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(recen)n(t)f(extensions)h(of)g(the)f(answ)n(er)h(set)f(\(stable)h(mo)r (del\))f(seman)n(tics)g(of)h(logic)374 2044 y(programs.)28 b(One)g(of)h(them,)e(due)h(to)h(Lifsc)n(hitz,)g(T)-6 b(ang)29 b(and)f(T)-6 b(urner,)28 b(allo)n(ws)i(the)e(b)r(o)r(dies)h (and)f(heads)374 2135 y(of)23 b(rules)g(to)g(con)n(tain)g(nested)g (expressions.)h(The)f(other,)g(due)f(to)h(Niemel\177)-38 b(a)23 b(and)f(Simons,)h(uses)g(w)n(eigh)n(t)374 2227 y(constrain)n(ts.)34 b(W)-6 b(e)33 b(sho)n(w)h(that)f(there)g(is)h(a)f (simple,)h(mo)r(dular)e(translation)j(from)e(the)g(language)i(of)374 2318 y(w)n(eigh)n(t)c(constrain)n(ts)h(in)n(to)e(the)h(language)h(of)g (nested)e(expressions)i(that)e(preserv)n(es)h(the)g(program's)374 2409 y(answ)n(er)g(sets.)h(Nested)f(expressions)h(can)f(b)r(e)g (eliminated)g(from)f(the)h(result)g(of)h(this)f(translation)h(in)374 2501 y(fa)n(v)n(or)e(of)g(additional)h(atoms.)f(The)g(translation)h (mak)n(es)e(it)h(p)r(ossible)h(to)f(compute)f(answ)n(er)h(sets)g(for) 374 2592 y(some)21 b(programs)h(with)g(w)n(eigh)n(t)h(constrain)n(ts)f (using)g(satis\014abilit)n(y)h(solv)n(ers,)g(and)f(to)g(pro)n(v)n(e)f (the)h(strong)374 2683 y(equiv)l(alence)j(of)h(programs)g(with)g(w)n (eigh)n(t)g(constrain)n(ts)h(using)f(the)f(logic)i(of)g (here-and-there.)374 2840 y Fu(KEYWORDS)p Fv(:)h(answ)n(er)h(sets,)h (cardinalit)n(y)f(constrain)n(ts,)h Ft(smodels)p Fv(,)f(stable)g(mo)r (dels,)g(w)n(eigh)n(t)h(con-)374 2931 y(strain)n(ts.)p 374 3084 V 1557 3371 a Fw(1)48 b(In)m(tro)s(duction)374 3534 y FB(The)30 b(notion)e(of)i(an)g(answ)n(er)e(set)i(\(or)g (\\stable)e(mo)r(del"\))g(w)n(as)h(de\014ned)h(in)f(\(Gelfond)h(and)f (Lif-)374 3642 y(sc)n(hitz)k(1988\))g(for)h(logic)d(programs)h(whose)h (rules)g(ha)n(v)n(e)h(simple)d(syn)n(tactic)i(structure.)h(The)374 3750 y(head)27 b(of)g(suc)n(h)g(a)h(rule)e(is)g(an)h(atom.)g(The)g(b)r (o)r(dy)h(is)e(a)h(list)f(of)i(atoms,)d(some)h(of)i(them)f(p)r(ossibly) 374 3858 y(pre\014xed)f(with)g(the)h(negation)d(as)i(failure)e(sym)n(b) r(ol.)g(In)j(this)f(pap)r(er,)g(w)n(e)g(compare)e(t)n(w)n(o)i(recen)n (t)374 3966 y(extensions)f(of)j(the)g(answ)n(er)e(set)i(seman)n(tics.) 457 4074 y(In)23 b(one)g(of)h(these)f(extensions,)f(the)i(head)f(and)g (the)h(b)r(o)r(dy)g(of)f(a)g(rule)f(are)h(allo)n(w)n(ed)d(to)j(con)n (tain)374 4182 y(negation)d(as)i(failure)e(\()p Fs(not)8 b FB(\),)23 b(conjunction)e(\(,\))i(and)g(disjunction)e(\(;\),)i (nested)g(arbitrarily)17 b(\(Lif-)374 4290 y(sc)n(hitz)23 b Fs(et)j(al.)g FB(1999\).)d(In)h(particular,)e(negation)g(as)i (failure)e(can)i(o)r(ccur)g(in)f(the)i(head)f(of)h(a)f(rule,)374 4398 y(as)j(prop)r(osed)f(in)h(\(Lifsc)n(hitz)f(and)h(W)-7 b(o)r(o)28 b(1992\).)e(F)-7 b(or)27 b(instance,)1730 4563 y Fr(a)p FB(;)14 b Fs(not)35 b Fr(a)1249 b FB(\(1\))374 4727 y(is)30 b(a)g(rule)g(with)g(the)i(empt)n(y)e(b)r(o)r(dy)-7 b(.)31 b(The)g(program)d(whose)j(only)e(rule)h(is)g(\(1\))h(can)g(b)r (e)g(sho)n(wn)374 4835 y(to)c(ha)n(v)n(e)f(t)n(w)n(o)h(answ)n(er)f (sets:)i Fq(;)f FB(and)g Fq(f)p Fr(a)p Fq(g)p FB(.)g(The)h(rule)1607 4999 y Fr(a)23 b Fq( )g Fs(not)36 b(not)f Fr(a)1127 b FB(\(2\))374 5164 y(is)34 b(another)h(example)e(of)j(a)g(rule)e(with)h (nested)h(expressions.)d(It)k(is)d(\\nondisjunctiv)n(e"|its)374 5272 y(head)e(is)f(an)h(atom;)f(but)j(its)d(b)r(o)r(dy)i(con)n(tains)d (nested)j(o)r(ccurrences)e(of)h(negation)f(as)h(failure.)374 5380 y(The)27 b(program)e(whose)i(only)f(rule)g(is)h(\(2\))g(has)g(the) h(same)e(answ)n(er)h(sets)g(as)g(\(1\).)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 374 432 a FB(2)946 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)457 631 y FB(According)c(to)i (the)h(second)f(prop)r(osal)e(\(Niemel\177)-42 b(a)25 b(and)k(Simons)d(2000\),)h(rules)f(are)i(allo)n(w)n(ed)374 739 y(to)39 b(con)n(tain)e(\\cardinalit)n(y)d(constrain)n(ts")j(and)i (more)e(general)g(\\w)n(eigh)n(t)g(constrain)n(ts.")f(F)-7 b(or)374 847 y(instance,)1616 982 y(0)23 b Fq(\024)f(f)p Fr(a;)14 b(b)p Fq(g)22 b(\024)g FB(1)1136 b(\(3\))374 1118 y(is)28 b(a)i(cardinalit)n(y)c(constrain)n(t.)i(This)g(expression) g(can)i(b)r(e)g(view)n(ed)e(as)i(a)f(logic)e(program)g(con-)374 1226 y(sisting)21 b(of)i(a)g(single)e(rule)h(with)h(the)h(empt)n(y)e(b) r(o)r(dy)-7 b(.)24 b(Its)g(answ)n(er)e(sets)h(are)g(the)g(subsets)h(of) f Fq(f)p Fr(a;)14 b(b)p Fq(g)374 1334 y FB(whose)27 b(cardinalities)22 b(are)27 b(b)r(et)n(w)n(een)g(0)g(and)h(1,)f(that)h(is)e(to)i(sa)n(y)-7 b(,)26 b(sets)i Fq(;)p FB(,)f Fq(f)p Fr(a)p Fq(g)g FB(and)g Fq(f)p Fr(b)p