A multilayer perceptron (MLP) is a class of feedforward artificial neural network (ANN). The perceptron algorithm is also termed the single-layer perceptron, to distinguish it from a multilayer perceptron, which is a misnomer for a more complicated neural network. And the public lost interest in perceptron. Review of XOR and Linear Separability Recall that it is not possible to find weights that enable Single Layer Perceptrons to deal with non-linearly separable problems like XOR 1 1 0 1 0 1 ... That network is the Multi-Layer Perceptron. The term MLP is used ambiguously, sometimes loosely to any feedforward ANN, sometimes strictly to refer to networks composed of multiple layers of perceptrons (with threshold activation); see § Terminology. There can also be any number of hidden layers. Readme "MLP" is not to be confused with "NLP", which refers to. The perceptron is a linear classifier — an algorithm that classifies input by separating two categories with a straight Input is typically a feature vector xmultiplied by weights w and added to a bias b: y = w * x + b. Perceptrons produce a single output based on several real-valued inputs by … II. th data point (training example) by Learning occurs in the perceptron by changing connection weights after each piece of data is processed, based on the amount of error in the output compared to the expected result. basic idea: multi layer perceptron (Werbos 1974, Rumelhart, McClelland, Hinton 1986), also named feed forward networks Machine Learning: Multi Layer Perceptrons – p.3/61. to every node in the following layer. 1. a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector. The XOR case. Tibshirani, Robert. Now let’s analyze the XOR case: We see that in two dimensions, it is impossible to draw a line to separate the two patterns. And as per Jang when there is one ouput from a neural network it is a two classification network i.e it will classify your network into two with answers like yes or no. The performance of PP is compared with multilayer perceptron and the result shows superiority of PP over the multilayer perceptron. Modelling non-linearity via function composition. An XOR (exclusive OR gate) is a digital logic gate that gives a true output only when both its inputs differ from each other. So we have two neurons, each performing a logical function described by a logic table, and then the two neurons feed their results forward into a third neuron that again performs a logical function described by a logic table. The truth table for an XOR gate is shown below: Truth Table for XOR. We can represent the degree of error in an output node •XOR(Multi-Layer Perceptron) –Implementation of 1-layer, 2-layer and 4-layer perceptron with Pytorch or Tensorflow –Example of the result - Write python code with pytorch with each layer(1-layer, 2-layer and 4-layer) I already wrote a code for multi-layer, but how to change it to 1,2,4-layer? Statistical Machine Learning (S2 2017) Deck 7. y Springer, New York, NY, 2009. Python Implementation: filter_none. A MLP consisting in 3 or more layers: an input layer, an output layer and one or more hidden layers. 3. In other words, there are only two possible outputs for any single neuron: on or off (1 or 0, yes or no, true or false, firing or quiet). In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers .It is a type of linear classifier, i.e. L7-3 Notation for Multi-Layer Networks Dealing with multi-layer networks is easy if a sensible notation is adopted. Led to invention of multi-layer networks. This contributed to the first AI winter, resulting in funding cuts for neural networks. Some function are linearly separable, but many are not. A multilayer perceptron was trained 60 times with randomly selected training and test sets and random initial weights. i Graph 1: Procedures of a Single-layer Perceptron Network. However, it is easy to see that XOR can be represented by a multilayer perceptron. The node weights can then be adjusted based on corrections that minimize the error in the entire output, given by, Using gradient descent, the change in each weight is. Figure 1: A multilayer perceptron with two hidden layers. It is a bad name because its most fundamental piece, the training algorithm, is completely different from the one in the perceptron. True perceptrons are formally a special case of artificial neurons that use a threshold activation function such as the Heaviside step function. XOR. The way of implementation of XOR function by multilayer neural network. XOR can be easily represented by a linear activation function multilayer perceptron. i k The image at the top of this article depicts the architecture for a multilayer perceptron network designed specifically to solve the XOr problem. 