nengo_spa.algebras

The following items are re-exported by nengo_spa.algebras:

base.AbstractAlgebra()	Abstract base class for algebras.
base.CommonProperties()	Definition of constants for common properties of vectors in an algebra.
base.ElementSidedness(value)	The side in a binary operation for which a special element’s properties hold.
hrr_algebra.HrrAlgebra()	Holographic Reduced Representations (HRRs) algebra.
hrr_algebra.HrrProperties()	Vector properties supported by the HrrAlgebra.
vtb_algebra.VtbAlgebra()	Vector-derived Transformation Binding (VTB) algebra.
vtb_algebra.VtbProperties()	Vector properties supported by the VtbAlgebra.
tvtb_algebra.TvtbAlgebra()	Transposed Vector-derived Transformation Binding (TVTB) algebra.
tvtb_algebra.TvtbProperties()	Vector properties supported by the TvtbAlgebra.

Base classes

AbstractAlgebra()	Abstract base class for algebras.
AbstractSign()	Abstract base class for implementing signs for an algebra.
CommonProperties()	Definition of constants for common properties of vectors in an algebra.
ElementSidedness(value)	The side in a binary operation for which a special element’s properties hold.
GenericSign(sign)	A generic sign implementation.

class nengo_spa.algebras.base.ElementSidedness(value)

Bases: enum.Enum

The side in a binary operation for which a special element’s properties hold.

LEFT = 'left'

RIGHT = 'right'

TWO_SIDED = 'two-sided'

class nengo_spa.algebras.base.AbstractAlgebra

Bases: object

Abstract base class for algebras.

Custom algebras can be defined by implementing the interface of this abstract base class.

is_valid_dimensionality(d)

Checks whether d is a valid vector dimensionality.

Parameters

d (int) – Dimensionality

Returns

True, if d is a valid vector dimensionality for the use with the algebra.

Return type

bool

create_vector(d, properties, *, rng=None)

Create a vector fulfilling given properties in the algebra.

Valid properties and combinations thereof depend on the concrete algebra. It is suggested that the properties is either a set of str (if order does not matter) or a list of str (if order does matter). Use the constants defined in CommonProperties where appropriate.

Parameters

• d (int) – Vector dimensionality

• properties – Definition of properties for the vector to fulfill. Type and specification format depend on the concrete algbra, but it is suggested to use either a set or list of str (depending on whether order of properties matters) utilizing the constants defined in CommonProperties where applicable.

• rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.

Returns

Random vector with desired properties.

Return type

ndarray

make_unitary(v)

Returns a unitary vector based on the vector v.

A unitary vector does not change the length of a vector it is bound to.

Parameters

v ((d,) ndarray) – Vector to base unitary vector on.

Returns

Unitary vector.

Return type

ndarray

superpose(a, b)

Returns the superposition of a and b.

This is commonly elementwise addition.

Parameters

• a ((d,) ndarray) – Left operand in superposition.

• b ((d,) ndarray) – Right operand in superposition.

Returns

Superposed vector.

Return type

(d,) ndarray

bind(a, b)

Returns the binding of a and b.

The resulting vector should in most cases be dissimilar to both inputs.

Parameters

• a ((d,) ndarray) – Left operand in binding.

• b ((d,) ndarray) – Right operand in binding.

Returns

Bound vector.

Return type

(d,) ndarray

binding_power(v, exponent)

Returns the binding power of v using the exponent.

For a positive exponent, the binding power is defined as binding (exponent-1) times bindings of v to itself. For a negative exponent, the binding power is the approximate inverse bound to itself according to the prior definition. Depending on the algebra, fractional exponents might be valid or return a ValueError, if not. Usually, a fractional binding power will require that v has a positive sign.

Note the following special exponents:

• an exponent of -1 will return the approximate inverse,

• an exponent of 0 will return the identity vector,

• and an exponent of 1 will return v itself.

The default implementation supports integer exponents only and will apply the bind method multiple times. It requires the algebra to have a left identity.

Parameters

• v ((d,) ndarray) – Vector to bind repeatedly to itself.

• exponent (int or float) – Exponent of the binding power.

Returns

Binding power of v.

Return type

(d,) ndarray

See also

AbstractAlgebra.sign

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Invert vector v.

A vector bound to its inverse will result in the identity vector.

Some algebras might not have an inverse only on specific sides. In that case a NotImplementedError may be raised for non-existing inverses.

Parameters

• v ((d,) ndarray) – Vector to invert.

