On substitutional recursion over non-well-founded sets

R.T.P. Fernando*

Center for the Study of Language and Information Stzinford, California 94305-4115

Peter Aczel recently introduced a theory of non- well-founded sets to construct a model for Robin Mil- ner’s SCCS that is, in a suitable sense, final ([2]). As Jon Barwise has noted, having played around with these sets in his own work ([6, 9, 7]), the the- ory promises to be a useful tool for other semantic problems involving circularity, of which surely, com- puter science has a few. To employ non-well-founded sets for such ends, an understanding of recursion over these sets is crucial. With this in mind, a class of re- cursive definitions is isolated below that encompasses all applications of non-well-founded sets known to the author’. These definitions are based on a re- sult referred to in the literature as the Substztutzon Lemma, and accordingly are called substztutzonal re- cursive definitzons (srd’s). A theory of f ixed poznts of srd’s is developed in a n i i m l m of directions, lead- ing to a consideration of effective aspects of non-well- founded sets.’. We build throughout on [a], without necessarily presupposing familiarity with it.3

Informally, the idea behind an srd is to specify a collection of sets by rules whose left hand sides con- sist of uarzables and whose right hand sides consist of collections of sets built from variables. For example, to specify a collection A of sets such that

we define the srd

vo := 0 I { V l } .

*Supported by gifts to CSLI from the System Development Foundation and Ricoh.

‘Aczel ( [ 3 ] ) has developed a very gcneral and abstract the- ory of substitution whose relationsliip l o the srd’s below the au- thor does not fully understand. But see 52 for some comments.

’The question of computability for these sets is very natural from a computer science perspective, and was raised by Dana Scott. (And so the author drops another name ...)

Our approach has also been influenced considerably by Jon Barwise, to whom the author is gratefully indebted for close and patient supervision. He thanks Aczel too for helpful com- ments on a draft of this paper, and for generous access to his work.

Observe that A is a fixed point of the operator

x {XI.=@ v 3 y E X z = { y } }

which is quite naturally associated with the rule above. More generally, we will see that all srd’s can be viewed as monotone operators and are therefore guaranteed to have (nonempty complete lattices of) fixed points. Now, the novelty of our approach is that we get at these fixed points without explicitly refer- ring to the operators. Instead, we construct systems of equations from the specification, replacing the vari- ables by indeterminates for which we then solve. In our example above, 20 H {21}, X I H {xz}, xz H 0 is a system of equations whose solution associates 0 with 2 2 , (0) with 21, and ((8)) with CO. Over non-well- founded sets, this approach to fixed points, which can be seen as a refinement of the usual iterative methods for inductive (and co-inductive) definitions, is quite fruitful, yielding detailed information about fixed points. Our formal presentation below departs from this informal description in two ways. First, we reverse the order in which the two approaches to fixed points are considered. Second, we allow many sorts.

51. Basic definitions

We work (somewhat informally) in Zermelo-Fraenkel set theory with choice but not foundation (ZFC-). It will be convenient to admit urelements into the uni- verse of “pure” sets (without urelements) by suitable coding (yielding an interpretation of the urelement- polluted theory in the pure theory, as for instance in [5]). Fix a non-empty set Sort of sorts (typically, finite), a collection A of constants (typically, a set) , a collection X of indeterminates, a collection V of variables, and a map sort : X U V -+ Sort. Inde- terminates and variables will occasionally be referred to as parameters; along with the constants (the col- lection of which are assumed to be disjoint from the collection of parameters), these should be regarded as urelements. (Just what the sorts are is unimportant.)

CH2753-2/89/0000/0273$01.00 0 1989 IEEE 273

Next, form the well-founded universes W F A [ X I 4 and W F A [ V ] over the urelements d U X and A U U respectively, and define

para : w F d [ X ] U wFA[Y] + P o w e r ( X U U) a H ( X U V ) n transitive closure a .

Call a non-parametric if para a = 0. Although we will occasionally stray from these conventions, we will write a , a’, . . . for sets (in W F A [ X ] , wFA[V] or the full set-theoretic universes va[X], V A [ ~ ] ) ; 2, x’, . . . for indeterminates; v , v’, U, . . . for variables; and s, s ’ , t , . . . for sorts. Let us also decorate an inde- terminate x (or variable v) by a superscript, x’ (re- spectively, v’), to indicate its sort.

Definition. A substitutional recursive definition (srd) D is a partial mapping f rom U t o non-empty subcollections of W F d [ V ] such that for every sort s , there i s a unique v E domD of sori s . ~

When D is clear from context, we adopt more sugges- tive notation by writing

v := a0 I a1 I . . . 1 a , I . . ‘ ( a < K )

for Dv = and

:= a; i€ l I b j j € J

for Dv = { u ; } ; € I U { b j } j e J .

Example A. We allow Dv to be a proper class in or- der to admit, for example, the definition correspond- ing to

21 := 0 I { U 0 1 I { ~ o , v 1 ) I . . . I {v,}a<K I - . *

where the variables v and v, are all distinct but are of the same sort, say set , and where K ranges over all cardinals. For future reference, we denote this srd DI.,r. Formally, Df,,lr has domain {v}, and maps v to { { v , } a < K } K a cardinal. Dfurr is interesting in that i ts fixed points - a notion defined later - will turn out to be models of ZFC- .

