7ECHN1CAL N 0 7 E

EXPRE551VENE55 0 F 5 7 A 8 L E M 0 D E L 5EMAN71C5 F 0 R D15JUNC71VE L 0 6 1 C P R 0 6 R A M 5 W17H F U N C 7 1 0 N 5

7 H 0 M A 5 E17ER AND 6 E 0 R 6 60•17L08*

1n th~ pape~ we 5tudy the e x p r e ~ e p0wer and recurf10n~he0ret~ c 0 m p ~ t y 0f d15jun~Ne 1091c pr09ram5 w1th funct10n5 5ym6015 0ver Her6rand m 0 d d ~ 1n p a ~ 1 c u ~ we c0nf1der the d1~unct1ve 5ta61e m0de1 5 e m a n t ~ and 5h0w that a rdat10n R 15 def1na61e 0ver the Her6rand un1ver5e 0f a d1~unct1ve 10~c pr09ram ff and 0n1y 1f R 15 H~ def1na6~. 7hu5, d1~un~Ne 10~c pr09ramm1n9 under the 5ta61e m0de1 5emant1c5 expre55e5 exact1y H~, and 15 thu5 H~ c0mp1ete 0ver the 1nte9e~. 7h15 re5u1t 15 5ttrpr1~n9 6ecau5e 1t 5h0w5 that d15jun~Ne 10~c pr09ramm1n9 15 n0t m0re expre5~ve than n0 rm~ 10~c pr09ramm1n9 under the 5ta61e 0r w ~ f 0 u n d e d 5emant1c5. 7h15 5harp~ c0ntra5t5 w1th the funct10n-f1ee ca5e. • E15e~er 5~ence 1nc., 1997

1N7R0DUC710N

L0~c pr09ramm1n9 ha5 turned 0ut t0 6e a f1u1tfu1 appr0ach t0 a de~ara t~e pr09ramm1n9 ~n9ua9e 6a5ed 0n 5tr0n9 m a t h e m a t ~ f0undat10n5. 1t ha5 attracted much a~ent10n 60th 0n the f1de 0f 5emant~5 a5 weH a55 c0mputat10n.

7he c 0 m p 1 e ~ and e x p r e ~ e p0wer 0f 1091c pr09ramm1n9 ha5 6een 1nve5t1- 9ated 6y 5ever~ re5earcher5 1n a num6er 0f paper~ e.9., [1, 7, 2, 11, 21, 22, 19, 8, 20, 22 32, 23, 34, 33, 12, 14, 15] (5ee ~3, 9] f0r a c0mprehenf1ve 5urvey), and 15 4u1te we11 under5t00d. 1n th15 re5earch pr09ram, var10u5 5emant~5 have 6een c0nf1dered, a5 we11 a5 d1fferent da~e5 0f pr09ram~

*Chf1~1an D0pp1er La6 f0r EXpea 5y5tem~ 1nf0rmat10n 5y5tem5 Depa~ment, 7U V1enna, A-1040 W1en, Pan1919a55e 1~ AU~f1~ Ema11: (e~e~ 90tt106)•d6a1.tUW1en.aC.at.

Addre55 C0~e5p0ndenCe t0 7h0ma5 E1~L 1n5f1tUt f1Jr 1nf0rmat1L Un1Ver~ 0f 61e8em D-35392 6~8e~ Arndt5tra8e 2, 6erman~ Ema11: e1ter~1nf0rmat1~Un1-91e55emde.

ReCeNed JU1y 1996; aCCepted JanUary 1997.

7HE J0URNAL 0F L061C PR06RAMM1N6 • E15e~er 5denCe 1nC., 1997 655 AvenUe 0f the Amer1Ca5, NeW Y0r~ NY 10010

0743-1066/97/$1200 P11 50743-1066(97)00027-7

168 ~ E17ER AND 6 . 6 0 3 W L 0 8

Eaf1y re5u1~ are 1n [7, 2] where the af1thmet~ da5f1f1cat10n 0f ~rat1f1ed 10~c pr09ram5 0ver 1nf1nRe Her6rand m0de~ 15 9Nen, wh~h ha5 6een extended t0 1 0 c ~ ~rat1f1ed pr09ram5 [8]. Ch0~k and 81a1r 5h0w that the 5et 0f 10c£~ ~rat1f1ed n0rm~ 1091c pr09ram5 15 H 1-c0mp1ete [8].

Marek, Ner0de, Remme1, and 0ther5 have 1nve~19ated the c 0 m p ~ t y and expre55Ne p0wer 0f ~a61e m0dd5 1n 5evera1 paper5 e9., [21, 22, 19, 20, 23]. 1ndependent1~ the expre55Ne p0wer 0f 5ta61e m0dd5 0f n0rm~ 1091c pr09ram5 (a5 we11 a5 many 0ther 5emant1c5) ha5 6een 1nve~19ated 1n the pr0f0und w0rk 6y 5ch11pf [34, 33], wh0 der~ed a num6er 0f 1mp0~ant re5uh5 f0r var10u5 5ett1n95 0f 10~c pr09ramm1n9

Recent~, part1cu~r a~ent10n ha5 6een pa1d t0 d14unctNe 1091c pr09ramm1n~ wh1ch ha5 6een 1ntr0duced 6y M1nker [24]. A5 ar9ued 1n [18, 3, 25], d1~unctNe 10~c pr09ramm1n9 15 a c0nven~nt extenf10n t0 10~c pr09ramm1n9 apt f0r advanced kn0w1ed9e repre5entat10n. Var10u5 5emant~5 f0r n0rm~ 10~c pr09ram5 (LP5) heve 6een extended t0 d1~unctNe 1091c pr09ram5 (DLP~ [18, 25]. 7he d0m1nat1n9 5emant1c5 f0r DLP5 are the m1n1ma1 m0de~ 5emant~5 f0r ne9at10n-~ee pr09ram5 [24] and the d1~unctNe 5ta61e m0de1 5emant~5 [29, 1~.

8e51de5 the 5 e m a n t ~ the c 0 m p ~ t y and expre55Ne p0wer 0f DLP5 a150 have receNed a~ent10n. 7he c 0 m p 1 e ~ and expre5f1vene55 0f 9r0und (pr0p051t10n~) and funct10n-~ee DLP5 have 6een determ1ned 1n 5evera1 paper~ e.9., [12, 14, 23, 27, 15, 32].