Fq(g)p FB(.)457 1442 y(Cardinalit)n(y)20 b(and)25 b(w)n(eigh)n(t)e(constrain)n(ts)f(are)i(imp)r(ortan)n(t)f (elemen)n(ts)g(of)h(the)i(input)e(language)374 1550 y(of)32 b Fp(smodels)g FB(|)h(a)f(soft)n(w)n(are)f(system)h(for)g(computing)e (answ)n(er)h(sets)i(that)g(can)f(b)r(e)h(used)g(to)374 1658 y(solv)n(e)22 b(man)n(y)h(kinds)g(of)i(com)n(binatorial)18 b(searc)n(h)23 b(problems.)2228 1628 y Fo(1)2288 1658 y FB(The)h(idea)f(of)i(this)f(programming)374 1766 y(metho)r(d,)c (called)f Fn(answ)n(er)h(set)h(programming)p FB(,)16 b(is)k(to)h(represen)n(t)f(the)h(giv)n(en)f(searc)n(h)f(problem)g(b)n (y)374 1874 y(a)25 b(logic)e(program)g(whose)i(answ)n(er)f(sets)i (corresp)r(ond)e(to)i(solutions.)d(Cardinalit)n(y)f(constrain)n(ts)374 1982 y(are)k(found)i(in)f(man)n(y)f(programs)e(of)k(this)f(kind.)457 2091 y(It)j(ma)n(y)e(app)r(ear)h(that)i(the)f(t)n(w)n(o)f(extensions)g (of)h(the)g(basic)e(syn)n(tax)i(of)g(logic)d(programs)f(|)374 2199 y(nested)i(expressions)d(and)j(w)n(eigh)n(t)e(constrain)n(ts)f(|)j (ha)n(v)n(e)f(little)f(in)h(common.)f(The)i(follo)n(wing)374 2306 y(observ)-5 b(ation)33 b(suggests)h(that)i(it)f(w)n(ould)f(not)i (b)r(e)g(surprising)c(actually)h(if)j(these)f(ideas)f(w)n(ere)374 2414 y(related)24 b(to)i(eac)n(h)f(other.)g(The)h(original)21 b(de\014nition)k(of)h(an)f(answ)n(er)g(set)h(is)e(kno)n(wn)i(to)f(ha)n (v)n(e)g(the)374 2522 y(\\an)n(ti-c)n(hain")g(prop)r(ert)n(y:)k(an)g (answ)n(er)f(set)i(for)f(a)g(program)e(cannot)i(b)r(e)h(a)f(subset)h (of)f(another)374 2630 y(answ)n(er)h(set)i(for)f(the)h(same)f(program.) e(Examples)f(\(1\))k(and)g(\(2\))g(sho)n(w)f(that)h(the)h(an)n(ti-c)n (hain)374 2738 y(prop)r(ert)n(y)i(is)h(lost)f(as)h(so)r(on)g(as)g (nested)h(expressions)d(are)h(allo)n(w)n(ed)e(in)j(rules.)f(Example)f (\(3\))374 2846 y(sho)n(ws)29 b(that)i(in)f(the)i(presence)e(of)g (cardinalit)n(y)d(constrain)n(ts)h(the)j(an)n(ti-c)n(hain)d(prop)r(ert) n(y)i(do)r(es)374 2954 y(not)d(hold)g(either.)457 3063 y(In)d(this)g(pap)r(er)g(w)n(e)g(sho)n(w)g(that)h(there)f(is)g(indeed)g (a)g(close)f(relationship)d(b)r(et)n(w)n(een)25 b(these)f(t)n(w)n(o)374 3171 y(forms)d(of)i(the)g(answ)n(er)f(set)h(seman)n(tics:)d(cardinalit) n(y)f(and)k(w)n(eigh)n(t)e(constrain)n(ts)g(can)i(b)r(e)g(view)n(ed)374 3279 y(as)i(shorthand)h(for)g(nested)h(expressions)d(of)i(a)g(sp)r (ecial)e(form.)i(W)-7 b(e)26 b(de\014ne)h(a)f(simple,)e(mo)r(dular)374 3387 y(translation)k(that)j(turns)g(an)n(y)f(program)f(\012)i(with)f(w) n(eigh)n(t)g(constrain)n(ts)f(in)n(to)g(a)i(program)d([\012])374 3494 y(with)34 b(nested)g(expressions)e(that)j(has)f(the)h(same)e(answ) n(er)g(sets)h(as)g(\012.)g(F)-7 b(urthermore,)33 b(ev)n(ery)374 3602 y(rule)k(of)j([\012])f(can)g(b)r(e)g(equiv)-5 b(alen)n(tly)36 b(replaced)h(with)i(a)g(set)g(of)g(nondisjunctiv)n(e)e(rules,)h(and)374 3710 y(this)27 b(will)f(lead)h(us)h(to)h(a)f(nondisjunctiv)n(e)e(v)n (ersion)g([\012])2091 3680 y Fm(nd)2199 3710 y FB(of)j(the)f(basic)f (translation.)e(Finally)-7 b(,)374 3818 y(w)n(e)26 b(will)e(de\014ne)j (a)f(\\nonnested)g(translation")d([\012])1958 3788 y Fm(nn)2044 3818 y FB(,)k(obtained)e(from)h([\012])2736 3788 y Fm(nd)2842 3818 y FB(b)n(y)h(eliminating)374 3926 y(nested)20 b(expressions)e(in)i(the)h(b)r(o)r(dies)f(of)h(rules)e(in)h (fa)n(v)n(or)f(of)i(additional)c(atoms.)i(The)i(nonnested)374 4034 y(translation)32 b(is)j(a)g(conserv)-5 b(ativ)n(e)33 b(extension)h(of)i(\012,)f(in)g(the)h(sense)g(that)g(dropping)d(the)j (new)374 4142 y(atoms)26 b(from)g(its)h(answ)n(er)f(sets)h(giv)n(es)e (the)j(answ)n(er)f(sets)g(for)g(\012.)457 4251 y(The)22 b(translations)d(de\014ned)k(in)f(this)g(pap)r(er)g(can)g(b)r(e)h(of)f (in)n(terest)f(for)h(sev)n(eral)e(reasons.)h(First,)374 4359 y(the)30 b(de\014nition)e(of)h(an)g(answ)n(er)f(set)i(for)f (programs)d(with)j(w)n(eigh)n(t)f(constrain)n(ts)f(is)i(tec)n(hnically) 374 4467 y(somewhat)36 b(complicated.)f(Instead)j(of)g(in)n(tro)r (ducing)e(that)j(de\014nition,)e(w)n(e)g(can)h(treat)g(an)n(y)374 4575 y(program)24 b(\012)k(with)f(w)n(eigh)n(t)f(constrain)n(ts)f(as)i (shorthand)g(for)g(its)f(translation)f([\012].)457 4683 y(Second,)36 b(the)i(de\014nition)d(of)i(program)d(completion)f(from)j (\(Clark)f(1978\))g(has)i(b)r(een)g(ex-)374 4791 y(tended)20 b(to)g(nondisjunctiv)n(e)f(programs)e(with)i(nested)i(expressions)c (\(Llo)n(yd)i(and)h(T)-7 b(op)r(or)19 b(1984\),)374 4899 y(and)h(this)g(extension)g(is)f(kno)n(wn)i(to)f(b)r(e)h(equiv)-5 b(alen)n(t)19 b(to)i(the)g(de\014nition)e(of)i(an)f(answ)n(er)g(set)h (when-)374 5007 y(ev)n(er)h(the)h(program)e(is)h(\\tigh)n(t")f(\(Erdem) h(and)h(Lifsc)n(hitz)e(2003\).)h(In)i(view)e(of)h(this)f(fact,)i(answ)n (er)374 5115 y(sets)34 b(for)g(a)g(tigh)n(t)g(logic)e(program)f(can)j (b)r(e)h(generated)f(b)n(y)g(running)f(a)h(satis\014abilit)n(y)c(solv)n (er)386 5356 y Fl(1)444 5380 y Fx(http://www.tcs.hut.fi/S)q(oft)q(ware) q(/sm)q(odel)q(s/)42 b(.)