Stability among r.e. quotient algebras

Annals of Pure and Applied Logic 59 (1993) 55-63

North-Holland

55

Stability among r .e. quotient algebras

John Love

Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia

Communicated by A. Nerode

Received 16 December 1991

Revised 20 June 1992

Abstract

Love, J., Stability among r.e. quotient algebras, Annals of Pure and Applied Logic 59 (1993)

55-63.

A recursive algebra % = (A; F,, F,, ) IS a structure for which A is a recursive set of numbers

and the F; are uniformly recursive operations. We define an r.e. quotient algebra ‘a/- to be the

quotient IY by an r.e. congruence =.

We say that % is recursively stable among r.e. quotient algebras if, for each r.e. quotient

algebra b/c’ and each isomorphism f from % onto %3/-l, the set {(a, b) ) a E A, b E B and

f(a) = [b],.} is r.e. We shall consider examples of recursive stability. Then, assuming that ?I has a recursive

existential diagram, we show that the task of determining its recursive stability among r.e.

quotient algebras can be reduced to a more routine consideration of syntactical conditions. To

this result, we provide a counter-example which demonstrates the necessity of YI having a

recursive existential diagram. This result and counter-example are on similar lines to ones

obtained by Goncharov (1975,1977), for the recursive stability of recursive structures (among

recursive structures).

1. Preliminaries

One will find the definitions of algebras, quotients and congruences in [5]. We require the following extensions to the definition of a recursive function.

Let A, B be recursive sets. Let = and =’ be equivalence relations on A and B, respectively. Then we say that a map f :A + B/G’ is recursive if the set {(a, b) 1 a EA, b E B and f(a) = [b],,} is r.e. A map f’ :A/ = + B is said to be recursive if the set {(a, b) 1 a E A, b E B and f([u]=) = b} is r.e. These definitions

Correspondence to: J. Love, Department of Mathematics, Monash University, Wellington Road,

Clayton, Victoria 3168 Australia.

0168~0072/93/$05.00 0 199s Elsevier Science Publishers B.V. All rights reserved

56 .I. Love

are motivated by the fact that a function f : A + B is recursive if and only if f has an r.e. graph. When defined, the composition of two recursive maps is clearly a recursive map.

We shall implicitly use the following straightforward corollary to the Homo- morphism Theorem [5].

Proposition 1. Let f be a homomorphism from a recursive algebra %?I onto a recursive algebra E3. Let = be the congruence on 2I induced by f, i.e., ai = ai if and only if f (a,) = f (ai). Then there exists an isomorphism f’ from a/= onto 58, defined by f ([a]_) = f (a). Furthermore, if f is recursive, then = is r.e. (indeed, recursive) and f’ Is recursive.

2. Examples of recursive stability

A recursive algebra which is easily shown to be recursively stable among r.e. quotient algebras is the natural numbers with successor. Next we construct a recursive algebra d which is not recursively stable among r.e. quotient algebras.

We define a recursive algebra to be recursively stable among recursive algebras if each isomorphism from it to another recursive algebra is recursive. Clearly any recursive algebra which is recursively stable among r.e. quotient algebras is also recursively stable among recursive algebras. However, the converse is not so. This fact, which is demonstrated by our construction of ‘$I, is essential in distinguishing our considerations, of recursive stability among r.e. quotient algebras, from the study of the recursive stability of recursive structures (among recursive structures), as undertaken, for example, in [l, 21.

Theorem 2. There exists a recursive algebra which is recursively stable among recursive algebras, but which is not recursively stable among r.e. quotient algebras.

Proof. Let 3 = (A; F) be the recursive algebra constructed as follows. Let

A = {ao, al, . . .} be the set of natural numbers. Let F be the unary operation on A defined as follows:

F(ai) = (z::: ii i iz ZJJ’

Now let f be an isomorphism from ‘3 onto a recursive algebra 3’. Then, by fixing the value off (ao) and considering the cases of i even and i odd separately, we can compute each f (a,). Thus f is recursive, and so ‘u is recursively stable among recursive algebras.

We now construct an r.e. quotient algebra %/= which is isomorphic, but not recursively isomorphic, to 3. Let B = N. Using a recursive pairing function (.;) , denote each element (i, j) of N by bi,p Let G be the recursive operation on B

Stability among r.e. quotient algebras 57

defined as follows. For each i, i define G(b,j) = bi,j+l. Let 58 = (B; G) be the

resulting recursive algebra. [Intuitively, B consists of infinitely many disjoint

strings, each extending infinitely in one direction. For each i, the ith string

corresponds to the application of the successor function to the ith section

@‘I= {(m, n) 1 ( m,n)EBandn=i}.]