7he 60tt0m 11ne 0f th05e re5u1t5 15 that, 1n the ca5e w1th0ut funct10n5, the c0mp1e~ty and expre55Nene55 0f DLP5 are at the 5ec0nd 1eve1 0f the p01yn0m1~ h~rarchy, and capture under 5evera1 5emant1c5 the c1a55 H~ (re5p., ~ ) [14], a5 0pp05ed t0 n0rm~ LP5, wh1ch capture under the 5ta61e and m0de1 5emant1c5 c0-NP (re5p., NP) D4]. 7hu5, 1n the funct10n-~ee ca5e, DLP5 are m0re expre5f1ve than n0rm~ LP5 (un~55 the p0~n0mh1 h~rarchy c01hp5e5L

0n the 0ther hand, 11tt1e 15 kn0wn a60ut the c0mp1ex1ty and expre55Ne p0wer 0f DLP5 1n the pre5ence 0f funct10n~ Eady w0rk 1n th15 d~ecf10n 15 [11], 1n wh1ch Ch0m~k1 and 5u6rahman1an have 5h0wn that the 6 e n e r ~ e d C105ed W0dd A55umpt10n (6CWA), wh1ch am0unt5 t0 the m1n1m~ m0de1 5 e m a n t ~ 15 H~- c0mp~te 0ver Her6rand m0de9, 7hu5, the 6CWA 15 ~r1ct1y m0re expre55Ne than the CWA, wh1ch 15 kn0wn t0 6e H~-c0mp1ete [1] f0r 60th p05RNe and 9enera1 LP5J H0wever, n0th1n9 15 kn0wn a60ut the c0mp1e~ty and expre5~vene55 0f 5ta61e m0dd~ the m~0r 5emant1c5 f0r 9enera1 DLP5, 1n the 1nf1n1te. 7h15 wa5 recent~ p01nted 0ut a5 an 0pen re5earch pr061em 6y M1nker [26].

1n th15 pape~ we 5hed 119ht 0n th15 155ue. We pre5ent the 1ntere~1n9 and unexpected re5uR that the d1~unct~e 5ta61e 5emant1c5 expre55e5 H 1 0ver Her- 6rand m0dd5 and 15 H1-c0mp1ete 0ver the 1nte9e~, and 15 thu5 a5 expre55Ne a5 the ~a61e m0de1 5emant~5 and the we11-f0unded 5emant~5 0f n0rm~ LP5. 7hu~ under 10~c pr09ramm1n9 w1th funct10n5, d1~unct10n up0n ~a61e m0dd5 d0e5 n0t add expre55~e p0wer. 7h15 5harp~ c0ntra5t5 w1th the re5uR5 f0r the d05ed w0dd a55umpt10n, where d1~unct10n 1ncrea5e5 the c0mp1e~ty ~0m H~ t0 H~.

A1th0u9h th15 paper 15 ma1n1y 0f the0ret1ca1 nature, we 6e11eve that 0ur re5uR5 are 0f p r a c t ~ 1ntere5L t00.7he p05f16111ty 0f de~1n9 exp11c1t1y w1th d1~unct10n5 ~

15mM~an ~5] a55erted a 005e~ re1ated ~ 1 t wNch ~m1N~ 5ay5 mat, 1n 0ur mrm~ the m1n1ma1 m0d~ ~mam1c5 0f 10~c pm9~mm~9 0v~ ar1~m~ ~dd5 ~c r.e. ~ .

E X P R E 5 ~ V E N E 5 5 0 F 5 7 A 8 L E M 0 D E L 5 E M A N ~ C 5 169

a ~ u r e that accur~e~ m~che5 n~u r ~ pa~ern5 0f kn0w1ed9e r e p r e 5 e n t ~ n and d15c0u~e. 7heref0re, ~ u n ~ e 1091c pr09ramm~9 ~10w5 0ne t0 f 0 r m ~ e many ta5k5 m0re c0mf0~a6~ than n0rm~ LP. 0 n the 0ther hand, 1n the pre5ence 0f f u n ~ n 5ym6~5, the c 0 m p u t ~ n ~ a5pe~5 0f n0rm~ LP are much 6etter unde~t00d than th05e 0f DLE Wh11e 5ever~ re5~ut~n-6a5ed pr0cedur~ m~h0d5 are ava11a~e f0r pr0ce5~n9 n0rm~ LP5 under the wdPf0unded 5emant1~ (5ee, e.9., [31, 10] and reference5 there~L 1t 15 a p~0~ undear h0w the 5ta~e 5emant~5 f0r DLP 5h0u1d 6e 1m~emen~d 1nn the ca5e 0f 1nf1n1te Her6rand un~e~e9

7he pr00~ 0f 0ur re5u1t5 1m~y that DLP under 5ta61e m0de1 5emant~5 can 6e recu~1ve~ ~an5f0rmed t0 n0rm~ 1091c p r 0 9 r a m m ~ 7hu~ 0nce a pr09rammer 15 a61e t0 501ve a pr0~em thr0u9h a ~ u n ~ e pr09ram (under the ~a~e m0de1 5emant1~L 6y 0ur r e 5 ~ , 5/he kn0w5 def1~tdy that there ~50 e ~ 5 a n0rm~ 10~c pr09ram 501v1n9 the 5ame pr061em. M0re0ve~ an aut0mat1c ~an~af10n ~0m DLP t0 n0rm~ LP 15 p05f1~e (the tran~af10n ~90r1thm 15 1m~16t ~ the pr00f5).

7he 0 r 9 a ~ 2 ~ n 0f the paper 15 a5 f0H0w~ 7he next 5ect10n ~ve5 50me pref1m~af1e5 and kn0wn re5u1t5. 5ect10n 3 pre5en~ the ma1n r e 5 ~ , and 5 e ~ n 4 ~ve5 a 5h0~ c0nduf10m 8ack9r0und m~ef1~ f0r the reader 1e55 fam~ar w1th recur510n t h e 0 r ~ c0ncept~ ~ d u ~ n 9 def1~t10n5 0f H~ and H~, 15 pr0~ded 1n the Append~.

2. PREL1M1NAR1E5 AND PREV10U5 RE5UL75

We a~ume that ~ e ~ader 15 h m ~ h r w1th ~ u n ~ e 10~c p~09ramm~9 (f0r a 6ack9r0und, 5ee [18~ and w1th the 5ta61e m0de1 ~mant1~ 0f n0rmM and ~ u n ~ f1ve 10~c pr09ram5 [18, 16, 29, 17].

We den0te p ~ c a ~ 5 6y cap~M~ ~d1v1dua1 vaf1a~e5 6y 10wer ca5e 1etter5 ~0m the end 0f the Mpha6~, and ~d1v1dua~ and c0n~ant5 6y 10wer ca5e 1ette~ ~0m the 6e~nn~9 0f the Mpha6~; a 6Cdhce ve~10n 0f a 1e~er den0te5 a tu~e cf c0~e5p0n~n9 Wm6C5.