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)690 b FB(3)374 631 y(on)26 b(the)i(program's)23 b(completion)h(\(Bab)r(o)n(vic)n(h)h Fs(et)k(al.)f FB(2000\).)d(Consequen)n(tly)-7 b(,)26 b(answ)n(er)f(sets)i(for)374 739 y(a)j(program)f(\012)i(with)g(w)n (eigh)n(t)e(constrain)n(ts)g(can)i(b)r(e)g(computed)g(b)n(y)g(running)e (a)i(satis\014abilit)n(y)374 847 y(solv)n(er)25 b(on)j(the)g (completion)d(of)i(one)h(of)g(the)g(translations)d([\012])2339 817 y Fm(nd)2419 847 y 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Fr(Z)i Fq(j)-14 b FB(=)23 b Fq(h)p Fr(F)1370 5294 y Fo(1)1408 5282 y Fr(;)14 b(:)g(:)g(:)f(;)h(F)1645 5294 y Fm(n)1691 5282 y Fq(i)23 b FB(:)g Fr(X)34 b FB(i\013)27 b Fq(f)p Fr(i)22 b FB(:)i Fr(Z)k Fq(j)-14 b FB(=)23 b Fr(F)2369 5294 y Fm(i)2397 5282 y Fq(g)g(2)g Fr(X)r(:)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 374 432 a FB(6)946 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)374 631 y(Pr)l(o)l(of)524 883 y Fr(Z)f Fq(j)-14 b FB(=)23 b Fq(h)p Fr(F)792 895 y Fo(1)830 883 y Fr(;)14 b(:)g(:)g(:)f(;)h(F)1067 895 y Fm(n)1113 883 y Fq(i)23 b FB(:)g Fr(X)90 b FB(i\013)110 b(for)27 b(some)f Fr(I)k Fq(2)23 b Fr(X)7 b FB(,)27 b(for)g(all)f Fr(i)p FB(,)h(if)g Fr(i)c Fq(2)g Fr(I)35 b FB(then)28 b Fr(Z)h Fq(j)-14 b FB(=)23 b Fr(F)3184 895 y Fm(i)1373 991 y FB(i\013)110 b(for)27 b(some)f Fr(I)k Fq(2)23 b Fr(X)7 b FB(,)27 b Fr(I)j Fq(\022)23 b(f)p Fr(i)f FB(:)h Fr(Z)29 b Fq(j)-14 b FB(=)23 b Fr(F)2689 1003 y Fm(i)2717 991 y Fq(g)1373 1099 y FB(i\013)110 b(for)27 b(some)f Fr(I)k Fq(2)23 b Fr(X)7 b FB(,)27 b Fr(I)j FB(=)23 b Fq(f)p Fr(i)f FB(:)h Fr(Z)29 b Fq(j)-14 b FB(=)23 b Fr(F)2689 1111 y Fm(i)2717 1099 y Fq(g)1373 1207 y FB(i\013)82 b Fq(f)p Fr(i)22 b FB(:)h Fr(Z)29 b Fq(j)-14 b FB(=)23 b Fr(F)1902 1219 y Fm(i)1930 1207 y Fq(g)g(2)g Fr(X)r(:)p 2246 1153 61 4 v 2246 1207 4 54 v 2304 1207 V 2246 1210 61 4 v 1384 1477 a Fj(2.4)48 b(Str)-5 b(ong)35 b(Equivalenc)-5 b(e)374 1640 y FB(Recall)22 b(that)k(programs)21 b(\005)1223 1652 y Fo(1)1286 1640 y FB(and)k(\005)1507 1652 y Fo(2)1569 1640 y FB(are)f(said)g(to)h(b)r(e)g Fn(strongly)e(equiv)-5 b(alen)n(t)22 b FB(to)j(eac)n(h)f(other)h(if,)374 1747 y(for)k(ev)n(ery)f(program)f(\005,)j(the)g(union)f(\005)1614 1759 y Fo(1)1672 1747 y Fq([)20 b FB(\005)30 b(has)f(the)h(same)f(answ) n(er)f(sets)i(as)f(\005)2953 1759 y Fo(2)3010 1747 y Fq([)20 b FB(\005.)30 b(This)374 1855 y(concept)j(is)f(essen)n(tial)e (b)r(oth)k(for)f(applications)c(of)34 b(our)e(translations)e(and)j(for) g(the)h(pro)r(of)f(of)374 1963 y(their)26 b(soundness.)457 2072 y(The)k(metho)r(d)g(of)h(pro)n(ving)d(strong)h(equiv)-5 b(alence)28 b(prop)r(osed)i(in)g(\(Lifsc)n(hitz)f Fs(et)j(al.)g FB(2001\))d(is)374 2180 y(particularly)i(simple)h(for)j(programs)d (that)k(do)f(not)h(con)n(tain)d(classical)e(negation.)j(W)-7 b(e)36 b(\014rst)374 2288 y(rewrite)g(b)r(oth)i(programs)d(in)i(the)h (syn)n(tax)f(of)h(prop)r(ositional)c(form)n(ulas)h(b)n(y)i(writing)f (ev)n(ery)374 2396 y(rule)31 b(\(4\))i(as)f(the)h(implication)27 b Fs(Bo)l(dy)40 b Fq(!)32 b Fs(He)l(ad)42 b FB(and)32 b(replacing)e(ev)n(ery)h(comma)f(in)i(the)h(rule)374 2504 y(with)23 b Fq(^)p FB(,)i(ev)n(ery)d(semicolon)e(with)k Fq(_)p FB(,)g(and)g(ev)n(ery)f(o)r(ccurrence)g(of)g(negation)g(as)g (failure)e(with)j Fq(:)p FB(.)374 2612 y(F)-7 b(or)27 b(instance,)f(rule)g(\(2\))i(in)f(this)g(notation)f(is)1693 2776 y Fq(::)p Fr(a)d Fq(!)g Fr(a:)374 2941 y FB(Then)32 b(w)n(e)f(c)n(hec)n(k)h(whether)f(the)i(rules)d(of)i(eac)n(h)g(of)g (the)g(programs)d(\005)2592 2953 y Fo(1)2629 2941 y FB(,)j(\005)2746 2953 y Fo(2)2816 2941 y FB(are)f(en)n(tailed)f(b)n(y)374 3049 y(the)j(rules)e(of)i(the)g(other)g(in)f(the)h(logic)d(of)j (here-and-there;)f(if)g(they)h(are,)f(then)i(\005)3026 3061 y Fo(1)3096 3049 y FB(and)f(\005)3325 3061 y Fo(2)374 3157 y FB(are)d(strongly)g(equiv)-5 b(alen)n(t)29 b(to)i(eac)n(h)g (other,)g(and)h(the)g(other)f(w)n(a)n(y)f(around)h(\(\(Lifsc)n(hitz)f Fs(et)j(al.)374 3265 y FB(2001\),)26 b(Theorem)f(1\).)457 3374 y(The)20 b(logic)e(of)j(here-and-there)e(w)n(as)h(originally)15 b(de\014ned)21 b(in)f(\(Heyting)g(1930\).)f(Its)i(de\014nition)374 3481 y(and)k(basic)f(prop)r(erties)f(are)h(discussed)g(in)h(\(Lifsc)n (hitz)f Fs(et)j(al.)g FB(2001,)c(Section)i(2\).)g(It)h(is)e(a)h(three-) 374 3589 y(v)-5 b(alued)22 b(logic,)e(in)n(termediate)g(b)r(et)n(w)n (een)j(in)n(tuitionistic)c(and)k(classical.)18 b(Recall)j(that)j(a)e (natural)374 3697 y(deduction)34 b(system)f(for)h(in)n(tuitionistic)c (logic)i(can)i(b)r(e)i(obtained)d(from)g(the)i(corresp)r(onding)374 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Fq(^)18 b Fr(G)p FB(\))24 b Fq($)f(:)p Fr(F)31 b Fq(_)19 b(:)p Fr(G)374 5380 y FB(\(the)28 b(dual)e(la)n(w)g(can)h(b)r(e)h(pro)n(v)n(ed)f(ev)n(en)g (in)n(tuitionisticall)o(y)o(\).)