Zeitschr. f. math. Log& und arundlagen d. Malh. Rd. 28, S. 1-5 (1982)

FIRST ORDER PROPERTIES OF RELATIONS WITH THE MONOTONIC CLOSURE PROPERTY

by GEORGE WEAVER in Bryn Maw, Pennsylvania (U.S.A.), and RAYMOND D. GUMB in Northridge, California and Philadelphia, Pennsylvania (U.S.A.) 1)

Introduction

We raise a problem suggested by an elementary theorem in the theory of relations: given A a nonempty set and RA a binary relation on A , there is a smallest transitive relation on A extending RA. This relation is called the transitive closure or proper ancestral of RdI. Arguments for this theorem are naturally seen as determining a func- tion Tr which, given nonempty set A and relation RA4, gives the transitive closure of RL4. This function is monoto& in that if A

Tr(B, RB). Our purpose here is to characterize those first order properties of rela- tions which, like transitivity, have the monotonic closure property.

B and Rli R B , then Tr(A, R,,)

1. Properties of Relations with the Closure Property

Let n be a natural number 2 2 , let T(n) be the similarity class of n-ary relational systems. Members of T ( n ) are pairs d = ( A , Rd) where A is a nonempty set (the doma.in of d ) and Rd is an n-ary relation on A . Given d, 99 E T(n), we let d 5? indicate that d is a subsystem of 99 and d z g indicate that d and 5? are isomorphic. d( V ) = ( A . A") ; and g is a relational expansion of d (d 5 g ) provided A = B and Rd R,.

fl2 is called a nonvoid intersection; Z is a property of relations when Z is closed under isomorphisms. Given L' a property and d, 9 in T(n), 99 is the 2-closure of d (99 = Cl(Z, d)) provided 99 is the smallest relational expansion of d in 2; Z(d ) denotes the class of relational expansions of d in 2; and Z has t,he closure property (CP) provided each member of T(n) has a 2-closure.

It is easily verified that if d has a L'-closure, Cl(Z , d) = n2(d); d' has a Z-closure provided nZ(d) E Z (d) ; and that d E 2 provided CZ(2, d) = d. Further, each of the following are necessary and sufficient for 2 to have CP: (1) for all d, nZ(d) E I(&); (2) for all d, d( V ) E E ( d ) arid flZ(d') E 2(d); and (3) for all $2,

Given Z a subclass of Y(n), n2 = (fl ( A : d E 2), n {ad: d E 2)); when fl2 E Y(n.),

Earlier drafts of the paper have been read at meetings of the Philitdelpllia Logic- clolloqium (1977), the Association for Symbolic Logic (1977), and a t the ACM Computer Science Conference (1978). The authors gratefully acknowledge the comments and criticisms offered by the participants of these meetings. Particular thanks goes to Professor WILLIAM DAVIDON (Haverford College) for pointing out some misleading terminology in an earlier draft,.

1 Ztschr. f . math. Logik

2 GEORGE WEAVER AND RAYMOND D. GUMB

a( V ) E E ( d ) and Z(d) is closed under non-void intersections. Thus, Z has CP pro- vided for all sets A , Z(A), the set of all n-ary relations on A having Z, is a closure system on A'& (cf. [2], pp.23-24); and hence, (Z(A), g) and ( ( Z ( A , A ) ) , &) are complete lattices.

2. Properties of Relations with the Monotonic Closure Property

Given property Z, Z has the monotonic closure property (MCP) provided Z has CP and for all &',9 if A g B and Rd 5 Ra, then €i'cl(=,&) 5 Rcl(2,B,. When L' has CP, each of the following is necessary and sufficient for Z to have MCP: (1) Z is closed under subsystems; (2) for all d, 9 if A 5 B, R, s; R, and 98 E Z, then RC1(=,&) 5 R, .

There are infinitely many properties which have CP but not MCP. For each m 2 1 let Ern consist of the following: (1) all systems d( V ) when A has a t most m-elements; and (2) all systems d where A has a t least m + 1 elements.

