# 1 Henry Selvaraj Henry Selvaraj Henry Selvaraj Henry

• Slides: 38

1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Binary Decision Diagrams A Binary Decision Diagram (BDD) is a graphical representation of a logic function, and often has a more compact representation than other methods. Thus, BDDs are very important in the design of logic networks using computers. Henry Selvaraj

2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; FHenry = A Selvaraj; + BC’ Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; If AHenry = 1 , Selvaraj; then f =Henry 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Let A = 0, Henry if B Selvaraj; = 0, then f = Selvaraj; 0. Henry Selvaraj; Henry Selvaraj; Let ASelvaraj; = 0, if B = 1, Selvaraj; if C = Henry 1 then. Selvaraj; f = 0 Henry Selvaraj; Henry Let ASelvaraj; = 0, if B = 1, Selvaraj; if C = Henry 0 then. Selvaraj; f = 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Example 0 A 1 1 B 0 1 0 C 1 0 0 1 Henry Selvaraj

3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; n Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henryn. Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; n+1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; In general, the complete binary decision tree for an n-variable function has 2 different paths, and there is one-to-one correspondence between the terminal nodes of BDDs and 2 elements of the truth table. The total number of nodes is 2 -1 Henry Selvaraj

4 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Binary Decision Diagrams (BDDs) BDDs are canonical, so if you correctly build the BDDs for two circuits, the two circuits are equivalent if and only if the BDDs are identical. This has led to significant breakthroughs in circuit optimization, testing and equivalence checking. Henry Selvaraj

5 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; BDDs are very effective in representing combinatorially large sets. This has led to breakthroughs in FSM (Finite State Machines) equivalence checking and in two level logic minimization. Henry Selvaraj

6 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1. taking complement is Selvaraj; difficult Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2. Taking AND of two or. Henry OR Selvaraj; of tro POS is Henry Selvaraj; Henry. SOP Selvaraj; Henry hard. Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Difficulties with SOP and POS representation • • EXOR representation is too big Changing from SOP to POS & vice versa is difficult (. . therefore, they Henry Selvaraj

7 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 1 1 2 2 1 1 0 0 3 3 1 0 0 1 Henry Selvaraj

8 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; a 1 f=a 0 . b 1 0 0 Henry Selvaraj b 1

9 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; a 0 f=a+b 1 b 0 1 1 Henry Selvaraj 0

10 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 2 Henry Selvaraj; Henry 2 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 1 Selvaraj; 0 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 Selvaraj; Henry Selvaraj; Henry 3 3 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 Selvaraj; Henry 1 Selvaraj; Henry Selvaraj; 1 Selvaraj; 0 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 Selvaraj; Henry 1 Selvaraj; Henry Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

11 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Two OBDDs G 1 Selvaraj; and G 2 are isomorphic if there exists Henry a one-to-one Henry Selvaraj; Henry Selvaraj; function σ from the vertices of G 1 onto the vertices of G 2 such that Henry Selvaraj; Henry Selvaraj; Henry v Selvaraj; Henry Selvaraj; for any vertex if σ(v) = w, then. Henry either both. Henry v and w are terminal Henry Selvaraj; Henry Selvaraj; vertices with value (v) = value (w), or both v and w are nonterminal Henry Selvaraj; Henry Selvaraj; Henry vertices with. Selvaraj; index. Henry (v) =Selvaraj; index. Henry (w), Selvaraj; σ (low (v)) = low (w)Selvaraj; and Henry Selvaraj; Henry Selvaraj; σ (high (v)) = high (w) Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Isomorphism between OBDDs Henry Selvaraj

12 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Every function is represented by a unique ROBDD for a given ordering of inputs to the function. Henry Selvaraj

Compliment 13 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 0 0 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; 2 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; Henry Selvaraj; Henry 1 Selvaraj; Henry 3 Selvaraj; 3 Henry Selvaraj; 0 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 0 1 Selvaraj; Henry Selvaraj; 0 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

14 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Complementing an ROBDD can be trivially accomplished by Interchanging the 0 and 1 terminal vertices. Henry Selvaraj

