leverage expert knowledge by diving the problem into lowlevel perception and high-level reasoning, and enabling neural networks to learn the former from data and emulate the latter from user defined rules (Xing et al. 2020).

Original: Yet, these approaches can lack expres- sivity at the symbolic level and can suffer from explainability issues.
Revised: Yet, these approaches can lack expressivity at the symbolic level and can suffer from explainability issues.

Original: Other approaches have addressed these limitations by still leveraging neural networks to process sub-symbolic (e.g. sensor) data, but relying on (probabilistic) logic languages to inject expert knowledge on higher level information, such as the rules of complex events (Vilamala et al. 2021;
Revised: Other approaches have addressed these limitations by still leveraging neural networks to process sub-symbolic (e.g. sensor) data, but relying on (probabilistic) logic languages to inject expert knowledge on higher level information, such as the rules of complex events (Vilamala et al. 2021;

Original: Apriceno, Passerini, and Serafini 2021).
Revised: Apriceno, Passerini, and Serafini 2021).

Original: Related HD computing/VSA approaches Recent works
Revised: Related HD computing/VSA approaches.

Original:  explore how VSAs and Holographic Reduced Representations can be used to propose novel neural network architectures capable of encoding heterogeneous knowledge, such asspatial information or analogy making together with pattern recognition  (Weiss, Cheung, and Olshausen 2016; Yılmaz 2015).
Revised: Recent works explore how VSAs and Holographic Reduced Representations can be used to propose novel neural network architectures capable of inferring heterogeneous knowledge, such as pattern recognition together with spatial information (Weiss, Cheung, and Olshausen 2016; Yılmaz 2015).

Revised: Analogical reasoning has been one of the tools used for such hybrid applications.

Revised: For a detailed overview of HD computing applications, we refer the reader to (Kleyko et al. 2021c).

Original: Aspects of our framework that differ from the work 
Revised: Aspects of our framework that differ from the work highlighted above.

Original: highlighted above The framework we propose is conceptually modular and we believe it has the potential of leveraging suitable machine learning and AI strategies at each layer.
Revised: The framework we propose is conceptually modular and we believe it has the potential of leveraging suitable machine learning and AI strategies at each layer.

Original: Thus, a potential future work direction would be to re-implement the perception and probabilistic reasoning layers through the neurosymbolic ap- proaches highlighted above.
Revised: Thus, a potential future work direction would be to re-implement the perception and probabilistic reasoning layers through the approaches highlighted above.

Original: An important different with end-toend hybrid and neurosymbolic implementations is the role of sym- bolic reasoning at the control layer in our framework.
Revised: An important difference with end-to-end hybrid and neurosymbolic implementations is the role of symbolic reasoning at the Control Layer in our framework.

Original: Unlike the techniques mentioned above, which exploit logical statements to improve the performance on the task of interest 5 , we plan to use symbolic reasoning to interface with elements outside the frame- work (actuation control), and to provide feedback to lower levels towards dynamic and context dependent adaptive and continual learning.
Revised: Unlike the techniques mentioned above, which exploit logical statements to improve the performance on the task of interest 5 , we plan to use symbolic reasoning to interface with elements outside the framework (actuation control), and to provide feedback to lower levels towards dynamic and context dependent adaptive and continual learning.

Original: This is further discussed in the following section.
Revised: This is further discussed in the following section.

Original: Discussion Implications of the proposed framework
Revised: Discussion Implications of the proposed framework

Original: - Note that the framework we proposed assumes that each layer is trained, learned or stated (i.e. symbolic knowledge) independently through the most appropriate technique.
Revised: - Note that the framework we proposed assumes that each layer is trained, learned or stated (i.e. symbolic knowledge) independently through the most appropriate technique.

Original: We envision that the inte- gration through HD computing can take place at  run-time.-
Revised: The believe that the HD-computing-enabled integration discussed in this work can be best exploited at inference-time or run-time.

Original:  The proposed probabilistic model within our framework is rather simple since it was trained with synthetic data sampled from an ideal model constructed with expert knowledge.
Revised: - The proposed probabilistic model within our framework is rather simple since it was trained with synthetic data sampled from an ideal model constructed with expert knowledge.

