Abstract
Full Text
MATHEMATICS
A. A. FRIDMAN
DEGREES OF UNSOLVABILITY OF THE IDENTITY PROBLEM IN FINITELY PRESENTED GROUPS
(Presented by Academician P. S. Novikov on 14 VI 1962)
In the present work we shall consider the reducibility (with respect to solvability) of algorithmic problems (a.p.) by means of unrestricted Post tables (¹). If an a.p. (A_1) is reducible (in the sense indicated above) to an a.p. (A_2), and the a.p. (A_2) is reducible to the a.p. (A_1), then one says that these problems have one and the same degree of unsolvability. It is known that there exist different degrees of unsolvability of algorithmic problems (¹).
P. S. Novikov constructed the first example of a finitely presented group with an unsolvable word identity problem (²). We use another proof of the existence of a finitely presented group with an unsolvable word identity problem, given by Boone in (³).
At the Fourth Mathematical Congress A. I. Mal’tsev posed the question: what degrees of unsolvability can the identity problem have in finitely presented groups. My attention to this problem was drawn by S. I. Adian. He conjectured that, using Boone’s construction (³), one can construct a finitely presented group having any prescribed degree of unsolvability of the identity problem. This conjecture has been confirmed. The main result of the work is Theorem 1, which answers the question posed by A. I. Mal’tsev.
Theorem 1. For every degree of unsolvability (\alpha) one can specify a finitely presented group (\mathfrak{B}_{\alpha}) in which the word identity problem has degree of unsolvability (\alpha).
An analogous result for associative systems was previously obtained by G. S. Tseitin *.
The proof of Theorem 1 and of the other assertions of this note is omitted on account of their length.
The first part of the work contains a result concerning Turing machines. Let (\mathfrak{M}_1) be a Turing machine with alphabet of external memory (tape): (s_0, s_1, \ldots, s_n) (where (s_0) is the blank symbol) and alphabet of internal states (e_0, e_1, \ldots, e_m) ((e_1) is the initial state, (e_0) the final state).
Consider on the tape the minimal neighborhood (E) of the scanned cell outside which the tape is blank. Suppose that in the cells of this neighborhood there are written, from left to right, the symbols
[
s_{i_1}s_{i_2}\ldots s_{i_r}\qquad (r \geqslant 1)
\tag{1}
]
((r) is the number of cells of the neighborhood (E), including the scanned one), that the machine (\mathfrak{M}1) is in the internal state (e\alpha) ((0 \leqslant \alpha \leqslant m)), and that the (\rho)-th cell of the neighborhood (E) is being scanned.
To this complete state ** of the machine (\mathfrak{M}_1) we associate the word
[
h s_{i_1}s_{i_2}\ldots s_{i_{\rho-1}} e_\alpha s_{i_\rho}\ldots s_{i_r} h
\tag{2}
]
* This result was reported on 3 XII 1961 at the seminar on constructive mathematical logic at Moscow State University.
** The complete state of the machine (\mathfrak{M}_1) at each given moment is determined by specifying: the neighborhood (E), the word (1), the internal state, and the scanned cell.
(where (h) is a symbol not belonging to the alphabet of the machine), called the Post word corresponding to the given complete state. The complete state of the machine at any given moment is uniquely determined by the corresponding Post word, and from the Post word the complete state of the machine (\mathfrak M_1) is uniquely recovered.
Theorem 2. Let (\alpha) be the degree of unsolvability of an enumerable set (M) of natural numbers. One can construct a Turing machine (\mathfrak M_2) for which the algorithmic problem: given an arbitrary complete state (Q), to recognize whether (\mathfrak M_2) will pass from (Q) to the internal final state (q_0) or not, has degree of unsolvability (\alpha).