15 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Neural Selvaraj; Networks Henry Selvaraj; Henry is Selvaraj; Henry Selvaraj; Artificial (ANNs) an abstract simulation Henry Selvaraj; Henrynervous Selvaraj; system Henry Selvaraj; of a real that contains a collection of neuron Henry Selvaraj; Henry Selvaraj; units. Henry communicating with each. Henry other. Selvaraj; via axon connections. Henry Selvaraj; Such. Henry a model bears a strong resemblance to Henry axons. Selvaraj; and Henry Selvaraj; Henry Selvaraj; dendrites in a nervous system. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henryfundamental Selvaraj; Henrymodeling Selvaraj; Henry Selvaraj; The first of neural nets was. Selvaraj; proposed Henry Selvaraj; Henry and Selvaraj; Henry Selvaraj; in 1943 by. Selvaraj; Mc. Culloch Pitts. Henry in terms of Henry a computational Henry Selvaraj; Henry Selvaraj; model of "nervous activity". Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Artificial Neural Networks Henry Selvaraj

16 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Types of ANN • • The first is the biological type. It encompasses networks mimicking biological neural systems such as audio functions or early vision functions. The other type is application-driven. It depens less on the faithfulness to neurobiology. For these models the architectures are largely dictated by the application needs. Many such neural networks are represented by the so called connectionist models. Henry Selvaraj

17 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Neurons and the interconnection synapses constitute the key elements for neural information processing. Henry Selvaraj

18 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • It is estimated that the human brain contains over 100 billion neurons and synapses in the human nervous system. Studies of brain anatomy of the neurons indicate more than 1000 synapses on the input and output of each neuron. Note that, although the neuron's switch time (a few milliseconds) is about a million fold times slower than current computer elements, they have a thousand fold greater connectivity than today’s supercomputers. Henry Selvaraj

19 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Three parts of a neuron • a neuron cell body, • branching extensions called dendrites for receiving input, and • an axon that carries the neuron's output to the dendrites of other neurons. Henry Selvaraj

20 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • A neuron collects signals at its synapses by summing all the excitatory and inhibitory influences acting on it. If the excitatory influences are dominant, then the neuron fires and sends this message to other neurons via the outgoing synapses. In this sense, the neuron function can be modeled as a simple threshold function f(. ). The neuron fires if the combined signal strength exceeds a certain threshold value. Henry Selvaraj • activation function f(. )

21 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; A neuron model Henry Selvaraj

22 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry neural Selvaraj; Henry Selvaraj; hinges Henry Selvaraj; • Henry The strength of application-driven networks Henry Selvaraj; Henry Selvaraj; upon three main characteristics: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry Adaptiveness and self-organization: it offers robust and Henry Selvaraj; Henry Selvaraj; adaptive processing capabilities by. Selvaraj; adopting learning Henry Selvaraj; Henryadaptive Selvaraj; Henry Selvaraj; Henry Selvaraj; and self-organization rules. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; processing: Henry Selvaraj; • Henry Nonlinear network it enhances the. Selvaraj; network's Henry Selvaraj; Henry Selvaraj; approximation, classification capabilities. Henry Selvaraj; and Henrynoise-immunity Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry Parallel processing: it usually employs a large number of Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; processing cells enhanced by extensive interconnectivity. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Application-Driven Neural Networks Henry Selvaraj

23 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henryneuron Selvaraj; Henry Selvaraj; • An. Selvaraj; artificial is a device with. Selvaraj; many. Henry inputs and. Henry one Selvaraj; Henry Selvaraj; Henry Selvaraj; output. The neuron has two modes of operation; the training Henry Selvaraj; Henry Selvaraj; mode, Henry Selvaraj; mode and Henry the using In the. Henry training the neuron can Henry Selvaraj; Henry Selvaraj; be trained to fire (or not), for particular input patterns. In the Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; using mode, when a taught input. Henry pattern is Henry detected at. Henry the input, Henry Selvaraj; Henry Selvaraj; its associated output becomes the current output. If the input Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; pattern does not belong in. Selvaraj; the taught list of. Henry input. Selvaraj; patterns, the Henry Selvaraj; Henry Selvaraj; firing rule is used to determine whether to fire or not. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; An engineering approach Henry Selvaraj

24 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Selvaraj; firing Henry Selvaraj; Henry Selvaraj; • Henry A simple rule can be. Selvaraj; implemented by. Henry using Hamming Henry Selvaraj; Henry Selvaraj; distance technique: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry Take. Selvaraj; a collection of training patterns for a node, some. Henry of which Henry Selvaraj; Selvaraj; Henry(the Selvaraj; Henry Selvaraj; and Henryothers Selvaraj; which Henry Selvaraj; cause it to fire 1 -taught set of. Henry patterns) Henry Selvaraj; Henry Selvaraj; prevent it from (the 0 -taught set). Then the patterns not Henry Selvaraj; Henry doing Selvaraj; so Henry Selvaraj; Henry Selvaraj; to. Henry Selvaraj; in the. Selvaraj; collection cause. Henry the node fire. Selvaraj; if, on Henry comparison , they Henry Selvaraj; Henry Selvaraj; have more input elements common with the 'nearest' pattern in Henry Selvaraj; Henryin Selvaraj; Henry Selvaraj; Henrythe Selvaraj; Henry Selvaraj; the 1 -taught set Selvaraj; than with 'nearest' pattern. Henry in the 0 -taught set. Henry Selvaraj; Henry Selvaraj; If. Henry there is a. Henry tie, then the pattern remains in the undefined state. Selvaraj; Henry Selvaraj; Firing rules Henry Selvaraj

