You are given an OR problem and an XOR problem to solve. Then, which one of…
2012
You are given an OR problem and an XOR problem to solve. Then, which one of the following statements is true?
Answer: D. OR problem can be solved using a single layer perceptron and XOR problem can be solved using a radial basis function. — ConceptA single-layer perceptron applies a threshold to a weighted sum of its inputs, so the boundary it draws between the two output classes is a single…
- A.
Both OR and XOR problems can be solved using a single layer perceptron.
- B.
OR problem can be solved using a single layer perceptron and XOR problem can be solved using self organizing maps.
- C.
OR problem can be solved using a radial basis function and XOR problem can be solved using a single layer perceptron.
- D.
OR problem can be solved using a single layer perceptron and XOR problem can be solved using a radial basis function.
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Show answer & explanation
Correct answer: D
Concept
A single-layer perceptron applies a threshold to a weighted sum of its inputs, so the boundary it draws between the two output classes is a single hyperplane — one straight line in two dimensions. It can therefore represent exactly those Boolean functions that are linearly separable.
A radial basis function (RBF) network first passes the inputs through localized basis functions, typically Gaussians centred at chosen points, and then combines those hidden activations linearly. This hidden mapping lifts the data into a space in which classes that no single straight line could separate become linearly separable.
A self-organizing map (SOM) is trained by unsupervised competitive learning and produces a topology-preserving grid of cluster centres; it fits the structure of the input distribution rather than a supervised mapping onto labelled target outputs.
Application
Tabulate OR on the four input pairs: it is 0 at (0, 0) and 1 at (0, 1), (1, 0) and (1, 1).
One straight line, x1 + x2 = 0.5, keeps (0, 0) on one side and the other three points on the other side. OR is therefore linearly separable, and a single-layer perceptron with weights w1 = w2 = 1 and threshold 0.5 realises it.
Tabulate XOR on the same four input pairs: it is 1 at (0, 1) and (1, 0), and 0 at (0, 0) and (1, 1).
The two 1-points lie on one diagonal of the unit square and the two 0-points on the other diagonal. Any straight line placing both 1-points on the same side also places at least one 0-point there, so XOR is not linearly separable and no single-layer perceptron can implement it.
Place two Gaussian basis units centred at (0, 1) and (1, 0). Each XOR-true input strongly activates one of them, while neither XOR-false input strongly activates either, so a linear output unit over those two hidden activations separates the classes. An RBF network therefore implements XOR.
Cross-check
Contrast the competing pairings by what each one claims:
Assigning XOR to a single-layer perceptron contradicts step 4 — no single hyperplane separates the XOR classes.
Assigning XOR to a self-organizing map asks an unsupervised clustering network to reproduce a labelled Boolean target, which competitive learning does not fit.
Assigning OR to a radial basis function network and XOR to a single-layer perceptron reverses the two capabilities: the linearly separable problem is the one a single hyperplane can handle.
Hence OR is solved by a single-layer perceptron and XOR by a radial basis function network.