MATHEMATICS

Yu. I. ZHURAVLEV

ON THE IMPOSSIBILITY OF CONSTRUCTING MINIMAL DISJUNCTIVE NORMAL FORMS OF FUNCTIONS OF THE ALGEBRA OF LOGIC IN ONE CLASS OF ALGORITHMS

(Presented by Academician S. L. Sobolev on 12 I 1960)

Algorithms for simplifying disjunctive normal forms (d.n.f.) make it possible to distinguish in (\mathfrak N) certain conjunctions that do not enter into (\mathfrak N_{\Sigma M}), and to mark certain conjunctions that do enter into (\mathfrak N_{\Sigma M}). Generally speaking, in (\mathfrak N) there exist conjunctions for which it is not possible to obtain information about whether they do or do not enter into (\mathfrak N_{\Sigma M}). It is natural to pose the question whether there exists a sufficiently effective algorithm that makes it possible, for every pair ((\mathfrak A,\mathfrak N)), where (\mathfrak N) is a reduced d.n.f. and (\mathfrak A) is a conjunction from (\mathfrak N), to establish whether (\mathfrak A) is contained in (\mathfrak N_{\Sigma M}) or is not contained. Formally, this problem admits the following formulation*:

Problem M. Indicate an algorithm (A), generated by a function (\varphi), such that:
1) if (\varphi) is monotone, then the image of an arbitrary reduced d.n.f. (\mathfrak N) in the algorithm (A) contains no conjunctions with the mark ((0));
2) if (\varphi) is nonmonotone, at least one of the images of an arbitrary reduced d.n.f. (\mathfrak N) in the algorithm (A) contains no conjunctions with the mark ((0)).

It turns out that, in the class of algorithms that do not use, in one form or another, a significant search, problem M is unsolvable. The aim of the present note is to prove this fact.

To every algorithm (A) and d.n.f. (\mathfrak N) one can assign a sequence (\mathfrak N,\mathfrak N_1,\ldots,\mathfrak N_\alpha) such that:
1) (A) successively transforms (\mathfrak N) into (\mathfrak N_1,\ldots,\mathfrak N_{\alpha-1}) into (\mathfrak N_\alpha);
2) neighboring terms of the sequence differ in their marks over exactly one conjunction;
3) if the algorithm (A) is generated by a monotone function (\varphi), then (\mathfrak N_\alpha) is the image of the d.n.f. (\mathfrak N) in the algorithm (A);
4) if the algorithm (A) is generated by a nonmonotone function (\varphi), then (\mathfrak N_1,\ldots,\mathfrak N_\alpha) are all the images of the d.n.f. (\mathfrak N) in the algorithm (A).

We shall call the algorithm (A) admissible if to (\mathfrak A) and an arbitrary admissible d.n.f. (\mathfrak N) there corresponds a sequence (\mathfrak N_1,\ldots,\mathfrak N_\alpha) of admissible d.n.f.’s. We shall call a function (\varphi) admissible if all simplification algorithms generated by the function (\varphi) are admissible. In what follows we shall consider only admissible functions.

Let us introduce the inductive notion of the principal neighborhood of order (k) of a conjunction (\mathfrak A^{(j)}) in a d.n.f. (\mathfrak N). The principal neighborhood of first order (S_1(\mathfrak A^{(j)},\mathfrak N)) of the conjunction (\mathfrak A^{(j)}) in the d.n.f. (\mathfrak N) will be called the set of all such conjunctions from (\mathfrak N) that the intervals corresponding to them ({}^{(2)}) have a nonempty intersection with the interval (N_{\mathfrak A}). Suppose that the principal neighborhood of order ((k-1)) of the conjunction (\mathfrak A^{j}) in the d.n.f. (\mathfrak N) has been defined. The principal neighborhood (S_k(\mathfrak A^{(j)},\mathfrak N)) of order (k) of the conjunction (\mathfrak A^{(j)}) in the d.n.f. (\mathfrak N)

* In the present note we shall use the terminology and notation of our note (1).

we shall call the totality of all conjunctions (\mathfrak A_k^{(l)}) from (\mathfrak N) for which one of the following conditions is fulfilled: 1) the interval corresponding to the conjunction (\mathfrak A_k^{(j)}) has a nonempty intersection with the interval corresponding to a conjunction from (S_{k-1}(\mathfrak A^{(j)},\mathfrak N)); 2) the interval corresponding to the conjunction (\mathfrak A_k^{(l)}) is contained in the sum of intervals, to each of which there corresponds a conjunction from (\mathfrak N) satisfying condition 1).

