Encoding Sets as Real Numbers

Encoding Sets as Real Numbers

Domenico Cantone¹ and Alberto Policriti²

1 Dept. of Mathematics and Computer Science, University of Catania, Italy

2 Dept. of Mathematics, Computer Science, and Physics, University of Udine, Italy

Abstract. We study a variant of the Ackermann encoding NA(x) :=

P

y∈x2^NA(y)of the hereditarily finite sets by the natural numbers, appli- cable to the larger collectionHF^1/2 of the hereditarily finite hypersets.

The proposed variation is obtained by simply placing a ‘minus’ sign be- fore each exponent in the definition of NA, resulting in the expression R^A(x) :=P

y∈x2^−RA(y). By a careful analysis, we prove that the encod- ingR^A is well-defined over the whole collection HF^1/2, as it allows one to univocally assign a real-valued code to each hyperset in it. We also address some preliminary cases of the injectivity problem forR^A.

1 Introduction

In 1937, W. Ackermann proposed the following recursive encoding of hereditarily finite sets by natural numbers:

NA(x) :=X

y∈x

2^NA(y). (1)

The encodingNA is simple, elegant, and highly expressive for a number of rea- sons. On the one hand, it builds a strong bridge between two foundational mathe- matical structures: (hereditarily finite) sets and (natural) numbers. On the other hand, it enables the representation of thecharacteristic function of hereditarily finite sets in terms of the usual notation for natural numbers as sequences of binary digits. That is: y belongs to x if and only if the NA(y)-th digit in the binary expansion ofNA(x) is equal to 1. As one would expect, the string of 0’s and 1’s representingNA(x) is nothing but (a representation of) the characteristic function ofx.

In this paper we study a very simple variation of the encodingNA, originally proposed in [Pol13] and discussed in [DOPT15] and [OPT17], applicable to a larger collection of sets. The proposed variation is obtained by simply placing a minus sign before each exponent in (1), resulting in the expression:

RA(x) :=X

y∈x

2^−RA(y).

As opposed to the encoding N^A, the range of R^A is not contained in the setN of the natural numbers, but it extends to the setRof real numbers. In addition,

the domain of RA can be expanded so as to include also the non-well-founded hereditarily finite sets, namely, the sets defined by (finite) systems of equations of the following form











ς1={ς1,1, . . . , ς1,m₁} ...

ςn={ςn,1, . . . , ςn,m_n},

(2)

withbisimilarity as equality criterion (see [Acz88] and [BM96], where the term hyperset is also used). For instance, the special case of the single set equation ς = {ς}, resulting into the equation (in real numbers) x= 2^−x, is illustrated in Section 3 and provides the code of the unique (under bisimilarity) hyperset Ω={Ω}.³

While the encodingN^Ais defined inductively (and this is perfectly in line with our intuition of the very basic properties of the collections of natural numbers N and of hereditarily finite sets HF—called HF⁰ in [BM96]), the definition of RA, instead, is not inductive when extended to non-well-founded sets, and thus it requires a more careful analysis, as it must beproved that it univocally (and possibly injectively) associates (real) numbers to sets.

The injectivity ofRAon the collection of well-founded and non-well-founded hereditarily finite sets—henceforth, to be referred to asHF^1/2, see [BM96]—was conjectured in [Pol13] and is still an open problem. Here we prove that, given any finite collection~1, . . . ,~n of pairwise distinct sets inHF^1/2 satisfying a system of set-theoretic equations of the form (2) in n unknowns, one can univocally determine real numbers RA(~1), . . . ,RA(~n) satisfying the following system of equations:











RA(~1) =Pm1

k=12^−RA(~1,k) ...

RA(~n) =Pm_n

k=12^−RA(~n,k).

This preliminary result shows that the definition of RA is well-given, as it associates a unique (real) number to each hereditarily finite hyperset in HF^1/2. This extends to HF^1/2 the first of the properties that the encoding NA enjoys with respect toHF. ShouldRA also enjoy the injectivity property, the proposed adaptation of NA would be completely satisfiying, and RA could be coherently dubbed anencoding forHF^1/2.

In the course of our proof, we shall also present a procedure that drives us to the real numbersRA(~1), . . . ,RA(~n) mentioned above by way of successive approximations. In the well-founded case, our procedure will converge in a finite number of steps, whereas infinitely many steps will be required for convergence in the non-well-founded case. In the last section of the paper, we shall also briefly hint at the injectivity problem forRA.