According to (4) there exists a Turing machine (\mathfrak M_{1-1}) computing the function (f(m)=1), whose domain of definition is (M). Let the external-memory alphabet of the machine (\mathfrak M_{1-1}) be (s_0, s_1), and the alphabet of internal states be (e_0, e_1,\ldots,e_n). The machine (\mathfrak M_2) is constructed from (\mathfrak M_{1-1}) as follows. The external-memory alphabet of the machine (\mathfrak M_2) consists of the alphabet of the machine (\mathfrak M_{1-1}): (s_0, s_1; e_0, e_1,\ldots,e_n), and the auxiliary symbols (\bar s_0, \bar s_1, \bar e_0, \bar e_1,\ldots,\bar e_n, h, H). In the alphabet of internal states of (\mathfrak M_2): (q_0, q_1, q_2,\ldots,q_k), three types of states are distinguished: printing, checking, and final.
Definition. A deductive chain of Post words (d.c.P.w.) is a word
[
h\bar Q_0h\bar Q_1h\ldots h\bar Q_mh,
\tag{3}
]
satisfying two conditions:
-
(h\bar Q_0h) is the initial Post word of the machine (\mathfrak M_{1-1}), i.e. the Post word corresponding to the recording of a natural number in the standard position (4).
-
(h\bar Q_i h) (where (i=1,2,\ldots,m)) is a Post word immediately following the Post word (h\bar Q_{i-1}h) (i.e. the complete state (Q_i) of the machine (\mathfrak M_{1-1}) is obtained from the complete state (Q_{i-1}) in one step of the operation of the machine (\mathfrak M_{1-1})).
The number (m) will be called the rank of the d.c.P.w. (3).
Suppose that on the tape there is printed a d.c.P.w. of rank (p), framed by two letters (H); (\mathfrak M_2) is in a printing state and scans the right-hand letter (H). Then we shall say that the machine (\mathfrak M_2) is in a canonical state of rank (p).
The commands of the machine (\mathfrak M_2) are constructed so that, if (\mathfrak M_2) is in a canonical state, then (\mathfrak M_2) erases the (H) located in the scanned cell, extends the d.c.P.w. of rank (p) to a d.c.P.w. of rank (p+1), writes the letter (H) on the right, and passes into one of the checking states—into the state of searching for the nearest letter (H) on the left. Having found it, (\mathfrak M_2) passes into a checking state: whether the word enclosed between the two letters (H) is a deductive chain of Post words. Having checked this, (\mathfrak M_2), in the case of a positive answer, passes into a canonical state of rank (p+1). Then the entire described cycle of operation is repeated. It is clear that, in further operation, (\mathfrak M_2) will successively pass into canonical states of ever higher rank with one and the same first Post word in the d.c.P.w. If the check gives a negative answer, then (\mathfrak M_2) stops.
For every complete state (P) one can specify such an (l(P)) that, if (\mathfrak M_2) began operation from the complete state (P) and after (l(P)) steps of operation has not stopped and has not passed into a canonical state, then (\mathfrak M_2) will not pass from the complete state (P) into the internal state (q_0). If (\mathfrak M_2) began operation from a canonical state (K), then (\mathfrak M_2) will arrive at the final state (q_0) if and only if the function (f(n)) is defined for the argument (n_0) represented in the first Post word of the d.c.P.w. of the canonical state (K).
Let (\mathfrak M_3) be an arbitrary Turing machine with external-memory alphabet (s_0, s_1,\ldots,s_{m_1}), internal-state alphabet (q_0, q_1,\ldots,q_k), and with some commands (we do not write them out). Using the construction
Boone’s (({}^{3})): from (\mathfrak M_3) one can construct the following associative system (\mathfrak A). The alphabet of the system (\mathfrak A) is obtained from the alphabet (\mathfrak M_3): (s_0, s_1,\ldots,s_{m_1}; q_0, q_1,\ldots,q_k), by adding the letters (s_{m_1+1}, q_{k+1}, q).
The defining relations of the system (\mathfrak A) are divided into two classes (A) and (B):
[
\text{(A)}\left{
\begin{aligned}
q_0s_j&=q_0,\
q_0s_{m_1+1}&=q_{k+1},\
s_jq_{k+1}&=q_{k+1},\
s_{m_1+1}q_{k+1}&=q,
\end{aligned}
\right.