Obviously,
[
M[S_1(\mathfrak A^{(j)},\mathfrak N)] \subseteq
M[S_2(\mathfrak A^{(j)},\mathfrak N)] \subseteq \cdots \subseteq
M[S_k(\mathfrak A^{(j)},\mathfrak N)].
]
Let the domain of definition of the function (\varphi) be a system of neighborhoods, and suppose that with each pair ((\mathfrak A^{(j)},\mathfrak N)), where (\mathfrak A^{(j)} \subseteq \mathfrak N) and (\mathfrak N) is an admissible d.n.f., there is associated a neighborhood (S(\mathfrak A^{(j)},\mathfrak N)). The function (\varphi) has index (k) if: 1) for every pair ((\mathfrak A^{(j)},\mathfrak N)), where (\mathfrak A^{(j)} \subseteq \mathfrak N), the relation
[
M[S(\mathfrak A^{(j)},\mathfrak N)] \subseteq
M[S_k(\mathfrak A^{(j)},\mathfrak N)]
]
holds; 2) there exists a pair ((\mathfrak A^{(j)},\mathfrak N)), where (\mathfrak A^{(j)} \subseteq \mathfrak N), such that
[
M[S(\mathfrak A^{(j)},\mathfrak N)] \nsubseteq
M[S_{k-1}(\mathfrak A^{(j)},\mathfrak N)].
]

We shall say that a simplification algorithm has index (k) if it is generated by a function (\varphi) of index (k). Let (D_k) be the set of all algorithms of index (k). An algorithm (A) has finite index if
[
A \subseteq \bigcup_{k=1}^{\infty} D_k.
]

We shall call a d.n.f. (\mathfrak N) absolutely irreducible in the algorithm (A) if there exists a unique image (\mathfrak N_1) of the d.n.f. (\mathfrak N) in the algorithm (A), and the d.n.f.’s (\mathfrak N) and (\mathfrak N_1) are equal in information.

Theorem. For each of the algorithms (A_k) of index (k) there is a reduced d.n.f. (\mathfrak N_{\varphi_k}^{(2)}), composed of unmarked conjunctions and absolutely irreducible in the algorithm (A_k).

Proof. Consider the function of the algebra of logic (f(x_1,\ldots,x_n)), (n \ge 2k+6), equal to one on the tuples
[
(0,0,\ldots,0,0),\quad (0,0,\ldots,0,1),\quad
(0,0,\ldots,0,1,1),\ldots,\quad (0,1,1,\ldots,1,1),
]
[
(1,1,\ldots,1,1),\quad (1,1,\ldots,1,0),\ldots,\quad
\ldots,(1,0,\ldots,0,0).
]
Write the reduced d.n.f. (\mathfrak N_{f_k}) of the function (f_k):
[
\mathfrak N_{f_k}
=
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-1}
\vee
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-2}\cdot x_n
\vee
]
[
\vee\,
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-3}\cdot x_{n-1}\cdot x_n
\vee \ldots \vee
x_2\cdot x_3\cdot \ldots \cdot x_{n-1}\cdot x_n
\vee
]
[
\vee\,
x_1\cdot x_2\cdot \ldots \cdot x_{n-2}\cdot x_{n-1}
\vee
x_1\cdot x_2\cdot \ldots \cdot x_{n-2}\cdot \bar x_n
\vee \ldots \vee
\bar x_2\cdot \bar x_3\cdot \ldots \cdot \bar x_n .
]

Let the algorithm (A) be generated by a function (\varphi) of index (k). We shall show that on the neighborhood (S(\mathfrak A,\mathfrak N_{f_k})), where (\mathfrak A) is an arbitrary conjunction from (\mathfrak N_{f_k}) and (S(\mathfrak A,\mathfrak N_{f_k})) is a neighborhood belonging to the domain of definition of the function (\varphi), the value of (\varphi) is equal to ((0)). For definiteness consider the conjunction
[
\mathfrak A_1=\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-1}.
]
Write out its principal neighborhood (S_k(\mathfrak A_1,\mathfrak N_{f_k})):
[
S_k(\mathfrak A_1,\mathfrak N_{f_k})
=
{\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-1},\
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-2}\cdot x_n,
]
[
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-3}\cdot x_{n-1}\cdot x_n,\ldots,\
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-k-1}\cdot x_{n-k+1}\cdot \ldots \cdot x_n;
]
[
\bar x_2\cdot \bar x_3\cdot \ldots \cdot \bar x_n,\
x_1\cdot \bar x_3\cdot \ldots \cdot \bar x_n,\
x_1\cdot x_2\cdot x_4\cdot \ldots \cdot \bar x_n,\ldots,\
x_1\cdot x_2\cdot \ldots \cdot x_{n-1}\cdot \bar x_{k+1}\cdot \ldots \cdot \bar x_n}.
]
From the definition of an algorithm of index (k) it follows that
[
M[S(\mathfrak A_1,\mathfrak N_{f_k})]\subseteq M[S_k(\mathfrak A_1,\mathfrak N_{f_k})].
]

