UDC 519.513

MATHEMATICS

V. A. NEPOMNYASHCHII

RUDIMENTARY PREDICATES AND TURING COMPUTATIONS

(Presented by Academician P. S. Novikov on 8 IV 1970)

1. Introduction. In important constructions of the theory of algorithms, such as Kleene normal form, protocols of Turing machines, etc., instead of the class of primitive-recursive predicates one often uses narrower classes. Such classes are obtained both by restrictions on recursive schemes for computing predicates (the classes of bounded-arithmetic (Ap) predicates \((^1)\), constructively arithmetic (K), rudimentary (R), \(s\)-rudimentary (\(\mathrm{R}_s\)) predicates \((^2)\)), and by restrictions on the parameters of Turing computations (the classes of predicates computable on Turing machines with logarithmic slowdown (Log) \((^3)\), with tags (M) \((^4)\), in real time \((T)\) \((^5)\)). In \((^6)\) it is indicated that, for analogous purposes, one may use the class \(\mathrm{R}_s^{\log}\), obtained by restrictions of both types. For related constructions in \((^7)\) (§ 33) the class of context-free languages \((\mathrm{L}_{\mathrm{kc}})\) is used. From \((^2)\) it follows that \(\mathrm{Ap} \supseteq \mathrm{K} \supseteq \mathrm{R} \supseteq \mathrm{R}_s\). In \((^2)\), p. 92, the following result of Bennett was announced: \(\mathrm{K} = \mathrm{R} \supset \mathrm{R}_s\). In \((^6)\) it is indicated that \(\mathrm{R}_s^{\log} \subset \mathrm{R}_s \subset \mathrm{M}\). From \((^{2,3,8})\) it follows that all the above-mentioned classes of predicates (except the class of primitive-recursive predicates) belong to the class of predicates computable on Turing machines with linear space, and also that K belongs to the smallest class of the Grzegorczyk hierarchy \((^9)\). In connection with the facts listed, a number of questions naturally arise. We shall note those of them which are investigated in the present work.

1. Since the classes R and Log are obtained by restrictions of different types and \(\mathrm{R} \nsubseteq \mathrm{Log}\), the hypothesis arises that they are incomparable (i.e. \(\mathrm{Log} \nsubseteq \mathrm{R}\)). Is this true? An analogous question for the cases when Log is replaced by M, \(T\), or \(\mathrm{L}_{\mathrm{kc}}\).

2. The class Ap is generated by more powerful operations than R. Is Ap broader than R?

3. Is it true that \((\mathrm{Log} \cap \mathrm{M} \cap T \cap \mathrm{L}_{\mathrm{kc}}) \subseteq \mathrm{R}_s\)?

4. All predicates from \(\mathrm{R}_s\) indicated in \((^2)\) are defined in nonarithmetic terms (for example, “\(x\) is a subword of \(y\),” etc.). Do the simplest arithmetic predicates belong to \(\mathrm{R}_s\) (for example, \(y = x + 1,\ x \leq y\))?

It turns out that the answers to all questions 1–4 are negative. In obtaining the answers to questions 1, 2, the main role is played by Theorem 1. It gives a sufficient condition for the rudimentarity of predicates in terms of restrictions on the parameters of computations of predicates on two-tape Turing machines from \((^{10})\).

2. Two-tape Turing machines. We consider Turing machines \(\mathfrak{M}\) with an input and a working tape \((^{10,11})\). The program of \(\mathfrak{M}\) consists of commands of the form
\[ q_i m_j m_k \to q_i' m_k' s_\rho' s_\mu'', \]
which are interpreted as follows: if \(\mathfrak{M}\) is in state \(q_i\) and reads the symbol \(m_j\) on the input tape and the symbol \(m_k\) on the working tape, then it passes into state \(q_i'\), writes \(m_k'\) on the working tape and shifts the input (working) head in the direction \(s_\rho'\) (\(s_\mu''\)) by one cell: \(s_\rho', s_\mu'' = \Pi\) (to the right), Л (to the left), Н (no shift). \(\mathfrak{M}\) is, generally speaking, a nondeterministic machine, since its program may contain two commands with the same left-hand side. Different applications of these commands yield different processing of the input word. During any processing of an input word the input head does not go beyond its bounds. \(\mathfrak{M}\) accepts the input word \(p\) if there exists

such a processing \(p\), under which \(\mathfrak M\) passes into the distinguished state, \(\mathfrak M\) computes the predicate \(\omega(x_1,\ldots,x_r)\) *, if \(\mathfrak M\) admits all and only such words
\[ p = *x_1*\ldots*x_r* \]
for which \(\omega(x_1,\ldots,x_r)\) is true.