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)690 b FB(7)457 631 y(As)27 b(an)h(example)d(of)i(the)h(use)g(of)f(this)g(idea,)g(note)g(that)h (the)g(absorption)d(la)n(ws)1595 824 y Fr(a)19 b Fq(_)f FB(\()p Fr(a)h Fq(^)g Fr(b)p FB(\))k Fq($)g Fr(a)1595 932 y(a)c Fq(^)f FB(\()p Fr(a)h Fq(_)g Fr(b)p FB(\))k Fq($)g Fr(a)374 1128 y FB(are)h(pro)n(v)-5 b(able)23 b(in)h(the)h(logic)e(of)i(here-and-there)e(\(their)h(usual)g(pro)r(ofs) g(in)h(the)g(natural)f(deduc-)374 1236 y(tion)e(formalization)c(of)23 b(prop)r(ositional)c(logic)h(do)j(not)g(use)g(the)h(la)n(w)e(of)h(the)g (excluded)g(middle\).)374 1344 y(It)28 b(follo)n(ws)e(that,)j(in)f(an)n (y)f(program,)f Fr(a)p FB(;)14 b(\()p Fr(a;)g(b)p FB(\))28 b(and)g Fr(a;)14 b FB(\()p Fr(a)p FB(;)g Fr(b)p FB(\))29 b(can)f(b)r(e)h(replaced)d(b)n(y)j Fr(a)f FB(without)374 1452 y(c)n(hanging)23 b(the)k(program's)c(answ)n(er)i(sets.)h(In)g (particular,)d(if)i(w)n(e)h(tak)n(e)g(a)f(program)e(con)n(taining)374 1560 y(a)f(m)n(ultiple)e(disjunction)i(of)h(the)g(form)f(\(8\))h(and)g (restrict)f(this)g(disjunction)g(to)h(the)g(sets)g Fr(I)30 b FB(that)374 1668 y(are)c(minimal)d(in)k Fr(X)7 b FB(,)27 b(then)h(the)g(answ)n(er)e(sets)i(of)f(the)h(program)d(will)f(remain)h (the)j(same.)457 1788 y(As)23 b(another)f(example,)f(let)h(us)h(v)n (erify)f(that)h(\(1\))g(is)f(strongly)f(equiv)-5 b(alen)n(t)21 b(to)i(\(2\))g(b)n(y)g(pro)n(ving)374 1896 y(the)28 b(equiv)-5 b(alence)1508 2068 y Fr(a)18 b Fq(_)h(:)28 b Fr(a)23 b Fq($)g(::)p Fr(a)h Fq(!)f Fr(a)374 2240 y FB(in)i(the)i(logic)d(of)i (here-and-there.)f(The)i(pro)r(of)f(left-to-righ)n(t)d(is)i(straigh)n (tforw)n(ard,)e(b)n(y)j(consid-)374 2348 y(ering)i(the)j(cases)e Fr(a)p FB(,)h Fq(:)p Fr(a)p FB(.)h(Righ)n(t-to-left,)c(use)k(the)f 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(tain)e(an)n(y)h(of)g(the)h(pairs)374 3115 y Fq(f)p Fr(a;)14 b(a)541 3085 y Ff(0)563 3115 y Fq(g)23 b FB(are)g(in)g(a)g(1{1)f (corresp)r(ondence)g(with)h(the)h(answ)n(er)e(sets)h(for)g(\005)h (\(Lifsc)n(hitz)e Fs(et)k(al.)e FB(2001,)374 3223 y(Section)34 b(5\).)h(If)g(the)h(rules)d(of)i(eac)n(h)f(of)h(the)h(programs)c(\005) 2235 3193 y Ff(0)2235 3244 y Fo(1)2272 3223 y FB(,)j(\005)2392 3193 y Ff(0)2392 3244 y Fo(2)2465 3223 y FB(can)g(b)r(e)g(deriv)n(ed)e (from)h(the)374 3331 y(rules)23 b(of)i(the)g(other)g(program)d(and)i (the)i(form)n(ulas)c Fq(:)p FB(\()p Fr(a)13 b Fq(^)g Fr(a)2237 3301 y Ff(0)2261 3331 y FB(\))25 b(in)g(the)g(logic)d(of)j (here-and-there)374 3439 y(then)32 b(\005)629 3451 y Fo(1)698 3439 y FB(and)g(\005)926 3451 y Fo(2)995 3439 y FB(are)f(strongly)e(equiv)-5 b(alen)n(t)30 b(to)h(eac)n(h)g(other,)g (and)h(the)g(other)f(w)n(a)n(y)f(around)374 3547 y(\(\(Lifsc)n(hitz)c Fs(et)j(al.)g FB(2001\),)d(Theorem)g(2\).)1089 4084 y Fw(3)48 b(Programs)30 b(with)i(W)-8 b(eigh)m(t)31 b(Constrain)m(ts)1639 4257 y Fj(3.1)48 b(Syntax)374 4431 y FB(A)38 b Fn(rule)f(elemen)n(t)f FB(is)h(a)h(literal)d(\()p Fn(p)r(ositiv)n(e)h FB(rule)g(elemen)n(t\))h (or)g(a)h(literal)c(pre\014xed)k(with)g Fs(not)374 4539 y FB(\()p Fn(negativ)n(e)25 b FB(rule)i(elemen)n(t\).)f(A)i Fn(w)n(eigh)n(t)e(constrain)n(t)f FB(is)h(an)i(expression)d(of)i(the)h (form)1256 4740 y Fr(L)23 b Fq(\024)f(f)p Fr(c)1501 4752 y Fo(1)1561 4740 y FB(=)h Fr(w)1708 4752 y Fo(1)1745 4740 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)1966 4752 y Fm(m)2052 4740 y FB(=)22 b Fr(w)2198 4752 y Fm(m)2262 4740 y Fq(g)h(\024)f Fr(U)785 b FB(\(9\))374 4941 y(where)498 5233 y Fq(\017)41 b FB(eac)n(h)25 b(of)g Fr(L)p FB(,)g Fr(U)34 b FB(is)24 b(\(a)h(sym)n(b)r(ol)e(for\))i(a)g(real)e(n)n(um)n(b)r(er)h(or)g(one)h (of)h(the)f(sym)n(b)r(ols)e Fq(\0001)p FB(,)i(+)p Fq(1)p FB(,)498 5380 y Fq(\017)41 b Fr(c)617 5392 y Fo(1)654 5380 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)875 5392 y Fm(m)965 5380 y FB(\()p Fr(m)24 b Fq(\025)e FB(0\))28 b(are)e(rule)h(elemen)n (ts,)e(and)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 