URL: http://www.elsevier.nl/locate/entcs/volume67.html 14 pages

A Completeness Result for Relation Algebra with Binders

R.P. de Freitas1

COPPE-UFRJ, Rio de Janeiro-RJ Brasil

J.P. Viana2

COPPE-UFRJ, Rio de Janeiro-RJ IM-UFF, Niteroi-RJ Brasil

Abstract

This paper presents an axiom system to Relation Algebra with Binder. This is a hybrid formalism introduced to fix the equipollence problems in Tarski's Relational Calculus. RAB does not fill the requirements imposed by Tarski, but as it was showed, that is a very interesting alternative to First-Order Logic when concerning the symbolization of statements about binary relations. The presentation of an ax-iomatization of RAB contributes to its development as a formalism to the inference of facts about relations. Relational calculi have been extensively applied as much in Computer Science as in Natural Language studies, and the classical theory has been extended to many directions. Since RAB seems to provide new perspectives on the application of both Relation Algebra and Hybrid Logic, the work in this paper could be considered as a contribution to that development.

Key words: relation algebra, hybrid logic, relation algebra with

binders, completeness

1 Relational calculi and implicit sorting

In this section Tarski's Relational Calculus (TEC) is reviewed. The important equipollence problems related to it are discussed as well as how sorts have been implicitly used in connection with TEC. A comprehensive review of TEC is

1 Email: naborgesQcos .ufrj .br

2 Email: petrucioQcos .ufrj .br

©2002 Published by Elsevier Science B. V.

Relational calculi are adequate formal systems to talk and reason about relations. Two interesting classes of relational calculi about homogeneous binary relations were introduced by A. Tarski in [21].

Consider as fixed a set {Ri : i E /}, whose symbols are intended to denote binary relations. The first calculus, that we call Elementary Theory of Binary Relations (ETBR), is just an extension by definition of the first-order language with identity taking {Rjt : i E 1} as a set of binary predicate symbols. Details can be found in [22]. Observe that in ETBR there are, in a sense, two sorts of variables: the individual and the relational variables.

The second, which we call Tarski's Relational Calculus (TRC) is a version of the equational theory of relation algebras having {Ri : i E 1} as a set of variables, the Booleans U and - (union and complementation) and the Peirceans | and (composition and conversion) as operations, and Id (identity) as a distinguished element. For the theory of relation algebras we suggest [10,15], for TRC, see [22]. Observe that in TRC there is only one sort of variables.

One important feature of TRC is the way in which it is possible to rewrite the sentences of ETBR without referring to individuals. For example, we can say that a relation R C u x u is a bijeetive function saying that it satisfies the equations3 l> 1 | R C ld„ (R is functional), R \ E = E (R is total), R \ l> 1 C Idu (R is injeetive), and R| E = E (R is surjeetive). Unfortunately, not all first-order sentences of ETBR can be expressed as an equation of TRC. In 1915, L. Lowenheim [11] reported a Korselt's proof that the assertion UR domain has at least four elements" is not equivalent to any equation of TRC. Thus, TRC is weaker than ETBR in means of expression.

Tarski started investigating if this inequipollenee was a consequence of his having selected the wrong relation algebraic formalism, i.e., if it was possible to change the chosen operations in order to obtain equipollenee in means of expression with ETBR. He imposed a very general requirement on the semantical interpretation of the operators allowed in enriching expressive power. He wanted the chosen operators to be "logical" in the following sense: an operation on relations on a set S is logical if it is invariant under all permutations of S; a symbol of operation of a relational calculus RC is logical if it denotes a logical operation in all models of RC. Around 1940, Tarski proved the following strong negative result [22].

Theorem 1.1 Given any formalism obtained by adjoining finitely many logical operators to TRC, there exist sentences of ETBR that are not equivalent to any equation of TRC.

Thus, the choice of TRC operations was not misguided: there is no finitarv formalism similar to TRC which captures the full expressive power of ETBR.

Besides being weaker than ETBR in means of expression, another drawback of Tarski's system is that it is incomplete, i.e., there are true equations

3 Note that inclusion R C S is equivalent to equation R ■ S = 0.

about relations (involving the Booleans and the Peireeans operations) that can not be proved in TEC, In 1950, E, Lyndon [12] showed that the inclusion:

flii^nSi&nTiTa C Rl(R^lSlf]R2S2lf](R^lTlf]R2T^l)(T^lSlf]T2R^l))R24 ,

though true for all binary relations /t'i. R2. >',. ,s'2. I]. and T2, can not be proved in TEC.

