Learning a circuit by injecting values

We propose a new model for exact learning of acyclic circuits using experiments in which chosen values may be assigned to an arbitrary subset of wires internal to the circuit, but only the value of the circuit’s single output wire may be observed. We give polynomial time algorithms to learn (1) arbitrary circuits with logarithmic depth and constant fan-in and (2) boolean circuits of constant depth and unbounded fan-in over AND, OR, and NOT gates. Thus, both AC0 and NC1 circuits are learnable in polynomial time in this model. Negative results show that some restrictions on depth, fan-in and gate types are necessary: exponentially many experiments are required to learn AND/OR circuits of unbounded depth and fan-in; it is NP-hard to learn AND/OR circuits of unbounded depth and fan-in 2; and it is NP-hard to learn circuits of bounded depth and unbounded fan-in over AND, OR, and threshold gates, even when the target circuit is known to contain at most one threshold gate and that threshold gate has threshold 2. We also consider the effect of adding an oracle for behavioral equivalence. In this case there are polynomial-time algorithms to learn arbitrary circuits of constant fan-in and unbounded depth and to learn boolean circuits with arbitrary fan-in and unbounded depth over AND, OR, and NOT gates. A corollary is that these two classes are PAC-learnable if experiments are available. Finally, we consider an extension of the model called the synchronous model. We show that an even more general class of circuits are learnable in this model. In particular, we are able to learn circuits with cycles.