Circuit complexity of symmetric Boolean functions in antichain basis

Consider the set of gates E ⊆ {e₁, …, e_s}. Consider an arbitrary input α ∈ {0, 1}ⁿ of the circuit S. Let er1,…,erl be all circuit gates such that for any j ∈ [l]

1. hrj(α)=1;

2. erj∈{e1,…,es}\E.

We call the gate e_mfirst non-zero gate on the tupleαwith respect to the set E if it has the minimal index among the gates er1…,erl.

Sometimes when we discuss a first non-zero gate on a given input we will not specify with respect to what set it is considered as far as it will be clear from the context.

We construct the chain in such a way that for any i after the i-th step the following properties are fulfilled.

1. For all t, p ≤ i: F_t ∩ T_p = ∅ (i.e. we can not put a variable to be equal to 1 and later change it to 0; it was mentioned earlier informally).

2. For any gate e_j ∈ {e₁, …, e_i}\E_i and for a function h_j computed on the output of the gate e_j the following equality takes place:

   hj(x[n]\Fi)=hj(xTi)=0.

Property 2 plays a crucial role in the construction process. Informally, this property means that if we input the top and the bottom tuples of a current subcube after the i-th step to the circuit inputs then all circuit gates except the gates from the set E_i output 0 on these two tuples.

There are two stages in the construction of the chain.

Stage 1. The stage consists of s steps, by the number of circuit gates. Suppose i steps are done, we describe the (i + 1)-th step.

At the end of the i-th step the first i circuit gates e₁, e₂, …, e_i are considered. We have constructed a certain set F_i which includes the indices of the input variables that we have put be equal to 0. Similarly, we have constructed a set T_i consisting of the indices of the input variables that we have put be equal to 1. The chain C contains the tuples 0, 1 ∈ {0, 1}ⁿ as well as the tuples that we have obtained simultaneously with constructing F_i and T_i. We have also constructed a set E_i. Now we consider the (n − |F_i| − |T_i|)-dimensional subcube with the top and the bottom tuples x[n]\Fi and xiT respectively.

Starting the (i + 1)-th step, we proceed to the gate e_i+1. All possible cases are considered below. Note they can not occur simultaneously.

1. The inputs of the gate e_i+1 are connected directly to none of the free circuit inputs with respect to the set F_i ∪ T_i. Let E_i+1 = E_i ∪ {e_i+1}, F_i+1 = F_i, T_i+1 = T_i. The (i + 1)-th step is finished.

2. An input of the gate e_i+1 is connected directly to a free circuit input with respect to the set F_i ∪ T_i. Let a variable x_m, where m ∈ [n] \ (F_i ∪ T_i), be assigned to this circuit input. Let us verify whether Property 2 is true for the gate e_i+1, i.e. the equation hi+1(x[n]\Fi)=hi+1(xTi)=0 takes place.

   1. If Property 2 is true, then we put E_i+1 = E_i, F_i+1 = F_i, T_i+1 = T_i. The (i + 1)-th step is finished.

   2. Assume Property 2 for the gate e_i+1 is not true for the top tuple of the current subcube (the bottom tuple case will be described later). That is, hi+1(x[n]\Fi)=1. For convenience let us divide this case into two substeps.

      1. At first we start with the gate e_i+1. By the condition of the case 2.2, the gate e_i+1 is a first non-zero one on the tuple x[n]\Fi with respect to the set E_i. Let the above-mentioned variable x_m be equal to 0.

         We will use some auxiliary notations: Fi+1r,Ei+1r and later Ti+1r, where r is a natural parameter.

         For r = 1 let Fi+11=Fi∪{m}. Let also T_i+1 = T_i. That is, we proceed to a subcube of a smaller dimension n−|Fi+11|−|Ti+1|. We add the top tuple x[n]\Fi+11 of this subcube to the chain C. Then we put Ei+11 be equal to the union of the set E_i and the set of gates e_j, where j ⩽ i + 1, such that when we put the value of the variable x_m to be equal to some value, their inputs will be no longer connected to any free circuit inputs with respect to the set Fi+11∪Ti+1.

      2. Further, we proceed to the gates e_j, where j ⩽ i + 1. At first, we still consider the value of r to be equal to 1.

         Consider the gates e_j, where j ⩽ i + 1 and ej∉Ei+1r. Let us verify whether Property 2 is true for them on the tuple x[n]\Fi+1r. Suppose we found gates such that Property 2 does not take place on them. Let e_l be a gate with the minimal index among these gates.

         Lemma 3

         At least one input of the above-mentioned gate e_lis connected directly to a free circuit input with respect to the setFi+1r∪Ti+1.

