The Reliability of Circuits in the Basis Anticonjunction with Constant Faults of Gates

We consider the realization of Boolean functions by asymptotically optimal reliable circuits with constant faults at the outputs of the gates in the basis {x|y} (where x|y п»„anticonjunction i.п»„. x|y = x&y). It is proved that almost all Boolean functions can be realized by asymptotically optimal reliable circuits that operate with unreliability asymptotically equal to 2ε0 + ε1 at ε0, ε1 → 0, where ε0 – probability of faults of type 0 at the output of basis gate, ε1 – probability of faults of type 1 at the output of basis gate.


Introduction
J.von Neumann [1] was the first who considered the problem of synthesis of reliable circuits from unreliable gates. He assumed that all gates of the circuit are exposed to inverse faults independently with probability ε (ε < 1/2), when functional gate with Boolean function e(x) in the faulty state realizes functionē(x)). Using an iterative method J.von Neumann established, that any Boolean function can be realized by a circuit at ε < 1/6 for which the error's probability at the output for any input set of variables is not greater than cε (c is the some constant, c depends from the basis). With a extension of iteration number the circuit complexity increases exponentially.
The circuit of unreliable gates is characterized by two important parameters: the probability of error at the circuit output (unreliability) and the complexity of the circuit. The optimization of circuit complexity was given the main attention in S.I.Ortyukov's [2], D.Ulig's [3] and some other authors' papers. The problem of constructing circuits that operate with the lowest (or near the lowest) probability of error was solved by M.A. Alekhina [4] on the assumption, that all circuit gates independently pass into faulty states, or only of the type 0 at the outputs (inputs) or only one type of 1 at the outputs (inputs).
In this paper we solve the problem of constructing asymptotically optimal reliable circuits in the basis {x|y} ( x|y = x&y), and in contrast to [4] a circuit gate can be independent of the other circuit gates pass into or faulty state type 0 on the output or faulty state type 1 on the output. Such faults of gates are considered the first, previously it was not studied.
Previously the problem of construction of asymptotically optimal reliable circuits in the basis {x|y} have been solved for various faults of gates (for example,at faults only of the type 0 at the outputs or only type of 1 at the outputs (briefly in [5], detail in [6]), at output inverse faults [7]) by the same method: at first it is constructed reliable circuits, then their reliability is improved by some circuit (that generally depends on the considered basis), and it is proved an upper bound for the unreliability of circuits. Then it is proved the lower bound for the unreliability of circuit realizing functions of a certain class K. If the obtained upper and lower bounds for the unreliability of circuits are asymptotically equal, then the functions of the class K not just reliable circuits are built , but circuits that are asymptotically optimal reliability.
The same method will be used in this work, however, this time its application is complicated by the presence of both parameters. The problem of constructing asymptotically optimal reliability circuits in which the gates are subjected to two types of faults with different probabilities, is considered the first time.

Concepts and Definitions
We consider the realization of Boolean functions by the circuits with unreliable functional gates in the basis {x|y}. We assume that the circuit with unreliable functional gates realizes the function f (x 1 , ..., x n ) (n ≥ 1) if at receipt on the circuit inputs the setã = (a 1 , ..., a n ) without faults in the circuit the value f (ã) appears on the circuit output. It is assumed that in each tact operation of the circuit at the outputs of all its gates can be independently constant faults or type 0 with probability ε 0 or type 1 with probability ε 1 (but not simultaneously). It is supposed ε 0 ∈ (0, 1/2) п»" ε 1 ∈ (0, 1/2).
Faults of type 0 on the outputs of gates are characterized by the fact that in good working order a functional gate realizes Boolean function x|y, and in a 52 The Reliability of Circuits in the Basis Anticonjunction with Constant Faults of Gates faulty constant it does 0 (zero).Faults of type 1 on the outputs of gates are characterized by the fact that in good working order a functional gate realizes Boolean function x|y, and in a faulty constant it does 1 (one).
We denote the probabilities of the occurrence 0 and 1 on the output of the gate with P 0 (E, (x 1 x 2 )), P 1 (E, (x 1 x 2 )) for the input set (x 1 x 2 ) of this gate.
Let P f (ã) (S,ã) be the probability of a appearance f (ã) on the output of the circuit S, realizing a Boolean function f (x) when the input set isã. Unreliability P (S) of the circuit S is defined as the maximum of the Obviously the unreliability P (E) of the basis gate E is equal to P (E) = max{ε 0 , ε 1 }. Denote ε = max{ε 0 , ε 1 }, than the reliability of the basis gate E is equal to 1 − ε.
Example. Consider a numerical example of calculating the unreliability of a circuit C (see picture 1).
Let value a 3 of the variable x 3 is equal to a 3 = 0. Then the correct value at the output of the circuit C is equal to 1, and the probability of correct value occurrence is not affected by working the basis gate E 1 . Then the probability of error P 0 (C,ã) is equal to P 0 (C, (a 1 , a 2 , a 3 )) = ε 0 Let value a 3 of the variable x 3 is equal to a 3 = 1.Then two cases are possible a) it is true a 1 &a 2 = 1 for value a 1 and a 2 of the variable x 1 and x 2 then it is true a 1 &a 2 = 0 for value a 1 and a 2 of the variable x 1 and x 2 then Thus the unreliability of the circuit C is equal to , where infimum is taken over all circuits S with unreliable gates realizing function f .
From the Theorem 1 follows the Theorem 2 if, instead α, β, δ, τ , we substitute in computed above error probability on the output basis gate.
From the Theorem 3 follows the Theorem 4 to prove that should instead µ substitute in max{ε 0 , ε 1 } = ε (the unreliability of the basis gate).