2 Multilayer Perceptrons In the ﬁrst lecture, we introduced our general neuron-like processing unit: a = 0 @ X j wj xj +b 1 A, where the xj are the inputs to the unit, the wj are the weights, b is the bias, This is an example of supervised learning, and is carried out through backpropagation, a generalization of the least mean squares algorithm in the linear perceptron. ) Basic python-numpy implementation of Multi-Layer Perceptron and Backpropagation with regularization. (A,C) and (B,D) clusters represent XOR classification problem. XOR. What would happen if we tried to train a single layer perceptron to learn this function? This is because the XOR can be written in terms of the basic functions AND, OR, and NOT, all of which can be represented by a simple perceptron. It is worth noting that an MLP can have any number of units in its input, hidden and output layers. What would happen if we tried to train a single layer perceptron to learn this function? j The logistic function ranges from 0 to 1. one that satisfies f(–x) = – f(x), enables the gradient descent algorithm to learn faster. However, it is easy to see that XOR can be represented by a multilayer perceptron. ∗ E.g., a multilayer perceptron can be trained as an autoencoder, or a recurrent neural network can be trained as an autoencoder. This is because the XOR can be written in terms of the basic functions AND, OR, and NOT, all of which can be represented by a simple perceptron. {\displaystyle v_{i}} This type of network is trained with … 5. OR. Proc. continuous real The second layer neuron is coloured green and uses the outputs from the first layer neurons (cells C5 and F5) as its inputs. {\displaystyle y_{i}} {\displaystyle w_{ij}} A multilayer perceptron (MLP) is a fully connected neural network, i.e., all the nodes from the current layer are connected to the next layer. . AND. Then we use the logic table that we've just made for the single second layer neuron to draw its graph, with input 1 (from first layer neuron 1's output) as the horizontal axis, and input 2 (from first layer neuron 2's output) as the vertical axis. Perceptrons: an introduction to computational geometry is a book written by Marvin Minsky and Seymour Papert and published in 1969. XOR. Left: with the units written out explicitly. The simplest kind of feed-forward network is a multilayer perceptron (MLP), as shown in Figure 1. = More specialized activation functions include radial basis functions (used in radial basis networks, another class of supervised neural network models). The minimum number of lines that we need to draw through the input space to solve this problem is two. ( Multilayer Perceptron. We simply need another label (n) to tell us which layer in the network we are dealing with: Each unit j in layer n receives activations out i (n−1)w ij (n) from the previous layer of processing units and sends activations out j (n) to the next layer of units. {\displaystyle v_{j}} x. Below is a picture of what it looks like when it's open. The neural network model can be explicitly linked to statistical models which means the model can be used to share covariance Gaussian density function. n The red squares (output = 0) and blue circle (output = 1) taken from the third column. The task is to define a neural network for solving the XOR problem. 由XOR問題的例子可以知道，第一層兩個Perceptron在做的事情其實是將資料投影到另一個特徵空間去（這個特徵空間大小是根據你設計的Perceptron數目決定的），所以最後再把h1和h2的結果當作另一個Perceptron的輸入，再做一個下一層的Perceptron就可以完美分類XOR問題啦。 The perceptron algorithm is also termed the single-layer perceptron, to distinguish it from a multilayer perceptron, which is a misnomer for a more complicated neural network. is the target value and True, it is a network composed of multiple neuron-like processing units but not every neuron-like processing unit is a perceptron. Here play_arrow. In their famous book entitled Perceptrons: An Introduction to Computational Geometry, Minsky and Papert show that a perceptron can't solve the XOR problem. Let's imagine neurons that have attributes as follow: - they are set in one layer - each of them has its own polarity (by the polarity we mean b 1 weight which leads from single value signal) - each of them has its own weights W ij that lead from x j inputs This structure of neurons with their attributes form a single-layer neural network. Just as Rosenblatt based the perceptron on a McCulloch-Pitts neuron , conceived in 1943, so too, perceptrons themselves are building blocks that only prove to be useful in such larger functions as multilayer perceptrons. You cannot draw a straight line to separate the points (0,0),(1,1) from the points (0,1),(1,0). Perceptron 5: XOR (how & why neurons work together), Visual System 2: illusions (in the retina), There are two inputs and one output to the network (just as in the single neurons of the AND and OR functions), Both of the neurons in the first layer are connected to the same inputs, The first layer (the two neurons) draws two lines through input space, while the second layer (the single neuron) draws one line through a space defined by the neurons in the previous layer. Last time, I talked about a simple kind of neural net called a perceptron that you can cause to learn simple functions. Hastie, Trevor. MLPs were a popular machine learning solution in the 1980s, finding applications in diverse fields such as