Original: As such, we were able to store it in superposition without significant performance degradation.
Revised: As such, we were able to store it in superposition without significant performance degradation.

Original: For real life applications, it will be necessary to per- form more complete analyses on the capacity of the models and related trade-offs.
Revised: For real life applications, it will be necessary to perform more complete analyses on the capacity of the models and related trade-offs.

Original: - Overall, the considered use-case benefits from the three layers in the proposed framework.
Revised: - Overall, the considered use case benefits from the three layers in the proposed framework.

Original: This opens the door for further investiga- tion into the adaptation of this hierarchical framework for different use-cases, which may benefit from a different configuration of our framework in terms of,for example, the classifierused in the perception layer, the probabilistic model and  algorithm used for the middle layer, or the type of symbolic reasoning or actuation control for the highest layer.
Revised: This opens the door for further investigation into the adaptation of this hierarchical framework for different use cases, which may benefit from a different configuration of our framework in terms of, the classifier in the perception layer, the probabilistic model and inference algorithm in the middle layer, or the type of symbolic reasoning or actuation control for the top layer.

Original: Future work
Revised: Future work

Original: - An interesting direction would be to explore the use of tractable probabilistic models, such as Probabilistic Circuits (Choi, Vergari, and den Broeck 2020) for the probabilistic inference layer.
Revised: - An interesting direction would be to explore the use of tractable probabilistic models, such as Probabilistic Circuits (Choi, Vergari, and den Broeck 2020) for the probabilistic inference layer.

Original: This would involve evaluating whether such a model could be mapped to the HD space, similar to the DBN example discussed herewith.
Revised: This would involve evaluating whether such a model could be mapped to the HD space, similar to the DBN example discussed herewith.

Original: - Beyond analogical reasoning, this framework can benefit from other forms of symbolic and logical reasoning and the various state- of-the-art techniques that enable its efficient execution.
Revised: - Beyond analogical reasoning, this framework can benefit from other forms of symbolic and logical reasoning and the various state-of-the-art techniques that enable its efficient execution.

Original: - As mentioned in the Analogical Reasoning section, we believe the control layer could coordinate a process of active learning.
Revised: -Asmentioned in the Analogical Reasoning section, we believe the control layer could coordinate a process of active learning.

Original: Thus we need to review whether existing strategies in the fields of transfer learning, meta learning or reinforcement learning could be appropriate for this, or whether a different technique may be required.
Revised: Thus we need to review whether existing strategies in the fields of transfer learning, meta learning or reinforcement learning could be appropriate for this, or whether a different technique may be required.

Original:  Another future work direction would be to derive bounds on the error propagation caused by the hypervector representations, and especially when storing complex probabilistic models with highdimensional superposition.
Revised: - Another future work direction would be to derive bounds on the error propagation caused by the hypervector representations, and especially when storing complex probabilistic models with high-dimensional superposition.

Original: - Finally, an important future work direction consists on comparing the performance of the proposed framework with other hybrid strategies, such as neurosymbolic computing.
Revised: - Finally, an important future work direction consists on comparing the performance of the proposed framework with other hybrid strategies, such as neurosymbolic computing.

Original: 5 Although, as mentioned, we believe these strategies could also be leveraged by our framework.

Original: Acknowledgements
Revised: Acknowledgements

Original: We thank the reviewers of this paper for their helpful comments and Denis Kleyko for his feedback on the camera-ready version.
Revised: We thank the reviewers of this paper for their helpful comments and Denis Kleyko for his feedback on the cameraready version.

Original: We also thank Andy Zhou for sharing the datasets used in this work and Jennifer Ruffing for useful discussions.
Revised: We also thank Andy Zhou for sharing the datasets used in this work and Jennifer Ruffing for useful discussions.

Original: This work was supported in part by the VIP and HYDDENN programs sponsored by DARPA.
Revised: This work was supported in part by the VIP and HYDDENN programs sponsored by DARPA.

Original: This work was also supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1752814.
Revised: This work was also supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1752814.

Original: Support was also received from sponsors of Berkeley Wireless Research Center.
Revised: Support was also received from sponsors of Berkeley Wireless Research Center.

Original: Any opinions, findings, and conclu- sions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Na- tional Science Foundation, DARPA or the U.S. government.
Revised: Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, DARPA, or the U.S. government.