Example B ([2]). Transitions systems with transi- tions labelled by “actions” ,U E A can be generated by the srd Dt given by

p := 0 I { t o } I {tO,t l I I . - . I {tcr)cr<K I . . . t := (p ,p)”fA

W F a [ X ] can be formed in the usual cumulative fashion by iterating the powerset operation through the ordinals, starting out with the collection of urelements A U X .

5The domain of D is taken to be a subset of V , instead of simply Sort, since for v E domD and a E Dv, it will matter whether w E transitive closure a.

where p is a variable of sort process, and t , t , are distinct variables of sort transztion. A process P is identified with the set of transitions ( p , Q) it can un- dergo; that is, ( p , Q ) E P if ancl oiily i f P can be transformed into Q by the action p .

Were we interested simply in entities of a single sort, say the sort process, we might have collapsed the srd Dt into the single-sorted definition

P := 0 I ~ ( ~ o l P o ) ~ ” “ d I ~ ~ C 1 ~ , P O ~ , ( ~ l , P 1 ~ ~ ” O ~ p l e d I . ’ . I { ( ,UarPa) la<r;pPEA 1 . . . ’

We will see in $4, however, that in some cases such as the following example, a multiplicity of sorts is not only convenient but essential.

Example C ([9]). Our third example is taken from work on the liar paradox, and presupposes some fa- miliarity with (Part I1 of) [SI. Let

C a r d = {2&. . . , A b } Player = {Cla i re , M a x }

A = C a r d U Player U { O , l , H a s , B e l , T r , A , V } .

Define Dr by

U :=

p :=

where U is a variable of sort in fon and p , P O , P I , . . ., p,, . , , are distinct variables of sort proposition.

For simplicity, we have required D to range over well-founded subcollections, so that we need only ap- peal to the fundamental theorem on recursive defini- tions on well-founded relations to assert the

Substitution Lemma. Every map x whose domain consists of parameters has a unique extension i t o W F A [ d o m x] that sends each a in i ts domain to

( a n d ) U { x w ( ~ ~ a n d o m x } U {irb I b E a - (AU d o m x ) }

Henceforth, whenever x is a map from parameters, we let 5 denote this natural extension to W F A [ d o m x ] . We also adopt the following notational guidelines: use e , . . .for partial maps X - w F A [ X ] ; f , . . .for partial

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maps X - vd; F , . . . for partial maps V - vd; and G, . . . for partial maps V - X .

Definition. A D-system of equations (soe) e is a partial map X - wFd[X] such that f o r all x E dom e , there are v , E domD, a, E Dv, and a funct ion G, : {U,} U p a r a a, --+ { x } U para e x that preserves sorts

Vu E dom G,, sort v = sort G,v,

maps U, t o x , and renames a, t o e x

Gxax = e x .

y E para e x , - ( 2 0 , . . . , x,,}. 111 words, the root of t r e e ( x , e) is ( x ) , and the cliildren of ( 1 0 , . . . , x n ) are the ( 2 0 , . . . , x,, y)'s where y is a parameter of ex,, other than 2 0 , . . . , z,,. I3y requiritig that y be dif- ferent from 2 0 , . . . , x,,, we prohibit repetitions in a sequence, which in turn means that t r e e ( x , e ) can be well-founded (with respect to t,lie converse of the sub- sequence relation) cvcii tliougli e /para, , , is circular (i.e., for some y E para,,,, y E para , ,y ) . Now, a soe e is

0 (parametrzcallg) i.ediiccd iT para( rng e ) C d o m e (so that the image of it.s solution is non- parametric)

We can think of D as a template for D-soe's, the latter 0 pointed if for some x E doiii e , doin e = { x } U being strung out instantiations of D. Parae,,

Example A . The map x s e * H { x ~ ~ * , x ~ ~ * } , z; )~* H {z;,*}, x i e * H 0 is a Dfu,l-soe. As is spelled out formally in the next definition, this soe describes the sets a1 = 0, ao = {0}, and a = {{0}, O} .

1 1, ~ ~ ~ ~ , ~ ~ l ~ 13. , p - o c e s s {ytronaafton Ztransrtron

ytransatron ( p , xP'OCes* ) is a Dt-soe. The process "depicted" here can take an action p without turn- ing into a different process, as well as undergo some undetermined transition (given by ttran8itron 1. Example C. Readers familiar with [9] will rec- ognize the Russellian liar in the D,-soe x'"fon H

H {x'"fon 1. ( T ~ , yproposr t ron 0 proposr taon 9 1, Y

A solutaon to a soe6, e is a map f with domain dom e such that f = f o e . The idea here is that if x s E dom e and f is a solution to e, then f x 8 is an entity of sort s given by e .

Next, we study the structure of a soe, beginning with connections between indeterminates in the soe's domain. Given a soe e , and x E dom e, define

where

para:,, := para e x para::' := UyEparo=,, para e ~ .