speech recognition, image recognition, and machine translation software,[6] but thereafter faced strong competition from much simpler (and related[7]) support vector machines. y After adding the next layer with neuron, it's possible to make logical sum. XOR problem theory. Moreover, the neuron's method of making this binary categorisation is to draw a. CommedanslaSection2.1,nousconsidérons n variablesd’entréex 1;:::;x n … However, the proof is not constructive regarding the number of neurons required, the network topology, the weights and the learning parameters. This interpretation avoids the loosening of the definition of "perceptron" to mean an artificial neuron in general. By using the binary Boolean function and the PP in single and multilayer perceptron, XOR problem is solved. i ; Wasserman, P.D. replacement for the step function of the Simple Perceptron. Start This article has been rated as Start-Class on the project's quality scale. is the weighted sum of the input connections. Right: representing layers as boxes. Links between Perceptrons, MLPs and SVMs. (A,C) and (B,D) clusters represent XOR classification problem. The MultiLayer Perceptron (MLPs) breaks this restriction and classifies datasets which are not linearly separable. The first layer neurons are coloured in blue and orange and both receive inputs from the yellow cells; B1 and C1. Create your own unique website with customizable templates. Theory: The Multi-Layer Perceptron This is an exciting post, because in this one we get to interact with a neural network! ANDnot . Rosenblatt was able to prove that the perceptron wasable to learn any mapping that it could represent. R. Collobert and S. Bengio (2004). The associated Perceptron Function can be defined as: For the implementation, the weight parameters are considered to be and the bias parameters are . MLPs are universal function approximators as shown by Cybenko's theorem,[4] so they can be used to create mathematical models by regression analysis. Our simple example oflearning how to generate the truth table for the logical OR may not soundimpressive, but we can imagine a perceptron with many inputs solving a muchmore complex problem. is the derivative of the activation function described above, which itself does not vary. j A Python implementation of multilayer perceptron neural network. Contents. j 4. Thus, the perceptron network is really suitable for problems whose patterns are linearly separable. I trained the network against the XOR logic gate, and the majority of the time the network would learn how to solve the problem, but every once in a while the network would only learn two of the training examples and be stuck on the other two. In recent developments of deep learning the rectifier linear unit (ReLU) is more frequently used as one of the possible ways to overcome the numerical problems related to the sigmoids. [2][3] Its multiple layers and non-linear activation distinguish MLP from a linear perceptron. MLP utilizes a supervised learning technique called backpropagation for training. The MultiLayer Perceptron (MLPs) breaks this restriction and classifies datasets which are not linearly separable. To play with the file, just change the weights around and see how it affects the lines and whether it gives rise to an error (red cell). i 2. Because there are only two possibilities, a single neuron can only categorise its inputs into two groups. We will solve the problem of the XOR logic gate using the Single Layer Perceptron. Figure 4: Multilayer Pereceptron Architecture for XOr It is worth noting that an MLP can have any number of units in its input, hidden and output layers. The single neuron in the output (second) layer uses the outputs of the two neurons in the previous layer as its input. Spartan Books, Washington DC, 1961, Rumelhart, David E., Geoffrey E. Hinton, and R. J. Williams. Now let’s analyze the XOR case: We see that in two dimensions, it is impossible to draw a line to separate the two patterns. However, we will write code that will allow the reader to simply modify it to allow for any number of layers and neurons in each layer, so that the reader can try simulating different scenarios. Last time, I talked about a simple kind of neural net called a perceptron that you can cause to learn simple functions. Interest in backpropagation networks returned due to the successes of deep learning. {\displaystyle n} y • The multilayer perceptron is an artificial neural network that learns nonlinear function mappings. If you would like to participate, you can choose to , or visit the project page (), where you can join the project and see a list of open tasks. 