]
where (j=0,1,\ldots,m_1).
The class (B) contains the relations:
[
\left.
\begin{aligned}
s_mq_is_j&=q_{i'}s_ms_j,\
s_{m_1+1}q_is_j&=s_{m_1+1}q_{i'}s_0s_{j'}
\end{aligned}
\right}
\quad
\begin{gathered}
\text{if among the commands of the machine } \mathfrak M_3 \text{ there is}\
\text{the command } q_is_j\Rightarrow q_{i'}s_{j'}L,\quad j'\ne0,
\end{gathered}
]
where (m=0,1,\ldots,m_1);
[
\left.
\begin{aligned}
s_mq_is_js_t&=q_{i'}s_ms_0s_t,\
s_{m_1+1}q_is_js_t&=s_{m_1+1}q_{i'}s_0s_0s_t,\
s_mq_is_js_{m_1+1}&=q_{i'}s_ms_{m_1+1},\
s_{m_1+1}q_is_js_{m_1+1}&=s_{m_1+1}q_{i'}s_0s_{m_1+1}
\end{aligned}
\right}
\quad
\begin{gathered}
\text{if among the commands of the machine } \mathfrak M_3 \text{ there is}\
\text{the command}\
q_is_j\Rightarrow q_{i'}s_{j'}L,\quad j'=0,
\end{gathered}
]
where (t,m=0,1,\ldots,m_1);
[
q_is_j=q_{i'}s_{j'},\quad
\text{if among the commands of the machine } \mathfrak M_3 \text{ there is the command } q_is_j\Rightarrow q_{i'}s_{j'}C,
]
[
\left.
\begin{aligned}
q_is_js_m&=s_jq_{i'}s_m,\
q_is_js_{m_1+1}&=s_jq_{i'}s_0s_{m_1+1}
\end{aligned}
\right}
\quad
\begin{gathered}
\text{if among the commands of the machine } \mathfrak M_3 \text{ there is}\
\text{the command } q_is_j\Rightarrow q_{i'}s_{j'}R,\quad j'\ne0,
\end{gathered}
]
where (m=0,1,\ldots,m_1);
[
\left.
\begin{aligned}
s_tq_is_js_m&=s_ts_0q_{i'}s_m,\
s_{m_1+1}q_is_js_m&=s_{m_1+1}q_{i'}s_m,\
s_{m_1+1}q_is_js_{m_1+1}&=s_{m_1+1}q_{i'}s_0s_{m_1+1},\
s_mq_is_js_{m_1+1}&=s_ms_0q_{i'}s_0s_{m_1+1}
\end{aligned}
\right}
\quad
\begin{gathered}
\text{if among the commands of the machine } \mathfrak M_3 \text{ there is}\
\text{the command}\
q_is_j\Rightarrow q_{i'}s_{j'}R,\quad j'=0,
\end{gathered}
]
where (t,m=0,1,\ldots,m_1)
(i.e., to each command of the machine (\mathfrak M_3) there corresponds in (\mathfrak A) a group of defining relations).
Lemma 1. Suppose the recognition problem: whether (\mathfrak M_3), from an arbitrary complete state (Q), will pass into the internal final state (q_0), has degree of unsolvability (\alpha). Then the identity problem for the fixed word (q) in (\mathfrak A) has degree of unsolvability (\alpha).
Definition. A word (Z) of the system (\mathfrak A) is called normal if
[
Z \underset{}{\supset} Z_1q_\xi Z_2
\quad
(\xi \underset{}{\supset} \Lambda,0,1,\ldots,k+1),
]
where (Z_1,Z_2) are words not containing the letters (q_\xi) ((\xi \underset{}{\supset} \Lambda,0,1,\ldots,k+1)).
