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We can view the perceptron as representing a hyperplane decision surface in the n-dimensional space of instances (ie, points) The perceptron outputs a 1 for instances lying on one side of the hyperplane and outputs a -1 for instances lying on the other side, as illustrated in Figure 43 The equation for this decision hyperplane is iir i! = 0 Of course, some sets of positive and negative examples cannot be separated by any hyperplane Those that can be separated are called linearly separable sets of examples

A single perceptron can be used to represent many boolean functions For example, if we assume boolean values of 1 (true) and -1 (false), then one way to use a two-input perceptron to implement the AND function is to set the weights wo = -3, and wl = wz = 5 This perceptron can be made to represent the OR function instead by altering the threshold to wo = -3 In fact, AND and OR can be viewed as special cases of m-of-n functions: that is, functions where at least m of the n inputs to the perceptron must be true The OR function corresponds to rn = 1 and the AND function to m = n Any m-of-n function is easily represented using a perceptron by setting all input weights to the same value (eg, 05) and then setting the threshold wo accordingly Perceptrons can represent all of the primitive boolean functions AND, OR, NAND ( 1 AND), and NOR ( 1 OR) Unfortunately, however, some boolean functions cannot be represented by a single perceptron, such as the XOR function whose value is 1 if and only if xl # xz Note the set of linearly nonseparable training examples shown in Figure 43(b) corresponds to this XOR function The ability of perceptrons to represent AND, OR, NAND, and NOR is important because every boolean function can be represented by some network of interconnected units based on these primitives In fact, every boolean function can be represented by some network of perceptrons only two levels deep, in which

You should now be able to test your application on your iPhone. Back in Xcode, set the correct SDK and device (at the time of writing, this is iPhone Device 3.1.2). See Figure 15 26.

The decision surface represented by a two-input perceptron ( a ) A set of training examples and the decision surface of a perceptron that classifies them correctly (b) A set of training examples that is not linearly separable (ie, that cannot be correctly classified by any straight line) xl and x2 are the Perceptron inputs Positive examples are indicated by "+", negative by "-"

STUDIUM Studiumaufnahme an der Universit t des Saarlandes in den Hauptf chern Germanistik und Anglistik und in dem Nebenfach Amerikanistik 1994 1996 1996 1997 DAAD-Stipendiat am Ohio University in Athens, Ohio, USA 1 Staatsarbeit: ,,Water Imagery in James Joyce s Ulysses 1 Staatsexamen: Note, sehr gut 2 Staatsarbeit: ,,The Grotesque in Flannery O Connor s Wise Blood and Selected Short Stories 2 Staatsexamen: Note, sehr gut Referendariat am Gymnasium am Krebsberg in Neunkirchen/Saarland

the inputs are fed to multiple units, and the outputs of these units are then input to a second, final stage One way is to represent the boolean function in disjunctive normal form (ie, as the disjunction (OR) of a set of conjunctions (ANDs) of the inputs and their negations) Note that the input to an AND perceptron can be negated simply by changing the sign of the corresponding input weight Because networks of threshold units can represent a rich variety of functions and because single units alone cannot, we will generally be interested in learning multilayer networks of threshold units

While you are here, change the Active Configuration to Release. You re nearly done. Now just Build and Run. Assuming all went well with installing your certificate, you will see a prompt asking for access to it. Click on Always Allow (unless you want to see this every time!). The status bar will tell you that Show Message is wending its way to your iPhone. The application will start up on your iPhone, as you can see from the screenshot in the Organizer window (see Figure 15 27).