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We consider the problem of learning an acyclic discrete circuit with n wires, fan-in bounded by k and alphabet size s using value injec- tion queries. For the class of transitively reduced circuits, we develop the Distinguishing Paths Algorithm, that learns such a circuit using (ns)O(k) value injection queries and time polynomial in the number of queries. We describe a generalization of the algorithm to the class of circuits with shortcut width bounded by b that uses (ns)O(k+b) value injection queries. Both algorithms use value injection queries that fix only O(kd) wires, where d is the depth of the target circuit. We give a reduction showing that without such restrictions on the topology of the circuit, the learning problem may be computationally intractable when s = n(1), even for circuits of depth O(log n). We then apply our large- alphabet learning algorithms to the problem of approximate learning of analog circuits whose gate functions satisfy a Lipschitz condition. Finally, we consider models in which behavioral equivalence queries are also available, and extend and improve the learning algorithms of (7) to handle general classes of gate functions that are polynomial time learnable from counterexamples.

Learning large-alphabet and analog circuits with value injection queries