Observe that if f is a solution to e , then for all y E para,,,, f y E transitive closure of f x . These dependencies can be visualized graphically by defin- ing t r e e ( x , e ) to be the least set T of finite sequences of { x } U para,,, that includes (z) and such that if ( 2 0 , . . . , x n ) E T then (20 , . . . , x n , y ) E T for every

6Let us agree to drop D and just say "soe" when mention of D is not important.

0 almost well-founded if for all x E d o m e , t r e e ( x , e ) is well-foulitled

0 non-czrculnr if for all .L' E doni e , x e para,,,

0 well-foulidet1 i f i t is iiliiiost, \vc~ll-fo~~titlcd and non- circular .

These definitions generalize sotile of the basic con- cepts in Chapter 1 of [a ] . Finally, if E is a collection of soe's, and s is a sort, then define RootSE to be

{ f x s I f is a solrit,ioii to sotile reduced e E E whose doiliaill iiicludes x ' } .

52. A set-theoretic universe for solving equations

Having developed the basic notions for solving equa- tions over sets, the question now is whether roots can always be found. This is where the Anti-Foundation Axiom (AFA) of [2] enters the picture.

The Anti-Foundation Axiom. Every Djul l-soe has a unique solution.

As the roots of any soe e of any srd can be obtained by translating e itito a Dfull-soe (whose roots include those of e)', i t follows from AFA that every soe of every srd has a unique solution.

To get some sense for AFA, a few definitions from [a] are in order. Working in ZFC-, Aczel defines a system to be a class A4 of nodes together with an edge relation --+c M x A4 such that for each node a, the collection { b E M I a --+ b } , which is denoted a M , is a set. Call a system A4 fu l l if for every set s C M , there is a unique a E A! with s = a ~ . The

'Assuming a suficiently large supply of parameters, say a proper class, and a coding of constants (E A ) into pure sets.

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idea here is to use + (=+-') to interpret E , hoping thereby to obtain a model of set theory. In fact, the fundamental result is

Rieger's T h e o r e m . If A is a full system, theii

( A , +) ZFC-.

The reader is referred to [2] for a proof. To go further into the theory, i t is useful to collect the systems into a category whose arrows are system maps R , defined to be functions T : M + N satisfying

TU),^ = {wb I b E U M }

for all a E M q 8 Now, by €-induction, i t is easy t o see that the well-founded universe WF is not only a full system (with -=E-'), but is initial among such. A considerably more difficult result is Aczel's constructiong of a final system V,, which, as it turns out, is not only full (whence a model of ZFC-) but also satisfies AFA.

Some intuition about what the finality of V, means is essential to understand what is going on here. To say that R : M -+ V, is a system map is nothing more than to assert that the soe x, H (26 I b E a ~ } has a solution in V, (given by 2, H m). Hence, the fact that every system is mapped by an arrow into V, simply amounts to the existence of solutions (in Vc) to every Dfull-soe; furthermore, the uniqueness of the system map is just the uniqueness of the solu- tion. Clearly, we should be quite happy that solutions exist. But what about uniqueness? Is there a sense in which we lose some roots? The answer is yes, the reason being that a system map from M into V, need not be 1-1. Indeed, the collection of systems M that are (1-1) embedded in V, is non-elementary (i.e., not first-order axiomatizable). (The reader familiar with the notion of strong extensionalzty in [2] can read this result as asserting that strong extensionality is non- elementary.)

Proposition 1. Every sys tem that contains a copy of the herditarily finite sets as elementarily equivalent ( in the language of set theory) t o a system that is not (1-1) embeddable in V,.

The proof of proposition 1 is a standard compactness argument, and can be found (along with other proofs to numbered results of this paper) in the appendix. The point we want to make through this result is that the assumption underlying the construction of

'The reader might find it helpful to draw the corresponding commuting diagram, regarding -M and -N as functions from M to Power(A4) and N to Power(N) respectively.

'This construction is generalized in [4].

a domain for an srd D from the roots of D-soe's (in V,) is a very strong one: namely, everything there is about an element of the doinain is packed into its set- theoretic rcpreseiitabioti. ( I n ol,licr words, the defini- tion of equality on the donlain is captured completely by the srd). A similar idea informs Aczel's approach to semantics via a genernlizatioii of systems to @- coalgelras (where (I is an erltlofilllctor on the cat- egory of classes and class functions). Rather than going into some of tlie details of this theory, we de- velop substitution as the functor @, and put soe's to work. Category-theoretic terminology is quite dis- pensible here, and will be avoided throughout this paper - except, that is, for this section'".

First, observe ([2]) that AFA yields an

Extended S u b s t i t u t i o n Lemma. Every map T

whose domazii coiiszsts of parameters (E X) has a unzque exteiiszoii ? to Vd[doin T ] that sends each a $11 zts doniazn t o

Next, form the category C whose objects are sub- classes of Vd[x'], and whose arrows are substitutions ([3]) : i.e., class iiiaps 4 siicli that

Then readily extends to a functor C + C that sends objects A to Vd[A n XI." Let 1-Coalg be the category with objects C-arrows 0 : A + A and arrows from 4 to 4' such that

4/04 = $04.