Lemma 1. Given property Z. Z has MGP provided Z contains d ( V ) for all d, and i s closed under subsystems and nonvoid intersections.

Proof. Suppose Z has MCP; then .Z is closed under subsystems and contains d( V ) for each d. Let 2' E 2, and suppose 027 is nonvoid. It suffices to show that RCl(z,nr) 5 Rnrr. Let d €2, Rcl(a,nzr) E Rd; hence Rcl(r,nzp) E Rnz,. The argu- ment in the other direction is immediate from earlier remarks.

3. First Order Properties of Relations

For each n 2 2, let R(n) by an n a r y relational constant, and Lob) be the first order language (with equality) whose only nonlogical constant is R(n). L(n) is interpretable in T(n). When no confusion results, we let T, R, 2; denote T(n), R(n) and L(2i). For a)

a sentence, M ( y ) denotes the models of y , and for T a set of sentences, M ( T ) drnotes the models of T. Given Z g T, Z is ECA provided ,Z = M ( T ) ; Z is EC provided 2 = M ( T ) , T finite; and Z is a first order property provided Z is ECA.

There are infinitely many Z which are EC, but which do not have MCP (e.g., the properties Z,,, above). Further, there are properties with MCP which are not first order. Consider the class Z* which contains all systems with finite domain, and all infinite systems where Rd contains tne diagonal of A". Z* is easily seen to have MCP. Suppose Z* is EC,. Let Z* = M ( T ) . Then for each finite cardinal k, there is a model of T, -3xRx . . . x of cardinality k; thus, 27, -3x Rx. . . x has an infinite model, con- tradictioii (cf. [l], Corollary 2.1.5, p. 67).

By a monotonic 17," sente~ce we mean any sentenccl of the form Vx, . . . x,,lO(xl, . . . , .r,,J where B(x,, . . ., z,,,) is either (i) an atomic formula other than x, = xJ when 2 $. 1 , or (ii) (pl + , . . + y t ) 3 (y, + . . . + ys) where q ~ , ~ is either an atomic forintila or x, =!= xJ when i =+ j and yir is an atomic formula other than x, = xJ when i $. 3 . Notice that a monotonic I?: sentence whose matrix is an atomic formula is a Horn sentence; and that all monotonic IT," sentences are special Horn sentenccs (cf. [ll, p. 340). Given Z g Y, Z i s a monotonic n," class provided 2 is the class of models of a set of niono- tonic I7," sentences. Each of the following are monotonic .@ classes: (i) the syirimetric

FIRST ORDER PROPERTIES OF RELATIONS WITH THE MONOTONIC CLOSURE PROPERTY 3

systems; (ii) the reflective systems; (iii) the quasi-ordered systems; and (iv) the protoordered (or pre-ordered) systems (cf. [6], p. 80). The main result of this section is that among first order properties, the inonotonic f l classes are exactly those with MCP.

The following are easily verified: (1) for all d, sl( V ) is a model of all monotonic Z7," sentences, and ( 2 ) the class of models of any monotonic 17," sentence is closed under nonvoid intersections. Thus, every set of monotonic sentences is satisfiable, no set is complete, and the entire set of monotonic IT: sentences is categorical in all powers. Moreover, since each monotonic I7," class is closed under subsystems (cf. [l], Theorem 3.2.2, p. 124), every monotonic IT: class has MCP.

For T a set of sentence, p a sentence T k p indicates that y is a logical consequence of T ; Cn(T) denotes the set of logical consequences of T; .T n m z denotes the set of monotonic Y, I (Z) denotes the result of closing C under nonvoid intersections; Sub(2) the result of closing L' under subsystems. For a2 E T, let D ( d ) denote the diagram of d (cf. [l], p. 68).

sentences in T . For Z

Theorem 1. Given T a set of sentences.