25 Example Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; is. Henry Selvaraj; Henrythe Selvaraj; Henry Selvaraj; • ASelvaraj; 3 -input neuron taught to output 1 when input (X 1, X 2 Henry Selvaraj; Henry Selvaraj; and. Selvaraj; X 3) is 111 or 101 and to output 0 when the. Selvaraj; input. Henry is 000 or Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 001. Then, before applying the. Henry firing rule, Henry the Selvaraj; truth table is: Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; X 1: Selvaraj; Henry 0 0 Henry Selvaraj; 0 0 Selvaraj; 1 Henry Selvaraj; 1 1 Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; X 2: 0 0 1 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; X 3: Selvaraj; Henry 0 1 Henry Selvaraj; 0 1 Selvaraj; 0 Henry Selvaraj; 1 0 Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry OUT: Selvaraj; Henry 0 0 Henry Selvaraj; 0/1 Henry Selvaraj; 1 0/1 Selvaraj; 1 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

26 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Simple ANN Henry Selvaraj

27 Simple neuron Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

28 A simple neurons Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

29 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; PERCEPTRONS • The architecture of a Perceptron consists of a single input layer of many neurodes, and a single output layer of many neurodes. The simple "networks" illustrated at the beginning, to produce logical "AND" and "OR" operations have a Perceptron architecture. But to be called a Perceptron, the network must also implement the Perceptron learning rule for weight adjustment. Henry Selvaraj

30 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj

31 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • In Backprop, a third neurode layer is added (the hidden layer) and the discrete thresholding function is replaced with a continuous (sigmoid) one. But the most important modification for Backprop is the generalized delta rule, which allows for adjustment of weights leading to the hidden layer neurodes in addition to the usual adjustments to the weights leading to the output layer neurodes. Using the generalize delta rule to adjust the weights leading to the hidden units is Henry Selvaraj backpropagating the error-adjustment.

32 Backprob Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry a. Selvaraj; Each Henry unit in the network has. Henry connections other. Henry units. Selvaraj; in Selvaraj; Henryto. Selvaraj; the. Selvaraj; network. Each connection from. Selvaraj; unit i. Henry to unit j is. Henry weighted Henry Selvaraj; Henry Selvaraj; Henry. W(i, j). Selvaraj; Henry Selvaraj; with weight Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry b. Each unit requires an activation function which takes as Henry Selvaraj; Henry Selvaraj; input the. Henry sum. Selvaraj; of all. Henry inputs into. Henry the Selvaraj; unit (weighted by. Henry their. Selvaraj; Henry Selvaraj; respective weights). The. Selvaraj; output of Selvaraj; the activation function is Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; 1 Henry Selvaraj; Henrydoesn�t Selvaraj; usually a value between and 0 (unit fires/ Henry Selvaraj; Selvaraj; step function is used. Henry a lot in multilayer networks but. Henry for back Henry Selvaraj; Henry Selvaraj; propagation we will have to determine the derivative of the Henry Selvaraj; Henry Selvaraj; activation function that will. Henry not Selvaraj; work. Henry Sigmoid a better Henry Selvaraj; so Henry Selvaraj; is. Henry Selvaraj; activation function derivate is relatively simple. Henry Selvaraj; and Henryits Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry Selvaraj; Henry Selvaraj; • Henry Sigmoid(x) 1/(1+e(-x)) Selvaraj; Henry=Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj f

33 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Method of Back Propagation • a. To do back propagation we take a network with arbitrary weights to start and we begin by applying one of our training examples to the network. We calculate network all the way to the output. • b. Once we have an output we calculate the Err(i) vector representing the error between our desired value and the actual output. Henry Selvaraj