3 Notice that the solution to the equationx=e^−x is the so-calledomega constant, introduced by Lambert in [Lam58] and studied also by Euler in [Eul83].

2 Basics

LetNbe the set of natural numbers and letP(·) denote thepowersetoperator.

Definition 1 (Hereditarily finite sets).HF:=S

n∈NHF_n is the collection of all hereditarily finite sets, where

( HF0:=∅,

HF_n+1:=P(HF_n), forn∈N.

In this paper we shall introduce and study a variation of the following map introduced by Ackermann in 1937 (see [Ack37]):

Definition 2 (Ackermann encoding).

N^A(h) := X

h⁰∈h

2^NA(h0), forh∈HF.

It is easy to see that the mapNAis a bijection between HF.

From now on, we denote byhithe element ofHFwhoseNA-code isi, so that NA(hi) =i, fori∈N. Using the iterated-singleton notation

{x}⁰:=x {x}ⁿ⁺¹:=

{x}ⁿ , forn∈N, we plainly have:

h0=∅, h1={∅}, h2={∅}², h3={{∅},∅}, h4={∅}³, etc.

In addition, forj∈Nwe have:

h0∈hj iff j is odd, (3)

h₂j ={hj} and h₂j−1={h0, h1, . . . , h_j−1}. (4) The mapNAinduces a total ordering≺among the elements ofHF(that we shall call Ackermann ordering) such that, forh, h⁰∈HF:

h≺h⁰ iff NA(h)<NA(h⁰).

Thus,h_i is thei-th element ofHFin the Ackermann ordering and, fori, j ∈N, we plainly have:

h_i≺h_j iff i < j.

The following proposition can be read as a restating of the bitwise comparison between natural numbers in set-theoretic terms.

Proposition 1. Forhi, hj∈HF, the following equivalence holds:

hi≺hj iff hi6=hj and max

≺ (hi∆ hj)∈hj,

where∆ is the symmetric difference operator A ∆ B:= (A\B)∪(B\A).

It is also useful to define the following maplow:N → Nwhich, for i ∈ N, computes the smallest codej of a seth_j not present inh_i (or, equivalently, the position of the lowest bit set to 0 in the binary expansion of i):

low(i) := min{j|hj ∈/ hi}.

The following elementary properties, whose proof is left to the reader, will be used in the last section of the paper.

Lemma 1. Fori∈N, we have:

(i) low(i) = 0 iff i is even, (ii) h_low(i)∈/hi,

(iii) {h0, . . . , h_low(i)−1} ⊆hi,

(iv) hi+1= hi \ {h0, . . . , h_low(i)−1}

∪ {h_low(i)}.

Finally, we briefly recall that hypersets satisfy all axioms of ZFC, but the ax- iom of regularity, which forbids both membership cycles and infinite descending chains of memberships. In the hypersets realm of our interest, based on the Forti- Honsell axiomatization [FH83] as popularized by P. Aczel [Acz88], the axiom of regularity is replaced by the anti-foundation axiom (AFA). Roughly speak- ing, AFA states that every conceivable hyperset, described in terms of a graph specification modelling its internal membership structure, actually exists and is univocally determined, regardless of the presence of cycles or infinite descending paths in its graph specification. To be slightly more precise, a graph specification (or membership graph) is a directed graph with a distinguished node (point), where nodes are intended to represent hypersets, edges model membership re- lationships among the node/hypersets, and the distinguished node denotes the hyperset of interest among all the hypersets represented by the nodes of the graph. However, extensionality needs to be strengthened so as structurally in- distinguishable pointed graphs are always realized by the same hyperset. More specifically, we say that two pointed graphsG= (V, E, p) andG⁰ = (V⁰, E⁰, p⁰), where p∈V and p⁰ ∈V⁰ are the ‘points’ of Gand G⁰, respectively, arestruc- turally indistinguishable if the points p and p⁰ are bisimilar, namely, if there exists a relationRoverV ×V⁰ such that the following three properties hold, for allv∈V andv⁰∈V⁰:

(B1) (∀w∈V)(∃w⁰∈V⁰) (vRv⁰∧(v, w)∈E)→((v⁰, w⁰)∈E⁰∧wRw⁰)

; (B2) (∀w⁰∈V⁰)(∃w∈V) (vRv⁰∧(v⁰, w⁰)∈E⁰)→((v, w)∈E∧wRw⁰)

; (B3) pRp⁰.