• sidedness (ElementSidedness, optional) – Side in the binding operation on which the returned value acts as inverse.

Returns

Inverted vector.

Return type

(d,) ndarray

get_binding_matrix(v, swap_inputs=False)

Returns the transformation matrix for binding with a fixed vector.

Parameters

• v ((d,) ndarray) – Fixed vector to derive binding matrix for.

• swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.

Returns

Transformation matrix to perform binding with v.

Return type

(d, d) ndarray

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Returns the transformation matrix for inverting a vector.

Some algebras might not have an inverse only on specific sides. In that case a NotImplementedError may be raised for non-existing inverses.

Parameters

• d (int) – Vector dimensionality (determines the matrix size).

• sidedness (ElementSidedness, optional) – Side in the binding operation on which a transformed vectors acts as inverse.

Returns

Transformation matrix to invert a vector.

Return type

(d, d) ndarray

implement_superposition(n_neurons_per_d, d, n)

Implement neural network for superposing vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• n (int) – Number of vectors to superpose in the network.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of length n of inputs to the network, and output the network output.

Return type

tuple

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)

Implement neural network for binding vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• unbind_left (bool) – Whether the left input should be unbound from the right input.

• unbind_right (bool) – Whether the right input should be unbound from the left input.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of the left and the right input in that order, and output the network output.

Return type

tuple

sign(v)

Returns the sign of v defined by the algebra.

The exact definition of the sign depends on the concrete algebra, but should be analogous to the sign of a (complex) number in so far that binding two vectors with the same sign produces a “positive” vector. There might, however, be multiple types of negative signs, where binding vectors with different types of negative signs will produce another “negative” vector.

Furthermore, if the algebra supports fractional binding powers, it should do so for all “non-negative” vectors, but not “negative” vectors.

If an algebra does not have the notion of a sign, it may raise a NotImplementedError.

Parameters

v ((d,) ndarray) – Vector to determine sign of.

Returns

The sign of the input vector.

Return type

AbstractSign

See also

AbstractAlgebra.abs

abs(v)

Returns the absolute vector of v defined by the algebra.

The exact definition of “absolute vector” may depend on the concrete algebra. It should be a “positive” vector (see sign) that relates to the input vector.

The default implementation requires that the possible signs of the algebra correspond to actual vectors within the algebra. It will bind the inverse of the sign vector (from the left side) to the vector v.

If an algebra does not have the notion of a sign or absolute vector, it may raise a NotImplementedError.

Parameters

v ((d,) ndarray) – Vector to obtain the absolute vector of.

Returns

The absolute vector relating to the input vector.

Return type

(d,) ndarray

absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the standard absorbing element of dimensionality d.

An absorbing element will produce a scaled version of itself when bound to another vector. The standard absorbing element is the absorbing element with norm 1.

Some algebras might not have an absorbing element other than the zero vector. In that case a NotImplementedError may be raised.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – Side in the binding operation on which the element absorbs.

Returns

Standard absorbing element.

Return type

(d,) ndarray

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the identity element of dimensionality d.

The identity does not change the vector it is bound to.

Some algebras might not have an identity element. In that case a NotImplementedError may be raised.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – Side in the binding operation on which the element acts as identity.

Returns

Identity element.

Return type

(d,) ndarray

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Returns the negative identity element of dimensionality d.

The negative identity only changes the sign of the vector it is bound to.

Some algebras might not have a negative identity element (or even the notion of a sign). In that case a :py:class`NotImplementedError` may be raised.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – Side in the binding operation on which the element acts as negative identity.

Returns

Negative identity element.

Return type

(d,) ndarray

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the zero element of dimensionality d.

The zero element produces itself when bound to a different vector. Usually this will be the zero vector.

Some algebras might not have a zero element. In that case a NotImplementedError may be raised.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – Side in the binding operation on which the element acts as zero.

Returns

Zero element.

Return type

(d,) ndarray

class nengo_spa.algebras.base.AbstractSign

Bases: abc.ABC

Abstract base class for implementing signs for an algebra.

is_positive()

Return whether the sign is positive.

is_negative()

Return whether the sign is negative.

is_zero()

Return whether the sign neither positive nor negative (i.e. zero), but definite.

is_indefinite()

Return whether the sign is neither positive nor negative nor zero.

to_vector(d)

Return the vector in the algebra corresponding to the sign.

Parameters

d (int) – Vector dimensionality.

Returns

Vector corresponding to the sign.

Return type

(d,) ndarray

class nengo_spa.algebras.base.GenericSign(sign)

Bases: nengo_spa.algebras.base.AbstractSign

A generic sign implementation.

