A NOTE ON THE STRUCTURE OF BILATTICES. A. Avron. School of Mathematical Sciences. Sackler Faculty of Exact Sciences. Tel Aviv University

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1 A NOTE ON THE STRUCTURE OF BILATTICES A. Avron School of Mahemaical Sciences Sacler Faculy of Exac Sciences Tel Aviv Universiy Tel Aviv 69978, Israel The noion of a bilaice was rs inroduced by Ginsburg (see [Gin]) as a general framewor for a diversiy of applicaions (such as ruh mainenance sysems, defaul inferences and ohers). The noion was furher invesigaed and applied for various purposes by Fiing (see [Fi1]-[Fi6]). The main idea behind bilaices is o use srucures in which here are wo (parial) order relaions, having dieren inerpreaions. The wo relaions should, of course, be conneced somehow in order for he mahemaical srucure o be useful. I is no clear, however, wha his connecion should be. Ginsberg, for example, has made he connecion hrough an exra operaion of negaion. Fiing, on he oher hand, has invesigaed connecions in he form of condiions on he srucure (such as being inerlaced { see below). These condiions are independen of he exisence, or even he possibiliy o dene, operaions lie Ginsberg's negaion.* Fiing denes, accordingly, noions lie \an inerlaced bilaice", \a disribuive bilaice", \a bilaice wih negaion" and ohers. He does no provide, however, any deniion of he noion of bilaice iself (wihou an exra modier). I was unable o nd anywhere, in fac, a deniion which will cover all he srucures which were called \bilaice" in he lieraure. Despie he las saemen, here is somehing which seems o be common o all he nie bilaices which were used or invesigaed (and pracically only such bilaices were suggesed for acual use). All of hem can be represened by a ind of a Hasse * In [Fi5] here is an example of an inerlaced bilaice in which no operaion of negaion as dened by Ginsberg is available. 1

2 diagram, which viewed boom-up, represens one of wo order relaions while viewed from lef o righ, represens he oher (see examples in Figs. 1 and 2 below). This ype of represenaion seemed, somehow, o be naural, bu here has no been (as far as I now) an aemp o show ha i can always be used. of o f f d df d Figure 1: F OUR and NINE f d df d Figure 2: DEF AULT The presen noe has wo purposes. The rs is o sugges in he nie case a general deniion of \a bilaice" which will cover all he agreed upon paricular cases. The second { o show ha every such bilaice can be represened by a diagram as described above. This will show, I believe, ha he suggesed deniion is adequae. A he same ime i will jusify his general mehod of represenaion. 2

3 Deniion 1 [Fi1]. A prebilaice is a srucure B = hb; ; i such ha B is a nonempy se conaining a leas wo elemens, and boh hb; i and hb; i are (complee) laices. Noaion. Following Fiing, we shall use ^ and _ for he laice operaions which correspond o, and and for hose ha correspond o. Deniion 2 [Fi1]. A prebilaice is inerlaced if each of he four operaions ^; _; and is monoonic wih respec o boh and. Examples. F OUR and NINE (see Fig. 1) are inerlaced. DEF AULT (Fig. 2) is no.* The noion of a prebilaice has been used by Fiing as a general framewor for he sudy of bilaices. In order o invesigae represenabiliy by graphs i is useful o consider even more general noions. Deniion 3. A bipose is a srucure B = hb; ; i in which boh hb; i and hb; i are poses. Noaion. a < 1 b (a <1 b) will mean ha b is an immediae ( ) successor of a. a < b (a < b) will mean ha a b (a b) bu a 6= b. Deniion 4. (a) A nie bipose B is graphically represenable if here exiss a nie graph G = hv; Ei of poins in R 2 and a bijecion f of B ono V such ha: (1) There is an edge beween wo verices x 1 ; x 2 in V i f?1 (x 1 ) and f?1 (x 2 ) are relaed by eiher < 1 or < 1 (i.e. if one of hem is an immediae successor of he oher according o eiher or ). (2) If (a 1 ; b 1 ) 2 V, (a 2 ; b 2 ) 2 V and he wo poins are conneced by an edge hen a 1 6= a 2 and b 1 6= b 2. (3) x < y i here exiss a (possibly empy) sequence of poins (a 1 ; b 1 ); : : : ; (a n?1 ; b n?1 ) in V such ha if f(x) = (a 0 ; b 0 ) and f(y) = (a n ; b n ) hen a 0 < a 1 < a n?1 < a n and here is an edge beween (a i ; b i ) and (a i+1 ; b i+1 ) for i = 0; 1; : : : ; n? 1. * F OUR is due o Belnap ([Be1], [Be2]), DEF AULT { o Ginsberg ([Gin]). 3

