Reports of the Academy of Sciences of the USSR

1962, Volume 147, No. 4

MATHEMATICS

S. Yu. MASLOV

TRANSFORMATION OF ARBITRARY CANONICAL CALCULI INTO CANONICAL CALCULI OF SPECIAL TYPES

(Presented by Academician P. S. Novikov on 19 VI 1962)

1. Let \(H\) be an arbitrary alphabet. Every canonical calculus in the alphabet \(H\) (see \((^{1})\)) is given by a nonempty finite list of \(H\)-words, called the axioms of the given calculus, and by a finite set of rules—production schemes*, each of which has the form

\[ \begin{gathered} G_{1,1}p_{1,1}G_{1,2}p_{1,2}\ldots G_{1,m_1}p_{1,m_1}G_{1,m_1+1}\\ G_{2,1}p_{2,1}G_{2,2}p_{2,2}\ldots G_{2,m_2}p_{2,m_2}G_{2,m_2+1}\\ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\\ G_{n,1}p_{n,1}G_{n,2}p_{n,2}\ldots G_{n,m_n}p_{n,m_n}G_{n,m_n+1}\\ \hline G_1p_1G_2p_2\ldots G_mp_mG_{m+1} \end{gathered} \]

Here \(G_{i,j}\) \((i=1,2,\ldots,n;\ j=1,2,\ldots,m_i+1)\) and \(G_k\) \((k=1,2,\ldots,m+1)\) are designations of certain concrete \(H\)-words, while \(p_{i,j}\) \((i=1,2,\ldots,n;\ j=1,2,\ldots,m_i)\) and \(p_k\) \((k=1,2,\ldots,m)\) are designations of certain schematic variables. Schematic variables are letters of some fixed alphabet \(\{q,r,s,q',r',s',\ldots\}\). It is assumed that every variable occurring in the conclusion of a certain production scheme occurs in at least one of the premises of this scheme.

The concept of “derivation in a given canonical calculus” (see \((^{1})\)) is defined in such a way that the admissible values of any schematic variable are arbitrary \(H\)-words, where \(H\) is the alphabet of the given calculus. If, however, we take only \(A\)-words as admissible values of the variables, where \(A\subset H\), then the same axioms and rules of inference will define a quasicanonical calculus in the alphabet \(H\) with principal alphabet \(A\). A canonical (quasicanonical) calculus \(\mathfrak R\) is called a calculus over the alphabet \(A\) if the alphabet of the calculus \(\mathfrak R\) contains \(A\). We shall say that the calculi \(\mathfrak R_1\) and \(\mathfrak R_2\) are equivalent relative to the alphabet \(A\) if the \(A\)-word \(S\) is derivable in \(\mathfrak R_1\) if and only if it is derivable in \(\mathfrak R_2\). Let \(\mathfrak P\) be a canonical calculus over the alphabet \(A\). We shall say that it is a completely canonical calculus with principal alphabet \(A\) if \(\mathfrak P\) is equivalent relative to \(A\) to the quasicanonical calculus obtained from \(\mathfrak P\) by fixing \(A\) as the principal alphabet.

1. It is easy to construct a canonical calculus \(\mathfrak P\) over the alphabet \(A\) such that an \(A\)-word \(S\) is derivable in this calculus if and only if the length of \(S\) is expressed by a number of the form \(2^{2^n}\) \((n=0,1,2,\ldots)\). It can be shown that a canonical calculus in the alphabet \(A\) producing all words of this type and only them is impossible. Consequently, not for every canonical calculus over an alphabet \(A\) does there exist a canonical calculus in the alphabet \(A\), equivalent to the original one relative to \(A\).

* Words in the alphabet \(H\) (including the empty word) will be called \(H\)-words.

At the same time the following theorems hold:

Theorem 1. Whatever the alphabet \(A\) and the canonical calculus \(\mathfrak P\) over \(A\) may be, one can construct a canonical calculus \(\mathfrak D\) in the alphabet \(A \cup \{\xi\}\) \((\xi \notin A)\), equivalent to the calculus \(\mathfrak P\) relative to \(A\) and having the following properties: 1) \(\mathfrak D\) is a completely canonical calculus with principal alphabet \(A\); 2) every word \(S\) derivable in \(\mathfrak D\) contains at most one occurrence of the letter \(\xi\), and in the case when \(A\) has at least two letters, \(S\) either is an \(A\)-word, or begins with the letter \(\xi\) and contains no other occurrences of it; 3) \(\mathfrak D\) has a single axiom and every production scheme of the calculus has only one premise.

