Table Of Contenthttp://dx.doi.org/10.1090/trans2/064
AMERICAN MATHEMATICA L SOCIET Y
TRANSLATIONS
Series 2
Volume 6 4
NINE PAPERS ON LOGIC AND
GROUP THEORY
by
S. D. Berman M . I. Graev
G. S. Ceitin A . I. Kostrikin
D. K. Faddeev M . A. Naimark
I. M. Gel'fand V . A. Rohlin
A. V. Gladkii R . T. Vol'vacev
Published by the
AMERICAN MATHEMATICA L SOCIET Y
Providence, Rhode Island
1967
Copyright © 196 7 by the American Mathematical Society
Library of Congress Catalog Number A51-5559
Printed in the United States of America
All Rights Reserved
No portion of this book may be reproduced
without the written permission of the publisher
TABLE O F CONTENT S
Page
Ceitin, G . S. Algorithmi c operator s i n constructiv e metri c spaces .
[UewTiiH, T . C . AjiropM(J)MMqecKM e onepaTopb i B KOHCTpyKTMBHbix
MerpwqeCKMX npoCTpaHCTBax. ] Trud y Mat. Inst. Steklov. 6 7 (1962),
295-361 1
Gladkii, A. V. O n the recognitio n o f replaceabilit y i n recursiv e lan -
guages. [TjiaAKHM , A . B. 0 pacno3HaBaHM M 3aMemaeM0CT M B
peKypCMBHblX H3blKax. ] Algebr a i Logika Sem. 2 (1963) No. 3, 5-22 8 1
Faddeev, D . K. O n the semigroup of genera in the theory of integer repre -
sentations. [<f>aAaeeB , J\. K. 0 nojiyrpynn e POAO B B TeopwM
IjejioqMCJieHHblX npeACTaBJieHMM. ] Izv . Akad. Nauk SSSR Ser . Mat. 28
(1964), 475-478 9 7
Naimark, M. A, Th e structure of unitary representations o f a locally com -
pact group in a space Uy [HaMMapK , M . A. 0 CTpyKTyp e yHMTapHbi x
npeACTaBJieHMM jioKajibH o 6wK0MnaKTHbi x rpynn B npocTpaHCTBe n .]
i
Izv. Akad. Nauk SSSR Ser . Mat. 29 (1965), 689-700 10 2
Gel fand, I . M. and Graev, M. I. Finite-dimensiona l irreducibl e representa -
tions of the unitary and the full linea r groups, and related specia l func -
tions. [ Tejib^aHA, M . M. M TpaeB, M . 14. KoHeqHOMepHbi e Henpw -
BOAMMbie npeflCTaBJieHMfl yHMTapno w n nojmo w jMHeftHO M rpynnb i M
CBH3aHHbie c HMM M cnerjna/ibHbi e (JjyHKijnM. ] IZV . Akad. Nauk SSSR
Ser. Mat. 29(1965), 1329-135 6 11 6
Berman, S. D. Representation s o f finit e group s over an arbitrary fiel d an d
over rings of integers. [BepMaH , C . fl. ripeflCTaBJieHM H KOHeMHbl X
rpynn Ha # npon3BOJibHbi M nojie M w HaA KOJibijaMn ijejibi x qwceji. ]
Izv. Akad. Nauk SSSR Ser . Mat. 30 (1966), 69-132 14 7
Vol vacev, R. T . Sylo w p-subgroup s o f th e genera l linea r group .
[BojibBaqeB, P. T. p-noArpynn w Ck/ioB a nojmo H jinHeMHO M rpynnw. ]
Izv. Akad. Nauk SSSR Ser . Mat. 27 (1963), 1031-1054 21 6
Page
Rohlin, V. A. Metri c properties o f endomorphisms o f compact commuta -
tive groups. [POXJIMH , B . A. MeTpnuecKH e CBOHCTB a 3HAOMOP4)M3MO B
KOMnaKTHbix KOMMyTaTHBHbi x rpynn.] Izv . Akad. Nauk SSSR Ser . Mat.