For an input word \(p\), denote by \(T_{\mathfrak M}(p)\) \(\bigl(L_{\mathfrak M}(p)\bigr)\) the maximum number of steps (of scanned cells of the working tape) of \(\mathfrak M\) under any processing \(p\) **. Let
\[ L_{\mathfrak M}(n)=\max_{|p|=n} L_{\mathfrak M}(p)\ ***, \qquad T_{\mathfrak M}(n)=\max_{|p|=n} T_{\mathfrak M}(p). \]

Theorem 1. Suppose there exist a nondeterministic two-tape Turing machine \(\mathfrak M\) and integers \(\alpha>1\), \(\beta>1\), \(C\) such that the predicate \(\omega(x_1,\ldots,x_r)\) is computable on the machine \(\mathfrak M\) with
\[ T_{\mathfrak M}(n)\le n^\alpha,\qquad L_{\mathfrak M}(n)\le n^{1-1/\beta} \]
(for all \(n\ge C\)). Then \(\omega(x_1,\ldots,x_r)\) is a rudimentary predicate.

From this theorem it follows that

Corollary 1. The class of predicates computable on nondeterministic two-tape Turing machines \(\mathfrak M\) with
\[ L_{\mathfrak M}(n)\le C\log_2 n \]
(where \(C\) is a constant depending on the predicate) is contained in the class \(\mathbf R\).

3. Comparison of the class \(\mathbf R\) with other classes. By the method proposed by G. S. Tseitin and in \((^{11})\), the following is proved.

Theorem 2. Suppose there exist a nondeterministic two-tape Turing machine \(\mathfrak M\) and an integer \(\alpha>1\) such that the predicate \(\omega(x_1,\ldots,x_r)\) is computable on the machine \(\mathfrak M\) with
\[ T_{\mathfrak M}(n)\le n^{2-1/\alpha} \]
(starting from some \(n\)). Then there exists a nondeterministic two-tape Turing machine \(\mathfrak M'\) computing the predicate \(\omega(x_1,\ldots,x_r)\) with
\[ T_{\mathfrak M'}(n)\le n^4 \quad\text{and}\quad L_{\mathfrak M'}(n)\le n^{1-1/3\alpha} \]
(starting from some \(n\)).

From Theorems 1, 2 it follows that

Corollary 2. **** \(\mathbf{Log}\subset \mathbf R\).

Corollary 3. \(\mathbf T\subset \mathbf R\).

From Theorems 1, 2 \(\bigl((^{7}), \S 19, (^{13})\bigr)\) it follows that

Corollary 4. \(\mathbf M\subseteq \mathbf R\).

Corollary 5. \(\mathbf L_{\mathrm{kc}}\subset \mathbf R\) *.

Remark. Instead of Corollary 2, from the same references one obtains the following, stronger fact: the class of predicates computable on deterministic one-tape Turing machines \(\mathfrak M\) with
\[ T_{\mathfrak M}(n)\le C n^{2-1/\alpha} \]
(\(\alpha\) is an integer \(>1\), \(C\) is a constant, \(\alpha\) and \(C\) depend on the predicate) is strictly contained in \(\mathbf R\). Analogously for Corollary 3.

4. Bounded-arithmetic predicates. \(\mathbf{Ar}\) is the smallest class of predicates containing \(x=y\) and closed under: 1) the operations of the algebra of logic; 2) prefixing bounded quantifiers; 3) substitution, in place of variables, of polynomials with natural coefficients \((^{1})\). The definition of \(\mathbf K\) is obtained from the definition of \(\mathbf R\) \((^{6})\), if instead of the predicate \(C(x,y,z)\) one takes the predicates \(x+y=z\) and \(x\cdot y=z\). From Corollary 1 it follows that \(x+y=z\) and \(x\cdot y=z\) belong to \(\mathbf R\). Hence, and from Theorem 6 of Ch. IV \((^{2})\), we have

Corollary 6 (Bennett). \(\mathbf K=\mathbf R\).