A ~ u n ~ e 1091c pr09ram (DLP) P 15 a f1~te 5et 0f dau5e5

A ~ V ~ V A k ~ 8 ~ , . . . , 8 ~ , 1 ~ k , 0 ~ m (1)

where each A 1 15 an at0m and each 8j 15 a f1terM. F0r c0nven1ence, we a110w e4uM1ty and 1ne4ua11ty 11terN5 1n the 60dy. A DLP P 15 n0rma1 ff k = 1 f0r every dau5e 1n P. A Her6rand 1nterpretat10n M 0f a 9r0und DLP P 15 a 5ta6~ m0de1 0f P 1ff 1t 15 a m1n1mM Her6rand m0de1 M 0f the 6~f0nd-L1NchR2 reduct 0f P w.~C t0 M [16, 29], wh~h re5u1t5 6y rem0~n9 a11 ru1e5 ~0m P that have a ne9atNe f1tera1 81 1n the 60dy 5uch that M ~ 81, and 6y rem0~n9 N1 ne9at1ve f1terN5 ~0m the remNn1n9 ru1e5. 7he 5ta61e m0d~5 0f a n0n9r0und pr09ram P are th05e 0f 1t5 9r0und 1n5tant1at10n; the 1n5tanf1ated pr09ram ~ 1n 9enerM 1nf1n1te, 6ut we 5f111 c0n51der 1t a5 a pr09ram.

A 9r0und ~terM L 15 1nferred ~0m P under the ~a61e 5emant~5 (P ~ L) 1ff M ~ L f0r every 5ta61e m0de1 M 0f P.

F0r evMuat10n 0f a DLP P 0ver 1091ca1 5tructure~ we a55ume that the pred9 cate5 0f P are 5p11t 1nt0 the exten~0nM pred~ate5 (ED8P5~ wh~h mu5t n0t 0ccur 1n the ru1e head~ and the 1nten~0nN pred1cate5 HD8P5L wh1ch are a11 0ther pred1cate5. 0ne 0f 1he 1D8P5 15 a dNf1n9u~hed pred1cate R, wh~h c0nta1n5 the re5uR 0f the evMuat10n.

170 ~ E17ER AND 6. 607•FL08

We ca11 any 5tructure 5~¢ = ( A , -.. ) f0r a f1n~e f1r5t-0rder hn9ua9e a Her6rand dam6a5e (Her6rand D8, f0r 5h0n) 0ver A 1f c0n5tant5 and funct10n5 1n 5a¢ have the Her6rand 1nterpretat~n and the d 0 m a ~ A 0f 5~ ~ the 9ene r~ed Her6rand unNe~e . 7ha t 15, a Her6rand data6a5e expand5 a Her6rand u n ~ e ~ e , 9 e n e r ~ e d 6y f1nRe~ many c0n5tant5 and f u n ~ n 5 , 6y f 1 n 1 ~ many re1at10n5. 7hen m ~ n c0ncern 0f 10~c p r 0 9 r a m m ~ 9 are Her6rand D85, and hence we f0cu5 0n 5uch 5tructure5 ~ th~ pape~

A ~ r u ~ u r e 0n wh~h a DLP P ha5 t0 6e e v ~ u a ~ d mu5t 1ndude ~e1af10n5 f0r a11 ED8P5 0f P, 6ut mu5t n0t have a r d ~ n f0r any 1D8P. 7he ~ a n f 1 ~ n 0f P 0n 5uch a 5tructure ~¢, P(~¢), 15 the 5et 0f ~1 9r0und dau5e5 C0, where C 15 a dau5e ~ 0 m P and 0 15 any 5u65t1tut10n 0ver A, au9mented 6y the 5et 0f a11 9r0und fa~5 R~(a)~- 5uch that ~ R 9 ( a ) , f0r ~1 rdar10n5 R 1 1n ~¢; 0n 0ther ~ru~ure5 , the ~ a n t h t ~ n 0f P 15 undef1ned. N0t1ce that P 15 ~f1n~e ~ 9ener~.

REMARK 2.1. F0r p r a ~ c0ncern~ the re1at10n5 R~ are pr0ved 6y an 0rade . A1~rn~Ne1y, and perhap5 m0re re~vant , they c0u1d 6e pr0v1ded 6y 50me 0ther 10~c pr09ram (w~h the r e 5 p e ~ e funct10n and c0n~ant 5ym6015).

Def1n#~n 21. Let P 6e a n 0 r m ~ 0r ~ u n c t 1 v e ~ c pr09ram. 7he re1at10n def1ned 6y P 0n a 5tru~ure ~¢, Rp(~¢), 15 Rp(59) = {a [ P(~¢) ~ R(a)}, where R ~ the d15t1n9u15hed 0utput re1at10n 0f P.

7he f0110w1n9 pr0p0f1t10n 5tate5 the ma1n re5u1t 0f [34] c0ncern1n9 5ta61e m0de1 5emant1c5 0f n0rma1 1091c pr09ram5 0ver 1nf1n1te Her6rand un1ver5e~

Pr0p051t1~ 21 ~ . L~ D ~ ~e He~mnd u n ~ e ~ m t e d ~ a p 0 5 1 ~ f1n1te num6~ ~ ~ m ~ d a p051t1~, f1n~e num6~ ~ c0~mn~. Let R 6e a ~ n 0n D, and ~t ~ 6e a He~mnd D 8 0ver D. ~ e n , ~e f 0 1 ~ a~ ~u~a~m~:

1. R ~ def1na6~ ~ ~ ~ a n0rma1 LP under ~e 5~6~ m0dd ~ m a n ~ . 2. R ~ d ~ k 1 e ~ ~ ~ a n0rma1 LP un~r ~e ~ - f 0 ~ d ~ ~ m a n ~ ~ . 3. R ~ H1-def1na6~ 0n ~ 1.e., def1na6~ ~ a ~ r m u ~ ~P~ whe~ ~ ~ f1r5t-0rder

( ~ Ap~ena~). 4. R ~ ~duc~e 0n ~ ( ~ Append~ f0r ~ c t 1 v e d ~ ~ L

F0r 5e1f-c0ntNnment and the readeF5 c0nven1ence, we p r0~de here a ~an5f06 mar10n 0f each H~ f0rmu1a ~(x) ~ t 0 a n0rmN ~ c pr09ram P ( ~ 5uch that ~ c0mpute5 0n every ~ r u a u r e 5a¢ 1n k5 0utput r~at10n R under the 5ta61e m0de1 5emanr1~ the rdar10n def1ned 6y ~(x) and ~ , w ~ c h N R.(~)(~ ¢) = {a 159 ~ ~ ( a ~ .

1t N we11 kn0wn that every H~ f0rmu1a ~(x) wRh ~ e e vaf1aMe5 am0n9 x ~ e4uNNent t0 a H~ f 0 r m ~ a 0f the f0rm

V7 ~yV2. ~ (x, y, ~

Where ~ 15 4Uan t1 f1e~e~ th15 f0rm Can 6e 06ta1ned 6y 5eC0nd-0rder 5k01em~a- t10n [36, 5ect. 25.2]. F0r 1ater u5e, we rewr1te the ne9a t~e 0f ~ a5 a d1~unct10n ~

27he r e 5 ~ are 5tated exp11c1t1y 0n~ f0r Her6rand D85 w1th n0 e~ra re1at10n5. H0weve~ they 1mme~e~ e~end t0 9enerN Her6rand D85 0.5ch11pL prN~e c 0 m m u n ~ n ) .