Parameters

sign (-1, 0, 1, None) – The represented sign. None is used for an indefinite sign.

is_positive()

Return whether the sign is positive.

is_negative()

Return whether the sign is negative.

is_zero()

Return whether the sign neither positive nor negative (i.e. zero), but definite.

is_indefinite()

Return whether the sign is neither positive nor negative nor zero.

class nengo_spa.algebras.base.CommonProperties

Bases: object

Definition of constants for common properties of vectors in an algebra.

Use these for best interoperability between algebras.

UNITARY = 'unitary'

A unitary vector does not change the length of a vector it is bound to.

POSITIVE = 'positive'

A positive vector does not change the sign of a vector it is bound to.

A positive vector allows for fractional binding powers.

Holographic reduced representations (HRR)

HrrAlgebra()	Holographic Reduced Representations (HRRs) algebra.
HrrProperties()	Vector properties supported by the HrrAlgebra.
HrrSign(dc_sign, nyquist_sign)	Represents a sign in the HrrAlgebra.

class nengo_spa.algebras.hrr_algebra.HrrAlgebra

Bases: nengo_spa.algebras.base.AbstractAlgebra

Holographic Reduced Representations (HRRs) algebra.

Uses element-wise addition for superposition, circular convolution for binding with an approximate inverse.

The circular convolution \(c\) of vectors \(a\) and \(b\) is given by

\[c[i] = \sum_j a[j] b[i - j]\]

where negative indices on \(b\) wrap around to the end of the vector.

This computation can also be done in the Fourier domain,

\[c = DFT^{-1} ( DFT(a) \odot DFT(b) )\]

where \(DFT\) is the Discrete Fourier Transform operator, and \(DFT^{-1}\) is its inverse.

Circular convolution as a binding operation is associative, commutative, distributive.

More information on circular convolution as a binding operation can be found in [plate2003].

plate2003

Plate, Tony A. Holographic Reduced Representation: Distributed Representation for Cognitive Structures. Stanford, CA: CSLI Publications, 2003.

is_valid_dimensionality(d)

Checks whether d is a valid vector dimensionality.

For circular convolution all positive numbers are valid dimensionalities.

Parameters

d (int) – Dimensionality

Returns

True, if d is a valid vector dimensionality for the use with the algebra.

Return type

bool

create_vector(d, properties, *, rng=None)

Create a vector fulfilling given properties in the algebra.

Parameters

• d (int) – Vector dimensionality

• properties (set of str) – Definition of properties for the vector to fulfill. Valid set elements are constants defined in HrrProperties.

• rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.

Returns

Random vector with desired properties.

Return type

ndarray

make_unitary(v)

Returns a unitary vector based on the vector v.

A unitary vector does not change the length of a vector it is bound to.

Parameters

v ((d,) ndarray) – Vector to base unitary vector on.

Returns

Unitary vector.

Return type

ndarray

superpose(a, b)

Returns the superposition of a and b.

This is commonly elementwise addition.

Parameters

• a ((d,) ndarray) – Left operand in superposition.

• b ((d,) ndarray) – Right operand in superposition.

Returns

Superposed vector.

Return type

(d,) ndarray

bind(a, b)

Returns the binding of a and b.

The resulting vector should in most cases be dissimilar to both inputs.

Parameters

• a ((d,) ndarray) – Left operand in binding.

• b ((d,) ndarray) – Right operand in binding.

Returns

Bound vector.

Return type

(d,) ndarray

binding_power(v, exponent)

Returns the binding power of v using the exponent.

The binding power is defined as binding (exponent-1) times bindings of v to itself. Fractional binding powers are supported.

Note the following special exponents:

• an exponent of -1 will return the approximate inverse,

• an exponent of 0 will return the identity vector,

• and an exponent of w1cne will return v itself.

The following relations hold for integer exponents, and for unitary vectors:

• \(v^a \circledast v^b = v^{a+b}\),

• \((v^a)^b = v^{ab}\).

If \(a \geq 0\) and \(b \geq 0\), then the first relation holds also for non-unitary vectors with real exponents.

Parameters

• v ((d,) ndarray) – Vector to bind repeatedly to itself.

• exponent (int or float) – Exponent of the binding power.

Returns

Binding power of v.

Return type

(d,) ndarray

See also

sign

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Invert vector v.

This turns circular convolution into circular correlation, meaning that A*B*~B is approximately A.