Number the defining relations of the system (\mathfrak A) and write them briefly as
[
\Sigma_i=\Gamma_i,\qquad i=1,2,\ldots,\lambda.
\tag{4}
]
It is easy to see that (\Sigma_i,\Gamma_i) are normal words. Construct, from (\mathfrak A) according to (1), the Boone group (\mathfrak B).
The positive alphabet of the group is
[
\left{
\begin{array}{l}
s_0,s_1,\ldots,s_{m_1+1};\ q_0,q_1,\ldots,q\
t,k,x,y;\ r_1,r_2,\ldots,r_\lambda;\ l_1,l_2,\ldots,l_\lambda
\end{array}
\right}
]
Defining relations*
[
\begin{gathered}
\Sigma_i = l_i \Gamma_i r_i, \qquad (I)\
s_j l_i = y l_i y s_j, \qquad (II) \qquad
r_i s_j = s_j x r_i x, \qquad (IIa)\
s_j y = y^2 s_j, \qquad (III) \qquad
x s_j = s_j x^2, \qquad (IIIa)\
l_i t = t l_i, \qquad (IV) \qquad
r_i k = k r_i, \qquad (IVa)\
y t = t y, \qquad (V) \qquad
x k = k x, \qquad (Va)\
q^{-1} t q k = k q^{-1} t q, \qquad (VI)
\end{gathered}
]
where (i = 1, 2, \ldots, \lambda;\ j = 0, 1, \ldots, m_1 + 1).
In (⁴) the following theorem is proved.
Theorem 3. Let (\Sigma) be an arbitrary normal word. In order that (\Sigma = q) in (\mathfrak A), it is necessary and sufficient that
[
t \Sigma k \Sigma^{-1} t^{-1} \Sigma k^{-1} \Sigma^{-1} = 1
\quad \text{in } \mathfrak B .
]
Using Theorem 3, one can show that Lemma 2 holds.
Lemma 2. The identity problem for the fixed word (q) in (\mathfrak A) is reducible to the identity problem in the group (\mathfrak B).
Denote by (G) the group obtained from (\mathfrak B) by deleting relation (VI).
Theorem 4. In the group (G) the identity problem for words is decidable.
* Definition. Let (Z = Z_1 k^\sigma Z_2 k^{-\sigma} Z_3) be a word of the group (\mathfrak B) (where (\sigma = \pm 1), and (Z_2) contains no letters (k^\sigma)). We shall say that in the word (Z) the distinguished letters (k^\sigma) and (k^{-\sigma}) cancel each other in the group (\mathfrak B), if there exists a word (Z_4), containing no letters (k^\sigma), and such that
[
Z_1 k^\sigma Z_2 k^{-\sigma} Z_3 = Z_1 Z_4 Z_3
\quad \text{in } \mathfrak B .
]
Lemma 3. In order that (Z = 1) in (\mathfrak B), it is necessary that all letters (k^\sigma) of the word (Z) cancel each other in the group (\mathfrak B).
Let us call problem A the problem of finding an algorithm which determines, for each pair of adjacent mutually inverse letters (k^\sigma) and (k^{-\sigma}) ((\sigma = \pm 1)) of any word (Z), whether they cancel each other in (\mathfrak B) or not.
Lemma 4. The identity problem for words in (\mathfrak B) and problem A have one and the same degree of unsolvability.
Theorem 5. Problem A is reducible, by means of unrestricted Post tables, to the identity problem for the fixed word (q) in the system (\mathfrak A).
From Lemmas 1, 4 and Theorems 4, 5, Theorem 1 follows.
In conclusion I express my gratitude to S. I. Adian for his advice and attention to the work.
Received
2 VI 1962
References
- E. L. Post, Bull. Am. Math. Soc., 50, 284 (1944).
- P. S. Novikov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 44 (1955).
- W. W. Boone, Ann. Math., 70, No. 2 (1959).
- S. K. Kleene, Introduction to Metamathematics, Moscow, 1957.
* We omit the trivial relations of the group (\mathfrak B).