E X P R E 5 5 1 V E N E 5 5 0 F 5 7 A 8 L E M 0 D E L 5 E M A N 7 1 C 5 171

0f dementa ry c0~unct10n5 ~ , 1.e.,

~ ~ 0 ( x , y , 2 ) = ~/ ~ ( x , y , ~ . 1

C0nNder the f01~w1n9 pr09ram P ( ~ ) (ar1f1e5 0f tune5 are 1mp11c1t~ under- ~00d; R N the deN9n~ed 0utput p r e d ~ e ) :

7(X) ~ ~ ~

~ ~ ~ 7 ( x )

R * ( ~ ~ ~ k* (x )

K*(x) ~ ~ R*(x)

Q~ ) ~ ~ ~ D, R*~)

R ~ ) ~ ~ Q ~ ) , R*(x)

R ( ~ ~ k * ( x )

R(x) ~ R*(x), R * ~ x 1 ¢x~

f0r each 7 ~ 7

f0r each ~1 1n ~

f0r each 1 = 1 . . . . . n, where x = x1,. . . ,x n.

Here, M1 pred~ate5 that d0 n0t 0ccur 1n • are new; each pred~ate )~ ha5 the 5ame ar1ty a5 X. 7he f1r5t tw0 9r0up5 0f dau5e5 ch005e an exten~0n f0r the pred~ate5 7 ~ 7. 7he 1ntended r61e 0f pred1cate R*, wh1ch ~ ch05en 1n the next tw0 dau5e~ 15 t0 5e1ect a ~n91e tup1e a 0f the af1ty 0f the 0utput re~t10n R f0r te~1n9 whether ( ~ , 7 ) ~ ~ y V ~ ( a , ~ 2 ) h01d5. 1f th~ 15 true, then a N 1nduded 1n R; 0therw15e, 1.e., ff ( ~ , 7 ) ~ V y ~ 0 ( a , ~ 2 ) h01d~ then a 15 n0t 1nduded 1n R. 7h15 15 acc0mp115hed 6y the tw0 ru1e5 that 1nv01ve Q. 1f the 5e1ect10n 0f R* 15 pr0per (1.e., R * = {a}~ then R 15 a5N9ned the c0mp1ement 0f R*; 0therw15e, ~ the 5e1ect10n 0f R* 15 1mpr0pe~ 1.e., R* c0nta1n5 n0 0r m0re than 0ne tup~, R 15 a5~9ned the 5et 0f a~ tup1e5. 7h15 15 1mp1emented 6y the 1a5t tw0 ru1e5.

FN a ~ructure ~ . F0r each ch0~e 0f 7 and R*, there 15 a 5ta61e m0de1 P 0n ~ . 8y tak1n9 the 1nter5ect10n 0f M1 5ta61e m0de1~ the pr09ram P thu5 c0mpute5 preN5e1y the re1at10n R . ( ~ ) . We 1eave the deta115 t0 the reade~ and 0n1y n0te the f0~0w1n9 fact f0r 1ater reference.

Pr0p051~0n 2 2 0n every 5tructure ~ , Rp~)(~a~) = R ~ ( ~

3. RE5UL75

1n t~5 5 e ~ we der1ve 0ur mMn r e 5 ~ . We 5ta~ w1th an upper 60und ~ r the expre551vene~ 0f ~ u n ~ e 10~c pr09ram5 0ver ar61ffa~ 5tructure9

Let H~(60~) 6e the c ~ n 0f H~ ~ r m ~ a 5 wh05e f1r560rder pa~5 are 8 0 0 ~ a n c0m6~at10n5 0f e ~ e n t 1 M ~ r m ~ a 5 (cL Append~).

Lemma ~1. Let P 6e a DLP. 7hen, de r e ~ 0 n Rp (re5p., de 5~ 0f re~t10n5 ~p) def1ned ~ P 0ver any 5tmcmre ~ (re5p., any c1a55 0f 5tmcture5 ~ ) ~ def1na6~ ~ a H~(60~) ~ r m u ~ .

172 ~ E17ER AND 6 . 60371•L08

PR00E Let P 6e the 1D8P pred1cate5 1n P (wNch 1ndude the d e~9 n ~ed 0utput re1at10n R). Den0m 6y he(5) the un~e~M f1r~-0rder d05ure 0f the c 0 ~ u n ~ n 0f a11 dau5e5 ~ P, where each p ~ c ~ e ~ fr0m P ~ ~ N a c e d 6y a f1e5h p ~ d ~ e var1a61e 51 0f the 5ame af1~. M ~ e ~ e C den0te 6y ~(5~, ~ the u N v ~ 1 f1r~-0rder d05ure 0f the ~ ~ 0f N1 dau5e5 C 1n P where ~ e ~ 0ccurrence 0f ~ fr0m P 15 ~ N a c e d 6y a f1e5h p ~ d ~ a ~ var1a61e 51 0f the 5ame af1~, and ~ r ev e~ ne9at1ve hterN ~ ~ , ~ ~ ~ the ne9~Ne 1~erM ~ 51(0 15 added 1n the 60dy.

C 0 n 5 ~ the ~ ~rmu1a ~ :

• (x) = v5 5 .6 (5) [R(x) v (5• < 5))]

where (5• < 5) 5tand5 ~ r the ~rmu1a

A (Vx,.(~(x~) -- 5,(x,))) ~ A a~.(5,(~) ~ ~ 51(~)), 1 1

1.e., 1t 5tate5 that each p ~ c ~ e 51 mu5t 6e c0nt~ned 1n 51, and that ~ r 50me ~ , the c0mNnmem 15 pr0pe~

1t 15 e a ~ t0 5ee that the ~ r m ~ a ~(x) def1ne5 R e 0ver aW 5tructure ~ 1ndeed, 1t 5ay5 that ~ r ~ e ~ ch01ce ~ r 5 that 15 a m0de1 0f P and ~ wNch 15 e ~ ~ +e(5L e~her x mu~ 6e c0ntNned 1n R, 0r 50me 5mN~r m0de1 0f the 6 d ~ n d - ~ N ~ 1 ~ reduct 0f P 0ver .~ w.r.t. 5 mu~ e ~ ~h1ch 15 e ~ 5 ~ d ~ ~(5~, 5) A (5• < 5~. 3 7h15 ~ e m e m 15 ~ue 1f and 0n~ 1f R ~ 15 c0ntNned 1n e v e ~ ~aNe m0de1 0f P 0n ~