Examples

For the vector [1, 2, 3, 4, 5], the inverse is [1, 5, 4, 3, 2].

Parameters

• v ((d,) ndarray) – Vector to invert.

• sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the inverse is two-sided.

Returns

Inverted vector.

Return type

(d,) ndarray

get_binding_matrix(v, swap_inputs=False)

Returns the transformation matrix for binding with a fixed vector.

Parameters

• v ((d,) ndarray) – Fixed vector to derive binding matrix for.

• swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.

Returns

Transformation matrix to perform binding with v.

Return type

(d, d) ndarray

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Returns the transformation matrix for inverting a vector.

Parameters

• d (int) – Vector dimensionality (determines the matrix size).

• sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the inverse is two-sided.

Returns

Transformation matrix to invert a vector.

Return type

(d, d) ndarray

implement_superposition(n_neurons_per_d, d, n)

Implement neural network for superposing vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• n (int) – Number of vectors to superpose in the network.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of length n of inputs to the network, and output the network output.

Return type

tuple

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)

Implement neural network for binding vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• unbind_left (bool) – Whether the left input should be unbound from the right input.

• unbind_right (bool) – Whether the right input should be unbound from the left input.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of the left and the right input in that order, and output the network output.

Return type

tuple

sign(v)

Returns the HRR sign of v.

See AbstractAlgebra.sign for general information on the notion of a sign for algbras, and HrrSign for details specific to HRRs.

Parameters

v ((d,) ndarray) – Vector to determine sign of.

Returns

The sign of the input vector.

Return type

HrrSign

absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the standard absorbing element of dimensionality d.

An absorbing element will produce a scaled version of itself when bound to another vector. The standard absorbing element is the absorbing element with norm 1.

The absorbing element for circular convolution is the vector \((1, 1, \dots, 1)^{\top} / \sqrt{d}\).

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the standard absorbing element is two-sided.

Returns

Standard absorbing element.

Return type

(d,) ndarray

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the identity element of dimensionality d.

The identity does not change the vector it is bound to.

The identity element for circular convolution is the vector \((1, 0, \dots, 0)^{\top}\).

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the identity is two-sided.

Returns

Identity element.

Return type

(d,) ndarray

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the negative identity element of dimensionality d.

The negative identity element for circular convolution is the vector \((-1, 0, \dots, 0)^{\top}\).

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the identity is two-sided.

Returns

Negative identity element.

Return type

(d,) ndarray

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the zero element of dimensionality d.

The zero element produces itself when bound to a different vector. For circular convolution this is the zero vector.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the zero element is two-sided.

Returns

Zero element.

Return type

(d,) ndarray

class nengo_spa.algebras.hrr_algebra.HrrSign(dc_sign, nyquist_sign)

Bases: nengo_spa.algebras.base.AbstractSign

Represents a sign in the HrrAlgebra.

For odd dimensionalities, the sign is equal to the sign of the DC component of the Fourier representation of the vector. For even dimensionalities the sign is constituted out of the signs of the DC component and Nyquist frequency. Thus, for even dimensionalities, there is a total of four sub-signs excluding zero. The overall sign is considered positive if the DC component is positive and the Nyquist component is non-negative; the sign is considered negative if either component is negative; and the sign is considered zero if both are zero. Binding two Semantic Pointers with the same sub-sign will yield a positive Semantic Pointer. See the table below for details.

Table 1 Resulting Semantic Pointer signs from HRR binding two Semantic Pointers. (Only the upper triangle is given as the matrix is symmetric.)

Sign (DC, Nyquist)	+ (+1, +1)	− (+1, -1)	− (-1, +1)	− (−1, -1)	(0, 0)
+ (+1, +1)	+ (+1, +1)	− (+1, -1)	− (−1, +1)	− (−1, -1)	(0, 0)
− (+1, -1)		+ (1, +1)	− (−1, -1)	− (−1, +1)	(0, 0)
− (−1, +1)			+ (1, +1)	− (+1, -1)	(0, 0)
− (−1, -1)				+ (1, +1)	(0, 0)
(0, 0)					(0, 0)

Parameters

• dc_sign (int) – Sign of the DC component.

• nyquist_sign (int) – Sign of the Nyquist frequency component. Will be set to the dc_sign if zero.

dc_sign

nyquist_sign

is_positive()

Return whether the sign is positive.

is_negative()

Return whether the sign is negative.

is_indefinite()

Return whether the sign is neither positive nor negative nor zero.

to_vector(d)

Return the vector in the algebra corresponding to the sign.