28 (1964), 867-874 24 4
Kostrikin, A. I. O n the definition o f a group by generators and definin g
relations. [KocTpwKMH , A. 14. K 33.j\3.nn\o rpynn o6pa3yioiuMM M M
0npeflejlHK)mwMM COOTHOllieHMHMM. ] I7v . Akad. Nauk SSSR Ser . Mat. 2 9
(1965), 1119-1122 25 3
IV
http://dx.doi.org/10.1090/trans2/064/01
ALGORITHMIC OPERATORS IN CONSTRUCTIVE METRIC SPACES
G. S. CEITIN
CONTENTS
Chapter I. Statemen t of the problem and formulation of the fundamental theore m 1
Chapter II. Additiona l information abou t the theory of algorithms 9
Chapter III. Proof of the fundamental theore m 2 6
Chapter IV. Operator s on general recursive function s 4 5
Chapter V. Constructiv e function s o f a real variable 5 7
Bibliography 7 9
CHAPTER I
Statement of the problem and formulatio n
of the fundamental theore m
Many objects constructe d in constructive mathematic s ar e connected wit h
algorithmically developed processes. Thu s with a recursively enumerable se t
there is associated th e process o f generating its elements , wit h a partial recur -
sive function th e process o f computing its value for different value s of its argu -
ment (mor e precisely, th e process of generating equations of the form f{k, • ••,m)=ft,
where / i s a symbol for the function an d k, • • •, m, n ar e natural numbers) ,
with a constructive rea l number the proces s of computing its successiv e approxi -
mations. Th e same holds for constructive measurable sets , measurabl e functions ,
etc. introduced by N. A. Sanin [12> 14] , wit h which are associated processes fo r
approximating them in a special way by sets o r functions respectively . Ever y
such process i s characterize d b y some prescription, accordin g to which it i s
carried out; thi s prescription, code d in the form of a word in some alphabet, i s
considered t o be the transcription of that object with which the given process i s
associated (i n some cases th e object itsel f i s suc h a transcription) .
Two processes, define d generall y speaking b y different prescriptions , ca n
give the same or, i n a certain sense, simila r results. I n these cases th e corre -
sponding objects ar e usually considered t o be equal. Precisel y thi s property i s
possessed b y the equality of partial recursive functions , constructiv e rea l num-
bers, etc .
In this sectio n we shall use an imprecise notion of algorithm .
1
2 G. S. CEITIN
An operator on such objects is given in the form of some rule according t o
which, starting from one algorithmically developed process (calle d the given proc-
ess), w e construc t another such process (calle d the desired process). Moreover ,
we must satisfy th e requirement of correctness, namely that if the operator is ap-
plicable t o some object, the n it is also applicable to every object equal to it,
and for equal objects the operator gives equa l objects (operator s are not assume d
to be everywhere defined). I n the definition of such operators two approaches
occur.
The first approach consists in giving a prescription which enables us t o
carry out the desired process parallel to the given one, usin g the results of the
given process. Th e prescription itself of the given process i s not used here for
the performance of the desired process. On e can imagine that the desired proc-
ess i s carrie d out by two people such that the prescription of the given proces s
is know n to the first person, wh o carries out this process and communicates th e
results t o the second person; th e latter, using these results, carries out the de-
sired process, eve n though he does not know the prescription of the given proc-
ess. Partia l recursiv e operators \} , §63 ] give n by a scheme computing the
value of the desired function from values of the given one, hav e precisely thi s
character. I n exactly the same way, th e definition of the sum of two construc-
tive real numbers contains a rule for computing an approximation of the sum for
approximating values of the summands, and the definition of the measure of a con-
structive measurable set [12 , 14 ] contain s a rule for computing approximate val -
ues of the measure for approximations of the set .
If two given objects ar e assumed equal only when the results of correspond -
ing processes coincide , the n the requirement of correctness i s fulfilled automat -
ically here. I f for the equality of given objects onl y "similarity'' of results of
corresponding processes suffices , the n to guarantee correctness it is necessar y
to introduce some additional condition of a kind of continuity. Operator s of the
kind described will be called Kleene operators.
In the second approach an operator is give n by means of some algorith m
which, fro m the prescription for the development of the given process, give s a
prescription for the development of the desired process; th e correctness require -
ment is in this case a n additional condition imposed on the algorithm. Th e con-
structive function s introduce d by A. A. Markov [5] ar e operators of this kind.