Any relation R⁰⊆V ×V⁰ satisfying properties (B1) and (B2) above is called a bisimulation overV ×V⁰.

In this paper, we are interested in those hypersets that admit a representation as afinite pointed graph: these are the hereditarily finite hypersets inHF^1/2.

Given a (finite) pointed graph G= (V, E, p) whose nodes are all reachable from its pointp, the collection of hypersets represented by all its nodes form the transitive closure of{~}, denotedtrCl({~}), where~is the hyperset correspond- ing to the pointp.

3 The real-valued map R

Consider the following map RA obtained from NA by simply placing a minus sign before each exponent in (1):

RA(x) :=X

y∈x

2^−RA(y). (5)

From (5) it follows immediately that all (valid) RA-codes are nonnegative.

For instance, we have:

RA(∅) = 0, RA({∅}) = 1, RA({∅}²) = 1 2, RA({∅}³) = 1

√2, RA({∅}⁴) = 2^−√12, RA({∅}⁵) = 2⁻²

−√1

2, etc.

The definition ofR^Abears a strong formal similarity withN^A, but calls into play real numbers. As a further example, from the definition of RA, it follows that the hyperset defined by the set equationς ={ς}yields the equation inR

x= 2^−x. (6)

It is easy to see that the equation (6) has a unique solution in R, since the functionsxand 2^−xare, respectively, strictly increasing and strictly decreasing, so that the functionx−2^−xis strictly increasing. In addition, we have:

x−2^−x|_x=1 2 =1

2 − 1

√2 <0<1−1

2 =x−2^−x|x=1.

Thus, the solution Ω of (6) over R, namely, the code of the hyperset defined by the set equation x ={x}, satisfies ¹₂ < Ω < 1. Furthermore, much by the same argument used by the Pythagoreans to prove the irrationality of√

2, it can easily be shown thatΩ is irrational. In fact, Ω is transcendental. This follows from the Gelfond-Schneider theorem (see [Gel34]), which states that every real number of the form a^b is transcendental, provided that a and b are algebraic numbers such that 06=a6= 1, andbis irrational.⁴Indeed, ifΩwere algebraic, so would be−Ωand therefore, by the Gelfond-Schneider theorem, 2^−Ω=Ωwould be transcendental, contradicting the assumed algebraicity of Ω. Thus, Ω must be transcendental after all. As is easy to check, the RA-code 2^−1/

√2 of{∅}⁴ is transcendental as well.

Much as forNA, the encodingRAis easily seen to be well-defined overHF. As a consequence of the results to be proved in Section 4, we shall see that (5) allows one to associate univocally a code to each non-well-founded hereditarily finite set as well, thus showing that RA is also well-defined over the whole collection HF^1/2 of hereditarily finite hypersets.

4 We recall that the Gelfond-Schneider theorem, obtained independently in 1934 by A. O. Gelfond and Th. Schneider, solves completely the seventh in a well-celebrated list of twenty-three problems posed by David Hilbert at theInternational Congress of Mathematicians held in Paris, 1900 (see [Hil02]).

Remark 1 For every singleton {h⁰} ∈ HF, definition (5) gives RA({h⁰}) = 2^−RA(h0). Thus, for every h∈HF, we have

RA(h) =

n

X

h⁰∈h

2^−RA(h0)=

n

X

h⁰∈h

RA({h⁰}). (7)

Once we prove that RA is well-defined over HF^1/2, equation (7) can be im- mediately generalized toHF^1/2 as well.

As the following proposition shows, the codes of hereditarily finite sets can grow arbitrarily large.

Proposition 2. For anyn∈N, there exists an i∈Nsuch that RA(hi)> n.

Proof. Notice that for any odd natural number j, we have ∅ ∈ hj. Thus, by (7), we have RA(hj) > R({∅}) = 1, RA({hj}) = 2^−RA(hj) 6 2⁻¹ = ¹₂, and RA({{hj}}) = 2^−RA({hj})>2^−1/2> ¹₂.

Letk= 4nand consider the hereditarily finite seth:=

{hk⁰} : k⁰ 6k . Then, we have:

RA(h) =

k

X

k⁰=0

RA({{h_k⁰}})>

k

X

k⁰=0 k⁰is odd

RA({{h_k⁰}})> 1 2 ·k

2 =n .

4 R

on Hereditarily Finite Hypersets

The fully general case corresponds to considering systems of set-theoretic equa- tions such as the ones introduced by the following definition (see also [Acz88]).