DC sign	Nyquist sign	Vector
1	1	[ 1, 0, 0, …] (identity)
1	-1	[ 0, 1, 0, 0, …]
-1	1	[ 0, -1, 0, …]
-1	-1	[-1, 0, 0, 0, …] (negative identity)
0	0	[ 0, 0, 0, …] (zero)

Parameters

d (int) – Vector dimensionality.

Returns

Vector corresponding to the sign.

Return type

(d,) ndarray

class nengo_spa.algebras.hrr_algebra.HrrProperties

Bases: object

Vector properties supported by the HrrAlgebra.

UNITARY = 'unitary'

A unitary vector does not change the length of a vector it is bound to.

POSITIVE = 'positive'

A positive vector does not change the sign of a vector it is bound to.

A positive vector allows for fractional binding powers.

Vector-derived transformation binding (VTB)

VtbAlgebra()	Vector-derived Transformation Binding (VTB) algebra.
VtbProperties()	Vector properties supported by the VtbAlgebra.
VtbSign(sign)	Represents a sign in the VtbAlgebra.

class nengo_spa.algebras.vtb_algebra.VtbAlgebra

Bases: nengo_spa.algebras.base.AbstractAlgebra

Vector-derived Transformation Binding (VTB) algebra.

VTB uses elementwise addition for superposition. The binding operation \(\mathcal{B}(x, y)\) is defined as

\[\begin{split}\mathcal{B}(x, y) := V_y x = \left[\begin{array}{ccc} V_y' & 0 & 0 \\ 0 & V_y' & 0 \\ 0 & 0 & \ddots \end{array}\right] x\end{split}\]

with \(d'\) blocks

where

\[\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d - d' + 1} & y_{d - d' + 2} & \dots & y_d \end{array}\right]\end{split}\]

and

\[d'^2 = d.\]

The approximate inverse \(y^+\) for \(y\) is permuting the elements such that \(V_{y^+} = V_y^T\).

Note that VTB requires the vector dimensionality to be square.

The VTB binding operation is neither associative nor commutative. Furthermore, there are right inverses and identities only. By transposing the \(V_y\) matrix, the closely related TvtbAlgebra (Transposed VTB) algebra is obtained which does have two-sided identities and inverses.

Additional information about VTB can be found in

• Gosmann, Jan, and Chris Eliasmith (2019). Vector-derived transformation binding: an improved binding operation for deep symbol-like processing in neural networks. Neural computation 31.5, 849-869.

• Jan Gosmann (2018). An Integrated Model of Context, Short-Term, and Long-Term Memory. UWSpace.

See also

TvtbAlgebra

is_valid_dimensionality(d)

Checks whether d is a valid vector dimensionality.

For VTB all square numbers are valid dimensionalities.

Parameters

d (int) – Dimensionality

Returns

True, if d is a valid vector dimensionality for the use with the algebra.

Return type

bool

create_vector(d, properties, *, rng=None)

Create a vector fulfilling given properties in the algebra.

Creating positive vectors requires SciPy.

Parameters

• d (int) – Vector dimensionality

• properties (set of str) – Definition of properties for the vector to fulfill. Valid set elements are constants defined in VtbProperties.

• rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.

Returns

Random vector with desired properties.

Return type

ndarray

make_unitary(v)

Returns a unitary vector based on the vector v.

A unitary vector does not change the length of a vector it is bound to.

Parameters

v ((d,) ndarray) – Vector to base unitary vector on.

Returns

Unitary vector.

Return type

ndarray

superpose(a, b)

Returns the superposition of a and b.

This is commonly elementwise addition.

Parameters

• a ((d,) ndarray) – Left operand in superposition.

• b ((d,) ndarray) – Right operand in superposition.

Returns

Superposed vector.

Return type

(d,) ndarray

bind(a, b)

Returns the binding of a and b.

The resulting vector should in most cases be dissimilar to both inputs.

Parameters

• a ((d,) ndarray) – Left operand in binding.

• b ((d,) ndarray) – Right operand in binding.

Returns

Bound vector.

Return type

(d,) ndarray

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Invert vector v.

A vector bound to its inverse will result in the identity vector.

VTB has a right inverse only.

Deprecated since version 1.2.0: Calling this method with the default sidedness=ElementSidedness.TWO_SIDED returns the right inverse for backwards compatibility, but has been deprecated and will be removed in the next major release.

Parameters

• v ((d,) ndarray) – Vector to invert.

• sidedness (ElementSidedness) – Must be set to ElementSidedness.RIGHT.

