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This paper describes a new learning by example mechanism and its application for digital circuit design automation. This mechanism uses finite state machines to represent the inferred models or designs. The resultant models are easy to be implemented in hardware using current VLSI technologies. Our simulation results show that it is often possible to infer a well-defined deterministic model or design from just one sequence of examples. In addition this mechanism is able to handle sequential task involving long-term dependence. This new learning by example mechanism is used as a design by example system for automatic synthesis of digital circuits. Such systems have not previously been successfully developed mainly because of the lack of mechanism to implement them. From artificial neural network research, it seems possible to apply the knowledge gained from learning by example to form a design by example system. However, one of the problems with neural network approaches is that the resultant models are very difficult to be implemented in hardware using current VLSI technologies. By using the mechanism described in this paper, the resultant models are finite state machines that are well suited for digital designs. Several sequential circuit design examples are simulated and tested. Although our test results show that such a system is feasible for designing simple circuits or small-scale circuit modules, the feasibility of such a system for large-scale circuit design remains to be showed. Both the learning mechanism and the design method show potential and the future research directions are provided.