The current paper is a study in Recurrent Neural Networks (RNN), motivated by the lack of examples simple enough so that they can be thoroughly understood theoretically, but complex enough to be realistic. We constructed an example of structured data, motivated by problems from image-to-text conversion (OCR), which requires long-term memory to decode. Our data is a simple writing system, encoding characters 'X' and 'O' as their upper halves, which is possible due to symmetry of the two characters. The characters can be connected, as in some languages using cursive, such as Arabic (abjad). The string 'XOOXXO' may be encoded as '∨∧∧∨∨∧'. It is clear that seeing a sequence fragment '|∧∧∧∧∧|' of any length does not allow us to decode the sequence as '…XXX…' or '…OOO …' due to inherent ambiguity, thus requiring long-term memory. Subsequently we constructed an RNN capable of decoding sequences like this example. Rather than by training, we constructed our RNN “by inspection,” i.e., we guessed its weights. This involved a sequence of steps. We wrote a conventional program which decodes the sequences as the example above. Subsequently, we interpreted the program as a neural network (the only example of this kind known to us). Finally, we generalized this neural network to discover a new RNN architecture whose instance is our handcrafted RNN. It turns out to be a three-layer network, where the middle layer is capable of performing simple logical inferences; thus the name “deductron.” It is demonstrated that it is possible to train our network by simulated annealing. Also, known variants of stochastic gradient descent (SGD) methods are shown to work.
2010 Mathematics Subject Classification: 92B20, 68T05, 82C32.