Problem in design?

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The series inductor-resistor combo is the relay coil equivalent. At first glance, large value resistors from the PNP bases to 5V would seem necessary to keep them turned off, but not much of anything can happen until deliberate turn-on of one of the NPNs occurs, after which niether base is floating.
 
After the relay has been pulsed in one direction (either direction) the base of each PNP transistor is connected in such a way that any leakage from them will turn them on more. This is basically called "floating."
 
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