As S. Givant indicated in [7], there is a trivial way of enhancing the deductibility power of TEC in order to obtain a system that is complete to the true equations about relations. But the axiomatization involved is not given by a finite set of axiom schemata. In fact, E. Maddux in [14] proved the following strong result:

Theorem 1.2 There is no set of sentences of TRC obtained from instances of only a finite number of schemata that is complete to the true equations about relations.

So, it emerges the two important problems of extending TEC in means of expression and proof:

Expressivity problem. Find a natural and intrinsically interesting infinite sequence of logical operators to extend TEC so as to achieve equipollence which ETBE in means of expression.

Deductivity problem. Find finitely many logical operators and finitely many logical axiom schemata to extend the vocabulary of TEC so that the resulting formalism is equipollent with ETBE in means of proof.

These are very difficult problems. Some partial results are reported in [22] and [7]. An up-to-date reference containing a partial solution is [19].

The fulfillment of the restrictions in the statement of the expressivity and deductivity problems is very important to a possible solution to be considered as the solution [2]. But there are some interesting solutions to both problems if we consider an infinite number of non-logical operators. In fact, if we consider each individual variable as a zero-ary operator, whose meaning could change from an interpretation to another, one can see ETBE as a degenerated solution to both of the problems by the introduction of an implicit sorting in TEC.

A review of more restrictive solutions, that use sorted formalisms implicitly, is here presented.

In [24], W.W. Wadge presents a natural deduction system to the derivation of formulas of the form xRy, where x and y are individual variables and R is a relational term. Wadge proves that his system is complete to all true equations of TEC. The main features of Wadge's relational calculus are the use of individual variables and the absence of any kind of connective or quantifier.

In [8], M.C.B. Henessv presents an analytic tableaux to refutation of formulas of the form (a, /3) E T, where a and /3 denote the sort of a relational

4 RS is R I S

term T. Henessy proves that his system is complete to all true equations of a relational calculus to homogeneous relations of all arities. As a corollary he proves the compactness of his calculus. The main features of Henessy's relational calculus are the implicit use of sequences of variables and the presence of projection operators.

In [20], Schonfeld presents a sequent calculus to a version of ETBR, He proves completeness and an upper bound for proof length.

In [13], R, Maddux presents a sequent calculus to the derivation of sequents containing formulas of the form xRy, where x and y are individual variables and R is a relational term, Maddux proves that his system is complete to all true equations of TRC and studies semantical questions. The main features of Maddux's relational calculus are the use of individual variables and the absence of any kind of connective or quantifier.

Finally, in [16], M, Marx introduces the Relation Algebra with Binders (RAB) by extending TRC with the machinery of the down-arrow hybrid logic [1,4], The main features of Marx's system are the use of individual variables and the implicit use of quantification by the presence of the down-arrow binder operator, Marx presents a semantical proof of the equipollenee of his system with ETBR in means of expression and proof and leaves completeness as a problem.

Since TRC is a kind of algebraic system, RAB is introduced as an algebraic formalism, i.e., its formal expressions are terms and equations between terms. In RAB, the validity of equations can be reduced to a certain property of terms. Hence, afterwards, our objective is to support RAB with a set of axioms and derivation rules, defining a property b of provability, in such a way that for any term a, we have b a iff a has the referred property. An important point of the approach is that, instead of reducing the completeness problem to the first-order case, using results of [16], it was decided to present a "direct" proof, adapting techniques from Hybrid Logic [6], This choice is an attempt to take the best from the algebraic and modal worlds.

The remainder of this paper is organized as follows. In Section 2, a version of Relation Algebra with Binders [16] is presented. In Sections 3 and 4, axioms and rules are presented and their completeness is proved. Section 5 is a conclusion containing some specific themes and questions,

2 Relation Algebra with binders

In this section syntax and semantics of RAB are described. Also some simple facts that are useful in the statement of the axioms and in the completeness proof are presented.