374 432 a FB(8)946 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)498 631 y Fq(\017)41 b Fr(w)640 643 y Fo(1)678 631 y Fr(;)14 b(:)g(:)g(:)f(;)h(w)921 643 y Fm(m)1013 631 y FB(are)26 b(nonnegativ)n(e)f(real)h(n)n(um)n(b)r (ers)g(\(\\w)n(eigh)n(ts"\).)2550 601 y Fo(6)374 800 y FB(The)36 b(part)72 b Fr(L)37 b Fq(\024)72 b FB(can)36 b(b)r(e)h(omitted)d(if)i Fr(L)h FB(=)g Fq(\0001)p FB(;)f(the)g(part)72 b Fq(\024)37 b Fr(U)82 b FB(can)36 b(b)r(e)g(omitted)f(if)374 908 y Fr(U)c FB(=)23 b(+)p Fq(1)p FB(.)k(A)h Fn(rule)f(with)g(w)n(eigh) n(t)f(constrain)n(ts)f FB(is)h(an)i(expression)d(of)i(the)h(form)1562 1072 y Fr(C)1621 1084 y Fo(0)1682 1072 y Fq( )23 b Fr(C)1847 1084 y Fo(1)1885 1072 y Fr(;)14 b(:)g(:)g(:)f(;)h(C)2128 1084 y Fm(n)3215 1072 y FB(\(10\))374 1236 y(where)28 b Fr(C)674 1248 y Fo(0)712 1236 y Fr(;)14 b(:)g(:)g(:)g(;)g(C)956 1248 y Fm(n)1030 1236 y FB(\()p Fr(n)26 b Fq(\025)g FB(0\))j(are)f(w)n (eigh)n(t)g(constrain)n(ts.)f(W)-7 b(e)29 b(will)e(call)g(the)i(rule)f (elemen)n(ts)g(of)374 1344 y Fr(C)433 1356 y Fo(0)498 1344 y FB(the)g Fn(head)f(elemen)n(ts)f FB(of)h(rule)f(\(10\).)457 1452 y(Finally)-7 b(,)36 b(a)j Fn(program)d(with)i(w)n(eigh)n(t)g (constrain)n(ts)e FB(is)j(a)f(set)h(of)h(rules)d(with)i(w)n(eigh)n(t)e (con-)374 1560 y(strain)n(ts.)670 1530 y Fo(7)457 1669 y FB(This)31 b(syn)n(tax)g(b)r(ecomes)g(a)h(generalization)27 b(of)33 b(the)f(basic)f(syn)n(tax)g(of)i(logic)c(programs)g(for)374 1777 y(whic)n(h)24 b(the)i(answ)n(er)e(set)h(seman)n(tics)e(w)n(as)h (originally)c(de\014ned)26 b(\(Gelfond)f(and)g(Lifsc)n(hitz)f(1988\)) 374 1885 y(if)j(w)n(e)g(iden)n(tify)f(a)h(rule)f(elemen)n(t)g Fr(c)i FB(with)f(the)h(w)n(eigh)n(t)e(constrain)n(t)1645 2049 y(1)c Fq(\024)h(f)p Fr(c)f FB(=)h(1)p Fq(g)p Fr(:)374 2213 y FB(By)1622 2348 y Fq( )g Fr(C)1787 2360 y Fo(1)1825 2348 y Fr(;)14 b(:)g(:)g(:)f(;)h(C)2068 2360 y Fm(n)374 2483 y FB(w)n(e)27 b(denote)g(the)h(rule)1468 2618 y(1)22 b Fq(\024)h(f)k(g)c( )g Fr(C)1919 2630 y Fo(1)1956 2618 y Fr(;)14 b(:)g(:)g(:)g(;)g(C)2200 2630 y Fm(n)2245 2618 y Fr(:)457 2753 y FB(A)37 b Fn(cardinalit)n(y)d(constrain)n(t)h FB(is)h(a)i(w)n(eigh)n(t)d(constrain)n(t)h(with)h(all)e(w)n(eigh)n(ts)h (equal)g(to)h(1.)g(A)374 2861 y(cardinalit)n(y)23 b(constrain)n(t)1324 2996 y Fr(L)g Fq(\024)f(f)p Fr(c)1569 3008 y Fo(1)1629 2996 y FB(=)h(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(c)1979 3008 y Fm(m)2065 2996 y FB(=)22 b(1)p Fq(g)g(\024)h Fr(U)374 3131 y FB(can)k(b)r(e)h(abbreviated)d(as)1469 3266 y Fr(L)e Fq(\024)g(f)p Fr(c)1715 3278 y Fo(1)1751 3266 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)1972 3278 y Fm(m)2035 3266 y Fq(g)22 b(\024)h Fr(U:)948 b FB(\(11\))1567 3543 y Fj(3.2)49 b(Semantics)374 3706 y FB(The)34 b(de\014nition)e(of)j(an)e (answ)n(er)g(set)h(for)g(programs)d(with)j(w)n(eigh)n(t)e(constrain)n (ts)g(in)h(\(Simons)374 3814 y Fs(et)h(al.)h FB(2002\))d(uses)h(the)g (follo)n(wing)d(auxiliary)f(de\014nitions.)i(A)j(consisten)n(t)e(set)h Fr(Z)39 b FB(of)33 b(literals)374 3922 y Fn(satis\014es)g FB(a)j(w)n(eigh)n(t)e(constrain)n(t)f(\(9\))j(if)f(the)h(sum)f(of)g (the)h(w)n(eigh)n(ts)e Fr(w)2611 3934 y Fm(j)2682 3922 y FB(for)h(all)e Fr(j)41 b FB(suc)n(h)35 b(that)374 4030 y Fr(Z)28 b Fq(j)-14 b FB(=)23 b Fr(c)592 4042 y Fm(j)646 4030 y FB(is)17 b(not)i(less)e(than)i Fr(L)f FB(and)g(not)h(greater)e (than)i Fr(U)9 b FB(.)18 b(F)-7 b(or)18 b(instance,)g Fr(Z)24 b FB(satis\014es)17 b(cardinalit)n(y)374 4138 y(constrain)n(t)32 b(\(3\))i(i\013)g Fr(Z)40 b FB(con)n(tains)33 b(at)h(most)f(one)h(of)h(the)g(atoms)d Fr(a)p FB(,)j Fr(b)p FB(.)f(Ab)r(out)h(a)f(program)d(\012)374 4246 y(with)k(w)n(eigh)n(t)e(constrain)n(ts)h(w)n(e)h(sa)n(y)f(that)i Fr(Z)41 b Fn(satis\014es)34 b FB(\012)i(if,)f(for)g(ev)n(ery)f(rule)g (\(10\))h(in)g(\012,)h Fr(Z)374 4353 y FB(satis\014es)29 b Fr(C)742 4365 y Fo(0)811 4353 y FB(whenev)n(er)h Fr(Z)37 b FB(satis\014es)29 b Fr(C)1641 4365 y Fo(1)1678 4353 y Fr(;)14 b(:)g(:)g(:)g(;)g(C)1922 4365 y Fm(n)1967 4353 y FB(.)32 b(As)f(in)f(the)i(case)e(of)h(nested)g(expressions,)386 4589 y Fl(6)444 4613 y Fz(In)i(\(Simons)h FC(et)g(al.)f Fz(2002\),)g(w)n(eigh)n(ts)i(are)e(not)g(required)h(to)f(b)r(e)g (nonnegativ)n(e,)j(and)d(the)g(meaning)i(of)d(a)444 4687 y(program)24 b(with)i(negativ)n(e)g(w)n(eigh)n(ts)g(is)f(de\014ned)h(b) n(y)e(describing)j(a)d(metho)r(d)h(for)f(eliminating)k(them.)c(Unfor-) 444 4762 y(tunately)-6 b(,)31 b(this)e(prepro)r(cessing)h(step)g(leads) f(to)g(some)g(results)g(that)h(seem)f(unin)n(tuitiv)n(e.)j(F)-6 b(or)28 