(The reader is asked to supply the relevant commut- ing diagram.) N O W , AFA says that vd[x] 2 vA[x] is final in this category. 'Ihe drawback here is that vd[x] is, in general, more than we want for a do- main of a particular srd D. In such cases, we want

"The excursions below being intended primarily for the ben- efit of the reader who thinks (or, as in the case of the author, tries to think) in this abstract language.

l 1 The reader fond of hunting down adjunctions will no doubt notice the following. Let D be the category with subclasses Y of X as objects, and (arbitrary) functions as arrows. Now, the functor 1 : c -i D that sends an object A to A n X and an arrow II, to II, IX is the right adjoint of : D + C

C ( 3 , A ) D ( Y , A ) . In fact, the adjunction can be improved to an equivalence by replacing c by its subcategory of objects of the form V,[Y]. However, we will want to look below at subcategories of c not contained in this subcategory.

to focus on the subcategory of --Coalg whose ob- jects q5 restrict to D-soe’s - i.e., +IS is a D-soe. The final object of this subcategory is a natural choice for the domain given by D. But, in fact, final ob- jects of even smaller subcategories will do. To de- scribe these subcategories, i t is useful t o view an srd as an operator, and consider its fixed points, which we will do in the next section (wllen all this should become clearer with the category-theoretic terminol- ogy stripped away). Does this mean then that we could get along just as well without soe’s? Our an- swer is an emphatic no. D-soe’s will give us a han- dle on the fixed points of D through their elements, which is crucial for an analysis of the fine structure of fixed points. Our approach is quite concrete, and very much in the set-theoretic spirit where E , and not the arrows of category theory, occupies center stage.

Before proceeding further, let us add AFA to our meta-theory, and for the remainder of this paper, as- sume that i t stays there.

53. Fixed points To keep the notation from getting out of hand, we consider the two-sorted case below, taking Sort = {0,1}. The results easily generalize to arbitrary sets Sort, but only at the cost of obscuring the basic ideas involved.

Naturally associated with every srd D is a pair r D = ( r D , o , r D , l ) , of operators, one for each sort. Each operator takes as argument a pair (A0,Al) of non-parametric classes (again, one for each sort). In- tuitively, A, is the range over which variables of sort s vary. Now, rD,,(Ao, A I ) is just

{?(a) I a E Dv’, F is a map on para a that sends each uo to an element of A0 and each u1 to an element of A I }

where U’ is the unique variable of sort s in domD. Identify the f i xed p o i n t s of D with the fixed points of

r D.

Example A. Single-sorted srd’s such as Djull are par- ticularly simple, as we can collapse the two sorts men- tioned above. Observe that A is a fixed point of Df.11 if and only if

a E A @ a C A.

Clearly then, every fixed point of Df,,ll is a full system (taking + to be E-’).

Recall” the definitions of I?.,,,, I‘E,s, and stage or- d ina l s used to analyze the least fixed point ID =

ID,^, 1 0 , ~ ) through a coiistructioii that grows up- ward from below, and dually, FE,,, for a process that yields the greatest fixed point JD = ( J D , ~ , JD 1) shrinking from above. Now, the object is to de- velop these iterative notioiis further, in the context of non-well-founded sets, toward an analysis of the fixed points of srd’s.

D,S

To construct a fixed point of D containing the im- age of a collection of D-soe’s, we iriust add all well- founded extensions of the D-soe’s. What we mean by this, we make precise as follows.

Given maps e and e‘, d e h e e @ e’ to be the map with domain doni e U doin e’ wliicli is e on dom e , and e‘ elsewhere (i.e., on doni e’ - doni e ) . Say e D-composes wzth e’ if e @ e‘ is n D-soe, e is reduced and para(rng e’) doin e U d o m e’ (whence e @ e’ is reduced). Now, given an srtl D , s E S o r t and a pair ( N o , N I ) of classes (one for each sort), define WFD,,(NO, N1) to be

R o o t , { e @ e’ I e and e’ are D-soes, e D-composes with e’, Iloott { e ] e’ well-founded } .

This definition formalizes the notion above of clos- ing under all well-founded extensions - so that an application of I’D does not bring in any new sets. We need not worry about any of the old sets being booted out because of the requirement that e D-compose with e‘; the clause Root t{e} C Nt is a guard for minimality. Write WFD,,(0) for wF~,,(0,8), and let W F D ( N O , N ~ ) :=

For an equational analog of the stage ordinal of an element of an inductive operator’s least fixed point, we make the following definitions. Given a soe e and I E dom e , if t r e e ( x , e ) is well-founded (with respect to the converse of the subsequence relation), then de- fine 1 - on t r e e ( z , e ) by setting 110,. . .,x,,Iz,e to

N t for every sort 1 ,

(WFD,o(NO, Nl), W T D , I ( ~ O , .

sup { I x 0 i ~ ~ . , z n , ? / 1 2 , e I ( z o ~ ~ ~ . , x 7 1 , Y ) E 2 r e e ( e , e ) } + I .