Definition 3 (Set systems). A set systemS(ς1, . . . , ςn)in the set unknowns ς1, . . . , ςn is a collection of set-theoretic equations of the form:











ς1={ς1,1, . . . , ς1,m₁} ...

ς_n={ς_n,1, . . . , ς_n,mn},

(8)

with mi>0 fori∈ {1, . . . , n}, and where each unknownςi,u, fori∈ {1, . . . , n}

andu∈ {1, . . . , mi}, occurs among the unknownsς1, . . . , ςn.⁵ The index map I_S ofS(ς1, . . . , ςn)is the map

I_S:

n

[

i=1

{hi, vi |16v6m_i} → {1, . . . , n}

such that I_S(i, v)is the index of the unknown ςi,v in the list ς1, . . . , ςn, for i∈ {1, . . . , n} andv∈ {1, . . . , mi}, namely,ςI_S(i,v)≡ςi,v.

5 Whenmi= 0, the expression{ςi,1, . . . , ςi,m_i}reduces to the empty set{}.

A set systemS(ς₁, . . . , ς_n)is normalif there exist npairwise distinct (i.e., non bisimilar) hypersets ~¹, . . . ,~ⁿ ∈ HF^1/2 such that the assignment ςi 7→ ~ⁱ satisfies all the set equations ofS(ς1, . . . , ςn).

Observe that, by the anti-foundation axiom, the assignment ςi 7→ ~i satisfying the equations of a given normal set systemS(ς1, . . . , ςn) is plainly unique.

From now on, we shall write ~, with or without subscripts and/or super- scripts, to denote a generic (possibly well-founded) hyperset inHF^1/2.

Having in mind the definition (5) ofR^A, to each normal set system we asso- ciate a system of equations inR, called code system, as follows.

Definition 4 (Code systems). Let S(ς1, . . . , ςn) be a normal set system of the form











ς1={ς1,1, . . . , ς1,m₁} ...

ς_n={ς_n,1, . . . , ς_n,mn},

with index mapI_S. The code systemassociated withS(ς₁, . . . , ς_n)is the follow- ing systemC(x₁, . . . , x_n)of equations in the real unknownsx₁, . . . , x_n:











x1= 2^−x1,1+· · ·+ 2^−x1,m1 ...

x_n= 2^−xn,1+· · ·+ 2^−xn,mn,

(9)

wherexi,u is a shorthand forx_IS(i,u), fori∈ {1, . . . , n} andu∈ {1, . . . , mi}.

Normal set systems characterise all possible elements ofHF^1/2 and we shall see that the corresponding code systems characterise their R^A-codes. In fact, we shall prove that every code system admits a unique solution which can be computed as the limit of suitable sequences of real numbers. Terms in such se- quences approximate the final solution alternatively from above and from below.

In case of non-well-founded sets, such approximating sequences are infinite and convergent (to the codes of the non-well-founded sets); additionally, its terms eventually become codes of certain well-founded hereditarily finite sets which can be seen as approximations of the corresponding non-well-founded set.

We begin by formally defining thesetandmulti-set approximating sequences (of the solution) of set systems.

Definition 5 (Hereditarily finite multi-sets).Hereditarily finite multi-sets are collection of elements—themselves multi-sets—that can occur with finite mul- tiplicities.

Hereditarily finite multi-sets will be denoted by specifying their elements in square brackets, in any order, where elements are repeated according to their multiplicities. For instance, the setaoccurs in the multi-set [b, a, b, b] with mul- tiplicity 1, whereasboccurs with multiplicity 3.

Remark 2 A natural embedding of the hereditarily finite sets in the hereditarily finite multi-sets is the following: a multi-set µ can be regarded as a set if and only if each of its elements has multiplicity 1 and, recursively, can be regarded as a set. Thus, in particular, ∅ is both a set and a multi-set.

Definition 6. Let S(ς₁, . . . , ς_n)be a (normal) set system of the form (8), and let I_S be its index map. The set-approximating sequence for (the solution of ) S(ς₁, . . . , ς_n) is the sequence

h~^ji | 1 6 i 6 ni

j∈N of the n-tuples of well- founded hereditarily finite sets defined by

h~^ji |16i6ni:=







h∅ |16i6ni ifj= 0 h{~^j−1i,1 , . . . ,~^j−1i,m_i} |16i6ni ifj >0,

where~^j−1i,u is a shorthand for ~^j−1_IS(i,u), for i∈ {1, . . . , n} andu∈ {1, . . . , mi}.