~ e a ~ the f1r560rder pa~ 0f ~ 15 e4uNMem t0 a 8001ean c 0 m 6 ~ n 0f e ~ e n f 1 N f1r560rder ~ rmu1~ . 7h15 pr0ve5 the 1emma. []

7he a60ve ~ m m a 5ay5 ~ m ~ N n 9 ~ ~ 9 a60ut DLP under the ~aN e m0de1 ~ m a m ~ 5 , name~, ~ ~15 ~ m a m 1 ~ 15 ~ r m M ~ a N e ~ ~ e ~ c t ~ f1a9mem H~(60d) 0f the p r e ~ da55 H~ 0f 5ec0nd-0rder 10Nc. 1t 15 we11 kn0wn that 0n 1nf1nRe 5tructure~ th15 ffa9ment 15 a pr0per 5u65et 0f H~. ~ m R N e ~ the Nct that the 5ta~e m0de1 ~ m a m ~ 5 0f DLP can 6e ~ r m ~ d there1n may 6e due t0 the 5u60rd~me r01e that n e 9 ~ n p1ay5 ~ the d ~ u n ~ N e 5ta61e m0de1 ~m an t1 ~ w.~C N ~ . L005e~ 5pea~n~ the ~weakne5f~ 0f ne9at10n ~ th15 ~man t1~ 15 due t0 the f a~ that n e 9 ~ n 15 evNu~ed 1ndependenf1y ~ r each ch0~e 0f a cand1date 5taNe m0dd , and n0t w~h re5pe~ t0 the ena~ ~ m ~ 0f 5ta61e m0dd5. We th ~k that a var1ant 0f DLP ~ t h a 5~ ta6~ def1ned ~r0n9er ~ m a ~ 5 0f n e 9 ~ n c0u1d 6e m0re e ~ N ~ 6ut we 1eave th15 155ue ~ r further re5earch.

7he ~ 0 ~ mchNcN 1emma 15 the k ~ ~ r 0 ~ ma1n ~5Mt. CN1 a ~ r u ~ e ~ 0ver a f1n1te ~9nNum that ha5 n0 ~ n ~ 6ut ~ d u d ~ a Nnaf f r d ~ n 5 an ~ e ~ ~ ~ e ~ t ~ 1f ~ [ 5 ] = ( A , 5 ) ~ (w, 5), 1.e., ~ e ~ r 1 ~ n 0f ,~ t0 5 15 1 ~ m ~ p h ~ t0 w = {0,1,2 . . . . } w1th the u5u~ 5ucce550r r ~ n ~ x , y ) ~ y = x + 1 .

Lemma ~ 2 ~ r ~ e ~ ~ 5 ~ ~e ~ t ~ , ~ e ~ H~(60~) ~ u ~ (p05~6~ ~ ~ e p ~ c a ~ and ~ d u a 1 v a ~ a 6 ~ 6ut n0 ~ 0 m ) ~ e4u~a~m ~ a H~ ~ u ~ .

P~00v. 5ee ~ ~ []

3We c0nf1der h~e 0~y m0dd5 1n wh~h 1m~f1y ~e ED8P5 c~n~de w1~ ~e1r ~ 1 ~ ~ ~. 7~5 ~ n0 ~0~c0m~9 5~cc the c~nc~ence h~d5 1n ~c~ ~a~e m0dd,

E X P R E 5 ~ V E N E 5 5 0 F 5 7 A 8 L E M 0 D E L 5 E M A N 7 1 C 5 173

7he0rem ~1 (MaM re5uRL L ~ D 6e ~ e Her6mnd un~e~e 9enem~d 6y a p0~t1t,e, f1n1te num6er 0 f f u n ~ n 5 and a p051t~e, f1n1te num6er 0 f c0n~an~. L ~ R 6e a m~t10n cn D, and 1et ~9 6e a Her6rand D8 0 v ~ D. 7hen, ~ e f0110w1n9 am

e4u~a~nt:

1. R ~ def1na6~ 0n ~ 6y a D L P under ~ e 5m6~ m0de1 5emant1c5. 2. R ~ def1na6~ 0n 0~ 6y a n0rma1 L P under ~ e 5m6~ m0de1 ~ m a n ~ . 3. R ~ def1nak1e ~n ~ 6y a n0rma1 LP under ~ e we~-f0unded ~ m a n ~ . 4. R ~ H1~ef1na6~ 0n 0~. 5. R ~ ~du~t1ve1y def1na6~ 0n ~ .

P900~ 8y Pr0p0~t10n 2.1, 1t ~ma~5 t0 5h0w that 1. ~ 4.; 8y Lemma 3.1, 1t remMn5 f0r ~15 t0 5h0w ~hat eve~ H~(60~) f0rmMa • 15 0n ~ 15 e4~v~em t0 a H 1 f0rmM~ Ap~f1n9 Lemma 3.2, th15 ~ e 5 ~ h e d u51n9 5tandard c0d1n9 tech- n14ue5. 8a~ca11~ th15 am0unt5 t0 5h0w1n9 that the Her6rand unNe~e 1mp11df1y pr0~de5 the 1nte9er5.

Fff~, rem0ve ~1 fun~Mn5 fr0m 0~ 6y re~adn9 each n-a~ fun~Mn f w1th an n + 1-a~ re1at10n F, 5uch that F(x~ . . . . . x~+~)~x~+~ =f(x~ . . . . . x~); 1et .~* 6e 1he re5Mt~9 5tru~ure. Re~a~n9 fun~Mn5 f w1~ r~at10n5 F, f0rmuh ~ can 6e rewf1~en t0 a H~(60~) f0rmMa ~* 5uch ~a t ~ • 1f and 0n~ 1f ~* ~ •*. 70 5h0w that • 15 H~ 0n ~e~, 1t 5uff1ce5 t0 5h0w that ~* 15 H 1 0n .~*.

U5e a 5u65et 1 0f the Her6rand unNe~e D(=A) a5 a c0py 0f the 1n~9e~ ~1 . . . . wh~h, a5 u5uM, 15 ~du~N e~ def1ned a5 f0~0w5:0 =c, and 1 + 1 = f ( c . . . . , c, 1), where c 15 a c0n5tant and f 15 a fun~Mn ~m601 0f p0~t~e af1W (the c5 are u5ed a5 ~ f f 1 ~ " ~ the ~ducf10n 1f the fun~Mn ha5 af1W 9reater than 0ne); thu5, 1 15 def1ned 6y the f0rmMa

V x . 1 ( x ) , ~ , x = c v ~ y , 2 . x =2 A F ( c . . . . . c, y , 2 ) A 1(y). (2)

Let E 6e the 9raph 0f a 6~e~Mn 6etween 1 and the d0mMn D. 7h15 6~ect10n n ~ u r ~ 1nduce5 a 5ucce~0r 5 0n D, wh1ch 15 de5cr16ed 6y 5(x , y ) ,~ , ~2 ~ LE(2 , x) A E(2 + 1, y). (Here, 2 + 1 15 the 5ucce550r f ( c . . . . . c, 2) 0f 2.)