The concept of effective operatio n introduced in the paper [*6 ] o f Myhill and
Shepherdson is also of the same type. W e shall call such operators Markov oper-
ators.
* The concept of constructive functio n use d in this paper is a variant of this concep t
and is essentially equivalent to the concept employed by A. A. Markov in his later paper [7],
ALGORITHMIC OPERATOR S 3
It is not difficult t o see that every Kleene operator can be represented in the
form of a Markov operator. I n fact, i n order to obtain the prescription of the de-
sired process fro m the prescription of the given process, i t suffices t o add to it
the prescription of the operator (jus t as the addition of the scheme of a partial
recursive operato r to the scheme of the given function gives a scheme of the de-
sired function). Th e question arises a s to the truth of the converse, i . e. i s
every Markov operator representable a s a Kleene operator ?
In dealing with operators transforming a partial recursive function into a
partial recursive function, th e role of Kleene operators is played by partial re-
cursive operators, an d it is natural to present a Markov operator by some partial
recursive function / giving , fo r a Godel number of the given function, a Godel
number of the desired function, wher e the correctness conditio n has the followin g
form: i f k an d I are Godel numbers of one and the same partial recursive func -
tion and f(k) i s defined, the n f(l) i s defined and the function wit h Godel num-
ber f(k) i s equal to the function with Godel number /(/) . Myhil l and Shepherd-
son call such Markov operators effective operations . The y proved [*" ] tha t
every effective operation coincides with some partial recursive operator. Thi s
also follows fro m more general results of V. A. Uspenskii [9 , Theorem s 10 , 10 ].
Now we shall consider operators transforming a general recursive functio n
into a partial recursive function. A s before, partia l recursive operators remai n
in the role of Kleene operators; fo r Markov operators it is necessary to replace
in the preceding formulation of the correctness conditio n the words "partia l re-
cursive* ' by the words "genera l recursive", a s a result of which the conditio n
becomes weaker. I n [16 ] th e question is posed as to the equivalence of suc h
operators and partial recursive operators. I n the present paper it is prove d
(Theorem 2, Chapte r IV, §l ) tha t every such Markov operator can be extended ,
in a certain sense, t o a partial recursive operator. I n particular, if an operator
gives, fo r every general recursive function, anothe r general function, the n it co-
incides with some partial recursive operator. Th e problem as to what the do-
main of definition of such a Markov operator can be remains open.
In dealing with operators transformin g a constructive real number into a con-
structive real number, constructiv e function s play the role of Markov operators.
To formulate here the concept of Kleene operator is more complicated, bu t
it is clea r that, sinc e the approximate values of the function defined by a Kleene
operator are computed from approximate values of the argument, suc h a function
possesses certai n continuity properties. Doe s a n arbitrary constructive functio n
possess suc h properties?
A similar result has bee n obtained by Kreisel, Lacombe, and Shoenfield [15] .
4 G . S. CEITIN
It has been established b y A. A. Markov [5], [7] tha t a constructive functio n
cannot have constructive discontinuities , i . e. tha t there cannot be a constructiv e
sequence o f real numbers X , X X • • • constructively convergin g to som e
Q y v
number x suc h tha t the value of the function i s defined fo r all elements o f thi s
sequence an d for x, an d there exists a positive number y suc h that, fo r ever y
n, th e value of the function fo r X differ s fro m th e value of the function fo r x b y
R
more than y.
In the present paper this result is strengthene d an d the constructive continu -
ity of every constructive functio n a t every constructive point is proved in th e
"e-8 sense " (Theore m 3 , Chapter V, §l) . W e shall als o prove the uniform ap -
proximability of constructive function s b y so-called pseudopolygonal function s
(Theorem 4 , Chapter V, §2) . Apparently , th e concept of a function constructivel y
uniformly approximabl e by pseudopolygonal function s als o corresponds t o the
concept of a Kleene operator on constructive rea l numbers. Th e question as t o
what the domain of definition o f a constructive functio n ca n be remains, a s i n
the case o f operators on general recursive functions , open .