The multi-set approximating sequencefor (the solution of )S(ς₁, . . . , ς_n)is the sequence

hµ^j_i |16i6n}i

j∈N of the n-tuples of well-founded hereditarily finite multi-sets defined by

hµ^j_i |16i6ni:=







h∅ |16i6ni ifj= 0 h[µ^j−1_i,1 , . . . , µ^j−1_i,m

i]|16i6ni ifj >0, whereµ^j−1_i,u is a shorthand forµ^j−1_I

S(i,u), fori∈ {1, . . . , n} andu∈ {1, . . . , mi}.

Given a (normal) set system S(ς₁, . . . , ς_n), we say that two unknowns ς_i and ς_i⁰, with i, i⁰ ∈ {1, . . . , n}, are distinguished at step k > 0 by the set- approximating sequence

h~^ji |16i6ni

j∈NforS(ς₁, . . . , ς_n) (resp., multi-set approximating sequence

hµ^j_i | 1 6 i 6 ni _j∈

N for S(ς1, . . . , ςn)) if ~^ki 6= ~^ki⁰

(resp., µ^k_i 6= µ^k_i0). Further, we shall refer to ~^ji (resp., µ^j_i) as the (j-th) set- approximation value (resp.,multi-set approximation value) ofς_i at stepj.

Example 1. Consider the following normal set system:

S(ς₁, ς₂, ς₃, ς₄) =











ς₁={ς2, ς₃} ς₂={}

ς₃={ς₃} ς₄={ς₂}.

In this case, we have:m1= 2, m2= 0, m3= 1, andm4= 1; in addition,ς1,1,ς1,2, ς3,1, andς4,1areς2,ς3,ς3, andς2, respectively, so thatI_S(1,1) = 2,I_S(1,2) = 3, I_S(3,1) = 3, andI_S(4,1) = 2.

The first four elements of the set-approximating sequence forS are:

~⁰1,~⁰2,~⁰3,~⁰4

=h∅,∅,∅,∅i ~¹1,~¹2,~¹3,~¹4

=h{∅},∅,{∅},{∅}i ~²1,~²2,~²3,~²4

=h{∅,{∅}},∅,{{∅}},{∅}i ~³1,~³2,~³3,~³4

=h{∅,{{∅}}},∅,{{{∅}}},{∅}i.

Notice that, forj= 2 andj= 3, all the~^ji’s are pairwise distinct. In fact, as a consequence of the next lemma, this is true also for allj >3.

The first four tuples of the multi-set approximating sequence ofS are:

µ⁰₁, µ⁰₂, µ⁰₃, µ⁰₄

=h∅,∅,∅,∅i µ¹₁, µ¹₂, µ¹₃, µ¹₄

=h[∅,∅],∅,[∅],[∅]i µ²₁, µ²₂, µ²₃, µ²₄

=h[∅,[∅]],∅,[[∅]],[∅]i µ³₁, µ³₂, µ³₃, µ³₄

=h[∅,[[∅]]],∅,[[[∅]]],[∅]i.

Also in this case, forj = 2 and j= 3, all the µ^j_i’s are pairwise distinct and this holds also forj >3. Observe that the unknownsςi andςj are distinguished by the multi-set approximating sequence before than by the set-approximating sequence: indeed,µ¹₁6=µ³₁, whereas~¹1=~³1.

All sets in the tuples of a set-approximating system are well-founded hered- itarily finite sets. The initial tuples of a multi-set approximating sequence may containproper multi-sets (as in the above example). However, as a consequence of the following lemma (whose proof can be found in the extended version [CP18]

of this paper), after at most n steps all pairs of distinct unknowns are distin- guished and each tuple contains only proper sets.

