Abstract
We consider a problem of implementation of Boolean functions by self-correcting logic circuits of unreliable gates in different bases. A set of permitted faults for each gate is predefined, with no restrictions imposed, except that it should be non-void. The following statements are proved:
1) For any integer \(m\geq 3\) there is a basis consisting of Boolean functions of no more than \(m\) variables, in which any Boolean function can be implemented by a logic circuit of unreliable gates that self-corrects relative to certain faults in an arbitrary number of gates.
2) For any positive integer \(k\) there are bases consisting of Boolean functions of no more than two variables, in each of which any Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to certain faults in no more than \(k\) gates.
3) There is a functionally complete basis consisting of Boolean functions of no more than two variables, in which almost no Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to at least some faults in no more than one gate.
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Popkov, K.A. On Self-Correcting Logic Circuits of Unreliable Gates. Lobachevskii J Math 42, 2637–2644 (2021). https://doi.org/10.1134/S1995080221110172
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DOI: https://doi.org/10.1134/S1995080221110172