b(instance,)i(it)444 4837 y(turns)24 b(out)g(that)h(the)g (one-rule)f(programs)1358 4955 y(1)c Fy(\024)f(f)p Fh(p)h Fz(=)f(1)p Fy(g)h( )g Fz(0)f Fy(\024)h(f)p Fh(p)f Fz(=)h(2)p Fh(;)12 b(p)19 b Fz(=)g Fy(\000)p Fz(1)p Fy(g)444 5073 y Fz(and)1484 5150 y(1)g Fy(\024)h(f)p Fh(p)f Fz(=)h(1)p Fy(g)g( )f Fz(0)h Fy(\024)g(f)p Fh(p)f Fz(=)g(1)p Fy(g)444 5227 y Fz(ha)n(v)n(e)25 b(di\013eren)n(t)g(answ)n(er)f(sets.)386 5282 y Fl(7)444 5305 y Fz(In)29 b(\(Simons)g FC(et)g(al.)f Fz(2002\),)i(programs)d(are)h(not)h(allo)n(w)n(ed)h(to)f(con)n(tain)h (classical)h(negation.)f(But)f(classical)444 5380 y(negation)d(is)e (allo)n(w)n(ed)i(in)f(the)f(input)h(\014les)g(of)e(the)i(curren)n(t)f (v)n(ersion)g(of)g Fe(smodels)p Fz(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)690 b FB(9)374 631 y(w)n(e)32 b(will)e(use)j Fq(j)-14 b FB(=)33 b(to)f(denote)h(the)g(satisfaction)d (relation)g(for)j(b)r(oth)g(w)n(eigh)n(t)e(constrain)n(ts)g(and)374 739 y(programs)24 b(with)j(w)n(eigh)n(t)f(constrain)n(ts.)457 848 y(The)21 b(next)g(part)g(of)g(the)g(seman)n(tics)e(of)i(w)n(eigh)n (t)e(constrain)n(ts)g(is)h(the)h(de\014nition)f(of)h(the)h(reduct)374 956 y(for)27 b(w)n(eigh)n(t)f(constrain)n(ts)f(of)i(the)h(form)1333 1121 y Fr(L)22 b Fq(\024)h(f)p Fr(c)1578 1133 y Fo(1)1638 1121 y FB(=)f Fr(w)1784 1133 y Fo(1)1822 1121 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)2043 1133 y Fm(m)2129 1121 y FB(=)22 b Fr(w)2275 1133 y Fm(m)2339 1121 y Fq(g)p Fr(:)374 1286 y FB(The)27 b Fn(reduct)f FB(\()p Fr(L)d Fq(\024)g Fr(S)5 b FB(\))1089 1256 y Fm(Z)1169 1286 y FB(of)27 b(a)g(w)n(eigh)n(t)e (constrain)n(t)g Fr(L)d Fq(\024)h Fr(S)32 b FB(with)26 b(resp)r(ect)h(to)g(a)f(consisten)n(t)f(set)374 1394 y Fr(Z)33 b FB(of)27 b(literals)d(is)j(the)h(w)n(eigh)n(t)e(constrain)n (t)f Fr(L)1768 1364 y Fm(Z)1844 1394 y Fq(\024)e Fr(S)1988 1364 y Ff(0)2011 1394 y FB(,)k(where)498 1566 y Fq(\017)41 b Fr(S)637 1536 y Ff(0)685 1566 y FB(is)23 b(obtained)g(from)g Fr(S)29 b FB(b)n(y)24 b(dropping)e(all)g(pairs)h Fr(c)g FB(=)f Fr(w)27 b FB(suc)n(h)d(that)h Fr(c)f FB(is)f(negativ)n(e,)g(and) 498 1677 y Fq(\017)41 b Fr(L)638 1647 y Fm(Z)720 1677 y FB(is)28 b Fr(L)g FB(min)n(us)f(the)i(sum)f(of)h(the)g(w)n(eigh)n(ts) e Fr(w)32 b FB(for)c(all)f(pairs)g Fr(c)e FB(=)f Fr(w)32 b FB(in)c Fr(S)34 b FB(suc)n(h)28 b(that)h Fr(c)581 1785 y FB(is)e(negativ)n(e)e(and)j Fr(Z)g Fq(j)-14 b FB(=)23 b Fr(c)p FB(.)374 1957 y(F)-7 b(or)27 b(instance,)f(the)i(reduct)f(of)h (the)g(constrain)n(t)1388 2122 y(1)22 b Fq(\024)h(f)p Fs(not)35 b Fr(a)23 b FB(=)f(3)p Fr(;)14 b Fs(not)35 b Fr(b)23 b FB(=)f(2)p Fq(g)374 2287 y FB(relativ)n(e)i(to)j Fq(f)p Fr(a)p Fq(g)g FB(is)1693 2423 y Fq(\000)p FB(1)22 b Fq(\024)h(f)k(g)p Fr(:)457 2560 y FB(The)g Fn(reduct)h FB(of)f(a)h(rule)953 2725 y Fr(L)1010 2737 y Fo(0)1070 2725 y Fq(\024)23 b Fr(S)1209 2737 y Fo(0)1269 2725 y Fq(\024)g Fr(U)1414 2737 y Fo(0)1474 2725 y Fq( )g Fr(L)1637 2737 y Fo(1)1697 2725 y Fq(\024)f Fr(S)1835 2737 y Fo(1)1895 2725 y Fq(\024)h Fr(U)2040 2737 y Fo(1)2077 2725 y Fr(;)14 b(:)g(:)g(:)g(;)g(L)2319 2737 y Fm(n)2386 2725 y Fq(\024)23 b Fr(S)2525 2737 y Fm(n)2593 2725 y Fq(\024)g Fr(U)2738 2737 y Fm(n)3215 2725 y FB(\(12\))374 2890 y(with)k(resp)r(ect)g(to)h (a)f(consisten)n(t)f(set)h Fr(Z)34 b FB(of)27 b(literals)d(is)498 3062 y Fq(\017)41 b FB(the)28 b(set)g(of)f(rules)f(of)i(the)g(form)1383 3227 y Fr(l)d Fq( )e FB(\()p Fr(L)1628 3239 y Fo(1)1688 3227 y Fq(\024)f Fr(S)1826 3239 y Fo(1)1864 3227 y FB(\))1896 3193 y Fm(Z)1949 3227 y Fr(;)14 b(:)g(:)g(:)g(;)g FB(\()p Fr(L)2223 3239 y Fm(n)2291 3227 y Fq(\024)23 b Fr(S)2430 3239 y Fm(n)2475 3227 y FB(\))2507 3193 y Fm(Z)581 3393 y FB(where)35 b Fr(l)j FB(is)d(a)g(p)r(ositiv)n(e)e(head)j(elemen)n(t)e (of)i(\(12\))f(suc)n(h)h(that)g Fr(Z)42 b Fq(j)-14 b FB(=)37 b Fr(l)r FB(,)e(if,)h(for)f(ev)n(ery)g Fr(i)581 3500 y FB(\(1)23 b Fq(\024)g Fr(i)f Fq(\024)h Fr(n)p FB(\),)28 b Fr(Z)g Fq(j)-14 b FB(=)23 b Fr(S)1271 3512 y Fm(i)1322 3500 y Fq(\024)g Fr(U)1467 3512 y Fm(i)1494 3500 y FB(;)498 3612 y Fq(\017)41 b FB(the)28 b(empt)n(y)f(set,)g (otherwise.)374 3784 y(The)f Fn(reduct)g FB(\012)860 3754 y Fm(Z)939 3784 y FB(of)h(a)e(program)e(\012)k(with)e(resp)r(ect)h (to)g Fr(Z)32 b FB(is)25 b(the)h(union)g(of)g(the)g(reducts)g(of)g(the) 374 3892 y(rules)g(of)h(\012.)457 4001 y(Consider,)e(for)i(example,)f (the)i(one-rule)d(program)944 4166 y(1)d Fq(\024)h(f)p Fr(a)f FB(=)h(2)p Fq(g)f(\024)h FB(2)f Fq( )h FB(1)g Fq(\024)g(f)p Fs(not)34 b Fr(a)23 b FB(=)g(3)p Fr(;)14 b Fs(not)35 b Fr(b)22 b FB(=)h(2)p