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.

中文翻译：

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不可靠门自校正逻辑电路的研究

摘要

我们考虑通过不同基数的不可靠门的自校正逻辑电路来实现布尔函数的问题。每个门的一组允许的故障是预定义的，没有强加任何限制，只是它应该是非无效的。以下陈述得到证明：

1) 对于任何整数\(m\geq 3\)都有一个由不超过\(m\) 个变量的布尔函数组成的基，其中任何布尔函数都可以通过不可靠门的逻辑电路实现相对于任意数量的门中的某些故障进行更正。

2) 对于任何正整数\(k\)都有由不超过两个变量的布尔函数组成的基，其中每个布尔函数都可以通过不可靠门的逻辑电路来实现，该逻辑电路相对于某些故障进行自我纠正在不超过\(k\) 个门中。

3) 有一个由不超过两个变量的布尔函数组成的功能完备的基础，其中几乎没有布尔函数可以通过不可靠门的逻辑电路来实现，这些逻辑电路至少可以在不超过一个的时间内对某些故障进行自我纠正门。