In considering Markov operators on general recursive functions an d on con-
structive rea l numbers a great resemblance manifest s itself . I t is natural t o see k
some general concept, specia l case s o f which would be general recursive func -
tions, an d constructive real numbers suc h that one could then prove some genera l
theorem from which would follow the above-mentioned propositions abou t opera -
tors on general recursive function s an d on constructive real numbers. N . A. Sanin
suggested t o the author using for this purpose th e concept of constructive metri c
space, whic h was outlined in his paper [ 12] (cf . als o [14]). I n §3 o f this chap -
ter the definition o f this concep t will be given and the fundamental theore m wil l
be formulated .
Brief exposition s of the results of this paper were contained in [10] an d t11].
Except for small changes, th e present paper is a Candidate dissertation .
The author expresses his great appreciation t o his thesi s directo r A. A.Markov,
as well as t o N. A. Sanin and I. D. Zaslavskii fo r advice and remarks which wer e
of great help in the writing of this paper .
§2
We shall employ the apparatus of normal algorithms developed by A. A. Markov
[6].
If 2 1 is an algorithm in some alphabet A , an d P i s a word which is not a
word in A , the n we shall say that 2 1 is not applicable t o P, an d the expressio n
2I(P) wil l be considered meaningles s (cf . [ 6, Chapte r III, §1 . 5]).
The designation 2Ip , wher e 2 1 is an algorithm and P i s a word, wil l some -
ALGORITHMIC OPERATOR S 5
times replace the designation 2I(P) .
We shall assum e that, fo r every alphabet in which the objects we shall con-
sider are built up, w e have fixed some two-letter extension (calle d the standard
extension) fo r the construction of algorithms over this alphabet . W e also assum e
fixed a method of forming transcriptions of algorithms [6 , Chapte r IV, §3 - 3 J in
this standar d extension, wit h the difference that , instea d of the letters a an d b
used in [ 6L w e shall use the letters 0 an d | .
By the transcription of the algorithm 2 1 with respect to the alphabet A we
shall mean the word obtained in the following way : w e define th e translation op-
eration [ 6 , Chapte r I, §6 ] fro m the alphabet in which the algorithm 2 1 is con -
structed into the standard extension of the alphabet A , no t changing the letters
of A , the n we construct the translation of the algorithm 2 1 [6, Chapte r III, §7. l],
and, finally , w e construct the transcription of the resulting algorithm; i n order
for this definition t o be single valued, i t is necessary t o fix an order among those
letters in the alphabet of the scheme of 2 1 whic h do not belong to the alphabe t
A. Thu s the transcription of the algorithm 2 1 with respect to A is the transcrip-
tion of a certain algorithm in the standard extension of A whic h is equivalent to
the algorithm 2 1 wit h respect to A . Th e transcription of 2 1 with respect to A
will be denoted byc c 21, A I
We shall use the following notation for certain fixed alphabets: T. Q = {0, |},
\ = {0, |, - }, ^ 2 = {0, | , - , / i and ^ = {0, | , -, /,. §1 Transcription s of al-
gorithms are words in the alphabet ^Q . The letters D and * wil l be employed
as separating signs. W e shall assume that these letters will not be used for the
extension of alphabets in the construction of new algorithms.
Below we shall introduce the concepts o f words of type H , typ e T, typ e P,
and type $ . Th e symbols a^ , a^ . . . , wher e a i s one of the letters H , T P,
y
and /{ , wil l be used as variables whose admissible value s ar e words of type a .
We shall also introduce variables H an d H« fo r words in the alphabet ^L . Th e
letters i, j wil l play the role of "metamathematical " variables for natural num-
bers. I n the formulation of statements containing variables, w e shall sometime s
omit reference t o universal quantifiers, s o that if it is not clear from the tex t
that the variable in the given formulation is boun d by some quantifier or that it
occurs fre e (fo r example, afte r the words "le t us fix"), the n we shall assum e
that the variable is bound here by a universal quantifie r whose scope extend s t o
the whole formula .
The definitions give n in this paper of words of the indicated type s (natural, integral,
rational numbers, and duplexes) do not differ essentiall y fro m the corresponding defini -
tions in N. A. Sanin's paper [14],