The basic ideas underlying syntax is that RAB is an extension of Relation Algebra (RA) by introducing the hybrid I apparatus, RA has a set of variables ranging over binary relations together with a set of symbols of operations denoting the Boolean and Peircean operations on relations. The hybridization

of the RA language presented in [16] involves making two changes in RA syntax. First, take a new set of variables and consider them as atomic terms. Second, add the down-arrow operator, that is used to bind the new variables. The alphabet of RAB on a set {x¿ : i E N} consists of the following symbols:

(i) Relational variables r¿ : i E N.

(ii) Individual variables Xi : i e N.

(iii) Booleans + and

(iv) Peirceans ;, " ', and 1',

(v) Down-arrow binders \.x, where x is an individual variable. The terms of RAB are defined as follows:

a := x | r | 1' | o¿i + «2 | oT \ ai, | oc'" | J,xa.

The set of relational variables is denoted by RVar and the set of individual variables by IVar,

Following the hybrid paradigm, individual variables are considered as denoting relations instead of individuals. Also a distinction between free and bound occurrences of individual variables in terms is drawn, and substitutions of individual variables by individual variables in terms are performed. The set of free individual variables of a term a is denoted by free (a). When one makes substitutions for logical purposes, one has to guard against accidental binding. Once more the hybrid paradigm is followed, maintaining the intuition behind this essentially first-order.

Given a domain D of individuals, each term denotes a binary relation on D. The denotation of relational variables is unrestricted but that of individual variables is restricted to atoms below identity on D. This is an indirect manner to denote the elements of D themselves. To handle the fact that individual variables may become bound, their denotations are given by a separate assignment function in a manner familiar from first-order logic,

A structure is a pair S = (D,I), where D is a non-empty set, called the domain of S, and I : RVar —2DxD is a function, called the interpretation of RVar in S, An assignment of IVar in S is a function g : IVar —D. Given x E IVar and a E D, the a variant of g in x is the assignment (a/x)g : Ivar —D such that for each y E IVaR, (a/x)g(y) = a if x = y, and g(y) otherwise.

The denotation of a term a under a structure S = (D, I) and assignment g, denoted by d(a)g, is defined by:

(i) d(x)S = {(g(x),g(x))}.

(ii) d(r)^ = J(r).

(iii) d(l')f = IdD.

(iv) d{ot\ + a2)g = d(ai)® U d(a2)^.

(v) d(ai;a2)f = d(a¡i)f I d(a2

(vi) d(a = d(a)g.

(vii) d(a-")S =

(viii) d(lxa)S = {(o, b) E D x D : {a, b) E d(a)fa/x)g}.

The definition of denotation ensures that the new variables in IVar range over the set of all relations of the form {(0,0)}, where a E D. Also, given a term ^xa, the operator lx binds x E IVar storing the value a of the first coordinate of the pair (o, b) where belongness to a is being evaluated,

A term a is non-empty if there are S, g, and a,b E D such that (a, b) E d(a)g. We write d(T)g for : a E T}, A set of terms T is non-

empty if there are S, g, and a,b E D such that (a, b) E d(T)g. A term a is a consequence of a set of terms T, denoted by T |= a, if for any S and g, one has d(T)g C d(a)g. Two terms a and /3 are equivalent, denoted by n = >. if for any S and g, one has d(a)g = d(fi)g.

The following arithmetical properties are easvlv verified:

(i) 1*0 «0

(ii) \-x 1 ~ 1

(iii) I I- r*^/ 1 ' •i'X X ~ 1

(iv) ixVtt y

(V) \-x\-y & ~ \-y\-x &

(vi) •i'ï'i'a; ® ~ ■i'X ®

viii) lx(a + (3) ~ X+lxP

(ix) ix(a • p) & lx a ixfj

M \-x \-x P) ~ \-x

(xi) lx(ar,x) « lxa V

(xii) rp ' ry r>J ry JU ^ JU ,XJ

xiii) x\y ~ x • y

xiv) ry '—' r>j ry Ju r«£,

The following basic propositions hold directly.

Proposition 2.1 (Agreement Lemma) Let S be a structure. For all assignments gi and g2 in S and all term a, if g\ and g2 agree on free (a), then d(a)sgi = d(a)f2.

Proposition 2.2 (Substitution Lemma) Let S be a structure. For all assignment g, all x,y E IVar and all term a, if y is substitutable for x in a, then d(a)fg(y)/x)g = d((y/x)a)Sg.