Theorem 2. If \(A\) is a one-letter alphabet, then, whatever the canonical calculus \(\mathfrak P\) over \(A\) may be, one can construct a canonical calculus \(\mathfrak D\) in the alphabet \(A \cup \{\xi\}\) \((\xi \notin A)\), equivalent to \(\mathfrak P\) relative to \(A\) and having the following properties: 1) \(\mathfrak D\) is a completely canonical calculus with principal alphabet \(A\); 2) every word derivable in \(\mathfrak D\) either is an \(A\)-word or begins with the letter \(\xi\) and contains no other occurrences of it; 3) \(\mathfrak D\) has a single axiom and every production scheme of \(\mathfrak D\) has no more than two premises.

In the case when the alphabet \(A\) contains at least two letters, Theorem 1 is proved by a comparatively simple construction (using Post’s first reduction, see (1)). If \(A=\{| \}\), Theorems 1 and 2, by means of Post’s theorem (see (1)), are reduced to the case when \(\mathfrak P\) is a normal calculus in the alphabet \(\{|,\alpha\}\); in the proof the following lemma is used:

Lemma. Let \(A\) and \(B\) be alphabets having no letters in common, \(B=\{\xi,\xi_1,\xi_2,\ldots,\xi_L\}\). Let \(\mathfrak R\) be a canonical calculus in the alphabet \(A \cup B\) such that every axiom and every premise of any production scheme of this calculus contains exactly one occurrence of a letter from the alphabet \(B\), while the conclusion of every scheme contains no more than one occurrence of a letter from this alphabet. Then one can construct a completely canonical calculus \(\mathfrak S\) in the alphabet \(A \cup \{\xi\}\) with principal alphabet \(A\), equivalent to \(\mathfrak R\) relative to \(A\), and such that every word derivable in \(\mathfrak S\) contains at most one occurrence of the letter \(\xi\). Moreover, if \(\mathfrak R\) is such that every word derivable in \(\mathfrak R\) is an \(A\)-word or begins with a letter of the alphabet \(B\), then \(\mathfrak S\) has the same property.

Theorems 1 and 2 show that any algorithmically enumerable set of \(A\)-words \(\mathfrak M\) can be specified by a canonical calculus \(\mathfrak D\) with a single auxiliary letter \(\xi\) such that, in an arbitrary derivation, the letter \(\xi\) merely “distinguishes” the auxiliary derivable words from the basic words (i.e., the words belonging to \(\mathfrak M\)) and nowhere occurs as a delimiter.

3. A canonical calculus will be called local if it contains one axiom and every production scheme of this calculus has one of the following three forms:

\[ \text{a) }\frac{qGr}{qG'r}, \qquad \text{b) }\frac{qG}{qG'}, \qquad \text{c) }\frac{Gr}{G'r}. \]

A local calculus will be called narrowly local (superlocal) if in any production scheme each of the words \(G\) and \(G'\) contains no more than two letters (respectively, no more than one letter). It is easy to see that for any canonical calculus in an alphabet \(H\) one can construct a narrowly local calculus over \(H\), equivalent to the original one relative to \(H\). On the other hand, it can be proved that for every superlocal calculus there exists a decision algorithm. Therefore, not for every canonical calculus in an alphabet \(H\) can one construct an equivalent, relative to \(H\), superlocal calculus.

A production scheme \(\Sigma\) of a local calculus \(\Omega\) will be called reversible in \(\Omega\) if \(\Omega\) contains the scheme inverse to \(\Sigma\), i.e. the scheme obtained—

from $\Sigma$ by a permutation of the premise and conclusion. A local calculus will be called reversible if all its schemes are reversible. Producing schemes of types b) and c) will be called edge schemes. A reversible local calculus that contains no edge schemes will be called an associative calculus with a fixed axiom. If $\mathfrak A$ is an associative calculus with a fixed axiom, then the list of schemes of this calculus will be called the associative calculus corresponding to the given calculus $\mathfrak A$, and will be denoted by $\widetilde{\mathfrak A}$. It is clear that $\widetilde{\mathfrak A}$ is an associative calculus in the generally accepted sense (see $(^2)$). The class of words derivable in $\mathfrak A$ coincides with the class of words equivalent, in the sense of $(^2)$, to the axiom of the calculus $\mathfrak A$ in the calculus $\widetilde{\mathfrak A}$.