Lemma 2. Let S(ς1, . . . , ςn) be a normal set-system, and let

h~^ji | 1 6 i 6 n}i _j∈

N and

hµ^j_i | 1 6 i 6 ni _j∈

N be its set- and multi-set approximating sequences, respectively. The following conditions hold:

(a) fori, i⁰ ∈ {1, . . . , n} andj, k>0, we have

~^ji 6=~^j_i⁰ =⇒ (~^j+k+1i 6=~^j+k+1_i⁰ ∧µ^j_i 6=µ^j_i0)

(namely, if at stepj >0 two unknowns are distinguished by the set-approx- imating sequence, then they are also distinguished by the multi-set approx- imating sequence; in addition, they remain distinguished in all subsequent steps);

(b) forj, k>0 andi∈ {1, . . . , n}, we have

~^ji =~^j+1i =⇒ ~^ji =~^j+k+2i

(namely, as soon as~^ji =~^j+1i holds for some j >0, the set-approximation value ofςi remains unchanged for all subsequent steps);

(c) fori, i⁰ ∈ {1, . . . n} andj>n, we have

(i6=i⁰ =⇒~^ji 6=~^ji⁰) ∧ hµ^j_i |16i6ni=h~^ji |16i6ni

(namely, starting from stepn, all pairs of distinct unknowns are distinguished and the set- and multi-set approximating sequences coincide).

Point a) in the above lemma suggests that even though set and multi-set approximating sequences will eventually “separate” all solutions of a set sys- tem, multi-set can introduce inequalities first. This is our first motivation for extending the notion of R^A-code to multi-sets and use such code-extension to approximate regularRA-codes. The second motivation is given in Remark 3.

Definition 7. Given a normal set system S(ς₁, . . . , ς_n) and its multi-set ap- proximating sequence

hµ^j_i |1 6i6n}i

j∈N, we define the corresponding code approximating sequence

hR^µA(µ^j_i)| 16i6n}i _j∈

N by recursively putting, for i∈ {1, . . . , n} andj∈N:

(

R^µA(µ⁰_i) := 0 R^µA(µ^j+1_i ) :=Pm_i

u=12^{−RµA(µji,u)}. (10) We also define the corresponding code increment sequence

hδ^j_i |16i6ni

j∈N

by putting, for i∈ {1, . . . , n} andj∈N:

δ_i^j:=R^µA(µ^j+1_i )−R^µA(µ^j_i). (11) Plainly,R^µA(µ^j_i)>0, for alli∈ {1, . . . , n}andj ∈N.

Remark 3 Consider a normal set system S(ς₁, . . . , ς_n) and its solutions

~1, . . . ,~n. The values R^µA(µ¹_i)and R^µA(µ¹_i0) are equal if and only if |~i|=|~i⁰|.

The values R^µA(µ²_i) and R^µA(µ²_i0) are equal if and only if the multi-sets of the cardinalities of elements in~ⁱ and~ⁱ⁰ are equal. The valuesR^µA(µ³_i)andR^µA(µ³_i0) are equal if and only if the multi-sets of multi-sets of the cardinalities of elements of elements in ~i and~i⁰ are equal, and so on.

In preparation for the proof of the existence and uniqueness of a solution to the code system associated with any normal set system, we state the technical properties contained in the following lemma, whose proof can be found in the extended version [CP18] of this paper.

Lemma 3. Let S(ς1, . . . , ςn)be a normal set system of the form











ς1={ς1,1, . . . , ς1,m₁} ...

ςn={ςn,1, . . . , ςn,m_n}, with index map I_S, code approximating sequence

hR^µA(µ^j_i)| 16i 6n}i _j∈

N, and code increment sequence

hδ^j_i | 16i 6n}i _j∈

N. Then, for i ∈ {1, . . . , n}

andj∈N, the following facts hold:

(i) R^µA(µ^j+1_i ) =δ_i⁰+· · ·+δ_i^j, (ii) δ⁰_i =mi,

(iii) δ^j+1_i =Pm_i

u=12^{−RµA(µji,u)}·(2^−δi,uj −1), (iv) δ^2j+1_i 606δ^2j_i ,

(v) |δ_i^j+1|6|δ^j_i|, and (vi) lim_j→∞δ_i^j= 0.

We are now ready to prove the main result of the paper.

Theorem 4. For any given normal set system, the corresponding code system admits always a unique solution.

Proof. Let S(ς1, . . . , ςn) be a normal set system of the form (8), and let hR^µA(µ^j_i)|16i6n}i _j∈

Nand

hδ^j_i |16i6n}i _j∈

Nbe its code approximat- ing sequence and code increment sequence, respectively. Also, letC(x₁, . . . , x_n) be the code system











x1= 2^−x1,1+· · ·+ 2^−x1,m1 ...

xn= 2^−xn,1+· · ·+ 2^−xn,mn, associated withS(ς₁, . . . , ς_n).

We first exhibit a solution of the system C(x1, . . . , xn) and then prove its uniqueness.