Theorem 4.
Any Boolean function f can be realized by such a circuit A that for any ε ∈ (0, 1/160] the following inequality holds P (A) ≤ 4ε.
From the Theorems 2 and 4 follows the Theorem 5.
Let the circuit S, realizing the boolean non-constant function f , is such that both of inputs of its output element E are connected to the output of some subcircuit B. By P 1 (B,ã) and P 0 (B,ã) denote the probabilities of errors at the output of circuit B.
Lemma 2. The probabilities of errors at the output of circuit S are equal to To prove this it is sufficient to calculate the probabilities of errors.
Let h(x) be arbitrary Boolean function,(x = (x 1 , ..., x n )), and K(n) be the set of Boolean functions of the type f (x) = ( Theorem 6. Let ε ≤ 1/8, f ̸ ∈ K, and S be any circuit realizing f . Then P (S) Proof. Let f be the function satisfying the conditions of Theorem 6, and S be any circuit its realizing. We single out in the circuit S the functional gate E 1 containing the output S. As the f ̸ ∈ K(n), there are two cases.
1. The outputs of gate E 1 are connected to the inputs different gates E 2 and E 3 .
1.1.The output of one gate, for instance E 2 , is connected to the input of gate E 3 . Compute the error probability p 1 on the output subcircuit consisting of gates E 1 , E 2 and E 3 and obtain It is equal to p 1 . By lemma 1 taking into account Remark 1 at ε ≤ 1/8 the following inequality holds P (S) ≥ a.
1.2. Neither a output of the gate E 2 is not connected to any of inputs of the gate E 3 , or a output gate E 3 is not connected to any of inputs of the gate E 2 . Compute the error probability p 1 on the output subcircuit consisting of gates E 1 , E 2 and E 3 and obtain It is equal to p 1 . By lemma 1 taking into account Remark 1 at ε ≤ 1/8 the following inequality holds P (S) ≥ a.
2. The outputs of the gate E 1 is connected to the output of one element E 2 . There are two cases.
2.1. The inputs of gate E 2 are connected to the outputs different gates E 1 and E 3 . Then (see paragraph 1 of the proof), the probability p 1 of error at the output of subcircurt consisting of gates E 2 , E 3 and E 4 is equal to It is equal to p 1 . By lemma 2 error probability p 0 at the output of subcircurt consisting of gates E 1 , E 2 , E 3 and E 4 is equal Taking into account that ε ∈ (0, 1/8] and ε = max{ε 0 , ε 1 }, we obtain By lemma 1 taking into account Remark 1 at ε ∈ (0, 1/8] the following inequality is true P (S) ≥ b ′ . As the b ′ ≥ b inequality holds P (S) ≥ b.
The theorem 6 is proved.
From the Lemma 1 follows that any circuit satisfying the conditions of Theorem 5, and realizing the Boolean function f ̸ ∈ K, is asymptotically optimal on reliability and operates with unreliability asymptotically equal to 2ε 0 + ε 1 at ε 0 , ε 1 → 0.
It is easy to verify that the number of functions in the class K(n) is not more than 2n2 2 n−1 , which is small in comparison with the total number of 2 2 n Boolean functions of n variables. Therefore, almost all Boolean functions in this basis can be realized asymptotically optimal reliable circuits that operate with unreliability asymptotically equal to 2ε 0 + ε 1 at ε 0 , ε 1 → 0.