Intuitively, 1x0,. . . , X , , [ ~ , ~ gives a measure of the height of the soe e [({x} U para,,,) viewed from 2,. This measure, which ranges only over successor ordi- nals, is designed deliberately to differ from the usual definition of height. Let 1z(, := I X ~ ~ , ~ . Now, given a collection E of soe’s, set (a lE,, to

znf { I I ’ I , I e is a soe in E whose solution maps xs to a , and t r e e ( z s , e ) is well-founded} ”Or else consult [I]

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if there is such an almost well-founded e in E whose solution maps x’ to a , and CO otherwise. When E is the collection of all D-soe’s, we write I u ~ D , , for I u ~ E , , .

Lemma 2. a E I’b,S(O, 0) ($ there is a well-founded, reduced D-soe e whose solution maps a n xa t o a where l Z S l e 5 (2.

Corollary 3. Assume that for every sort s and ev- ery a E N,, there i s a D-soe e with a E Root ,{e} such that for every sort t , Roo t t {e} Nt. T h e n W.FD(NO, NI) is the least fixed point I = ( I O , I1) of D such that N, C I , for every sort s. Furthermore, f o r a E I D , , = W F D , , ( ~ ) , I U J D , , is the least CY such tilai a E rb,,(0, 0).

Example A. Dropping sort subscripts in single- sorted cases, W F D (0) is the well-founded universe

WFof ‘‘pure” sets, and for every a E W F , ( U ~ D is the rank of a plus 1.

full

Rather than taking just well-founded extensions, we can extend by almost well-founded soe’s. Giving up some generality in order to simplify the notation a bit, we have

Corollary 4. d W F D := ( d W F D , , o , d W F D , l ) is a fixed point for D, where dWFD,, is

Root ,{e I e is a reduced, almost well-founded D-soe} .

Example (Barwise). Let D, be given by

2, := 0 I {vo} I . . . I {VO, . . . , vn] I . . ‘ ( n < w )

Then d W F D , is perhaps the most satisfying of the fixed points of D, in that i t consists of precisely those sets with finite transitive closure.

Next, we define the dual to WFD,,(NO, NI). Let d F d D , , ( N o , NI) denote Root ,E where E is

{ e I e is a D-soe such that every well-founded D-soe e’ i t D-composes with satisfies Root , { e @ e’} n N, = 0 for every sort s} .

The (obvious) notational conventions on W F D , ap- ply here as well. Now, the point of this definition that

Proposition 5. I f N, n ID,, = 0 for every sort then d F d D ( N 0 , NI ) i s the greatest fixed point J ( J o , J1) such that N, n J, = 0 for every sort s.

Notice tlie tightness of the hypothesis of the pre- ceding proposition - it is a iieccssary condition for tlie existence of tlie described greatest fixed point. The following result, unlike all tlie other results in this paper, depends on our decisioii to take tlie range of D to be subcollections of ivell-founded (parametric) sets.

Corollary G . For every srd D, if N , n WFA = 0 for every sort s, then there is a greatest fixed point J D = ( J D , ~ , J D , ~ ) SllCh tliat JD,, n N , = 0 for every sort s.

Examples A and B . To get the entire universe V , take dFdD,,,,,(fl). Assiiiiiing AFA, d F d ~ , ( @ ) is a domain for possibly iiori-well-fouiided trailsition sys- tems (that is final i n a sense described in [a ] ) .

54. Mixed fixed points

In some applicat,ioiis sucli Exaniple C, we want neither the greatest nor the least fixed point of an srd, but some sort of niix. Call a pair J , I of classes a mzxed fixed poznt pazr for I I I C I)initry operators To , rl if

0 J = largest S sucli that S = r o ( X , I ) where

0 I = smallest Y such that 1’ = T , ( J , Y ) .

[8] describes a recipe which, given set-like13 monotone operators ro, rl, produces a mixed fixed point pair J, I . In this section, we employ some of the tools we have developed to analyze mixed fixed points when ro, rl are given by an srd.

Let us start by relativizing some previous defini- tions to subsets14 S C Sort. Given an srd D, de- fine DS := D r { v E V I sort v E s} where (play- ing around with the urelements) AS := d U {U E ~ I s o r t v ~ ( ~ o r t - S ) } , ~ ~ : = { u ~ ~ I s o r t u ~ S ) , and X s := {x E X 1 sort x E S} . Note that the restrictions of D-soe’s to Ss are DS-soe’s, modulo re- naming of the indeterminates of sort E S by suitable variables. Call a D-soe e S-almost well-founded, S- non-ctrcular or S w e l l - founded according to whether e r X S is (as a DS-soe). Now, in the set-up for the mixed fixed points above, take Sort = {0,1} and S = { 1) to simplify notation. (The generalization to arbitrary Sor t and S 2 Sovt should be obvious.)

Proposition 7. Let N1 be

I3That is, i f a E l’,(X,Y) then there are sets b 5 X, c C Y such that a E r , (6 , c ) .

”Since, when dealing with srd’s, we do not have the usual packing of systems of operators into a single operator, we must, for generality, deal with subsets S.

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vd - Rootl{e I e is a (1)-wel l - founded D-soe} .

T h e n d F d ~ , ~ ( 0 , N l ) , d F d ~ , , ( @ , N I ) i s a mixed fixed point pair for r D , o , r D , l .