C0n~der the ~ructure ~ = (A*, 1, E, 5). 51nce 1t 15 an ~ffee expan~0n 0f the 1nte9er~ 6y Lemma 3.2, f0rmMa • 15 e4u~Ment 0n . ~ t0 a H 1 f0rmMa • = V5~.

C1ear1y, E can 6e character12ed 6y a f1r~-0rder f0rm~a fr0m L 7heref0~, the re1at10n5 L E, and 5 can 6e ~ 1 m ~ e d fr0m the f0rmuh • 6y 1ntr0duc1n9 un~er5a11y 4uant1f1ed pre~c~e var1a61e5 and ~ht1v1Mn9 1~ f1r5t-0rder pa~ ~ ~ a f1~0rder chara~er12at10n e(L E, 5) 0f L E, and 5 ~mf1a~y t0 the pr00f Lemma 3.2 1.e., the f0rmMa

~1, E , 5 , 5 . ~ - ~ ~

15 0n ~ e 4 u ~ e n t t0 ~. 7hu5, ~* 15 0n ~* e4u~Ment t0 a H 1 f0rmM~ wh1ch 1mp1~5 that • 15 0n ~" e4~v~ent t0 a H 1 f0rmuh. 7h15 pr0ve 1 ) ~ 4L wh1ch c 0 m ~ e 5 ~ e ~5~t. []

REMAR~ 3.1. We 065erve th~ the ~an5f0rm~Mn ef a H] f0rm~a ~ 1~0 a n0rm~ 10~c~ pr09ram P , (5ee end 0f 5ect10n 2) can 6e m0d~ed ea51~ t0 pr0duce an

e4u~ena t n 6 R* 6y~raf1f1ed1 d~un~d1e~unct1c1 veau5e5 1°~7 c~)~r~;m,~ 6 ~ p ~ : ~ ~ h ~ : ~ c~;~;tf1~1~.

M0re0ve~ the re5u1t1n9 pr09ram 15 even head-cyc1e-free 1n the 5en5e 0f [6]. Hence, we 06tMn f0r 5~af1f1ed DLP5 (even w1th 11m~ed u5e 0f ~unct10n) the 5ame expre5~vene~ r e 5 ~ a5 f0r n0rmM 1091c pr09ram5 under the ~a61e m0de1 5e- mant~5.

174 ~ E17ER A N D 6 . 6 0 7 1 • L 0 8

A5 a c0n5e4uence, we 06ta1n the f01~w1n9 re5uR~

C0~11a~ ~1. A ~ t 1 0 n R ~ d•na61e 0n a ~mctu~ ~ ~at ~ p a n ~ ad~met1c (cf. Append~) ~ ~ t 1 0 m ~ u 9 h a DLP un~r ~e 5m6~ m0~1 5emant1c~ 1ff ~ ~ d ~ n a 6 ~ ~ a n0rma1 LP under ~e 5m6~ m0dd ~ m a n ~ , 1ff # ~ d ~ n a 6 ~ ~ a n0rma1 LP under ~e weH-f0un~d ~man~c~ 1ff ~ ~ H 1 d ~ n a 6 ~ 1ff ~ ~ ~duc~ve.

1f we v1ew, a5 u5ua1, the re1at10n Re(D) def1ned 6y a DLP P 0ver a Her6rand unNe~e D a5 a 5u65et 0f the ~ 9 e r 5 de5cr16ed 6y a 66de1 num6er1n9 0f Re(D~ then we have ~ e f0H0w~9 ~5dt .

C 0 r 0 1 ~ ~ 2 7he 5m6~ m0de~ ~ m a n ~ 0f DLP5 0ver Her6rand u n 1 v e ~ ~ H1-c0mp1ete 0ver the ~te9e~.

~ C0NCLU510N

7he a60ve r e 5 ~ 5h0w that under the 5ta~e m0de1 5emant1c5, ~ u n ~ e ~ c P~09ramm~9 15 ~5 expre~Ne a5 n0rm~ 10~c pr09ramm~9 0ver ~f1n1~ Her6rand m0dd5. 7h15 re5~t c0ntra5t5 w1th r e 5 ~ 0n the m1n1m~ m0dd 5emant1c5 (H~- c0m~e~ne~ 0f the 6CWA f0r DLP5 ver5u5 H~-c0m~etene55 0f the CWA f0r n0rm~ LP5 [11~. M0re0veL the r e 5 ~ 1mp~ that 0n ~f1~te Her6rand m0dd~ there 15 a hu9e 9ap 6etween the expre~Ne p0wer 0f ~ u n ~ N e ~ c pr09ram5 under the 6rave ~a61e m0de1 5emant~5 (c1ear~ E~m0mp1ete) and the 6rave m1~m~ m0de1 5emant~5 (E~c0m~e~, a5 f01~w5 ~0m the pr00f 1n [11~.

APPEND1X

7h15 Append1x pr0~de5 50me 6ack9r0und 0n rdevant recurf10n-the0ref1c c0ncep~ (f0r m0re, c0n5u1t, e.9., [3~) and the pr00f 0f Lemma 3.2.

7he prenex h1erarchy H~, E~, k ~ 0 0f f1r5~0rder f0rmu1a5 ~ 6u11t 0ver the c0Hect10n E~ = H 1 0f a11 4uanf1f1e~ee f1r5~0rder f0rmu1am 5uch that H~+ 1 ~e5p., E~+ ~) c0nt~n5 a11 f0rmu1a5 ~ x ~ ~e5p. ~x~) where ~ ~ ~0m E~ ~e5p., f1~). 7he h1erarchy H~, E~, k ~ 0 0f 5ec0nd-0rder f0rmu1a5 ~ f0rmed ~m11ar1y, where H~ = E~ 15 the 5et 0f a~ f1~t-0rder f0rmu1a5, and H~+ 1 ~e5p., E~+~) c0nt~n5 a11 f0rmu1a5 ~ P ~ e 5 ~ , ~P~) where P ~ a 11~ 0f pred1cate var1a61e5 0f f1xed af1ty and • 15 ~0m E~ (re5p., H~). (7he 5uper5cr1pt "1" 1nd1cate5 that 4uant1f1cat10n ~ 0ver pred~ate~ rather than 1ndN1du~5. N0te that, u 5 u ~ , 5ec0nd-0rder 4uant1f1cat10n ~50 1ndude5, re5p. ~ re~r1cted t0, fun~10n 5ym601~ th15 d0e5 n0t affect the e x p r e ~ e p0wer 0f E~, H~; cL [30, p.38~3 1n p a ~ u 1 a L H~ and H~ c0nt~n a11 f0rmu1a5 0f f0rm ~P+ and ~P~P2 ~, re5pe~e1y, where ~ ~ f1r5~0rde~

Every f0rmu1a ~(x) wh05e ~ee var1a61e5 are am0n9 x = xa,. . . , x~ def1ne5 0n any ~ru~ure 5e" a re1af10n R . , ~ ¢ ) = {a ~ A n [ ~ae~ ~(a)}. ff a r~af10n ~ def1ned 0n 5¢ 6y a f0rmu1a ~0m a da55 ~, ~ 15 ca11ed C-def1na61e.