Theorem 3. Let $\Omega$ be a local calculus over the alphabet $A$. One can construct a local calculus $\Omega'$, containing no edge schemes and equivalent to the calculus $\Omega$ relative to $A$. The calculus $\Omega'$ can be constructed so that all schemes of $\Omega'$, except possibly one, are reversible.

Theorem 4. Let $\Omega$ be a local calculus over the alphabet $A$. One can construct a reversible calculus $\Omega'$, equivalent to $\Omega$ relative to $A$ and containing only four edge schemes.

We give the construction of the calculus $\Omega'$. The set of $A$-words derivable in $\Omega$ is enumerable. Let $\mathfrak U$ be a normal algorithm in the alphabet $B$ ($A \cup \{\,|\,\} \subset B$), which transforms any word of the form $|^k$, to which it is applicable, into an $A$-word derivable in $\Omega$, and such that for every $A$-word $P$ derivable in $\Omega$ one can find an $l$ for which $\mathfrak U(|^l)=P$**. In a suitable extension of the alphabet $B$ one can construct an associative calculus $\mathfrak B$ such that, for any $B$-words $P$ and $Q$, the equality $\mathfrak U(P)=Q$ holds if and only if $\mathfrak B:\beta\alpha P\beta \parallel \beta\gamma Q\beta$ (see $(^2)$, p. 208). As the axiom of the calculus $\Omega'$ we take the word $\beta\delta_1|\beta$. As the producing schemes of the calculus $\Omega'$ we choose the schemes

\[ \frac{q\beta\delta_1 \mid r}{q\beta\delta_1 \parallel r}, \qquad \frac{q\delta_1 \mid r}{q \mid \delta_1 r}, \qquad \frac{q\delta_1\beta r}{q\delta_2\beta r}, \]

\[ \frac{q \mid \delta_2 r}{q\delta_2 \mid r}, \qquad \frac{q\beta\delta_2 \mid r}{q\beta\alpha \mid r}, \]

\[ \frac{\beta\gamma q}{\eta q}, \qquad \frac{q\eta x r}{q x \eta r}\ (x\in A), \qquad \frac{q\eta\beta}{q}, \]

the schemes inverse to them, and all schemes of the calculus $\mathfrak B$.

Theorem 5. There exists a local calculus $\mathfrak K$ over the alphabet $A$ such that any reversible calculus equivalent to it relative to $A$ necessarily contains edge schemes both of type b) and of type c).

As $\mathfrak K$ one may take, for example, a calculus over the alphabet $\{|\,\}$ such that a $\{|\,\}$-word $S$ is derivable in $\mathfrak K$ if and only if the length of $S$ is expressed by a number of the form $2^n$ ($n=0,1,2,\ldots$). The proof rests on the obvious assertion: if a reversible calculus $\mathfrak D$ does not contain schemes b) (does not contain schemes c)), then, whatever the words $P,Q,R$ in the alphabet of the calculus may be, if $\mathfrak D:P\parallel Q$, then $\mathfrak D:PR\parallel QR$ (respectively $\mathfrak D:RP\parallel RQ$).

An immediate consequence of Theorem 5 is the following assertion:

One can construct a canonical calculus over the alphabet $A$ for which there exists no associative calculus with a fixed axiom equivalent to it relative to $A$.

* Two mutually inverse schemes for each edge of a word.

** $|^n$ denotes the word $\underbrace{||\ldots|}_{n\ \text{times}}$.

At the same time, it follows from the proof of Theorem 4 that for any canonical calculus \(\mathfrak P\) over the alphabet \(A\), one can construct, over the alphabet \(A \cup \{\bar\beta\}\) \((\beta \notin A)\), an associative calculus \(\mathfrak D\) with a fixed axiom such that the word \(\beta S \beta\), where \(S\) is an \(A\)-word, is derivable in \(\mathfrak D\) if and only if \(S\) is derivable in \(\mathfrak P\).

In conclusion, the author expresses deep gratitude to N. A. Shanin for his attention to the work and for valuable advice.

REFERENCES

1. E. L. Post, Am. J. Math., 43, 163 (1943).
2. A. A. Markov, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 17 (1954).