Existence: By Lemma 3(i),(iv),(vi), using the Leibniz criterion for alternating series, it follows that each of the sequences {R^µA(µ^j_i)}j∈N is convergent, for i∈ {1, . . . , n}. Let us putα_i := limj→∞R^µA(µ^j_i), fori∈ {1, . . . , n}. From (10), we have

R^µ_A(µ^j+1_i ) =

m_i

X

u=1

2^{−RµA(µji,u)}, forj∈N.

Then, by taking the limit of both sides asj approaches infinity, it follows that

α_i=

m_i

X

u=1

2^−αi,u,

fori∈ {1, . . . , n}, whereαi,uis a shorthand forαI_S(i,u), withI_S the index map ofS(ς1, . . . , ςn), proving that then-tuplehα1, . . . , αniis a solution of the code systemC(x1, . . . , xn).

Uniqueness: Next we prove that the solution hα1, . . . , αni is unique. Let hα⁰₁, . . . , α⁰_ni be any solution of the code system C(x₁, . . . , x_n). To show that hα⁰₁, . . . , α⁰_ni=hα1, . . . , α_niit is enough to prove that

R^µ_A µ^2j_i

6α⁰_i6R^µ_A µ^2j+1_i

, forj∈Nandi∈ {1, . . . , n}, (12)

holds. Indeed, from (12), it follows immediately αi= lim

j→∞R^µA µ^2j_i

6α⁰_i6 lim

j→∞R^µA µ^2j+1_i

=αi, for everyi∈ {1, . . . , n}, showing thathα⁰₁, . . . , α⁰_ni=hα1, . . . , αni.

We prove (12) by induction onj, for alli∈ {1, . . . , n}.

For the base casej = 0, we observe that sinceα⁰_i =Pm_i

u=12^−α0i,u (where, as usual,α⁰_i,u stands forα⁰_I

S(i,u)), then R^µA µ⁰_i

= 06α⁰_i6m_i=R^µA µ¹_i

, fori∈ {1, . . . , n}.

For the inductive step, let us assume that R^µA µ^2j_i

6α⁰_i6R^µA µ^2j+1_i

, fori∈ {1, . . . , n}, (13) holds for some j ∈ N, and prove that it holds for j + 1 as well. From (13) and recalling that α⁰_i =Pm_i

u=12^−α0i,u, the following inequalities hold, for every i∈ {1, . . . , n}andu∈ {1, . . . , mi}:

R^µA µ^2j_i,u

6α⁰_i,u6R^µA µ^2j+1_i,u 2^−RµA(^µ2j+1_i,u )

62^−α0i,u62^−RµA(^µ2j_i,u)

m_i

X

u=1

2^−RµA(^µ2j+1i,u )6

m_i

X

u=1

2^−α0i,u6

m_i

X

u=1

2^−RµA(^µ2ji,u) R^µA µ^2j+2_i

6α⁰_i6R^µA µ^2j+1_i .

The inequalities on the last line (for i∈ {1, . . . , n}) imply in particular that we have

R^µA µ^2j+2_i,u

6α⁰_i,u6R^µA µ^2j+1_i,u ,

for everyi∈ {1, . . . , n}andu∈ {1, . . . , m_i}. Hence, by repeating the very same steps as above, one can deduce also the inequalities

R^µA µ^2(j+1)_i

=R^µA µ^2j+2_i

6α⁰_i6R^µA µ^2j+3_i

=R^µA µ^2(j+1)+1_i ,

for i ∈ {1, . . . , n}, proving that (13) holds for j+ 1 too. This completes the induction, and also the proof of the theorem.

Remark 5 To show that the codeRA is well-defined over the whole HF^1/2, we proceed as follows. Given a hereditarily finite hyperset~∈HF^1/2, let ~1, . . . ,~n

be the distinct elements of the transitive closuretrCl({~})of{~}, where ~1=~. Then we have











~1={~1,1, . . . ,~1,m1} ...

~n ={~n,1, . . . ,~n,m_n}

(14)

for suitable hypersets~i,j∈ {~1, . . . ,~n}, withi∈ {1, . . . , n}andj∈ {1, . . . , mi}.

Consider the set systemS(ς₁, . . . , ς_n)











ς1={ς1,1, . . . , ς1,m₁} ...

ς_n={ς_n,1, . . . , ς_n,mn},

associated with (14), where ςi,j =ς` iff ~i,j =~`, for all i, ` ∈ {1, . . . , n} and j ∈ {1, . . . , mi}. Plainly, S(ς1, . . . , ςn)is normal and ~1, . . . ,~n is its solution.