M(Cn(T) n mn:) = Xub( I (M(T) ) ) u {sl( V ) : d E Y). Proof. Let Z: = M ( C n ( T ) n mZ7?), Z' = ~S'ub(l(M(T)j) w (d( 7): d E Y}. Prom

our earlier remarks it follows that ,Z' s L'. Let d EL' and suppose d + d( V ) . Then there is (a,, . . . , a,) $ R,. Let

Tol, , . , a , = (91, + . . . + p,], + "Ra, . . . a, l : m 2 0, p l l is either an atomic sentence or inequality in D ( s l ) ) .

T w Tal,, ,~~ has a model. Otheruise, there is y E Tol, ,, n~ such that T , y has no model. Let y be v1 + . . . + vnL + "Ra , . . . a,. Then, T k ((p, + . . . + p,) 2 Ra, . . . u,J. Let T be the universal quantification of this consequence of T ; T k r and r is mono- tonic @. Hence, d k r ; contradiction. Let M a , . . . a, be a model of T such that A E a 'a , . . . a,, Rd and (a l , . . . , arL) 4 Rd,Gl. Let LZ? = n { d ' a , . . . a,,: (a, , . . . , a,) 6 R,}. 99 E I ( M ( T ) ) and d

Corol lary 1. Given L' a n ECA class. (1) ,Z has ,WCP provided Z is a monotonic f l class; and (2) if Z is EC and has MCP then 2 i s the class of models 01 a finite .wt of mono- tonic 17," sentences.

There are first order properties which have MCP but which are not EC. To see this, let n = 2 and for each WL 2 3 let vrn be

Rdra, 9?, hence, d E Z'.

vx1 . . . x,fz( (Rxlxz + Rx2x3 + . . f + Rx,f&-lx, + RXnlX,?<-Z + * x2 + . . . + xrji-1 * z j f~ ) 2 Rxr2.1) . Let 2 = (pn1: m 2 3). For each k 2 3, there is a model of { y 3 , . . . , pk} which is not a model of pKI1; hence, L' is not EC. Notice that the above also establishes that there are infinitely many EC classes with the MCP.

R, , Where ,Z is a chain of subsystems or weak subsystems U,Z denotes the union of the chain.

Corol lary 2. (i) All first order properties of relations with MCP are closed under uibions of chains of subsystems and weak subsystems; (ii) there are properties of relations

Let d, B E Y, d is a weak subsystem of B (,.Z G G ~ 98) provided A 5 B and R,

1*

4 GEORGE WEAVER AND RAYMOND D. GUMB

with, MCP which are ?lot closed under either unions of chains of subsystems or of weak subsystems .

Proof , (i) is immediate from the theorem. To see (ii) consider the property Z* discussed above.

4. An Application to Modal Logic

In [5] KRIPKE demonstrated by tableaux constructions the completeness and sound- ness of the faniily of normal modal logics: s,, s,, 13 and M. It is natural to ask whether or not KRIPKE'S techniques can be extended to other modal logics. As detailed in [4], the Kripke constructions can be applied to yield strong soundness and strong com- pleteness results whenever the accessibility relation is restricted to those with the computable Kripke closure property.

Given property 2, 2 has the hrripke closure property (KCP) provided (1) Z h a s MCP; (2) 2 is closed under unions of countable chains of weak subsystems; and (3) Z is closed under preiinages of strong homomorphisms. Z has the computable Kripke closure property provided Z has KCP and (4) it is decidable whether or not a finite relational system is in ,E.

is =-free provided the identity sign does not occur in p: ; and rp is = (+)-free provided the identity sign does not occur positively in p (cf. [l], p. 89). Given 2, Z is an -1 -free class provided 2 is the class of models of a set of =-free sentences; similarly we define ,E is an = (+)-free class.

Lemma 2. Given 2 a first order property. (i) z' is closed under yreimages of strong homomorphisms i f f L' is a n = (+)-free class; a i d (ii) 2 is closed under strong homo- morpiiisms and the preimages of stroiLg hornomorphisins iff Z i s a =-free class.

P r o of. The argument here is a straight forward generalization of analogous results for weak homomorphisms (cf. [l], pp. 126-127).