         Proof. For the function h_l computed on the output of the gate e_l we have hl(x[n]\Fi+1r)=1, hl(xTi+1)=0. The value of the antichain function g_l corresponding to the gate e_l is determined by values on some circuit inputs as well as by values on the gates of two types:

         1. gates e_u such that eu∉Ei+1r, where u < l,

         2. gates e_u such that eu∈Ei+1r, where u < l.

            Values on the outputs of the 2-type gates are determined by values on the outputs of the 1-type gates. Since l is minimal, all 1-type gates e_j output 0 on the top and the bottom tuples of the subcube. Therefore, 2-type gates output the same values on these two tuples. That is, the function g_l outputs two different values on a pair of tuples with the following property: in these tuples the components corresponding to the values of gates with indices smaller than l are equal. Therefore, the values of some other components in these tuples should be different. That is, there exists a free circuit input with respect to the set Fi+1r∪Ti+1 and it is connected directly to an input of the gate e_l. Lemma 3 is proved. □

            Let us proceed with an argument. By Lemma 3, the gate e_l is a first non-zero one on the tuple x[n]\Fi+1r with respect to the set Ei+1r. Consider any free circuit input with respect to the set Fi+1r∪Ti+1 such that it is connected directly with some input of the gate e_l. Put a variable assigned to this input to be equal to 0. That is, we add the index of this variable to the set Fi+1r. The set obtained we denote by Fi+1r+1. That is, we proceed to the subcube of a smaller dimension. We add to the chain C the top tuple of this subcube x[n]\Fi+1r+1.

            Let Ei+1r+1 be equal to the union of the set Ei+1r and the set of all gates e_j, where j ⩽ i + 1, such that their inputs are no longer connected directly to any circuit inputs free with respect to the set Fi+1r+1∪Ti+1.

         Further, we start a cycle: we perform the process described in 2.2.2 for all r = 2, …, q, where q is a value of the parameter r such that Property 2 is true for all gates e_j, where j ∈ [i + 1] and ej∉Ei+1q. Note an important property that will be used further.

         Lemma 4

         Any gate e_p, where p ∈ [i + 1], such that during the process described in 2.2 it was added to the set Ei+1r, where r ⩽ q, outputs 0 on all tuples of the (n−|Fi+1r|−|Ti+1|)-dimensional subcube on which gates e_a, where a < p, ea∉Ei+1r, output 0.

         Proof. No inputs of the gate e_p are connected to circuit inputs free with respect to the set Fi+1r∪Ti+1. That is, this gate outputs the same value on all above-mentioned tuples. Since T_i+1 = T_i, it outputs 0 on the bottom tuple of the subcube. Hence, this gate also outputs 0 on all tuples described. Lemma 4 is proved. □

         After all above actions are performed we put Fi+1=Fi+1q,Ei+1=Ei+1q. The (i + 1)-th step is finished. Now we consider the last possible case.

   3. Let Property 2 for the gate e_i+1 be not true on the bottom tuple of the subcube: hi+1(xTi)=1.

      Lemma 5

      Cases 2.2 and 2.3 cannot occur simultaneously.

      Proof. Suppose the opposite. For all gates e_j ∉ E_i, where j < i + 1, functions h_j output 0 on the tuples x[n]\Fi and xTi. For all gates e_j ∈ E_i with j < i + 1 functions h_j output the same on both tuples. The tuples x[n]\Fi and xTiare comparable. By the conditions of the case 2, there is an input of the gate e_i+1 which is connected to a circuit input such that it has different values in these two tuples. Hence, the function g_i+1 outputs 1 on a pair of comparable tuples that differ at least in one component. This contradicts the fact that g_i+1 is an antichain function. Lemma 5 is proved. □

      Now we return to the Case 2.3. It is symmetrical to the Case 2.2. So, we proceed similarly with the only difference that we let free variables be equal to ones instead of zeros. First, put a variable x_k be equal to 1 and Ti+11=Ti∪{k} (using the notation introduced in the Case 2.2). Put F_i+1 = F_i. That is, we proceed to a subcube of a dimension n − |F_i+1| − |T_i+1|. We add the tuple xTi+1 to the chain C. Then, similarly to the Case 2.2, we perform the cycle described in 2.2.2 to construct respectively the sets Ti+12,Ti+13 and so on, until for some v we obtain that Property 2 is true for all gates e_j, with j ⩽ i + 1 such that ej∉Ei+1v. Then we put Ti+1=Ti+1v,Ei+1=Ei+1v. The (i + 1)-th step is finished.

If as a result of Stage 1 the values of all input variables became fixed, then the chain C consisting of n + 1 tuples will be constructed and we finish the process.

Suppose we have performed s steps described in the Cases 1 and 2 of Stage 1. That is, we have considered all s gates of the circuit S but the chain is still not constructed. After the s-th step we has obtained the sets F_s, T_s and E_s. For simplicity, from now on we omit indices and write F, T and E respectively. Now we proceed to Stage 2.