By Theorem 4, letα1, . . . , αn be the solution to the code system associated with S(ς1, . . . , ςn). ThenRA(~i) =αi, fori∈ {1, . . . , n}, and, in particular,RA(~) = RA(~1) =α1. Further, it is immediate to check that, for every normal set system S⁰(x⁰₁, . . . , x⁰_m) containing ~ in its solution, say at position ¯k∈ {1, . . . , m}, if α⁰₁, . . . , α⁰_m is the solution to the corresponding code system, then α⁰_¯k = α1. In other words, the valueRA(~)computed by the above procedure is independent of the normal set system used. By the arbitrariness of ~, it follows that RA(~) is defined for every hyperset~∈HF^1/2.

5 A first step towards injectivity

As already remarked, the problem of establishing the injectivity of the map RA is still open. As an initial example, we provide here a very partial result.

However, the arguments used in the proof below do not seem to easily generalize even to the narrower task of proving the injectivity ofRA over thewell-founded hereditarily finite sets only.

Lemma 4. For alli∈N, we have:

(a) R^A(hi)6=R^A(hi+1), and (b) RA(hi)6=RA(hi+2),

wherehj is thej-th element of HFin the Ackermann ordering.

Proof. Leti∈N. From (iii) and (iv) of Lemma 1 and from (4), we have RA(hi+1) =RA(hi)−RA(h₂low(i)−1) +RA(h₂low(i)). (15) Ifiis even, thenlow(i) = 0, and therefore

R^A(h₂low(i)−1) = 06= 1 =R^A(h₂low(i)). On the other hand, ifiis odd, thenlow(i)6= 0, and so, by (3):

RA(h₂low(i))<1 =RA({h0})6RA(h₂low(i)−1).

In any case, we haveRA(h₂low(i))−RA(h₂low(i)−1)6= 0, proving (a), by (15).

Concerning (b), we begin by putting

∆j:=RA(h₂j)−RA(h₂j−1),

forj∈N. Let us show that∆_j6=−1, for allj∈N. To begin with, forj= 0,1,2, we have:

∆₀=RA(h₁)−RA(h₀) = 16=−1

∆₁=RA(h₂)−RA(h₁) =RA({h₁})−RA(h₁) = 2⁻¹−1 =−1 2 6=−1

∆2=R^A(h4)−R^A(h3) =R^A({h2})−R^A({h0, h1})

= 2⁻²⁻¹−

1 +1 2

= 1

√2−3 2 6=−1. In addition, forj >2, we have{h0, h1, h2} ⊆h₂j−1, and therefore:

RA(h₂j−1)>RA({h0, h1, h2}) = 2^−RA(h0)+ 2^−RA(h1)+ 2^−RA(h2)

= 2⁻⁰+ 2⁻¹+ 2⁻²

= 1 +1 2 + 1

√2 >2.

Hence, we have ∆j 6= −1 also for j > 2, sinceRA(h₂j) = 2^−RA(hj) < 1. But then, from (15), we have, for everyi∈N:

RA(hi+2)−R(hi) =RA(hi+2)−R(hi+1) +RA(hi+1)−R(hi)

=∆_low(i+1)+∆_low(i)6= 0,

since eitheriori+ 1 is even and so either∆_low(i)= 1 or∆_low(i+1)= 1, whereas, as we proved above, ∆_low(i)6=−16=∆_low(i+1). This proves (b), completing the proof of the lemma.

Remark 6 While the value of RA µ⁰_i

is 0 for any i ∈ {1, . . . , n}, the value of RA µ¹_i

—first approximation of RA(µ_i)—is the cardinality of µ_i, and the subsequent approximations oscillate within the interval [0,|µi|].

Conclusions

By turning labels into sets, the encoding proposed in this paper can be used on a variety of structures, going from labelled graphs to Kripke models. This can be done in many different ways and in [DPP04,PP04] the reader can find a rather general—albeit non optimised—technique to carry out this label elimination task. A label elimination performed to optimise code computation (or its form) is under study.

The algorithmic side ofRA is also under study. A possible direction towards its usage—for example in bisimulation computation—starts from the observation that only the computation of a bounded number of digits is actually necessary to realise all the inequalities in any given set system.

Acknowledgements

The authors thank Eugenio Omodeo for very pleasant and fruitful conversations, and an anonymous reviewer for his/her comments.

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