In some hybrid systems, albeit individual variables to denote individuals of the model, one also has a special operator that allows us to jump to a point denoted by a variable and see if some formula is true there [4], These

operators work as a bridge between syntax and semantics permitting, in a sense, the internalization of the satisfaction relation. In RAB, an operator like this can be introduced by definition, as one can see from the following proposition. The term V + is denoted by 1, Then = D x D, for all structure S and assignment g.

Proposition 2.3 Let i,|/G IVar and a be a term. Then, for all structure S, assignment g and a,b e D, (a, b) e d{ 1; x\ a; y; 1)® iff (g(x), g{y)) G d{a)^.

Based in proposition 2,3, the term 1; x; a; y; 1 is denoted by xay.

3 Axioms and rules

This section presents axioms and rules. The duality existing between RA and Arrow Logic [17,18] is used to formulate the axiom system. In this context each axiom can be read as a term of RAB and as a formula of the extended modal logic obtained from RAB by the known method of taking the algebra of complexes [5]. This permits the use of the hybrid modal machinery to prove the completeness result.

The axiom system is defined by the following axioms and rules. Explanation will be given afterwards. The term l- is denoted by 0, (a- + by a ■ [3 a~ + [3 by a < /3, (a < [3) • {[3 < a) by a « /3, (a-; [3by a J /3, by by fxa, and x\ 1; y by xy.

RA) RA axioms [23] and algebraic versions of axioms and rules for a normal

modal logic.

Below-ld) x < 1'.

Non-empty) l<l;a;;l.

Atom) (x ■ y « 0) + (x « y).

Identifier) xy • a < xay.

Rev) xa'-'y « yax.

Comp) xaz • zßy < x(a; ß)y.

Bind) x \.zoty ~ x(x/z)ay, ii x is substitutable by z in a.

Self-Dual) (^a-)- « \,xa.

Ql) -i-x (a < ß) < {a < ixß), if x free (a),

Q2) lxa • (y; 1) < (y/x)a, if y is substitutable for x in a.

Q3) (z; 1 < a) < Ix®.

xay■yßz < 7

Paste) -, if y is new,

x(a] ß)z < 7

Axioms Below-ld, Non-empty, and Atom say that individual variables have to be interpreted as atoms below identity. Axioms Rev, Comp, and Identifier say that we are interpreting terms in a representable relation algebra. Axiom

Bind expresses, syntactically, the down-arrow operator semantics. Axiom Self-Dual says that the down-arrow operator is the dual of itself. Finally, axioms Ql, Q2, and Q3 say that the down-arrow operator is a quantifier. Rule Paste is a version of the Paste Rule of Hybrid Logic [6] to the composition operator, A term a is a theorem, denoted by b a, if there exists a sequence of terms («i,,,,, an), called a proof of a, such that an is a and, for any i = 1,,,,, n, a>i is an axiom or ajt is a consequence of previous in the sequence by the use of some rule of inference, A term a is derived from a set of terms T, denoted by r b a, if there exists a sequence of terms («i,,,,, an), called a derivation of a from T, such that an is a and, for any i = 1,,,,, n, a>i is a theorem, c^ e T, or a>i is a consequence of previous in the sequence by the use of Modus Ponens,

Theorem 3.1 (Soundness Theorem) IfV b a, then T |= a.

The proof of the Completeness Theorem uses the results in the sequel. Results that are inherited directly from the duality Arrow Logic/Relation Algebra are referred to as RA,

The following terms are theorems: Reflex) xl'x. Simet) xl'y < yl'x. Trans) xl'y • yl'z < xl'z. Cong) xl'z ■ yl'w ■ xry < zrw. Var) xl'z ■ xl'y « xzy.

Q4) lxa pa ly (y/x)a, if y does not occur in a.

Q5) ^ (xy < a) < (1; y < lxa).

Q6) tx (1; y < a) <

Q7) ^xa « a, if x 0 free(a),

Q8) -fxa « a, if x £ free(a).

The following rule is derived: a

Nec f)-, where x is any individual variable,

The term obtained from a term a by the replacement of some occurrences of a term /3 in a by a term 7 is denoted by [7//?]a. Since one has a normal modal logic, the Replacement Lemma holds.

Lemma 3.2 (Replacement) J/b 7 « /3, then b [7//?] (X PH (X.

The Alphabetic Variant Lemma is derived using the Replacement Lemma and theorem Q4,

Lemma 3.3 (Alphabetic Variant) Let IVar and a be a term. There

exists an alphabetic variant a' of a such that:

(i) free (a) = free (a').