By prefixing quantifiers bounded by a polynomial \(\pi(\bar y)\) to a predicate
\[ \Omega(\bar x,\bar y,z)\quad (\bar x=x_1,\ldots,x_m;\ \bar y=y_1,\ldots,y_n), \]
we mean the operations
\[ (\exists z)\bigl(z\le \pi(\bar y)\ \&\ \Omega(\bar x,\bar y,z)\bigr), \qquad (\forall z)\bigl(z\le \pi(\bar y)\to \Omega(\bar x,\bar y,z)\bigr). \]
From the definition of \(\mathbf{Ar}\) the following is easily obtained.

Lemma. \(\mathbf{Ar}\) is the smallest class containing the predicates \(x+y=z\) and \(xy=z\) and closed under: 1) the operations of the algebra of logic; 2) linear transformations *; 3) prefixing quantifiers bounded by polynomials.*

* We do not distinguish between numerical and word (in the alphabet \(\{1,2\}\)) predicates, keeping in mind the known one-to-one correspondence between words and numbers (see, for example, \((^{6})\)).

** \(\mathfrak M\) must halt under any processing \(p\).

*** \(|p|\) denotes the length of the word \(p\).

**** For definitions of \(\mathbf R_3\), \(\mathbf R\), \(\mathbf{Log}\), \(\mathbf M\) see \((^{6})\), \(\mathbf T\) see \((^{5})\).

***** \(\mathbf L_{\mathrm{kc}}\) is the class of context-free languages over the alphabet \(\{1,2\}\). For the definition see \((^{7})\).

** For the definition see \((^{8})\).

With the aid of this lemma and Theorem 1 one establishes

Theorem 3. The classes of rudimentary and bounded-arithmetical predicates coincide.

Remark. Thus the class R is closed under such powerful operations as the prefixing of quantifiers bounded by polynomials.

5. \(s\)-Rudimentary predicates. Denote by \(s(x)\) the following predicate: “there exists an \(n=1,2,\ldots\) such that \(x=1\ldots 1\,2\ldots 2\)*.”

\[ \underbrace{1\ldots 1}_{n}\ \underbrace{2\ldots 2}_{n} \]

Theorem 4. \(s(x)\) does not belong to the class of \(s\)-rudimentary predicates.

From this follow

Corollary 7. \(\log \cap M \cap T \cap L_{\mathrm{кс}}\) does not enter the class \(\mathbf{R}_s\).

Corollary 8. The predicates \(x+1=y\) and \(x\leq y\) do not belong to the class \(\mathbf{R}_s\).

Remark. Since the predicate \(x\leq y\) belongs to the class R \((^2)\), Bennett’s result (announced in \((^2)\)) that \(\mathbf{R}_s\) is a proper subclass of R follows from Corollary 8.

Note added in proof. After obtaining the present results, we learned that Corollary 5 is also contained in \((^{14})\).

CITED LITERATURE

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\(^{2}\) R. M. Smullyan, Theory of Formal Systems, Princeton, 1963.
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\(^{8}\) R. W. Ritchie, Trans. Am. Math. Soc., 106, No. 1, 139 (1963).
\(^{9}\) A. Grzegorczyk, Some Classes of Recursive Functions, Rozprawy matematyczne, 4, 1953.
\(^{10}\) J. Hartmanis, P. M. Lewis II, R. E. Stearns, Proc. IFIP Congress 65, 1, 31 (1965).
\(^{11}\) J. E. Hopcroft, J. D. Ullman, J. Assoc. Comp. Math., 15, No. 3, 414 (1968).
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\(^{13}\) V. A. Nepomnyashchii, Computation on Turing machines with marks, Abstracts of scientific communications of the IX All-Union Algebraic Colloquium, Gomel, 1968, p. 145.
\(^{14}\) N. D. Jones, Math. Syst. Theory, 3, No. 2, 102 (1969).

\[ {}^{*}\quad l^{n}=l\ldots l,\quad |l^{n}|=n\quad (l=1,2). \]