DEEP Neural Networks

show how the brain is wired

## Advantages of FFDs

This is a brief summary of the advantages of FFDs. A more detailed description can be found in [3] in the **Related Articles** section.

A description of the advantages of electronic FFDs is in preparation.

The FFD architecture that minimizes the total cost of the number of gates, total connection length, and packing density is discussed in [3]. That architecture nearly minimizes each one of these cost functions separately. The architecture also has several other desirable properties.

**Computational complexity.** The FFDs' particular kind of fuzzy logic can identify the individual components and their relative strengths in mixed input patterns, such as the complex sensory receptor signals produced by mixtures of colors or odorants.

**Computation time.** Although the number of FFD outputs grows exponentially with the number of inputs (2^{n} outputs for n inputs), computation time grows linearly. The total time required to compute all 2^{n} outputs is the time it takes for a signal to pass through n – 1 AND NOT gates.

**Nonlinearities.** Nonlinearities in the AND NOT gates do not have serious adverse effects on the system. An accurate measure of the input difference is not necessary to generate useful fuzzy logic since such logic deals with largely subjective assignments of truth values. Useful fuzzy logic follows from the FFD interval measure property because each output increases as the length of the interval it measures increases. That is, as the component truth values move closer to the Boolean values that make the Boolean truth value of the conjunction true, this higher degree of truth is reflected in the FFD’s measure of the conjunction’s truth value.

**Information.** In spite of possible nonlinearities in the AND NOT function, FFD outputs retain all of the information in the inputs [2].

**Only one type of component.** The only component required is an AND NOT gate whose response provides a measure of the difference between the two input values.

**Resource requirements.** FFDs are capable of fuzzy logic as well as Boolean logic, and they require fewer components than conventional decoders that are only capable of Boolean logic. Fewer than five AND NOT gates per output are required regardless of the number of inputs. If not all of the 2n outputs are needed for the n inputs, fewer than 2.5 AND NOT gates per output may be needed. Unlike FFDs whose outputs share several AND NOT gates, conjunctions are constructed separately in conventional electronic decoders. This means the component requirement per output increases without bound as the number of inputs increases.

**Modular architecture.** The FFDs’ architecture is modular as a result of the recursive definitions.

**Connection length.** In a three-dimensional configuration, nearly all FFD connections are between adjacent AND NOT gates. This is a fortuitous feature of FFDs because minimum-connection-length configurations do not in general imply short connections. The connections are less efficient in a two-dimensional configuration but are still quite good.

**Packing density.** The packing density of an FFD in a three-dimensional configuration is more than two thirds of the maximum possible density. Like the short connections, dense packing is a fortuitous feature of FFDs because minimum-connection-length configurations do not in general imply high density.

**Layers.** In a three-dimensional configuration, FFDs have a limited number of layers. Five layers are sufficient for any number of inputs.

**Connection directions.** Since nearly all connections are either within layers or between adjacent layers and perpendicular to the layers, the architecture is conducive to a three-dimensional, layered construction. The simple connections and modular architecture simplify the assembly process.