Returns

Right inverse of vector.

Return type

(d,) ndarray

binding_power(v, exponent)

Returns the binding power of v using the exponent.

The binding power is defined as binding (exponent-1) times bindings of v to itself.

Fractional binding powers are supported for “positive” vectors if SciPy is available.

Note the following special exponents:

• an exponent of -1 will return the inverse,

• an exponent of 0 will return the identity vector,

• and an exponent of 1 will return v itself.

Be aware that the binding power for the VTB algebra does not satisfy the usual properties of exponentiation:

• \(\mathcal{B}(v^a, v^b) = v^{a+b}\) does not hold,

• \((v^a)^b = v^{ab}\) does not hold.

Parameters

• v ((d,) ndarray) – Vector to bind repeatedly to itself.

• exponent (int or float) – Exponent of the binding power.

Returns

Binding power of v.

Return type

(d,) ndarray

See also

sign

get_binding_matrix(v, swap_inputs=False)

Returns the transformation matrix for binding with a fixed vector.

Parameters

• v ((d,) ndarray) – Fixed vector to derive binding matrix for.

• swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.

Returns

Transformation matrix to perform binding with v.

Return type

(d, d) ndarray

get_swapping_matrix(d)

Get matrix to swap operands in bound state.

Parameters

d (int) – Dimensionality of vector.

Returns

Matrix to multiply with a vector to switch left and right operand in bound state.

Return type

(d, d) ndarry

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Returns the transformation matrix for inverting a vector.

VTB has a right inverse only.

Deprecated since version 1.2.0: Calling this method with the default sidedness=ElementSidedness.TWO_SIDED returns the right transformation matrix for the right inverse for backwards compatibility, but has been deprecated and will be removed in the next major release.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness) – Must be set to ElementSidedness.RIGHT.

Returns

Transformation matrix to invert a vector.

Return type

(d, d) ndarray

implement_superposition(n_neurons_per_d, d, n)

Implement neural network for superposing vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• n (int) – Number of vectors to superpose in the network.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of length n of inputs to the network, and output the network output.

Return type

tuple

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)

Implement neural network for binding vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• unbind_left (bool) – Whether the left input should be unbound from the right input.

• unbind_right (bool) – Whether the right input should be unbound from the left input.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of the left and the right input in that order, and output the network output.

Return type

tuple

sign(v)

Returns the sign of v defined by the algebra.

The exact definition of the sign depends on the concrete algebra, but should be analogous to the sign of a (complex) number in so far that binding two vectors with the same sign produces a “positive” vector. There might, however, be multiple types of negative signs, where binding vectors with different types of negative signs will produce another “negative” vector.

Furthermore, if the algebra supports fractional binding powers, it should do so for all “non-negative” vectors, but not “negative” vectors.

If an algebra does not have the notion of a sign, it may raise a NotImplementedError.

Parameters

v ((d,) ndarray) – Vector to determine sign of.

Returns

The sign of the input vector.

Return type

AbstractSign

See also

AbstractAlgebra.abs

abs(v)

Returns the absolute vector of v defined by the algebra.

The exact definition of “absolute vector” may depend on the concrete algebra. It should be a “positive” vector (see sign) that relates to the input vector.

The default implementation requires that the possible signs of the algebra correspond to actual vectors within the algebra. It will bind the inverse of the sign vector (from the left side) to the vector v.

If an algebra does not have the notion of a sign or absolute vector, it may raise a NotImplementedError.

Parameters

v ((d,) ndarray) – Vector to obtain the absolute vector of.

Returns

The absolute vector relating to the input vector.

Return type

(d,) ndarray

absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

VTB has no absorbing element except the zero vector.

Always raises a NotImplementedError.

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the identity element of dimensionality d.

VTB has a right identity only.

Deprecated since version 1.2.0: Calling this method with the default sidedness=ElementSidedness.TWO_SIDED returns the right identity for backwards compatibility, but has been deprecated and will be removed in the next major release.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness) – Must be set to ElementSidedness.RIGHT.

Returns

Right identity element.

Return type

(d,) ndarray

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the negative identity element of dimensionality d.

VTB has a right negative identity only.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – Must be set to ElementSidedness.RIGHT.

Returns

Negative identity element.

Return type

(d,) ndarray

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the zero element of dimensionality d.

The zero element produces itself when bound to a different vector. For VTB this is the zero vector.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the zero element of the VTB algebra is two-sided.

Returns

Zero element.

