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To construct of multi-layer network in a FPGA, we discuss simplified network constructions. We reexamined neuron functions for back-propagation learning. We made some improvements for the functions, but couldn't achieve drastic reduction. Therefore, we abandoned back-propagation learning, and proposed a new neural network, named as AND/OR-neural network, which is derived from the disjunctive normal-form of logical expressions. The network is defined in the binary logic only and has a conclusive learning, and can be implemented in a small size of FPGA. However, since it has not prediction, we expand it to multi-valued type. The extension is accomplished approximately by replacements of logical operators. We discussed the property, and implemented the multi-valued AND/OR-network in a 20,000 gates FPGA, and we solved 7-dimensional exclusive-OR problem in the microsecond level.