(ii) b a « a'.

(iii) y is substitutable for x in a!.

4 Completeness theorem

In this section it is proved that, when a is a consequence of T, then a can be derived from T in the system defined in Section 3, The Hybrid Logic techniques are used to derive the completeness result from a model existence lemma. The proof has two stages. First, it is proved that all sets of terms satisfying some special conditions are non-empty. Second, it is proved that all consistent set of terms can be extended in an adequated language to a set satisfying the special conditions,

A set of terms T is closed if for any a,rba:iffa:er, r is consistent if r \f 0, r is complete for minus if for any n. I'hn or I'hn . T has witnesses if for all x, y, «i, a2, if T b x(a>i; a2)y, then there is some variable z such that T b xaiz and T b za2y. A term xy is an identifier of T when xy E T, T is identified if there are variables x and y such that xy is an identifier of T,

Let T be a closed set of terms. An individual variable x is congruent to an individual variable y modulo T, denoted by x ~r y, or simply x ~ y, lixl'y E T, By theorems Reflex, Simet, and Trans, ~ is an equivalence relation. Hence, IVar is partitioned by ~ in equivalence classes modulo T, Given x E IVar the equivalence class of x is denoted by x.

The canonical structure associated to a closed, consistent, complete to minus, and with witnesses set of terms T is the structure Sr with domain D = {x : x E IVar} and interpretation I : RVar —2DxD such that J(r) = {(x,y) : xry E T}, for any r 6 RVar, The canonical assignment is the function g : IVar —D such that g(x) = x, for any x E IVar, Theorem Cong assures that Sr is well defined.

Lemma 4.1 (Truth Lemma) IfT is closed, consistent, complete to minus, and with witnesses, then for all x, y, (x, y) E d(a)^r iff xay E T.

Proof. By induction on the structure of a, using Substitution Lemma (SL), Alphabetic Variant Lemma (AV), Replacement Lemma (RL), axioms Rev, Comp, and Bind and theorem Var,

Case z) (x,y) E d(z)^r iff g(z) =x = y (def d)

iff z = x = y (def g)

iff xl'z,xl'y E T (def ~)

iff xzy E T (Var)

Case r) (x, y) E d(r)®r iff (x, y) E I(r) (def d)

iff xry E T (def Sr)

Case 1') (x,y)Ed(l')®r iff x = y (def d)

iff x ~ y (def x) iff xl'y E T (def

Case a-') (x, y) E iff {y,x)Ed(a)^r (def d)

iff yaxET (I.H.)

iff xoT'y E T (Rev)

Case a;/3) (x,y) E d(a\ /3)®r iff (x,z) E d(a)^r, (z,y) E d(/3)fr

iff ./■(>:. : >1/ E I' (I.H.) iff x(a; (3)y E T (Comp, T has witn.)

Case a) (x, y) E d(lza)s/ iff c r (rUf ri\

The proof of the next lemma follows closely the first-order logic analogous.

Lemma 4.2 If T is closed and consistent, then there is some A D T such that A is closed, consistent and complete to minus.

Lemma 4.3 If T consistent, then there are some set IVar' of individual variables and some set A of terms on IVar', such that IVar' D IVar, A D T, and A is closed, consistent and with witnesses.

Proof. Define (IVar„ : n E N) and (A„ : n E N) simultaneously by the following rules:

(i) IVari = IVaR and Ax = Cn(r);

(ii) IVar„+1 = IVar„ U {za : a E A„, a is of the form x(ai, a2)y} and A„+1 = Cn{A„ U {xctiza, zaa2y : ot E A„, a is of the form x(a>i; a2)y}).

Observe that IVar„+1 and A„+1 are constructed from IVar„ and A„ by considering each term a in A„ of the form x(a>i; a2)y and taking a new individual variable za not in IVar„ and the terms xa\Za and zaa2y.

For any n E N, A„ C A„+i, A„ is closed and consistent. In fact, Ai = T is consistent. If A„ is consistent and A„+1 is not, then there are terms Xiaiyi, ViPiZi,..., xnanyn, y„p„zn E A„+i \ A„ such that:

By Paste, since %ji is new, we have A„, xi(ai; (3i)zi,..., xn(an; (3n)zn b 0. Since [xi(ai] (3i)zi : 1 < i < n} C A„, then A„ b 0, a contradiction.