Return type

(d,) ndarray

class nengo_spa.algebras.vtb_algebra.VtbSign(sign)

Bases: nengo_spa.algebras.base.GenericSign

Represents a sign in the VtbAlgebra.

The sign depends on the symmetry and positive/negative definiteness of the binding matrix derived from the vector. For all non-symmetric matrices, the sign is indefinite. It is also indefinite, if the matrices’ eigenvalues have different signs. A symmetric, positive (negative) definite binding matrix corresponds to a positive (negative) sign (equivalent to all eigenvalues being greater than 0, respectively lower than 0). If all eigenvalues are equal to 0, the sign is also 0.

to_vector(d)

Return the vector in the algebra corresponding to the sign.

Parameters

d (int) – Vector dimensionality.

Returns

Vector corresponding to the sign.

Return type

(d,) ndarray

class nengo_spa.algebras.vtb_algebra.VtbProperties

Bases: object

Vector properties supported by the VtbAlgebra.

UNITARY = 'unitary'

A unitary vector does not change the length of a vector it is bound to.

POSITIVE = 'positive'

A positive vector does not change the sign of a vector it is bound to.

A positive vector allows for fractional binding powers.

Transposed vector-derived transformation binding (TVTB)

TvtbAlgebra()	Transposed Vector-derived Transformation Binding (TVTB) algebra.
TvtbProperties()	Vector properties supported by the TvtbAlgebra.
TvtbSign(sign)	Represents a sign in the TvtbAlgebra.

class nengo_spa.algebras.tvtb_algebra.TvtbAlgebra

Bases: nengo_spa.algebras.base.AbstractAlgebra

Transposed Vector-derived Transformation Binding (TVTB) algebra.

TVTB uses elementwise addition for superposition. The binding operation \(\mathcal{B}(x, y)\) is defined as

\[\begin{split}\mathcal{B}(x, y) := V_y^T x = \left[\begin{array}{ccc} V_y'^T & 0 & 0 \\ 0 & V_y'^T & 0 \\ 0 & 0 & \ddots \end{array}\right] x\end{split}\]

with \(d'\) blocks

where

\[\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d - d' + 1} & y_{d - d' + 2} & \dots & y_d \end{array}\right]\end{split}\]

and

\[d'^2 = d.\]

The approximate inverse \(y^+\) for \(y\) is permuting the elements such that \(V_{y^+} = V_y^T\).

Note that TVTB requires the vector dimensionality to be square.

The TVTB binding operation is neither associative nor commutative. In contrast to VTB, however, TVTB has two-sided identities and inverses. Other properties are equivalent to VTB.

See also

VtbAlgebra

is_valid_dimensionality(d)

Checks whether d is a valid vector dimensionality.

For TVTB all square numbers are valid dimensionalities.

Parameters

d (int) – Dimensionality

Returns

True, if d is a valid vector dimensionality for the use with the algebra.

Return type

bool

create_vector(d, properties, *, rng=None)

Create a vector fulfilling given properties in the algebra.

Creating positive vectors requires SciPy.

Parameters

• d (int) – Vector dimensionality

• properties (set of str) – Definition of properties for the vector to fulfill. Valid set elements are constants defined in TvtbProperties.

• rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.

Returns

Random vector with desired properties.

Return type

ndarray

make_unitary(v)

Returns a unitary vector based on the vector v.

A unitary vector does not change the length of a vector it is bound to.

Parameters

v ((d,) ndarray) – Vector to base unitary vector on.

Returns

Unitary vector.

Return type

ndarray

superpose(a, b)

Returns the superposition of a and b.

This is commonly elementwise addition.

Parameters

• a ((d,) ndarray) – Left operand in superposition.

• b ((d,) ndarray) – Right operand in superposition.

Returns

Superposed vector.

Return type

(d,) ndarray

bind(a, b)

Returns the binding of a and b.

The resulting vector should in most cases be dissimilar to both inputs.

Parameters

• a ((d,) ndarray) – Left operand in binding.

• b ((d,) ndarray) – Right operand in binding.

Returns

Bound vector.

Return type

(d,) ndarray

binding_power(v, exponent)

Returns the binding power of v using the exponent.

The binding power is defined as binding (exponent-1) times bindings of v to itself.

Fractional binding powers are supported for “positive” vectors if SciPy is available.

Note the following special exponents:

• an exponent of -1 will return the inverse,

• an exponent of 0 will return the identity vector,

• and an exponent of 1 will return v itself.

The following relations hold for integer exponents:

• \(\mathcal{B}(v^a, v^b) = v^{a+b}\),

• \((v^a)^b = v^{ab}\).