Note that
[
\mathfrak A_1 \subseteq (\mathfrak N_{f_k}){\Sigma M}.
]
Write a reduced d.n.f. (\mathfrak N) such that
[
S(\mathfrak A_1,\mathfrak N_{\psi_k})=S(\mathfrak A_1,\mathfrak N_{f_k})
\quad\text{and}\quad
\mathfrak A_1 \nsubseteq (\mathfrak N_{\psi_k}){\Sigma M}.
]
If (k) is even, then
[
\mathfrak N
=
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-1}
\vee
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-2}\cdot x_n
\vee
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-3}\cdot x_{n-1}\cdot x_n
\vee
]
[
\vee\,
\bar x_1\cdot \bar x_2\cdot \ldots \cdot \bar x_{n-k-1}\cdot x_{n-k+1}\cdot \ldots \cdot x_n
\vee
\bar x_2\cdot \bar x_3\cdot \ldots \cdot \bar x_n
\vee
x_1\cdot \bar x_3\cdot \ldots \cdot \bar x_n
\vee
]
[
\vee\,
x_1\cdot x_2\cdot x_4\cdot \ldots \cdot \bar x_n
\vee \ldots \vee
x_1\cdot x_2\cdot \ldots \cdot x_{k-1}\cdot \bar x_k\cdot x_{k+1}\cdot \ldots \cdot x_n
\vee
\bar x_1\cdot \bar x_2\cdot \ldots
]
[
\ldots \cdot x_{n-k+1}\cdot \bar x_{n-k}\cdot x_{n-k+2}\cdot \ldots \cdot x_n
\vee
x_1\cdot x_2\cdot \ldots \cdot x_{k-1}\cdot x_k\cdot \bar x_{k+2}\cdot \ldots \cdot \bar x_n .
]
If (k) is odd, then
[
\mathfrak N_{\psi_k}=\bigvee_{i=1}^{2k+1}\mathfrak A_i,
]
where (\mathfrak A_i) are all the conjunctions entering into
[
M[S(\mathfrak A_1,\mathfrak N_{f_k})].
]
It is not difficult to see that both for even (k) and for odd (k)
[
\mathfrak A_1 \nsubseteq (\mathfrak N_{\psi_k})_{\Sigma M}.
]
The neighborhoods belonging to the domain of definition of (\varphi) possess

the following property: if (M[S(\mathfrak A,\mathfrak N)] \subset M(\mathfrak N_1)) and (M(\mathfrak N_1) \subset M(\mathfrak N)), then (M[S(\mathfrak A,\mathfrak N)] = M[S(\mathfrak A,\mathfrak N_1)]) (1). Obviously, (M(\mathfrak N_{\psi_k}) \subseteq M(\mathfrak N_{f_k})) and (M[S(\mathfrak A_1,\mathfrak N_k)] \subseteq M(\mathfrak N_{\psi_k})). Therefore
[
M[S(\mathfrak A_1,\mathfrak N_{f_k})] = M[S(\mathfrak A_1,\mathfrak N_{\psi_k})].
]
Since all conjunctions in (S(\mathfrak A_1,\mathfrak N_{f_k})) are unmarked, (S(\mathfrak A_1,\mathfrak N_{f_k})) and (S(\mathfrak A_1,\mathfrak N_{\psi_k})) are equal in information, and
[
\varphi[S(\mathfrak A_1,\mathfrak N_{f_k})]
=
\varphi[S(\mathfrak A_1,\mathfrak N_{\psi_k})]
=
\widetilde{\varphi}.
]
From the last equality it follows that
[
\varphi[S(\mathfrak A_1,\mathfrak N_{f_k})]=(0).
]
Indeed, if (\widetilde{\varphi}\ne(0)), then the function (\varphi) generating the algorithm (A) is not admissible.

The theorem is proved. We have proved the unsolvability of problem M in the class of algorithms with finite index.

Moscow State University
named after M. V. Lomonosov

Received
7 I 1960

References Cited

1. Yu. I. Zhuravlev, DAN, 132, No. 1 (1960).
2. S. V. Yablonskii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 51, 5 (1958).