Notice the following: (1) all monotonic II," Horn sentences are equivalent to = (+)- free monotonic rr," sentences ; ( 2 ) every = ( + )-free monotonic n;" sentence is equivalent to a conjunction of monotonic I?," Horn sentences ; (3) every = ( + )-free sentence whose matrix is a conditional is equivalent to a special Horn sentence; (4) every = (+)-free monotonic 17," sentence is equivalent to a strict Horn sentence; and ( 5 ) each = ( + )-free monotonic II: sentence is equivalent to an =-free monotonic II: sentence.

Given Z, Z is a monotonic 17: H o m class provided 2 is the class of models of a set of monotonic rr," Horn sentences; thus, by the above, classes which are the models of a set of = (+)-free monotonic 17," sentences are exactly the monotonic ll: Horn classes. Thus. we obtain:

Theorem 2. Given 2 a first order pr0perty.Z has KCP iff 2 is a monotonic n,O Horn

Thus, first order properties having KCP are closed under both direct and reduced products (cf. [l], Proposition 6.2.2, p. 329). Further, any property having KCP which is closed under reduced products is an ECA class Let Z be such a class. 2 is closed under ultrapowers and isomorphisms; further if d' is in the complement of 22 and ~ 4 ?

Given p a formula in L,

class.

FIRST ORDER PROPERTIES OF RELATIONS WITH THE MONOTONIC CLOSURE PROPERTY 5

is some ultrapower of d, then d is isomorphic to a subsystem of 98 and 3? is not in Z; hence Zis ECA (cf. [l], Corollary 6.1.16, p. 322).

The above result seems to delimit the applicability of the Kripke tableaux constmc- tions (as generalized in [4]) to intensional logics and more generally evolving theories which have a Kripke-style semantics. By theorem 2 , an a t least sufficient condition for the Kripke constructions to yield strong soundness and strong completeness results is that the accessibility relation be restricted by a set of first order sentences equivalent t o a set of monotonic 17," sentences and that condition (4) be satisfied.

Further, it appears that soundness and strong completeness proofs could not be obtained, unless the rules concerning tableaux were changed, if the accessibility relation did not have the computable KCP. Let Z* be that property discussed above. Consider the Kripke semantics where the accessibility relations are members of Z*. In this Iogic, the following infinite set of seiitences is unsatisfiabIe but no finitary modification of the Kripke tableaux method will reveal this (i.e. strong completeness is lost):

(o"(--A,, + n ( A , + . . . + An) ) : n 2 1, A,, is the n-th sentence parameter).

Let Zo be the models of Vxyz((Rxy + Rxx + y + x ) 3 Rxx) . Consider the Kripke semantics where the accessibility relations are members of Zo. I n this logic, the follow- ing sentence is not valid but a tableaux for it closes (hence the logic is not sound) - (OA + O B + U C + W C ) . Zo is not a monotonic @ Horn class and Kripke's techniques are not applicable to every logic in which the same restrictions are placed on both tableaux and interpretations (cf. [5] , p. 76).

References

[l] CHANG, C. C., and H. J. KEISLER, Model Theory. North-Holland Publ. Comp., Amsterdani 1973. [2] GRATZER, GEORGE, Universal Algebra. D. van Nostrand Company Inc., Princeton 1968. [3] GUMB, R. D., A mechanized proof procedure for free intensional logics. In: Proceedings of the

Fifth International Joint Conference on Artificial Intelligence, Vol. 1 (Cambridge, Mass.) 1977, p. 567.

[4] GUMB, R. D., Evolving Theories. Haven Publ. Corporation, Flushing, N.Y. 1979. [5] KRIPKE, S. A., Semantical Analysis of Modal Logic 1, Normal Propositional Calculi. This Zeitschr.

[6] KURATOWSKI, K., and A. M o s T o w s K r , Set Theory. North-Holland Publishing Comp., Amster- 9 (1963), 67-96.

dam 1968.

(Eingegangen am 21. Februar 1980)