(Technically, these relations also hold for positive unitary vectors, but the only such vector is the identity vector.)

Parameters

• v ((d,) ndarray) – Vector to bind repeatedly to itself.

• exponent (int or float) – Exponent of the binding power.

Returns

Binding power of v.

Return type

(d,) ndarray

See also

sign

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Invert vector v.

A vector bound to its inverse will result in the identity vector.

Parameters

• v ((d,) ndarray) – Vector to invert.

• sidedness (ElementSidedness) – This argument has no effect because the inverse of the TVTB algebra is two-sided.

Returns

Inverse of vector.

Return type

(d,) ndarray

get_binding_matrix(v, swap_inputs=False)

Returns the transformation matrix for binding with a fixed vector.

Parameters

• v ((d,) ndarray) – Fixed vector to derive binding matrix for.

• swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.

Returns

Transformation matrix to perform binding with v.

Return type

(d, d) ndarray

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Returns the transformation matrix for inverting a vector.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness) – This argument has no effect because the inverse of the TVTB algebra is two-sided.

Returns

Transformation matrix to invert a vector.

Return type

(d, d) ndarray

implement_superposition(n_neurons_per_d, d, n)

Implement neural network for superposing vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• n (int) – Number of vectors to superpose in the network.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of length n of inputs to the network, and output the network output.

Return type

tuple

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)

Implement neural network for binding vectors.

Parameters

• n_neurons_per_d (int) – Neurons to use per dimension.

• d (int) – Dimensionality of the vectors.

• unbind_left (bool) – Whether the left input should be unbound from the right input.

• unbind_right (bool) – Whether the right input should be unbound from the left input.

Returns

Tuple (net, inputs, output) where net is the implemented nengo.Network, inputs a sequence of the left and the right input in that order, and output the network output.

Return type

tuple

sign(v)

Returns the sign of v defined by the algebra.

The exact definition of the sign depends on the concrete algebra, but should be analogous to the sign of a (complex) number in so far that binding two vectors with the same sign produces a “positive” vector. There might, however, be multiple types of negative signs, where binding vectors with different types of negative signs will produce another “negative” vector.

Furthermore, if the algebra supports fractional binding powers, it should do so for all “non-negative” vectors, but not “negative” vectors.

If an algebra does not have the notion of a sign, it may raise a NotImplementedError.

Parameters

v ((d,) ndarray) – Vector to determine sign of.

Returns

The sign of the input vector.

Return type

AbstractSign

See also

AbstractAlgebra.abs

absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

TVTB has no absorbing element except the zero vector.

Always raises a NotImplementedError.

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the identity element of dimensionality d.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness) – This argument has no effect because the identity of the TVTB algebra is two-sided.

Returns

Identity element.

Return type

(d,) ndarray

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the negative identity element of dimensionality d.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the negative identity of the TVTB algebra is two-sided.

Returns

Negative identity element.

Return type

(d,) ndarray

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)

Return the zero element of dimensionality d.

The zero element produces itself when bound to a different vector. For VTB this is the zero vector.

Parameters

• d (int) – Vector dimensionality.

• sidedness (ElementSidedness, optional) – This argument has no effect because the zero element of the VTB algebra is two-sided.

Returns

Zero element.

Return type

(d,) ndarray

class nengo_spa.algebras.tvtb_algebra.TvtbSign(sign)

Bases: nengo_spa.algebras.base.GenericSign

Represents a sign in the TvtbAlgebra.

The sign depends on the symmetry and positive/negative definiteness of the binding matrix derived from the vector. For all non-symmetric matrices, the sign is indefinite. It is also indefinite, if the matrices’ eigenvalues have different signs. A symmetric, positive (negative) definite binding matrix corresponds to a positive (negative) sign (equivalent to all eigenvalues being greater than 0, respectively lower than 0). If all eigenvalues are equal to 0, the sign is also 0.

to_vector(d)

Return the vector in the algebra corresponding to the sign.

Parameters

d (int) – Vector dimensionality.

Returns

Vector corresponding to the sign.

Return type

(d,) ndarray

class nengo_spa.algebras.tvtb_algebra.TvtbProperties

Bases: object

Vector properties supported by the TvtbAlgebra.

UNITARY = 'unitary'

A unitary vector does not change the length of a vector it is bound to.

POSITIVE = 'positive'

A positive vector does not change the sign of a vector it is bound to.

A positive vector allows for fractional binding powers.