Now, take IVar' = (J;) . IVar,, and A = (JneN A, Then Ivar' D Ivar, A D T, A is closed, consistent, and with witnesses: if A b x(a\;a2)y, then there is

A„, xiaiyu yifczi,..., xnanyn, yn(3nzn b 0.

some n E N such that An b x(ai; a2)y. Since An is closed, x(ai; a2)y e An. Hence, .rn i . :,,n-.// e An+i C A, □

Lemma 4.4 If T is consistent, then there are some set IVar' of individual variables and some set A of terms on IVar', such that A D T and A is identified and consistent.

Proof. Define IVar' = IVar U {x, y}, with x, y 0 IVar and x ^ y. Define A = ru{a;y}. One has that IVar' D IVar, A D T, and A is identified and consistent.

In fact, if A b 0, then, by EA, h xy < (ji.....jnwith 7l,... ,7n E I'.

By Nec b \.x (xy < (7l.....7n)-). By Q5. b 1:// < (-,.....7n)-. By

Nec f, b % (1; y < 4 (7l.....7n)-). By Q6, b f, (4 (7l.....7n)-). By Q7

and Q8, since x and y do not occur in T, h (7l.....By EA, T h 0, a

contradiction, □

Now Lemmas 4,2, 4,3, and 4,4 are put together to get the desired result about the extension of consistent sets.

Theorem 4.5 If T is consistent, then there is some A D T such that A is identified, closed, consistent, complete to minus and with witnesses.

Proof. Given T, apply Lemma 4,4 and take r'DT identified and consistent. Then define (A„ : n E N) by the following rules:

(i) Ai = Cn(r'),

(ii) A2„+i is the result of apply Lemma 4,2 to A2„,

(iii) A2„+2 is the result of apply Lemma 4,3 to A2n+i.

Now take A = (J;) ..A,,. One has that A is closed, consistent, and complete to minus. In fact, let a be a term such that A Ifa. Hence, a 0 A, So, given n E N, a 0 A2„, Since A2„ is complete to minus, a- e A2„, So, n e A2„ and n e A. Finally, Aha", Moreover, A has witnesses. In fact, let x, y, ai, a2 be such that A b x(ai; a2)y. Hence, there is some n E N such that A2„+1 b x(ai;a2)y. Since A2„+1 has witnesses, there is some individual variable z such that A2„+i b xa\z and A2„+i b za2y. Hence, A b xa\z and A b za2y. □

Corollary 4.6 (Model Existence Theorem) If T is consistent, then T is non-empty.

Proof. Let A be an identified, consistent, closed, complete to minus, and with witnesses set of terms including T, Let xy be an identifier of A, Let a be a term in T, Then a E A. Hence, by axiom Identifier, xay E A. Therefore, by the Truth Lemma, (x,y) E d(a)^A. □

Finally, it follows the Completeness Theorem,

Corollary 4.7 (Completeness) IfV \= a, then T b a.

5 Perspective

Relation Algebra with Binders (EAB) is an extension of the classical language of relation algebras with the machinery of the down-arrow hybrid binder. This extension increases both the expressive and proof powers of the language. In [16] it was proved that any first-order property of binary relations can be expressed in the extended language. This paper provides a completeness result, showing that any such property that is valid can also be proved. The aim was not to reduce the completeness problem to that of first-order logic, using results of [16], but to present a "direct" proof, adapting the well-known method of canonical models. In fact, the axioms presented here are very simple, rendering to the system a very elegant formulation. The examples discussed in [16] provide new perspectives on both Relation Algebra and Hybrid Logic, The work in this paper is an attempt to contribute to this development.

Now we present some specific themes and questions on RAB that could be developed.

The most immediate question about the axioms presented in this paper is about their independence. How many axioms and rules could be deleted without losing in proof power?

As a corollary of the expressivity result, it follows a version of Craig's interpolation theorem for RAB, A future task is the definition of a sequent calculus for RAB in order to obtain a constructive proof of this interpolation theorem.

One important feature of RAB is the absence of formulas, i.e., RAB is a system of terms that is equipollent to ETBR in means of expression and proof. In [9] a system of terms that is equipollent to first-order logic was introduced, A very interesting topic to investigate could be the relationship between RAB and that system.

Acknowledgment

Thanks are due to M, Marx for many stimulating discussions and for the suggestion of a lot of improvements.

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