We now construct the r.e. quotient = on B, in stages s = 0, 1, 2, . . . . At stage

s we will have defined a recursive congruence =S on 5-B. The sS form a chain,

from which we define =, i.e., we let = = lJ, zS.

Stage s = 0. Let =” be the identity congruence on %.

Stage s + 1 = 2n + 1. Let i > 0 be the least such that, for all i >O, H(bj,,) f

s bo,+ [Informally, i > 0 is the least for which the 0th string has had no unit branch

added at position i.] Let k be the least for which [‘@“I]_, tl [‘&“I]_, = 0 [i.e.,

assume that the kth string has not previously been used in the construction].

Extend sS to the congruence ~~+i on B such that, for each i =

0, 1, . . . ) bk,j+l =s+l bo,i+j and for which no further identifications are made.

[That is, add a unit branch to the 0th string at position i, using the kth string.]

Stage s+1=2n+2. Let Ko, K,, . . . denote a standard enumeration of the

r.e., nonrecursive, set K = {i 1 &(i) converges}, i.e., KS E Ks+l, IK,,, - KS1 6 1

and UseN KS = K. Then for each i E KS+, and i > 0 such that b,,i+, =s h(bj.0) and

such that, for all k > 0, bk,O#s bo,i, we do the following. Extend=, to the

congruence zS + 1 on B such that bj,o~,+1 bo,iy and for which no further

identifications are made. [Informally, if i E Kscl, if the 0th string has a unit

branch at position i + 1 and if no similar branch was removed at an earlier stage,

then remove this branch.]

(End of construction)

It is evident from the construction that = is r.e. Furthermore, there is an

isomorphism g : %?I = !-B/G, completely determined by setting g(ao) = [bo,o]_, i.e.,

the head of the 0th string. Moreover, one may show that if g is recursive, then K is recursive. Contradiction. Thus g is a nonrecursive isomorphism. Hence ‘?I is not

recursively stable among r.e. quotient algebras. 0

3. Conditions for recursive stability

We now establish syntactical conditions equivalent to the recursive stability of a

recursive algebra among r.e. quotient algebras. However, in doing so, we require

the recursive algebra to have a recursive existential diagram.

Our proof is inspired by results concerning the recursive stability of recursive

structures (among recursive structures), due to [3,6].

First, we require the following terminology. For an arbitrary effective first-

order language, i.e., possibly including relation, function and constant symbols,

we define a sequence of (finitary) formulas to be an r.e. sequence offormulas if its

corresponding sequence of Gbdel numbers is r.e.

58 J. Love

For a structure (21, let Y,,(E, v), lu,(ii, v), . . . be a sequence of formulas for

which there exists a finite sequence a from ‘3 of length G such that, for each i,

Y;(d, V) has at most one solution in % and that each a’ from I?! satisfies at least

one Yi(E, V) in 3. Then we say that Y,,, Yr, . . . is a sequence of defining formulas for I?f.

We shall refer to the notion of a recursive algebra having a recursive existential diagram, which is to say that there exists an effective procedure to determine

satisfaction in the algebra of any given existential formula by any given finite

sequence of elements. We shall also use the term positive existential formula, i.e.,

a formula built up from atomic formulas using only the connectives v, A and the

quantifier 3.

Theorem 3. Let 8 be a recursive algebra having a recursive existential diagram. Then the following are equivalent:

(i) !?I is recursively stable among r.e. quotient algebras; (ii) !?l has an r.e. sequence of positive existential defining formulas.

Proof. We can assume that %?I is infinite, say, having universe A = {ao, a,, . . .}.

For otherwise, % would clearly be recursively stable among r.e. quotient

algebras, and a suitable r.e. sequence of defining formulas would consist of

formulas of the form u = v,, for each a E A, and so the result would be obvious.

(i)+ (ii) Suppose that !?I is recursively stable among r.e. quotient algebras, and

that the required r.e. sequence of defining formulas does not exist. Then we

construct an r.e. quotient algebra and a nonrecursive isomorphism from this

algebra onto !?I. Let L be the language of %?I. Let E be the recursive term algebra

for L, having infinite recursive generating set X = {x0, x1, . . .}. Then one may

show that each map from X onto A extends uniquely to a homomorphism from 5Z

onto % and that this extension is recursive if and only if the original map is

recursive. Thus, it suffices to construct a nonrecursive map f from X onto A for

which the congruence =, induced by the extension off to a homomorphism from

E onto ?I, is r.e. To do this, we construct f to meet the following requirements

for each i = 0, 1, 2, . . . :

4: xiEdomf, Qi: aiEranf, Ri: f # pi.