1t 15 kn0wn that H1-def1na6111 ~ 15 d05e~ re1ated t0 1nductNe def1na6111ty [28, 4]; th15 c0nnect10n wa5 ~ud1ed ~ready 6y K1eene. A rdaf10n R(x) 15 p051tNe e~men- tary 1nduct1ve~ def1na61e (~mp~, 1 n d u ~ e ) 0n a 5tructure 5¢ ff there ~ a f1r~-0rder f0rmu1a ~(x,~ 5) 1n the 1an9ua9e 0f 5¢ where the pred~ate vaf1ah~ 5

E X P R E 5 ~ V E N E ~ 0 F 5 7 A 8 L E M 0 D E L 5EMAN71C5 175

0ccur5 p0f1t~e 1n ~, and there are demen~ a 1n 5~ 5uch that R(x )~ (x,a)~ 1¢, where 1~ 15 the 5 m ~ f1xp01nt 0f the 0perat0r F def1ned 6y F (5 )= {(x,y) [(A, 5 ) ~ ~(x,~ 5) ~8, 5]. 7he f0110w1n9 1mp0nant re5uR5 11nk H~ def1na6f1- 1ty and 1 n d u ~ e def1na6f11ty.

Pr0p0~t10n A.1 [28, 4]. Let ~ 6e a ~ 5tmcm~. 7hen, a ~ n ~ H 1 def1na6~ 0n ~ ~ # ~ ~ d u ~ ~ ~ .

P r 0 p 0 ~ n A . 2 ~, 5]. L ~ ~ 6e a c0unm61e 5tmcm~ (1.e., A ~ c 0 u n m 6 ~ ~ A ha5 a f1r5t-0rder 0r ~duct1ve~ def1na6~ paMn9 ~nc~0n, 1.e., a 0 n ~ n e map ~ A ×A ~ A , ~ a ~ t 1 0 n • H~ def1na6~ 0n ~e" ~ and 0n~ ~ # ~ ~duct1ve cn ~.

Def1na6111ty 0ver the 1nte9e~ ha5 6een ea~y rec09n~ed a5 an 1mp0~ant t0p1c, a5 many 5e~ 0f f0rma1 06ject5 can 6e effect1ve~ enc0ded t0 a 5u65et 0f the 1nte9er5 6y 66d~-num6ef1n~ 7h15 15 deaf1y p05~61e f0r any Her6rand un1ver5e H 9ene~ ated 6y c0unta6~ many c0n5tant5 c~,c2,...and funct10n 5ym6015 f~,f2 . . . . . E.9., a5519n each c0n~ant c~ the c0de 9(c~)=U, and 1nduc t~y each term t = ~(t~ . . . . . ~) the c0de 9 ( t ) = ~ . 5 ~ ) . 7 9 ~ ) . . . . . p ~ , where p~,p2, . . . ,are a11 pr1me5. 06~0uM~ the enc0d1n9 9 and the dec0d1n9 9-~ are c0mputa61~ where 9 ~(n)= • ff n 15 n0t a vM1d c0de. U~n9 9, any Ca~e~an pr0duct H k= H × --- × H can 6e ea51~ enc0ded t0 a 5u65et 0f the 1nte9e~.

Den0te 6y ~V= (w, 0, 5, + , . ) ar1thmet1c 0n w = {~ 1,...}, 1.e., the 5tructure wh05e d0m~n are the 1nte9er5 and wh1ch pr0v1de5 the fam111ar 5ucce550r funct10n t09ether w1th add~10n and mu1t1p1~at10n.

A f1~-def1na61e re1at10n R 0n , ¢ 15 11~-c0mp1ete 1f, f0r every 11~-def1na61e re1at10n 5, there e~5t5 a recur5Ne (1.e., c0mputa61e) funct10n f 5uch that f0r M1 tup1e5 0f 1nte9e~ n, 5(n) ,~, R(f(n)); a da55 cf r~at10n5 15 H~-c0mp1ete 1f a11 1~ r~at10n5 are f1~-def1na61e and at 1ea~ 0ne 0f them 15 11~-c0mp1ete. 7he def1n1t10n 0f E~-c0mp1ete r~at10n5 and da55e5 15 anM090u5.

~ 15 we11 kn0wn that the da55e5 0f 17~, re5pect1ve~ E~, re1at10n5 0ver~V, k ~ 1, f0rm 5tr1ct h1erarch1e~ and that each da55 ha5 c0mp1ete re1at10n5 (cL M50 ~6]). N0te that def1na6111ty 0f a re1at10n 1n 11~, 2~, 0r H~(6001) rem~n5 unaffected ff we ad0pt af1thmet~ 1n a rdat10nM 5ett1n~ 1n wh~h the funct10n5 are pr0v1ded 6y theft 9raph5 (1.e., a5 re1at10n5).

Lemma ~ 2 ~ r ~ e ~ m ~ ~e ~ , e ~ H ~ 6 0 ~ ) ~ u ~ ( p 0 ~ 6 ~ ~ ~ ~ ~ d ~ ~ ~ 6 ~ , ~ t ~ ~ m ) ~ ~ a ~ t W a H~ ~ u ~ .

PR00~ 1t 5Uff1Ce5 t0 C0n~der ~ru~ure ~¢ that expand (w, 5 ) 6y add1t10nM re1at10n5. (N0te that C0n~ant5 are n0t an 1~ue ~nCe they Can 6e pr0~ded 6y de5~nated re1at10n5.)

7he 1emma 15 a c0n5e4uence 0f the f0H0W1n9 re5uR. 0ver af1thmet~ ~V 1n a re1at10na1 5ett1n9 w1th exp0nent1at10n (exp), every f0rmu1a V7Vx3~ ~, where ~ 15 4uant1f1e~ffee and may have 0cCurrence5 0f f1ee pred1Cate and 1nd1v1du~ var1- a61e5, 6ut n0t 0f fun~10n5, ~ e 4 u ~ e n t t0 a f1r5t-0rder f0rmu1a ~.