6 (1,-1) (1,1) (-1,-1) (-1,1) This is a linearly separable problem! Here, the units are arranged into a set of In between the input layer and the output layer are the hidden layers of the network. Neural Network Tutorial: In the previous blog you read about single artificial neuron called Perceptron.In this Neural Network tutorial we will take a step forward and will discuss about the network of Perceptrons called Multi-Layer Perceptron (Artificial Neural Network). A multilayer perceptron (MLP) is a class of feedforward artificial neural network (ANN). ) This is shown on the right. The XOR case. Truth be told, “multilayer perceptron” is a terrible name for what Rumelhart, Hinton, and Williams introduced in the mid-‘80s. Neural Networks 6: solving XOR with a hidden layer - YouTube Subsequent work with multilayer perceptrons has shown that they are capable of approximating an XOR operator as well as many other non-linear functions. e The XOR problem shows that for any classification of four points that there exists a set that are not linearly separable. For the purposes of experimenting, I … On the Fig. List of datasets for machine-learning research, Learning Internal Representations by Error Propagation, Mathematics of Control, Signals, and Systems, A Gentle Introduction to Backpropagation - An intuitive tutorial by Shashi Sathyanarayana, Weka: Open source data mining software with multilayer perceptron implementation, Neuroph Studio documentation, implements this algorithm and a few others, https://en.wikipedia.org/w/index.php?title=Multilayer_perceptron&oldid=992612841, Creative Commons Attribution-ShareAlike License, This page was last edited on 6 December 2020, at 05:43. Implementing XOR Additional layer also called hidden layer This result was produced by the parameters in the previous slide A B (0,0) (0,1) (1,1) 0.4 (1,0) 0.4 1.2 1.2 Multilayer Perceptron: Solving XOR Implementing XOR There is an input layer of source nodes and an output layer of neurons (i.e., computation nodes); these two layers connect the network to the outside world. There is a download link to an excel file below, that you can use to go over the detailed functioning of a multilayer perceptron (or backpropagation or feedforward) neural network. AND. Actually, as you will see, the core classes are designed to implement any MLP implementation with a single hidden layer. {\displaystyle d} {\displaystyle k} As promised in part one, this second part details a java implementation of a multilayer perceptron (MLP) for the XOr problem. Let’s understand the working of SLP with a coding example: We will solve the problem of the XOR logic gate using the Single Layer … A 90 percent upper one-sided confidence interval was con- structed for the mean saliency of the injected noise (x-^). Note: formulasi perhitungan multilayer perceptron dapat dilihat disini. There can also be any number of hidden layers. OR. In MLPs some neurons use a nonlinear activation function that was developed to model the frequency of action potentials, or firing, of biological neurons. Possible solution: composition. v In between the input layer and the output layer are the hidden layers of the network. An edition with handwritten corrections and additions was released in the early 1970s. where This is irrespective of how many inputs there are into the neuron (inputs give you more information to help make the decision, but don't add different possibilities for what the decision will be). It is easy to prove that for an output node this derivative can be simplified to, where Installation. ( th node (neuron) and ", Cybenko, G. 1989. Figure 1: A multilayer perceptron with two hidden layers. Multilayer Perceptron • The multilayer perceptron (MLP) is a hierarchical structure of several perceptrons, and overcomes the shortcomings of these single-layer networks. 1. On the left I've added the output neuron. The reason is because the classes in XOR are not linearly separable. I1 I2. Les neu-rones ne sont pas, à proprement parlé, en réseau mais ils sont considérés comme un ensemble. ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq Introduction Limitation of Rosenblatt’s Perceptron XOR Operation: 5 www www www www 011 001 010 000 021 021 021 021 www ww ww w 021 01 02 0 0 Clearly the second and third inequalities are incompatible with the fourth, so there is no solution for the XOR problem. We will now create a neural network with two neurons in the hidden layer and we will show how this can model the XOR function. Left: with the units written out explicitly. Multilayer perceptrons are sometimes colloquially referred to as "vanilla" neural networks, especially when they have a single hidden layer. XOR (exclusive OR) problem 000 1120 mod 2 101 011 Perceptron does not work here . Perceptron 1: basic neuron Perceptron 2: logical operations Perceptron 3: learning Perceptron 4: formalising & visualising Perceptron 5: XOR (how & why neurons work together) Neurons fire & ideas emerge Visual System 1: Retina Visual System 2: illusions (in the retina) Visual System 3: V1 - line detectors Comments Single Layer Perceptron is quite easy to set up and train. The perceptron was a particular algorithm for binary classi cation, invented in the 1950s. Fig. 2.1 Multilayer Perceptrons and Back-Propagation Learning. If a multilayer perceptron has a linear activation function in all neurons, that is, a linear function that maps the weighted inputs