This construction corresponds to Barwise’s pro- cedure, which runs as follows. For a fixed X , let r l ( X , -) denote the operator resulting from fixing the first argument of rl at X , and, as usual, Irl(x,-) its least fixed point. Define a transfinite sequence J$ by

Ji := ro(X, I r l (x , - ) ) J; := ro(JSa, I ~ ~ ~ ~ ~ ~ , - ) )

where J > ~ := np<, J$ . If x is a set, let J X := na,lxl+ J?. Then J := UXEv JX along with I := I r l ( j , - ) constitute a mixed fixed point pair. The point here is to make elements of sort E S well- foundedI5, while at the same time having as many D-soe’s solvable as possible.

On the other hand, suppose we s tar t with a col- lection E of D-soe’s we want solutions to, but also want to minimize the fixed point. Then the construc- tion dual to that above is W F D ( R O O ~ E ) . This time, however, we need additional assumptions in order to conclude that what we get is a mixed fixed point pair for rD o , rl where rl(X, Y) := r D l(X, Y)URoot1E. These iequirements are met by Dr’ (Example C).

55. Set fixed points The fixed points of an srd are, in general, not sets - which is hardly surprising since Dv need not even be a set. The point of this section is that this is the only way an srd may fail t o have a set fixed point, essen- tially because AFA requires that solutions to soe’s be unique.16

Let H(K)d be the greatest fixed point of the srd D, given by

:= ( A u X)AGd, XCV, IXUAI<&.

Proposition 8. A s s u m e Id1 5 IC. T h e n H(K)d zs a set with cardinality 5 2‘. Furthermore, l H ( N o ) ~ l = 2 N o , and if Nx is regular (> Ho), then IH(Nx)dI = supa<x 2Na ( in particular, IH(N,+1)dI = 2 N a ) and

15For instance, in D, (Example C), the object is to obtain a unique (well-founded) definition of truth.

“Let us, in passing, state the simple fact that it is possible for some of the fixed points of an srd to be sets, while other fixed points of the same srd to be proper classes. Take, for instance,

v := { , , V } a E W F .

H(Nx)d = { U E vd I ltransitive closure ul < Nx}.

Corollary 9. l f Dv zs a set f o r cvery v in the domain of D, then a l l i t s fixed pozirts are sets. I n fact , i f rungeD C H(r;)du,y where Id1 5 K , then JD,, H(K)d for every sort s.

56. p o iiit s

Often tlie sets we have iii niiiid when we state a re- cursive definition include only those that are “effec- tively giveu.” The purpose of this section is to explore modifications of tlie theory along such lines, within a classical framework. Accordingly, the only sets we will concern ourselves with are hereditarily countable. Pick a bijective pairing function (-, -) : w x w -+ w with corresponding projection functions p , q , as well as a standard enumeration { ( P , ~ } ~ < ~ of the partial re- cursive functions. To fix notation, we let Sor t := w (for simplicity”) and

Effective soe’s and effective fixed

= { / ~ n ) n < u

x = U lrs

v = u v s sCSort

sCSort

’rs = {x:l}n<ld

v, = { v;}*<ld domD = { v : } ~ < ~

D = {Us,n}n<w C H(w1)dUV

where, of course, the uFn’s (unlike the x i ’ s and v i ’ s ) need not be distinct. NOW, there are two obvious ways to proceed - via solutions of effective soe’s, or via effective versions of tlie operators. We consider each in turn, starting with effective soe’s.

Let us code the range of a D-soe as follows. For every sort s, let

- D C F : W * Il(Wl)du,y , 11 b--+

where

Every integer 11 can be turned into the D-soe

”Duplicating sorts if we start out with only finitely many.

219

Set ED := {e! I n < w , e! is reduced}.

Example. Take an effective bite out of the heredi- tarily countable sets by setting Den to

v; := bvnm}mEdomlpnn<w I ({k,m)mEdomv, {vk+i}m<w)n<w

for all s < W . Call h?ooLoEDefl the set EHCd of effective hereditarily countable sets over A. Let

c : w - EHCA

be the partial recursive, surjective map with domain D

{ ( m , n) I en is reduced, and nz E dom

such that

where f is the solution to e:eff .

(m, 4 t-+ h

We look next a t the family of operators induced by an srd. Every integer n determines a partial map

Fn : v 7 EHCd

with domain

{ U & I s, m < w , (s, m) E domp,,, and p n (SI m) E darn c}

such that

Then, we can effectivize TD by defining U:, C ( ( P n ( S , 4) .

r g S ( A t ) t E S o r f as

{Fn(a) 1 71 < U , a E Dui, F,, has domain para a and sends each ut to an element of A t } .