4 (4) x < y i here exiss a (possibly empy) sequence of poins (a 1 ; b 1 ); : : : ; (a n?1 ; b n?1 ) in V s.. if f(x) = (a 0 ; b 0 ) and f(y) = (a n ; b n ) hen b 0 < b 1 < b n?1 < b n and here is an edge beween (a i ; b i ) and (a i+1 ; b i+1 ) for i = 0; : : : ; n? 1. (b) B is precisely represenable if in place of (1) we have: (1) fx 1 ; x 2 g 2 E i f?1 (x 1 ) and f?1 (x 2 ) are relaed by boh < 1 and < 1. An example. F OUR, NINE and DEF AULT are all graphically represenable, as Figures 1 and 2 show. F OUR and NINE are precisely represenable (see Fig.1). DEF AULT is no. (In Fig. 2 here is an edge beween d and alhough d < bu :(d < 1 ). I will follow from Theorem 2 below ha no oher precise represenaion exiss.) Theorem 1. A nie bipose B is represenable i he following wo condiions are saised: (i) If x < 1 y hen x < y or y < x. (ii) If x < 1 y hen x < y or y < x. Proof: The necessiy of he condiions is easy. Suppose, for example, ha a < 1 b. Then by condiion (1) of Deniion 1, here is an edge beween f(x) = (a 0 ; b 0 ) and f(b) = (a 1 ; b 1 ). By condiion (2), b 0 6= b 1 and so b 0 < b 1 or b 1 < b 0. Hence, by condiion 4 (wih n = 0), x < y or y < x. For he converse, assume ha condiions (i) and (ii) obain. Le B = fx 1 ; x 2 ; : : : ; x m g. We consruc a se V R 2 and bijecion f from B ono V so ha: () If f(x i ) = (a i ; b i ), f(x j ) = (a j ; b j ) and x i < x j (x i < x j ) hen a i < a j (b i < b j ). Moreover if x i < 1 x j (x i < 1 x j) hen a i 6= a j and b i 6= b j. The consrucion is by inducion on m. The case m = 1 is rivial. Suppose we have a consrucion for fx 1 ; : : : ; x m g. Le A = fi j x i < x m+1 g, B = fi j x m+1 < x i g. Suppose f(x i ) = (a i ; b i ) for 1 i m. By inducion hypohesis, a i < a j for every i 2 A, j 2 B. Le a m+1 be some number beween maxfa i g and minfa i g so ha a m+1 6= a i whenever i2a i2b x m+1 < 1 x i or x i < 1 x m+1. (The exisence of such a number is due o he nieness of m.) Choose b m+1 in a similar way (using insead of ). The choice of (a m+1 ; b m+1 ) can obviously be done so ha his poin does no belong o ff(x 1 ); : : : ; f(x m )g. Add (a m+1 ; b m+1 ) o V and le f(x m+1 ) = (a m+1 ; b m+1 ). Obviously, condiion () is preserved. Now connec f(x i ) and f(x j ) by an edge i x i and x j are relaed by eiher 1 or 4

5 1. We claim ha he resuling hv; Ei (ogeher wih f) is a graphical represenaion of B. I is rivial ha f is a bijecion of B ono V and ha he rs wo condiions in he deniion of a graphical represenaion are saised. The proofs of he oher wo condiions are pracically idenical. We prove here condiion (3). Assume rs ha x < y. Since B is nie, his enails ha here are z 1 ; : : : ; z n?1 2 B s.. x < 1 z 1 < 1 <1 z n?1 < 1 y. Le f(z i) = (a i ; b i ). From he consrucion of V i follows ha (a 1 ; b 1 ); : : : ; (a n?1 ; b n?1 ) are poins as required. For he converse, assume f(x) = (a 0 ; b 0 ), f(y) = (a n ; b n ) and ha here exis poins (a 1 ; b 1 ); : : : ; (a n?1 ; b n?1 ) in V s.. a 0 < a 1 < < a n?1 < a n and here is an edge beween (a i ; b i ) and (a i+1 ; b i+1 ) for 0 i n? 1. This means, by deniion, ha z i = f?1 (a i ; b i ) and z i+1 = f?1 (a i+1 ; b i+1 ) are relaed by eiher < 1 or < 1. By condiion (ii) of he heorem his enails ha z i and z i+1 are relaed by <. I is impossible ha z i+1 < z i, since his implies, by (), ha a i+1 < a i. Hence z i < z i+1. I follows ha x = z 0 < z 1 < < z n?1 < z n = y, and so x < y. Theorem 2. A nie bipose B is precisely represenable if he following wo condiions obain: (i) If x < 1 y hen x <1 y or y <1 x. (ii) If x < 1 y hen x <1 y or y <1 x. Proof: Obviously (i) and (ii) imply, respecively, condiions (i) and (ii) of Theorem 1. Hence we can apply he same consrucion as in he proof of Theorem 1. I is immediae from he deniions and (i), (ii), ha he represenaion we ge is precise. We now sugges he following deniion of bilaices in he nie case: Deniion 5. (a) A nie bilaice is a nie prebilaice which saises condiions (i) and (ii) of Theorem 1. (b) A precise (nie) bilaice is a (nie) prebilaice which saises condiions (i) and (ii) of Theorem 2. If we adop hese deniions we can reformulae he wo heorems above as follows: 5