For these requirements, we choose, and fix, the following decreasing priority

listing: PO, Q,, RO, PI, Ql, RI, . . . . The construction must also ensure that = is

r.e. The construction proceeds recursively, in stages s = 0, 1, 2, . . . _ At stage s, we

construct a map fs from X into A and, in addition, we may define a witness of the

form yi,, or zi,s to a requirement Qi or Ri, respectively. With each fs, associate a

finite partial congruence es on & defined as follows. Let to, tl, . . . be a systematic

enumeration of the terms of L. Then, for each pair of elements t,(X), t,(Z’) from D _, where n, m cs and X, X’ E domf,, define t,(Z) =s t,(Z’) if and only if


M33) = wx~‘))~ N ow, the construction ensures that the =s form a chain and, moreover, that = = lJ, ss is r.e. Furthermore, the fs converge pointwise on X to the required function, i.e., we let f = lim,f,.

Stage 0. Let fO = 0. Stage s + 1. We attack the requirement of highest priority which requires

attention and which can be attacked without injury occurring to higher-priority requirements, where the underlined phrases are defined as follows. Let i be an arbitrary natural number.

fi requires attention if xi $ dom f,. 4 is attacked as follows. Extend fs to fs+l so that xi E domf,+i. For each defined

witness y+ and z,,,, define updated witnesses yi,s+l = yj,s and Zj,,+l = Zj,s. e is injured if xi E domf,, but either xi 4 domf,+l or fs+l(xi) #fs(Xi). Qi requires attention if a witness yi,, to Qi has not been specified, i.e., for which

f,(Yi,s) = ai. Qi is attacked as follows. Choose a witness yi,s+l E X - domf, and extend fs to a

map fs+l for which f,+,(yi,,+J = ai. For each defined witness Yj,s and Zj,s, define updated witnesses Yj,s+l = yj,s and Zj,s+l = Zj,s.

Q is injured if a witness yi,, has been defined, but either yi,, $ dom f,,, or

L+l(Yi,s> # ai. Ri requires attention if a witness Zi,s to Ri has not been specified, that is, for

which fs(2i.s) # &Azi,AV Ri is attacked as follows. Choose a witness z~,~+~ E dom &+i fl X. Define a map

f s+l so that domA+ G domf, U {~~+l), ~0 that fs+l(ti,s+l) # #i,s+l(zi,s+l) and ~0

that = s+1 c =s. For each j G i, update each defined witness yj,s and .z~,,~ by setting

Yj,s+l = Yj,, and zj,,+l = zj,s* Ri is injured if a witness 2i.s has been defined but zi,, $ domf,,, or fs+l(zi,s) #

L(zi,s)* (End of construction)

We now show that the construction gives the required map J Proceeding by induction on the position of the requirements in the priority listing, it is clear by the form of the construction that: each requirement is attacked only finitely often; each requirement is injured only finitely often; and each c and Qi is attacked, never to be later injured, at some finite stage, i.e., is met. Consequently, the fs converge pointwise on X to a total function from X onto A. It remains to show that = is r.e. and that each Ri is met.

To show that = is r.e., it suffices to show that the construction is recursive. Suppose that Ri requires attention at stage s + 1. Then we need only show that we can effectively decide whether or not Ri can be attacked without injuring higher-priority requirements. First, for each z E X, we construct a positive existential formula Yz as follows. Let X0 consist of the numbers xi, yj,,, z~,~ fixed for higher-priority requirements. Let X1 be such that {X0, X1} =domf,. Let e(i& V, W) be a positive, quantifier-free, formula constructed by taking the conjunction of positive atomic formulas representing the equalities between the

60 J. Love

elements of X0, 2, Xi, and of atomic formulas, of the form t,, = t,, representing

each equivalence t,(j) ss t,(Z’). In an atomic formula representing an equiv-

alence &(Z) =s t,(i), we naturally require that each of the variables (U)j, V, (W)k

be located in such a way as to represent the numbers (Q, z, (X& respectively,

in the computations of tn(Z) and t,(2). We set Yz(U, V) = 3W f3(ii, 21, W). Now,

in view of the definition of the Yz, we can attack R, without causing injury to

higher-priority requirements if and only if there exists z E dom +i,s+l rl X for

which YG+i(4, b) h as a solution a such that &,+,(z) #u. This last condition

is equivalent to finding z E dom $i,s+l n X for which ~i,s+l(Z) is not a unique

solution of Yz(fs+l(&,), v). N ow, since the existential diagram of $!I is recursive,

we can effectively decide, for each z E X, whether or not Yz(fs+,(Z,), v) has a

unique solution, by deciding whether or not 2l l=3v, V’ [V #v’ A Yz(f7+,(_Q,

V) A Yz(fs+i(ZO), v’)]. Furthermore, we can effectively enumerate all solutions to

Yz(fs+l(&,), v), and so, if a unique solution exists, we can effectively find it.