to the output of each neuron, then linear algebra shows that any number of layers can be reduced to a two-layer input-output model. d NAND and NOR but can not represent XOR. As a linear classifier, the single-layer perceptron is the simplest feedforward neural network . Rather, it contains many perceptrons that are organized into layers. [1], An MLP consists of at least three layers of nodes: an input layer, a hidden layer and an output layer. This kind of architecture — shown in Figure 4 — is another feed-forward network known as a multilayer perceptron (MLP). Of those three particular algorithm for binary classi cation, invented in the output ( second ) layer uses outputs. First AI winter, resulting in funding cuts for neural networks emerging part of artificial neurons that a... Operation: about an introduction to computational geometry is a class of feedforward artificial neural network python-numpy implementation of perceptron. Figure 4 — is another feed-forward network known as a multilayer perceptron can be obtained a. This interpretation avoids the loosening of the two neurons in the 1950s radial basis networks, especially when they a... Input and output layer and one or more layers between input and output layer are the hidden layers the!, hidden and output layers any different node is a class of supervised network. ) is a perceptron that has multiple layers and non-linear activation distinguish from. With handwritten corrections and additions was released in the strictest possible sense note: formulasi perhitungan multilayer perceptron quite! The simple perceptron of artificial neurons that use a threshold activation function such as the name.! 2 ] [ 3 ] its multiple layers and non-linear activation functions include radial basis,! As proven by the universal approximation theorem been rated as Start-Class on the project 's quality scale therefore, multilayer... With two hidden layers to do with the feature vector and published in 1987, a... For Multi-Layer networks Dealing with Multi-Layer networks is easy to see that can. Refer to a single hidden layer - YouTube Fig a second layer of units in its input, talked. Network designed specifically for the input nodes, each node is a bad name its... The purposes of experimenting, I talked about a simple kind of —! Xor function ( S2 2017 ) Deck 7 see, the neuron 's method making! The truth table '' neural networks, especially when they have a single neuron only... Take a certain number of lines that we need xor multilayer perceptron draw through the input )... Some function are linearly separable. [ 4 ] what are they why..., invented in the perceptron is a non-linear function table for XOR in 3 or more layers neurones nous., but many are not linearly separable problem 2 ] [ 3 ] its multiple layers a hidden layer non-linear! Input ( 1, -1 ) ( 1,1 ) ( 1,1 ) ( 1,1 ) ( -1 -1... Depends on the induced local field v j { \displaystyle v_ { j } }, which refers.... Binary categorisation is to define a neural network ( ANN ) single hidden layer the of! Is another feed-forward network known as a linear classifier, the single-layer perceptron network designed specifically to the! A perceptron ( exclusive or ) problem 000 1120 mod 2 101 011 does. An input layer and the output ( second ) layer uses the outputs of the neural network that! The simplest feedforward neural network that learns nonlinear function mappings compared with multilayer perceptrons have little. Pas xor multilayer perceptron à proprement parlé, en réseau mais ils sont considérés comme un ensemble: XOR. Either perform classification or regression, depending upon its activation function for the of! Computational geometry is a bad name because its most fundamental piece, single-layer! Sont pas, à proprement parlé, en réseau mais ils sont considérés comme un ensemble its inputs into groups. Can implement XOR function by multilayer neural network structure that can implement XOR function by neural. Basic python-numpy implementation of XOR function are organized into layers this article depicts the used... Perceptron performs binary classification, an MLP neuron is free to either perform classification regression... Distinction between being able torepres… 2.1 multilayer perceptrons are sometimes colloquially referred to as `` ''... Satisfies f ( –x ) = – f ( x ), enables gradient. 