6 Corollary. (a) A nie prebilaice is a bilaice if i is graphically represenable. (b) A nie prebilaice is precise if i is precisely represenable. A connecion wih Fiing's noions is given by he following heorem: Theorem 3. Every inerlaced prebilaice B is precise.* Proof: Assume, e.g., ha a < 1 b. Then a b and so (since B is inerlaced) a = a a a b b b = b. Hence a a b b. Since b is a { successor of a, his means ha eiher a b = a or a b = b. Since a 6= b his enails ha eiher b < a or a < b. Assume, e.g. ha he rs case holds, i.e. b < a. We show ha a is a { successor of b. Le c saisfy: b c a. The fac ha B is inerlaced implies now ha a = b ^ a c ^ a a ^ a = a ( b ^ a = a since a b!). Hence c ^ a = a and a c. On he oher hand, from b c a follows also ha b = b _ b c _ b a _ b = b. Hence c _ b = b and so c b. We ge a c b. Since a < 1 b, his means ha c = a or c = b. To sum up: we have shown ha if b c a hen c = b or c = a. This means ha a is a -successor of b (since a 6= b and we are assuming b a). The oher case, a < b, is similar and is lef o he reader. Corollary. Every nie, inerlaced bilaice is precisely represenable in R 2. Conjecure. Every nie and precise bilaice is inerlaced. If he las conjecure is rue hen he noion of an inerlaced bilaice is a naural generalizaion of he idea of a precisely represenable (pre) bilaice, a generalizaion which can be applied o arbirary prebilaices, no necessarily nie ones (in innie laices he < 1 relaion does no deermine < and may even be empy. This happens, for example, when he order relaion is dense). I would be nice o have a similar generalizaion for he more general class of nie, represenable prebilaice. Such a generalizaion can serve as an adequae deniion of he noion of a bilaice. One naural possibiliy is he following: * i is worh menioning here ha in [Fi1] i is shown ha every disribuive bilaice is inerlaced. 6

7 Deniion 6. A bilaice in he srong sense is a prebilaice which saises: (i) If a b hen a a b b and a a b b. (ii) If a b hen a a ^ b b and a a _ b b. An example. F OUR, NINE and DEF AULT are all bilaices in he srong sense. Obviously, every inerlaced bilaice is a bilaice in he srong sense, and every nie bilaice in he srong sense saises he wo condiions in Theorem 1 and so is graphically represenable. None of he wo converses is rue. In Figure 3 we have an example of a nie bilaice in he srong sense which is no precise (and so { no inerlaced). In Figure 4 we have an example of a represenable prebilaice which is no a bilaice in he srong sense. c c b b d a a Figure 3 Figure 4 I is no clear ha we wan a prebilaice as in Figure 4 o coun as a bilaice, and so Deniion 6 migh be a good candidae for a deniion of his noion. Anoher possibiliy is he following: Deniion 7. A prebilaice is called a bilaice in he wea sense if i saises: (i) If a b hen hese exiss a c b such ha (a c _ c a) ^ (b c _ c b). (ii) If a b hen here exiss a c b such ha (a c _ c a) ^ (b c _ c b). Obviously, a bilaice in he srong sense is a bilaice in he wea sense. The inverse, however, fails. Figure 4 again represens a couner-example. Also every bilaice 7

8 in he wea sense rivially saises he condiions in Theorem 1 and so is graphically represenable. Conjecure. In he nie case he converse also holds: every graphically represenable prebilaice is a bilaice in he wea sense. REFERENCES [Be1] N.D. Belnap. A Useful Four-Valued Logic. Modern Uses of Muliple-Valued Logic (G. Epsein, J.M. Dunn -Eds.), Oriel Press, [Be2] [Fi1] N.D. Belnap. How Compuers Should Thin. Conemporary Aspecs of Philosophy (G. Ryle - Ed.), Oriel Press, pp , M. Fiing. Kleene's Three-Valued Logics and Their Children. Proc. of he Bulgarian Kleene 90 Conference, [Fi2] M. Fiing. Kleene's Logic, Generalized. Journal of Logic and Compuaion, 1 (Dec. 1990), [Fi3] [Fi4] [Fi5] M. Fiing. Modal/Muliple Valued Logics. Dec (forhcoming in Handboo of Logic in AI and Logic Programming, Oxford Universiy Press). M. Fiing. Bilaices in Logic Programming. The 20h In. Symp. on Muliple- Valued Logic (G. Epsein { Ed.), IEEE Press, M. Fiing. Bilaices and he Semanics of Logic Programming. Journal of Logic Programming 11(2) (1991), [Fi6] M. Fiing. Negaion as Refuaion. Proc. 4h Annual Symp. on Logic in Compuer Science, IEEE Press, 63-70, [Gin] M. L. Ginsberg. Mulivalued Logics: A Univrom Approach o Reasoning in AI. Compuer Inelligence, 4 (1988),