A 5e1~c0nt~ned pr00f 0f th~ re5u~ 15 ~ven 1n [13, Lemma 4.~. ~ appear5 1n m0re c0mpa~ ff0m 1n [4] (5tr1~ H~ f0rmu1a52 8arw15e a~r16ute5 a 5~0n9er f0rm t0 Kre15~ (wh~h a110w5 60unded u n ~ e ~ 4uant1f1cat10n 1n the f1~60rder pan 0f

176 ~ E17ER AND 6. 60•1WL08

~, and re4~re5 that a11 unNe~N 4uant~er5 are 60unded ~ 0~ and pr0ve5 a 9enera112at10n 0f ~ t0 c0unta61e a d m N 5 ~ 5et5.

Ut11121n9 ~he re5u1t, the pr00f 0f 1emma 15 a5 f0H0w~ Any H~6001) f0rmu1a • can 6e wf1~en a5

V537~xVy. ~

wh~h, 6y ~ r c h a n ~ n 9 e ~ e n t ~ 1 4uan t~er~ 15 e 4 u ~ e n t t0

~ 5 ~ x ~ 7 ~ y . ~ .

5upp05e, f0r the m0ment, that the c0n5tant 0 and the a f 1 t h m ~ r ~ n 5 are ava~a6~ ~ ~ . 7he 4 u 0 ~ d re5uh 1 m ~ 5 t h ~ 0n ~ , th~ f0 rm~a ~ e 4 u ~ e n t t0

v5~x.~ (3) where ~ N f1~t-0rde~ C1eadN 0 and the af1thmef1cM r~af10n5 +,•,exp can 6e de5c~6ed 6y u51n9 an 1ndN~uN va~a61e x~, re5p. u n N e ~ N ~ 4uant1f1ed p reNc~e va~a61e5 P, M, E, and def1n1n9 the var1a61e5 u~n9 a f1r5~0rder f0rmMa ~(x 0, P, M, E). Henc~ 0 and the a~thmef1cN re1af10n5 can 6e ~ 1 m ~ e d fr0m the prev10u5 f0 rm~a 6y unNe~N 4uant~caf10n 0ver the va~aNe5 and r~1v12af10n 0f the f1r5t-0rder part t0 ~, 1.e., the f0rmMa

VP, M, E, 5Vx0. ~ -~ 3x .0

15 0n 5e• e4~vMent t0 the f0rmu1a 1n (3). 1f f0~0w5 that 4~ 15 0n ~ e4~vMent t0 a n1 f0rmu1~ wh~h pr0ve5 the ~mma. []

7he auth0r5 w0dd f1ke t0 thank J. 5ch11pf f0r v~ua6~ c0mment5 and dar~c~n5 0f the extent 0f h~ re5uR5 ~ ~4], M. F1tt~9 f0r 6ack9r0und 1nf0rm~n a60ut ~5~ a5 we11 a5 the part1c~ant5 0f the Da9~uh1-5em1nar 0n D15junct1ve L09~ Pr09ramm1n9 and D~a6a5e~ N0nm0n0t0~c A5pe~ Ju~ 1-5 1996, 1n part1c~ar Jack M~keC wh0 ha5 ~1m~ed th15 re5earch 6y p~nt1n9 0ut 0pen ~5ue5 0n ~ u n ~ e ~ c Pr09ramm~9 ~ a keyn0te t~k. Fu~herm0~, we appred~e the he~fu1 c0mmen~ and 5u99e5t10n5 0f the an0nym0u5 re~ree5 f0r 1mpr0v1n9 t~5 pape~

1. Andreka, H. and Nemet1, 1, 7he 6en~MNed C0mN~ene~ 0f H0rn Pred1cate L0Nc5 a5 Pr0~amm~9 Lan9ua9e, Acm Cy6emet1~ ~ D ~ - ~ (1978L

2. A N, K. and 81Nr, H., Ar1thmet1c C N ~ c ~ n ~ P~%~ M0deN d 5 ~ e d Fr0~am~ ~: Pr0c. 5~ J0~t ~t . C0nf. and ~),mp. 0n L091c Pr0~amm1n9 (J1C5LP-88~ R. K0wN5M and K. 80uwen (ed~L M17 Pre5~ 1988, pp. 766-779.

3. 8~N, C. and 6e1f0nd, M., L0Nc Pm~amm~9 and Kn0w1ed9e Rep~mat10n, J. L091c Pr0~amm1n9 19/2~73-148 (1994).

4. 8arw15e, J., A d m ~ 6 ~ 5e~ and 5t~cmre5, 5p~n9eL 1976. 5. 8arw15e, J. and M ~ c h 0 v a ~ Y., 6106a1 1ndu~Ne Def1na6111~, J. 5ym6011c L091c 43:

521-534 (1978L 6. 8en-E11yahu, R. and D e c ~ , R., Pr0p0f1f10na1 5emam~5 ~ r D~un~Ne L0Nc Pr0-

9ram, Anna~ 0f Ma~. and A~1f1c1a1 ~te11. 12:53-87 (1994~ 7. 81N~ H., 7he Recur~0n-7he~et1c C 0 m N e ~ 0f the 5emam~5 0f Pred1cate L0Nc a5 a

Pr0~amm~9 Lan9ua9~ 1nf0rm. C0n~. 25-47 ~ y / A u ~ 1982~ 8. NNr, H. and Ch0N~ C., 7he C0m~e~ff 0f L0cM 5~af1f1c~n, Fundamenm 1nf0rma~

cae 21:333-344 (199~.

ExPRE5~vENE~ 0F 57A8LE M0DEL 5EMAN~c5 177

9. C a d ~ M. and 5chaerL M., A 5u~ey and C0m~ex1~ R e 5 ~ ~ r N0n-M0n~0n~ L 0 ~ , 2 L091c Pr09ramm1n9 1~127-160 (1993L

10. Chen, W. and Warre~ D. 5., 7a6~d E v ~ u ~ n w1th De1ay1n9 ~ r 6ener~ L0~c Pr09ram6 ~ ACM 43(1~20-74 (199~.

11. Ch0m1ck1, J. and 5u6rahma~aK V., 6 e n e r ~ e d C105ed W0dd A~umpf10n5 15 U~ C0m~ete, 1nf0rm. Pr0ce551n9 Lett. 34:289-291 (199~.

12. E1te~ 7. and 60 t t~K 6., 0n the C0mputat10n~ C0~ 0f D15junct1ve L0~c Pr09ram- m1n9: Pr0p05~0n~ Ca5e, Anna~ Ma~. and A~1f1c1a1 1 n ~ . 150/~:289-323 (1995L

13. E 1 ~ 7., 60t t66, 6., and 6u re~c6 Y., N0rm~ F0rm5 f0r 5ec0nd-0rder L0~c 0ver F1n1te 5 W u ~ u ~ and C1~f1cat10n 0f NP 0pt1m~at10n Pr06~mK Anna~ Pu~ and AppL L091c 78:111-125 (1996L

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