Thus, we can effectively decide whether or not Ri can be attacked without

injuring higher-priority requirements. Hence, the construction is recursive, and so

= is r.e.

We now show that each Ri is met. Suppose, for a contradiction, that there is an

Rj which is not met. Suppose that i is the least such. Then we construct an r.e.

sequence of positive existential defining formulas for 3. Let s be a stage after

which all requirements of higher priority than Ri have been attacked for the last

time. Now, since & is necessarily defined from all of X onto A, we can effectively

find stagess<sO<s,<--. such that, for each n, there is z,, E dom +i,s, n X for

which &,,(z,) = a,. For each IZ, let Yz. be the existential formula constructed for

Ri at stage s,. Observe that, for each s’ as, the elements of domf,. fixed for

higher priority constitute the same fixed set. Let _& be a sequence consisting of

these elements. Observe that, for each s’ as, f&&) =f&). Let E0 =f&). Now,

each lu,,(Z,, V) has a, as a solution. Furthermore, this solution is unique, for

otherwise Ri would be attacked at stage s, and, consequently, met. Thus,

YZ”, Yz,, Yz,, . . . is an r.e. sequence of positive existential defining formulas for

53, where ti,, is the required finite sequence. Contradiction. Hence each Ri is met.

(ii)+(i) Let YJU, v), Yi(U, v), . . . be an r.e. sequence of positive existential

defining formulas for 5?l, where tie is a finite sequence from %?I associated with such

Yi. Then we show that ‘$?I is recursively stable among r.e. quotient algebras. Let f

be an isomorphism from an r.e. quotient algebra %/ = onto %?I. Then it suffices to

show that, for each b E B, we can effectively find f([b],), i.e., show that f is a

recursive map. First, assume that f-l(&) = [&I,. Now, since each element of A

satisfies at least one of the Yn(&,, v), by isomorphism [b], satisfies at least one of

the Y~([d,,]~, v). M oreover, since we can enumerate finite sequences of numbers

whose corresponding sequence of equivalence classes constitute the existential

diagram of B/E, we can enumerate the Yx to find Ym([6,J,, V) for which [b], is

a solution. Next, effectively find the unique solution a of Ym(ti,,, v). Now, by

isomorphism, f([b],) is also a solution. Thus f([b],] = u, and so we have


effectively found f([b],). H ence, f is recursive, and so ‘$?I is recursively stable among r.e. quotient algebras. 0

4. Counter-example

We now adapt a result due to Gonocharov [4, Theorem 51 to provide a counter-example to Theorem 3. That is, we construct a recursive algebra,

recursively stable among r.e. quotient algebras, which does not have an r.e. sequence of positive existential defining formulas. However, we must first mention the following.

Let 3 be a family of recursive functions. Then we say that a listing

Ycl, Y1, . . . of 3, such that the Yn are uniformly recursive, is an enumeration of 8. If, in addition, ‘y, # Yn whenever m # n, then we say that ‘I/“, Y1, . . . is a one-one enumeration. Now, let Y& Yi, . . . be another enumeration of 3. Let there exist a recursive map f such that, for each n, YL = Y&,. Then we say that Y& Y;, . . . reduces to YO, Yi, . . . . If, in addition, YO, Yi, . . . reduces to Y& Yu;,..., then we say that the two enumerations are equivalent.

In [7, Theorem 31, a family G of recursive functions is constructed which has a one-one enumeration pO, pi, . . . , unique, up to equivalence. Using this enumeration, define, for the first-order language having unary predicate symbols

PO, p1, * * * 7 a recursive structure (N; PO, P1, . . . ) such that Pci,j,(n) t-, pn(i) = j, for each n E k4.

This structure, as shown in [4, Theorem 51, does not have an r.e. sequence of existential defining formulas. This result is fundamental to the following counter-example.

Theorem 4. There exists a recursive algebra which is recursively stable among r.e. quotient algebras, but for which there does not exist an r.e. sequence of positive existential defining formulas.

