I just looked up deterministic finite automata and I don't think that, in general, they are a good fit for support vector machines.
I am also not sure that you have described your problem that well. It appears that your input data is variable in length. Is there a limit? DFA's can operate with sequences of arbitrary length. The number of elements in the input vector of an SVM is finite and fixed.
*IF* there is a maximum limit to the input string size in your problem, then you could define a third state for each element which means "no data". Let's say that you never wanted to look at input strings longer than ten binary digits. Give your SVM a ten-element input of floating-point numbers. If your string is "0110", train and test "0110xxxxxx", where x is distinct from both 0 and 1. If your string is "011101110", encode it as "011101110xx".
You would use this same strategy you were training an artificial neural network (ANN) architecture.
You may be interested in recurrent neural networks (RNN), the machine learning approach which is designed for sequential data streams. I am just starting to study them myself. The RNN architecture that I am finding most interesting is the "long short term memory" (LSTM) architecture.