34 Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry c. Then for each output unit i we update the weights of the Henry Selvaraj; Henry Selvaraj; connections Unit to the Henry unit Selvaraj; i as so: Henry Selvaraj; Henryfrom Selvaraj; Henryj Selvaraj; • Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry W(j, i) = Henry W(j, i) + *ai * Selvaraj; err(i) *Henry g(input(i)) Selvaraj; Henry Selvaraj; • Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry Where Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry ai Selvaraj; = the Henry output of i. Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • Henry Input(i) the. Selvaraj; sum of all Selvaraj; inputs. Henry to unit(i) Selvaraj; = Henry Selvaraj; weighted Henry Selvaraj; appropriately Henry Selvaraj; Henry Selvaraj; Henry • Henry g =Selvaraj; the derivative of. Henry the Selvaraj; activation function of. Selvaraj; unit(i)Henry Selvaraj; Henry Selvaraj; • Henry err(i) = the error defined above. Henry Selvaraj; Selvaraj; Henry Selvaraj; Henryas Selvaraj; Henry Selvaraj; • Henry = the learning constant (passed a parameter to the Henry Selvaraj; Henry Selvaraj; supervised learning algorithm. All this does is control how Henry Selvaraj; Henry Selvaraj; much of Henry the error is. Henry taken into Henry consideration update Henry Selvaraj; Henrywhen Selvaraj; we Henry Selvaraj; the weight. For example =0. 1 will not change W(j, i) as much as =0. 5). Henry Selvaraj

35 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • After we update all the weights to the output layer of the network we can propagate the error back a layer. The error for a unit(j) in this layer is just the sum of all the errors of the units(i) that unit(j) connects to weighted by the weight of the connection. • • So, err(j) = W(j, i)*err(i) • (And now we can update the weights Henry Selvaraj connecting to unit(j) just as before).

36 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; • e. Next we repeatedly propagate the error back using equation 1 until all weights have been updated. • f. Now that we have updated the network to this training example we can start over and apply another training example. We should repeat examples over and over again until the weights of the network converge. Henry Selvaraj

37 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Before Decomposition After Decomposition Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; EX Time[s] Errors 9 syn Henry Selvaraj; Henry Selvaraj; 391 0 68 0 Henry Selvaraj; Henry adr 4 Selvaraj; Henry Selvaraj; 1186 Henry Selvaraj; 0 208 Selvaraj; 0 Henry Selvaraj; Henry Bbtas Henry Selvaraj; 21 Henry Selvaraj; 0 19 0 Henry Selvaraj; beect Selvaraj; Henry Selvaraj; Henry 72 0 48 0 Henry Selvaraj; Henry bulm Selvaraj; Henry Selvaraj; 8 Henry Selvaraj; 0 12 0 Henry Selvaraj; Henry Selvaraj; clip Henry Selvaraj; Henry 2238 Henry Selvaraj; 642 612 Selvaraj; 0 Henry Selvaraj; ex 7 Selvaraj; Henry Selvaraj; Henry 156 0 38 0 Henry Selvaraj; Henry misex 1 Selvaraj; Henry Selvaraj; 841 0 162 0 Henry Selvaraj; Henry Selvaraj; opus 5931 Henry Selvaraj; 262 430 Selvaraj; 0 Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Neural network Henry Selvaraj

38 NN Results continued. . Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj; Before Decomposition After Decomposition Henry Selvaraj; Henry Selvaraj; Selvaraj; Henry Selvaraj; Time. Henry [s] Errors Henry Selvaraj; Time [s] Errors Henry Selvaraj; Henry Selvaraj; EX Henry Selvaraj; Henry Selvaraj; rd 53 17 0 9 0 Henry Selvaraj; Henry Selvaraj; 233 Henry Selvaraj; 0 29 0 Selvaraj; Henry Selvaraj; rd 73 Henry Selvaraj; rd 84 Henry Selvaraj; Henry Selvaraj; 1200 5 53 0 Henry Selvaraj; Henry Selvaraj; 1200 Henry Selvaraj; 5 661 0 Selvaraj; Henry Selvaraj; root Henry Selvaraj; s 8 Henry Selvaraj; Henry Selvaraj; 165 0 26 0 Henry Selvaraj; Henry Selvaraj; 4004 106 586 0 Henry Selvaraj; sao 2 Henry Selvaraj; Henry Selvaraj; sqrt 8 Henry Selvaraj; Henry 470 Henry Selvaraj; 0 52 0 Selvaraj; Henry Selvaraj; Henry Selvaraj; 432 0 67 0 Henry Selvaraj; xor 5 Henry Selvaraj; Henry Selvaraj; z 4 Henry Selvaraj; Henry 108 Henry Selvaraj; 0 56 0 Selvaraj; Henry Selvaraj; Henry Selvaraj; Henry Selvaraj