Fq(g)f(\024)h FB(4)p Fr(:)422 b FB(\(13\))374 4331 y(Since)30 b(the)h(only)e(head)h(elemen)n (t)f(of)i(\(13\))f(is)g Fr(a)p FB(,)h(the)g(reduct)f(of)h(this)f(rule)g (with)g(resp)r(ect)g(to)h(a)374 4439 y(set)c Fr(Z)33 b FB(of)28 b(atoms)e(is)g(empt)n(y)h(if)g Fr(a)c Fq(62)h Fr(Z)6 b FB(.)27 b(Consider)f(the)i(case)e(when)i Fr(a)23 b Fq(2)g Fr(Z)6 b FB(.)28 b(Since)1285 4604 y Fr(Z)h Fq(j)-14 b FB(=)22 b Fq(f)p Fs(not)35 b Fr(a)23 b FB(=)g(3)p Fr(;)14 b Fs(not)34 b Fr(b)23 b FB(=)g(2)p Fq(g)f(\024)g FB(4)p Fr(;)374 4769 y FB(the)28 b(reduct)f(consists)f(of)h(one)h(rule) 1231 4934 y Fr(a)23 b Fq( )g FB(\(1)g Fq(\024)f(f)p Fs(not)35 b Fr(a)23 b FB(=)g(3)p Fr(;)14 b Fs(not)34 b Fr(b)23 b FB(=)f(2)p Fq(g)p FB(\))2429 4899 y Fm(Z)2482 4934 y Fr(:)457 5100 y FB(It)j(is)f(clear)g(from)g(the)h(de\014nition)f(of)h (the)h(reduct)f(of)h(a)e(program)f(ab)r(o)n(v)n(e)h(that)h(ev)n(ery)f (rule)g(in)374 5208 y(a)j(reduct)g(satis\014es)f(t)n(w)n(o)h (conditions:)498 5380 y Fq(\017)41 b FB(its)27 b(head)g(is)g(a)g (literal,)d(and)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 374 432 a FB(10)904 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)498 631 y Fq(\017)41 b FB(ev)n(ery)36 b(mem)n(b)r(er)g(of)h(its)g(b)r(o)r(dy)h (has)f(the)h(form)e Fr(L)j Fq(\024)g Fr(S)j FB(where)37 b Fr(S)43 b FB(do)r(es)37 b(not)g(con)n(tain)581 739 y(negativ)n(e)26 b(rule)g(elemen)n(ts.)374 897 y(A)j(rule)f(satisfying) e(these)j(conditions)d(is)i(called)f(a)h Fn(Horn)h(rule)p FB(.)e(If)j(a)e(program)e(\012)j(consists)f(of)374 1005 y(Horn)g(rules)f(then)j(there)e(is)g(a)h(unique)f(minimal)c(set)29 b Fr(Z)34 b FB(of)29 b(literals)c(suc)n(h)k(that)g Fr(Z)i Fq(j)-14 b FB(=)25 b(\012.)j(This)374 1113 y(set)f(is)g(called)e(the)j Fn(deductiv)n(e)f(closure)e FB(of)i(\012)h(and)g(denoted)f(b)n(y)g Fs(cl)9 b FB(\(\012\).)457 1220 y(Finally)-7 b(,)26 b(a)i(consisten)n (t)g(set)h Fr(Z)34 b FB(of)29 b(literals)c(is)j(an)h Fn(answ)n(er)e(set)i FB(for)g(a)f(program)e(\012)j(if)f Fr(Z)j Fq(j)-14 b FB(=)25 b(\012)374 1328 y(and)i Fs(cl)9 b FB(\(\012)695 1298 y Fm(Z)749 1328 y FB(\))23 b(=)g Fr(Z)6 b FB(.)457 1436 y(T)-7 b(o)26 b(illustrate)e(this)i (de\014nition,)f(assume)h(that)h(\012)g(is)f(\(3\).)h(Set)g Fq(f)p Fr(a;)14 b(b)p Fq(g)26 b FB(is)f(not)i(an)g(answ)n(er)e(set)374 1544 y(for)h(\012)h(b)r(ecause)f(it)g(do)r(es)h(not)f(satisfy)g(\012.)h (Let)g(us)f(c)n(hec)n(k)g(that)h(ev)n(ery)f(prop)r(er)f(subset)i(of)g Fq(f)p Fr(a;)14 b(b)p Fq(g)374 1652 y FB(is)22 b(an)h(answ)n(er)e(set.) j(Clearly)-7 b(,)20 b(ev)n(ery)i(suc)n(h)h(subset)g(satis\014es)e (\012.)j(It)f(remains)d(to)j(sho)n(w)g(that)g(eac)n(h)374 1760 y(of)k(these)h(sets)f(is)g(the)h(deductiv)n(e)e(closure)g(of)h (the)h(corresp)r(onding)d(reduct)i(of)h(\012.)498 1918 y Fq(\017)41 b FB(\012)641 1888 y Ff(;)707 1918 y FB(is)26 b(empt)n(y)-7 b(,)27 b(so)g(that)h Fs(cl)9 b FB(\(\012)1502 1888 y Ff(;)1540 1918 y FB(\))24 b(=)e Fq(;)p FB(.)498 2024 y Fq(\017)41 b FB(\012)641 1994 y Ff(f)p Fm(a)p Ff(g)777 2024 y FB(consists)26 b(of)h(the)h(single)d(rule)i Fr(a)p FB(,)g(so)g(that)h Fs(cl)9 b FB(\(\012)2254 1994 y Ff(f)p Fm(a)p Ff(g)2363 2024 y FB(\))23 b(=)g Fq(f)p Fr(a)p Fq(g)n FB(.)498 2131 y Fq(\017)41 b FB(\012)641 2101 y Ff(f)p Fm(b)p Ff(g)770 2131 y FB(consists)26 b(of)i(the)g (single)d(rule)h Fr(b)p FB(,)h(so)g(that)h Fs(cl)9 b FB(\(\012)2239 2101 y Ff(f)p Fm(b)p Ff(g)2341 2131 y FB(\))23 b(=)g Fq(f)p Fr(b)p Fq(g)n FB(.)457 2289 y(T)-7 b(o)40 b(giv)n(e)f(another)h(example,)f(let)h(\012)h(b)r(e)g(\(13\).)g (Set)g Fq(f)p Fr(b)p Fq(g)f FB(is)g(not)h(an)g(answ)n(er)e(set)i(for)f (\012)374 2397 y(b)r(ecause)31 b(it)h(do)r(es)g(not)g(satisfy)f(\012.)i (The)f(other)g(subsets)g(of)g Fq(f)p Fr(a;)14 b(b)p Fq(g)31 b FB(satisfy)f(\012.)j(Consider)d(the)374 2505 y(corresp)r(onding)24 b(reducts.)498 2662 y Fq(\017)41 b FB(\012)641 2632 y Ff(;)707 2662 y FB(is)26 b(empt)n(y)-7 b(,)27 b(so)g(that)h Fs(cl)9 b FB(\(\012)1502 2632 y Ff(;)1540 2662 y FB(\))24 b(=)e Fq(;)p FB(.)498 2769 y Fq(\017)41 b FB(\012)641 2739 y Ff(f)p Fm(a)p Ff(g)777 2769 y FB(is)1724 2902 y Fr(a)23 b Fq( )g(\000)p FB(1)f Fq(\024)h(fg)p Fr(:)581 3035 y FB(Consequen)n(tly)-7 b(,)26 b Fs(cl)9 b FB(\(\012)1272 3004 y Ff(f)p Fm(a)p Ff(g)1380 3035 y FB(\))24 b(=)e Fq(f)p Fr(a)p Fq(g)o FB(.)498 3141 y Fq(\017)41 b FB(\012)641 3111 y Ff(f)p Fm(a;b)p Ff(g)826 3141 y FB(is)1768 3274 y Fr(a)23 b Fq( )g FB(1)f Fq(\024)h(fg)581 3407 y FB(Consequen)n(tly)-7 b(,)26 b Fs(cl)9 b FB(\(\012)1272 3377 y Ff(f)p Fm(a;b)p Ff(g)1429 3407 y FB(\))24 b(=)e Fq(;)h(6)p FB(=)g