Stage 2. The original n-dimensional cube is restricted to a subcube of a dimension n − |F| − |T| such that all the circuit gates e₁, …, e_s, except those from the set E, output 0 on the top tuple x^[n]\F and on the bottom one x^T of this subcube. Let T⁰ = T. Then we add to T⁰ the indices of variables from the set [n] \ (F ∪ T). That is, we obtain the sets T¹, T² and so on as follows:

1. if i ⩾ 0 and there is no first non-zero gate on the tuple xTi with respect to the set E, then we add to the set Tⁱ the index of any variable from the set [n] \ (F ∪ Tⁱ). We obtain a set Tⁱ⁺¹ and a tuple xTi+1 which we add to the chain;

2. if i ⩾ 0 and there exists a first non-zero gate on the tuple xTi with respect to the set E, then, similarly to the proof of Lemma 3, it is easy to show that there exists an input of this gate connected to a circuit input free with respect to the set [n] \ (F ∪ Tⁱ). We put the free input variable corresponding to that circuit input to be equal to 1. Thus we obtain a set Tⁱ⁺¹ and a new tuple xTi+1 which we add to the chain.

If during the above process when we add a variable to the set Tⁱ and obtain a gate e_d such that its inputs are no longer connected to circuit inputs free with respect to the set F ∪ Tⁱ, then we put E = E ∪ {e_d}. Clearly, all such gates have a property similar to one from Lemma 4. Note that each time we consider a first non-zero gate with respect to a current set E.

We repeat the above process until all free input variables became fixed to some values, i.e. until we construct the chain required. Note that in Stage 2 we choose the set T to fix the idea, we could also choose the set F and put variables to be equal to zeros respectively.

When constructing the chain C we feed to the circuit inputs only tuples of types x^T and x^[n]\F (for brevity, indices are omitted). These tuples are comparable, since on each step T ⊆ [n] \ F. Thus, the set C is indeed a chain.

None of the gates of the circuit S occur to be first non-zero (with respect to the corresponding sets) twice on the tuples of the chain C.

Proof. The proof is by contradiction. Without loss of generality that assume that the tuple x^p is added to the chain on the step b and the tuple x^p′ is added on the step c, where b < c. On these steps the sets E_b and E_c are constructed. Moreover, E_b ⊆ E_c. Assume there is an index t ∈ [s] such that the gate e_t is a first non-zero gate on the tuples x^p and x^p′ with respect to the corresponding sets. In particular, h_t(x^p) = h_t(x^p′) = 1. For any gate e_j ∉ E_c, where j < t, by the definition of first non-zero gate (with respect to the set E_c and hence, the set E_b) the following equality takes place: h_j(x^p) = h_j(x^p′) = 0. Also for any gate e_j ∈ E_c \ E_b, where j < t, by the property from Lemma 4, we obtain: h_j(x^p) = h_j(x^p′) = 0. Values of gates e_j ∈ E_b, where j < t, are determined by values of other gates with smaller indices. Therefore, values of any gates with the indices less than t are equal on the tuples x^p′ and x^p. By the construction, the tuple x^p′ differs from the tuple x^p by the value of at least one component corresponding to an input variable such that it is connected directly with an input of the gate e_t.

That is, the antichain function g_t corresponding to the gate e_t outputs 1 on a pair of comparable tuples. This contradicts the fact that g_t is an antichain function. Lemma 6 is proved. □

Let us finish the proof of Lemma 1. Recall that the constructed chain C contains n + 1 tuplessssssssss of the cube {0, 1}ⁿ. The circuit outputs the same value on all tuples of C such that there are no first non-zero gates on them with respect to a corresponding set. Denote this value by a ∈ {0, 1}. There exists a first non-zero gate on any tuple of the chain on which the circuit outputs 1 − a. The function f(x₁, …, x_n) takes value 1 on k(f) tuples of this chain and value 0 on n + 1 − k(f) tuples. There are two possible cases.

1. If a = 0, then for any tuple on which the circuit outputs 1 there exists a first non-zero gate in the circuit. That is, there are first non-zero gates on k(f) tuples of the chain C. By Lemma 6, all these gates are different. Therefore, L(S) ⩾ k(f).

2. If a = 1, then for any tuple on which the circuit outputs 0 there exists a first non-zero gate in the circuit. That is, there are first non-zero gates on n + 1 − k(f) tuples of the chain C. Similarly, by Lemma 6, we obtain: L(S) ⩾ n + 1 − k(f). Moreover, the last circuit gate e_s is not the first non-zero one, i.e. if there is a first non-zero gate on a tuple, then the circuit outputs 0 on it, and the gate e_s as well. Therefore, L(S) ⩾ n + 2 − k (f).

We have shown that L(S) ⩾ min k(f), n − k(f) + 2. Since the circuit S is arbitrary, we obtain the similar inequality for L(f). Lemma 1 is completely proved.