Proof. For a language L having unary function symbols F,, F1, . . . , define a recursive algebra 2I = (N; F,, F,, . . . )

function Fci,j) defined on each n E N by

1, F(i,j)(n) = {o, if p”(i) = j,

otherwise.

We now show that ‘?I is recursively stable among r.e. quotient algebras. It suffices to show that each isomorphism f from an r.e. quotient algebra ‘?I’/ = onto 2l is a recursive map. Clearly %?I’/ = is recursively isomorphic to an r.e. quotient of a recursive algebra having universe N, and so we can assume that %!I’ has universe N. It suffices to show that the homomorphism g from 2l’ onto $2, defined

by g(n) =f (bl=), is recursive. To do this, we first show that pgcO), ,~r), . . . is an’

consisting of, for each F~i,j), a unary

62 J. Love

enumeration of a. Let n,, be such that g(no) = 1. Then, for each i, j and n,

P&2)(9 =i c, F(i,j)(g(n)) = l f, F(i,j)(g(n>) =g(nO) @ F;i,j)(n) e no.

Thus, since = is r.e. and each pgCn) is total, the psCn) are recursive uniformly in n. Hence, since po, ,v~, . . . is an enumeration of C% and since g is onto,

P&?(O)7 P&?(l), . . . is an enumeration of CY. Now, since the enumeration po, pt, . . . is unique up to equivalence, there exists a recursive function h such that, for each IZ,

P,(,) = phCn). Thus, since po, pl, . . . is a one-one enumeration, for each n, g(n) = h(n), and so g = h is recursive. Hence f is a recursive isomorphism, and so ‘3 is recursively stable among r.e. quotient algebras.

Now it remains to show that % does not have an r.e. sequence of positive existential defining formulas. Suppose, for a contradiction, that @o(fio, v),

@I(fiO, n), . . . is such a sequence; we can assume that each @,(fi,, V) has unique solution II in X Then, to establish the contradiction, it suffices to obtain an r.e. sequence ‘Iv,, YI, . . . of existantial defining formulas for the structure 23 =

(N; PO, PI, * * . ) discussed just prior to stating Theorem 4. For a fixed IZ, assume that @,, = 3W @(ii, I.J, W). We can assume that 0 is the conjunction of atomic formulas. Moreover, letting ? = ii, V, W, since each atomic formula t(Fi(q)) = t’ can be replaced by the equivalent formula 3; [t(ri) = t’ A ri = F,(q)], we can assume that 0 is the conjunction of atomic formulas of the form Fi(ri) = r, and the form rj = rk. Next, find a solution fi = Sz,, n, 6zr to @ in % and let @‘(zZ, u’, V, KJ) be the formula of L obtained from 0 as follows. Replace each Fi(q) = rk by

(i) Fi(q) = U’ A rk = u’, if mk = 0; (ii) Fi(q) # U’ h rk = u’, if mk = 1.

Then 3W O’(tio, 0, II, W) has unique solution n in 5%. Now, using the language for 93, let @‘(fi, u’, V, VP) be the formula obtained from 0’ as follows. Replace each F,(q) = U’ by P,(q) and each Fi(q) #u’ by lpi(q). Let YJn(ii, u’, V) = 3~ @“(ii, u’, V, W). Then Y(fio, 0, V) has unique solution n in 23. Thus

yo, VI,. . * is an r.e. sequence of defining formulas for ‘23. But, as previously indicated, no such sequence exists for ‘23. Hence, ‘2l does not have an r.e. sequence of positive existential defining formulas. q

References

[l] C.J. Ash, Stability of recursive structures in arithmetical degrees, Ann. Pure Appl. Logic 32 (2) (1986) 113-13s.

[2] C.J. Ash, Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees, Trans. Amer. Math. Sot. 289 (1986) 497-514; Corrections, ibid. 310 (1988) 1.

[3] S.S. Goncharov, Autostability and computable families of constructivizations, Algebra and Logic 14 (1975) 392-409.


[4] S.S. Goncharov, The quantity of nonautoequivalent constructivizations, Algebra and Logic 16 (3)

(1977) 169-185. [S] G. Grltzer, Universal Algebra (Van Nostrand, Princeton, NJ, 1968).

[6] A.T. Nurtazin, Strong and weak constructivizations and computable families, Algebra and Logic,

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[7] V.L. Selivanov, Enumerations of families of general recursive functions, Algebra and Logic 15

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Stability among r.e. quotient algebras