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I need to know for each of the $2^{2^3}$ boolean functions with $3$ inputs the smallest boolean circuit made only of NAND gates computing it (smallest in terms of the number gates).

I would be glad if someone could tell me a source where I can look this up or a clever procedure to find the minimal circuits myself.

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For 3 inputs it's not $2^3$ functions but $2^{2^3}$ functions. Circuit minimization is generally hard. You could try using the aiger package http://fmv.jku.at/aiger/, which will give you a circuit but not necessarily minimal.

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