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  Concept of Threshold Logic, Models of Computation

Threshold logic, perceptrons and artificial neural networks:

The threshold gate or cell is a logic element with multiple inputs which is characterized by its threshold value t. It counts or adds up the present input values and fires, that is, generates an output of 1, if the sum equivalents or surpasses t. The threshold gates are more helpful when they are generalized to comprise two types of inputs: excitatory and inhibitory inputs. The cell’s output fires if at least t excitatory inputs are on and no inhibitor is on. In this model that is used in the given illustrations all inhibitor has the veto power! (That is, a variant termed ‘subtractive inhibition’ adds input values with negative or positive sign).

We suppose synchronous operation, with a gate delay of 1 clock tick. The threshold gate with inhibitory inputs is a universal logic element: the figure below shows how to implement and, or, not gates; delay and memory elements.

1487_threshold logic.jpg

Note: Logic design: How to assemble the primitive elements to systems with a preferred behavior. The theory of computation mainly deals with ‘black box behavior’. However it insists that such black boxes can be built, at least in principle, from simplest possible building blocks. Therefore, the theory of computation comprises an intricate interplay between the abstract postulates and detailed ‘micro-programming].

Illustration: The black-box behavior of the circuit above is specified by finite state machine (that is, output is s):

2009_black box behaviour.jpg

Artificial neural nets applied to picture recognition:

We consider digital pictures that is, 2-d arrays of black and white pixels. We define the devices, termed as perceptrons, which recognize some classes of pictures which share some specific property, like ‘convex’ or ‘connected’. Perceptrons are the illustration of parallel computation which is based on joining local predicates. What geometric properties can be identified by these devices? The simple variant of ‘perceptrons’ termed as conjunctively local, is described in the figure below. Assume that f1... fn be predicates with a bounded fan-out, that is, each fi looks at ≤ k pixels. The perceptron employs AND as the threshold operator that means that the output F = AND (f1, ..., fn).

2062_artificial neural nets.jpg

Convexity can be recognized. Introduce an f for all triple a, b and c of points on a straight line, with b between a and c. Define f = 0 if a and c are black, b white; f = 1 or else.

Connectedness can’t be recognized by any perceptron of the bounded fan-out k. Proof by contradiction: let consider the 2 pictures above labeled C and ¬C which extend for > k pixels. When perceptron P can differentiate them, there must be an f* which yields f* = 0 on ¬C, and f* = 1 on C. As f* looks at ≤ k pixels, there is a pixel p in center row not looked by P. By blackening this pixel p, we modify ¬C into a joined picture C’. However f* ‘doesn’t notice’ this modify, keeps voting f* = 0, and causes F to output the wrong value on the C’.

Conclusion: The convexity can be found out as a cumulation of local characteristics, while connectedness can’t.

Note: Combinational logic alone is very weak. The universal model of computation requires unbounded memory and something such as feedback loops or recursion.

Illustration: Define an interesting model appropriate for the computing pictures on an infinite 2-d array of pixels. Read its properties. Is your model ‘universal’, in the sense that it is capable to calculate any ‘computable picture’?

Solution: Let N = {1, 2,..} be natural numbers. Consider the infinite pixel array N x N, and pictures P over array, defined as functions P: N x N -> {0, 1}. Let consider 4 primitive operations, each of which acquires an arbitrary row or column and color it with a single color, either 0 or 1: Set [row, column] k to [0, 1]. Characterize the class C of pictures produced by finite programs that is finite sequences of operations of the kind above. Give a decision procedure which decides whether a specific picture P is in C; and if it is, gives lower and upper bounds on the complexity of P, that is, on the length of its shortest program which produces P.

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