  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
'${\vee}{\wedge}\kern-1.5pt{\wedge}{\vee}\kern-1.5pt{\vee}{\wedge}$'. It
follows that we may need to know arbitrarily long past to decode a current
character, thus requiring long-term memory. Subsequently we constructed an RNN
capable of decoding sequences encoded in this manner. 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 3 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.