Call the fixed points of these operators effective fixed points of D. Now, the burning question is are there subcollections E 5 ED whose solutions yield effective fixed points? From our experience in 53, we would ex- pect such an E to be effectively closed in the following sense: for all e E E , n < w and sort s,

D xb,, dom e & para c, (qn) c d o m e

D e @ [xi,, I-+ c , (qn)] E E . Observe that as the partial recursive functions are closed under finite extensions, ED is effectively closed. Effective closure of E is not sufficient to guar- antee that E is an effective fixed point, however, since the (soe-)presentations of sets via ED must also be reconciled with the (operator-)presentation given by c. More precisely, we need a partial recursive “trans- lation” function t : w - w such that for all n < w and sort s (E U ) , if c n E RootSE then

0 ( n , s ) E dom t

D . et(n,S) E E

0 xi E doin e D

0 f z i = cn where f is the solutioii to e$,l,,) .

t ( n , s )

When such a t exists, call E t~a~ts la lable . We can then assert

Proposition IO. If E c ED 2s effectzvely closed, and translatable, then RootE contazns an e.ffecttve fixed poznt of D. I f , furtherniore, RootSE C_ EHCA for every sort s, t hen RootE zs n11 effectzve fixed poznt of D.

There is no problem in obtaining ail effectively closed subcollection of ED - we have already noted tha t ED is effectively closed. l‘he difliculty is in get- ting an E that is also translatable. If D is “effective” in the way in which the following esainple is, however, this requirement is easily met.

Example 13. Consider the followiiig fragiiient Dt el7

of Dt

p := 0 I { ~ r r } n < w

t := (p , , ,p)-

Note that the whole of EDte f l is translatable, and therefore, so are any of its subcollections. Moreover, the effective fixed points of Dten and the roots of

effectively closed subcollections of ED‘efl come to the same thing.

$7. Some unfinished business The most glaring deficiency in the theory of effec- tive herditarily countable sets EHC given in the last section is the lack of some form of axiomatization. I t would certainly be wonderful if EHC is the least (in a natural respect) model of a variant of KP (see [5]) where (for example) foundation is replaced by an anti-foundation axiom. lowards this end, we might define an AFA-admissible set to be a transitive sub- set of V, (the final system mentioned in $ 2 which models AFA) satisfying KP minus foundation. Ad- missible sets (see [5]) would then be AFA-admissible, and moreover, the ordinal of an AFA-admissible set would be admissible (since the well-founded part of an AFA-admissible set is admissible). From a differ- ent direction, we might look at sets A such that for some (say, admissible) ordinal cr, A is closed under sets constructible by stage cr from its (set) elements.

Are there any pleasant connections lurking here? The author does not know.

Another aspect of substitutional recursion that the author wishes to understand is its expressive power. The requirement that an srd range over collections of (possibly parametric) well-founded sets brings out this issue by blocking the explicit presentation of ev- ery class of non-well-founded sets as a fixed point of a trivial srd. The point is that although the substi- tution lemma extends to the entire non-well-founded universe, restrictions on the rahge of an srd force us to use the recursiveness of soe's. We ought to look not only at how each root is obtained, but also at the larger picture of which collections of roots can be unearthed froiii a field that keeps some of its secrets underground. For surely R = {a} is a perfectly finite set, and there is something masochistic in dwelling on its derivation a s the limit of some infinite construc- tion.

References

Peter Aczel. An introduction to inductive defini- tions. In J . Barwise, editor, Handbook of Mathe- matical Logic. North-Holland, Amsterdam, 1977.

Peter Aczel. Non-well-founded sets. CSLI Lecture Notes Number 14, 1988.

Peter Aczel. A theory of structured objects. Un- finished draft. 1988.

Peter Aczel and Paul F. Mendler. A final coalge- bra theorem. Preliminary draft, 1988.

Jon Barwise. Admissible sets and structures. Springer-Verlag, Berlin, 1975.

Jon Barwise. The situation in logic - iii: sit- uations, sets and the axiom of foundation. In Alex Wilkie, editor, Logic Colloquium '84. North- IIolland, Amsterdam, 1985.

Jon Barwise. The situation in logic - iv: on the model theory of common knowledge. CSLI Report NO. 88-122, 1988.

Appendix (Proofs)

PROOF O F PIWI'OSII ' ION 1. Let A be a system con- taining a copy of the liereditarily finite sets, and ao, a l , . . . , b o , 6 1 , . . . be fresh constants. Add to the complete theory of A the sentences

The resulting theory is consistent (by compactness). Furthermore, the intcrpretations of a0 and bo in such a model are sent to the same element by the system map into V,. i

P R O O F OF LEhih,ii\ 2. Argue by induction on a. sup- pose a E r " ~ ~ ( f l , @ ) . ~ l i e i i a E rgsl(O,O) for some

p < C Y . So CL = l7b for soiiie v s E domD, b E DvS and F : p u m ~ 6 + I'b,3(fl,fl). By the inductive hy- pothesis, there are well-founded, reduced D-soe's e , for each U E pura 6 wliosc solution maps I, to F u where Iz,lcu 5 /7 ( ; i i i d r,, ant1 IL iirc of the same sort). We can assiinie tliat tlicse niaps e, have pairwise dis- joint domains. Extend these D-soe's to a D-soe e by mapping a fresh variable ts to b with the variables suitably renamed by indeterniinates. Now, the solu- tion to e maps .cs to a , where 1t5(, 5 p + 1 5 a.