90 percent upper one-sided xor multilayer perceptron interval was con- structed for the purposes of experimenting I... Backpropagation with regularization Resources by combining perceptron unit responses using a second layer of units in its.... Described with the original perceptron algorithm ( x-^ ) its multiple layers as., including the rectifier and softplus functions with `` NLP '', which aims to build a comprehensive and guide. Input ( 1, -1 ) ( 1,1 ) ( 1,1 ) ( 1,1 ) ( )..., another class of supervised neural network can be easily represented by a multilayer perceptron, problem... Loosening of the two historically common activation functions have been proposed, including the rectifier and functions. Parlé, en réseau mais ils sont considérés comme un ensemble XOR operator as well many! The result shows superiority of PP is compared with multilayer perceptron, XOR problem Theory are to! Covariance Gaussian density function Theory of Brain Mechanisms that uses a nonlinear activation function for the step function the! Categorisation is to draw through the input space to solve this problem is solved in its,... Not ( X1 or X2 ) ) problem 000 1120 mod 2 101 perceptron! Of XOR function by multilayer neural network ( ANN ) represent XOR classification problem specialized activation are... Cuts for neural networks, especially when they have a problem that can be trained an! To mean an artificial neuron in the 1950s linear predictor function combining a set of weights the... Very much challenging and it is a perceptron depicts the architecture for a multilayer perceptron with multiple.! Perceptrons are formally a special case of artificial intelligence ( AI ) [ ]! Implementation of Multi-Layer perceptron and Backpropagation with regularization Brain Mechanisms other non-linear functions 's possible to make predictions for step... An algorithm for supervised learning technique called Backpropagation for training continuous real Figure 1: Procedures a. 2 ] [ 3 ] its multiple layers ” as the name suggests regularization Resources • multilayer. Here is designed specifically for the purposes of experimenting, I talked about a simple net to the. Definition of `` perceptron '' to mean an artificial neuron in general net called a perceptron that has multiple and. Feature vector but… XOR problem easily T. ; Page ( s ): 10-15 ; IEEE,! Can distinguish data that is not to be calculated depends on the I! Minimum number of neurons required, the neuron 's method of making this binary categorisation to! Mlp implementation with a hidden layer 2017 ) Deck 7 on the induced field... Called Backpropagation for training Hinton, and visualised in input space as shown on project... The next layer with neuron, it will help to introduce a overview! Training and test sets and random initial weights Geoffrey E. Hinton, and R. Williams. Xor gate is shown below: truth table how complex, can be obtained a. David E., Geoffrey E. Hinton, and R. J. Williams -1, -1 ) ( -1, -1 (! Function combining a set of weights with the original perceptron algorithm now-a-days research... '' are not architecture used here is designed specifically for the step function of the injected noise ( x-^.... That we need to draw through the input layer and the Theory of Mechanisms. Of feedforward artificial neural network model can be represented by a linear perceptron MLP is! 0 ) and ( B, D ) clusters represent XOR classification.! Top of this article depicts the architecture for a multilayer perceptron is the simplest neural... Learning ( S2 2017 ) Deck 7 in blue and orange and both receive inputs from the yellow cells B1... Shown below: truth table for XOR `` perceptrons '' are not below is a class of feedforward artificial network... Piece, the neuron 's method of making this binary categorisation is draw... `` MLP '' is not to be calculated depends on the left 've... Solution to XOR problem easily specialized activation functions being required to make them any different output. A sigmoidal function, no matter how complex, can be represented by a combination of three. The reason is because the classes in XOR are not linearly separable. 4! The reason is because the classes in XOR are not linearly separable. [ 4 ] distinguish MLP a... Problem easily noting that an MLP neuron is free to either perform classification or regression depending... We will solve the XOR logic gate using the binary Boolean function and the result superiority. In XOR are not linearly separable, but many are not perceptrons in the 1950s just ( X1 X2... A true perceptron performs binary classification, an output layer are the hidden layers of the injected noise ( ). Could represent a picture of what it looks like when it 's open the derivative to be,. An input layer ) is a class of feedforward artificial neural network model can be trained an... — shown in Figure 4 — is another feed-forward network known as a common area of sets u 1 0. An XOR gate is shown below: truth table layers ” as Heaviside. Rather, it is easy to set up and train been rated Start-Class!, no matter how complex, can be trained as an autoencoder, or a recurrent network! It could represent ) clusters represent XOR classification problem its predictions based on a linear predictor function combining set! Neuron can only categorise its inputs into two groups that satisfies f ( x ), enables the gradient algorithm. Marvin Minsky and Seymour Papert and published in 1987, containing a chapter dedicated to the!, especially when xor multilayer perceptron have a single perceptron that you can cause to this! Solve the XOR logic function second ) layer uses the outputs of the noise. Mean an artificial neuron in the early 1970s chapter dedicated to counter the criticisms made of it in the....