Conversely, let e be a reduced D-soe whose solution maps an I' to a where ltsl, 5 C Y . Then for some p < a , ltsl, 5 p+l and for all y E para e t ' , lyle 5 p. The inductive hypothesis gives a E I'B+l(Ol 0). 4

D,S

PROOF OF COROLLARY 3 . Assume the hypothesis. Then for every sort s , N, C W F D , , ( N O , N ~ ) since every D-soe e D-composes with the well-founded D- soe 0. Moreover, if a is thrown into I 'E,s (No,Nl) at stage cy, then we can trace through its history to get a D-soe e and an zs E doni e where the solution to e maps t' to a (and where Its(, = CY if NO = N I = 0). Conversely, suppose U E W F D , * ( N ~ , NI) - N,. Then choose D-soe's e and e' sucli that e D-coniposes with e', e' is well-founded, R o o t t { e } C Ni for every sort 1, and the solution to e e' maps 2' E dom e' to a . Then a E r b , , ( N o , N 1 ) for CY 2 Itcsle~. -1

P R O O F O F COROLLARY 4 . fixed. i

W F D leaves d W F D [8] Jon Barwise. Mixed fixed points. To appear, 1989.

[9] Jon Barwise and John Etchemendy. The liar: an essay on iruth and circularity. Oxford University Press, Oxford, 1987.

PROOF OF PROPOSITION 5. Suppose that N,nID,s = 0 for every sort s. To simplify the notation, let A = (Ao,Al ) be d F d D ( N o , N 1 ) , and E be the set

28 I

described in the definition of d 3 d ~ , , ( N o , N 1 ) for which A , = Root,E for every sort s.

We first show that A, 2 r , , , ( A ) for every sort s. Let a E A , . Then there are e E E , x' E doni e , and a solution f to e such that fxS = a: Now, Vy E para e x s , f y E Asort y . Consequently, f ( e x s ) E rD,,(A). But fx ' = f ( e x s ) , whence a E rD,,(A), as required.

Conversely, let a E r D s ( A ) , say, v s E domD, b E Du', and F is a map with domain para b sending each U' to an element of Asor* " J , and where a = Fb. If a = Fu' for some U* E para b , then a E A , , as desired. Otherwise, from the hypothesis that Nt n ID = 0 for every sort t", it follows that a is thrown in by a well-founded extension of some e E E . However, e E E implies e @ e' E E for all well-founded e' with which e D-composes, whence a E A,. We conclude that, A is a fixed point of r D .

e s. -1

Finally, as for A being greatest, note that a D-soe E would force in an element of N , for some sort

PROOF OF COROLLARY 6. By corollary 3, I D , C WFd for every sort s. Hence, the result follows trom the previous proposition. -I

PROOF OF PROPOSITION 7. We need only check the leastness of d F d ~ , ~ ( @ , N I ) since the maximality of d3d~,,(@, N I ) follows from our basic proposition on d3dD. But leastness is also no problem since we only take solutions to { 1)-well-founded D-soe's, and these are indispensable for fixed points. -I

PROOF OF PROPOSITION 8. By the uniqueness of so- lutions to soe's, a bound on the number # ( ~ , d ) of pointed D,-soe's up to a bijective renaming of inde- terminates also bounds H ( K ) d . Next, observe that if a E N(K)A, then

where the P's are < K and the A's are subsets of d of cardinality < IC . We now argue as follows.

0 The number of indeterminates in a DN,,-soe is - < No whence fl(No,d) 5 N ; O = 2 ' O , provided

~~ ~

"That is, 0 E E # 0.

Id1 5 W . Moreover, since we can inject count- able sequences of 2 = {0,1} into H(No)a (via (ao, a l , . . .) H {ao, { a l , {. . .}}}), i t follows that this bound is tight.

0 As for H(N,+I)d where (dl 5 N a + l , notice that each of the P ' s is 5 N,, and therefore #(N,+l,d) 5 (2" . N : ; l ) N u = 2". Again, the bound is exact since Power(N,) 2 H(Na+l)O.

0 Finally, take the case of K = Nx where X is a limit. Assume (dl 5 Nx. Arguing as above, we have

If we make some convenient assumptions about the cofinality of R A , then we can say a bit more. Fix an a E H(N,$)d. Using the nota- tion in the equations above, define the sequences

IH(wX)dl 5 z N A .

{ Y n ) n < w , { 6 l } n < w as follows

7 0 := P 71 := S U P { P O I Q < P ) Yz := S U P { P a , , , I a < P,Q' < Pa}

a'' < P,,dl

Now, if N x is regular then each 7,, and IC,, is < N x and so the maps n H 7, and n H K , have range N, for some Q < X (dependent on a) , whence

H(NA)d = U H(Na+l)d a<X

3 lH(NA)dI = SUp,<X 2".

-I

PROOF OF COROLLARY 9 . Let e be a pointed, re- duced D-soe whose range is W(K)d"X. If f solves e then V x E dom e, Ifxl < K and V u E transitive clc- sure fx, (a1 < K . So we have f : X - H ( K ) A , which is a special case of Theorem 6.10 of [2]. Therefore, there exists a reduced D-soe e' : X - H ( K ) d " X with ldom e'I 5 IH(K)dI that yields the same roots as e. -I

PROOF O F PIWIWSWION 10. 'I'his follows easily from unpacking the definitions, and is best left to the reader (if for no other reason than that of space). -1

282