Fq(f)p Fr(a;)14 b(b)p Fq(g)p FB(.)374 3565 y(W)-7 b(e)28 b(conclude)e(that)i (the)g(answ)n(er)e(sets)h(for)g(\(13\))h(are)e Fq(;)h FB(and)h Fq(f)p Fr(a)p Fq(g)p FB(.)1565 3830 y Fw(4)48 b(T)-8 b(ranslations)1408 3991 y Fj(4.1)49 b(Basic)34 b(T)-7 b(r)i(anslation)374 4151 y FB(In)40 b(this)f(section,)g(w)n(e)h (giv)n(e)e(the)j(main)d(de\014nition)g(of)i(this)g(pap)r(er)f(|)h(the)h (description)d(of)374 4259 y(a)f(translation)d(from)i(the)i(language)c (of)k(w)n(eigh)n(t)d(constrain)n(ts)g(to)j(the)f(language)e(of)i (nested)374 4367 y(expressions)31 b(|)j(and)g(state)f(a)h(theorem)e(ab) r(out)i(the)g(soundness)g(of)f(this)h(translation.)c(The)374 4475 y(de\014nition)c(of)h(the)h(translation)d(consists)g(of)j(4)f (parts.)374 4630 y Fn(1.)g(The)h(translation)c(of)j(a)h(constrain)n(t)d (of)j(the)g(form)1344 4790 y Fr(L)23 b Fq(\024)f(f)p Fr(c)1589 4802 y Fo(1)1649 4790 y FB(=)h Fr(w)1796 4802 y Fo(1)1834 4790 y Fr(;)14 b(:)g(:)g(:)f(;)h(c)2054 4802 y Fm(m)2140 4790 y FB(=)23 b Fr(w)2287 4802 y Fm(m)2350 4790 y Fq(g)823 b FB(\(14\))374 4951 y Fn(is)26 b(the)i(nested)g (expression)1287 5112 y Fq(h)p Fr(c)1355 5124 y Fo(1)1393 5112 y Fr(;)14 b(:)g(:)g(:)f(;)h(c)1613 5124 y Fm(m)1676 5112 y Fq(i)24 b FB(:)1778 5045 y Fi(\010)1826 5112 y Fr(I)30 b FB(:)23 b Fr(L)g Fq(\024)2105 5050 y Fi(P)2193 5137 y Fm(i)p Ff(2)p Fm(I)2313 5112 y Fr(w)2372 5124 y Fm(i)2400 5045 y Fi(\011)3215 5111 y FB(\(15\))374 5272 y Fn(where)j Fr(I)34 b Fn(ranges)26 b(o)n(v)n(er)f(the)i(subsets)g (of)g Fq(f)p FB(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(m)p Fq(g)p Fn(.)26 b(W)-7 b(e)27 b(denote)g(the)h(translation)c(of)j Fr(L)22 b Fq(\024)h Fr(S)374 5380 y Fn(b)n(y)k FB([)p Fr(L)c Fq(\024)f Fr(S)5 b FB(])p Fn(.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)648 b FB(11)374 631 y Fn(2.)27 b(The)h(translation)c(of)j(a)h(constrain)n(t)d(of)j(the)g(form)1340 802 y Fq(f)p Fr(c)1418 814 y Fo(1)1477 802 y FB(=)23 b Fr(w)1624 814 y Fo(1)1662 802 y Fr(;)14 b(:)g(:)g(:)f(;)h(c)1882 814 y Fm(m)1968 802 y FB(=)23 b Fr(w)2115 814 y Fm(m)2178 802 y Fq(g)g(\024)g Fr(U)827 b FB(\(16\))374 973 y Fn(is)26 b(the)i(nested)g(expression)1143 1144 y Fs(not)1309 1077 y Fi(\000)1347 1144 y Fq(h)p Fr(c)1415 1156 y Fo(1)1453 1144 y Fr(;)14 b(:)g(:)g(:)f(;)h(c)1673 1156 y Fm(m)1736 1144 y Fq(i)23 b FB(:)1838 1077 y Fi(\010)1886 1144 y Fr(I)30 b FB(:)23 b Fr(U)32 b(<)2175 1082 y Fi(P)2262 1169 y Fm(i)p Ff(2)p Fm(I)2382 1144 y Fr(w)2441 1156 y Fm(i)2470 1077 y Fi(\011\001)2570 1144 y Fr(:)3215 1143 y FB(\(17\))374 1314 y Fn(where)25 b Fr(I)34 b Fn(ranges)24 b(o)n(v)n(er)h(the)i(subsets)f(of)g Fq(f)p FB(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(m)p Fq(g)p Fn(.)25 b(W)-7 b(e)27 b(denote)f(the)h (translation)c(of)j Fr(S)i Fq(\024)23 b Fr(U)374 1422 y Fn(b)n(y)k FB([)p Fr(S)h Fq(\024)22 b Fr(U)9 b FB(])p Fn(.)374 1600 y(3.)27 b(The)h(translation)c(of)j(a)h(general)d(w)n (eigh)n(t)h(constrain)n(t)f(is)i(de\014ned)h(b)n(y)1286 1771 y FB([)p Fr(L)23 b Fq(\024)g Fr(S)k Fq(\024)c Fr(U)9 b FB(])23 b(=)f([)p Fr(L)h Fq(\024)g Fr(S)5 b FB(])p Fr(;)14 b FB([)p Fr(S)27 b Fq(\024)c Fr(U)9 b FB(])p Fr(:)457 1945 y FB(Recall)31 b(that)k Fr(L)e Fq(\024)g Fr(S)39 b FB(is)33 b(shorthand)g(for)h Fr(L)f Fq(\024)h Fr(S)k Fq(\024)c(1)p FB(,)g(and)g Fr(S)k Fq(\024)c Fr(U)43 b FB(is)33 b(shorthand)g(for)374 2053 y Fq(\0001)28 b(\024)g Fr(S)34 b Fq(\024)28 b Fr(U)9 b FB(;)31 b(translations)d(of)j(w)n(eigh) n(t)e(constrain)n(ts)f(of)j(these)h(sp)r(ecial)c(t)n(yp)r(es)j(ha)n(v)n (e)f(b)r(een)374 2160 y(de\014ned)j(earlier.)e(It)i(is)g(easy)f(to)i (see)f(that)h(the)g(old)e(de\014nition)g(of)h([)p Fr(L)g Fq(\024)g Fr(S)5 b FB(])33 b(giv)n(es)e(a)i(nested)374 2268 y(expression)25 b(equiv)-5 b(alen)n(t)25 b(to)j([)p Fr(L)22 b Fq(\024)h Fr(S)28 b Fq(\024)22 b(1)p FB(],)28 b(and)f(similarly)22 b(for)27 b([)p Fr(S)h Fq(\024)23 b Fr(U)9 b FB(].)374 2447 y Fn(4.)32 b(F)-7 b(or)33 b(an)n(y)f(program) e FB(\012)j Fn(with)g(w)n(eigh)n(t)e(constrain)n(ts,)g(its)h (translation)e FB([\012])j Fn(is)f(the)i(program)374 2555 y(with)27 b(nested)g(expressions)e(obtained)i(from)f FB(\012)i Fn(b)n(y)f(replacing)d(eac)n(h)j(rule)f(\(10\))i(with)1004 2725 y FB(\()p Fr(l)1061 2737 y Fo(1)1098 2725 y FB(;)14 b Fs(not)35 b Fr(l)1312 2737 y Fo(1)1349 2725 y FB(\))p Fr(;)14 b(:)g(:)g(:)g(;)g FB(\()p Fr(l)1623 2737 y Fm(p)1661 2725 y FB(;)g Fs(not)35 b Fr(l)1875 2737 y Fm(p)1914 2725 y FB(\))p Fr(;)14 b FB([)p Fr(C)2065 2737 y Fo(0)2102 2725 y FB(])23 b Fq( )h FB([)p Fr(C)2337 2737 y Fo(1)2374 2725 y FB(])p Fr(;)14 b(:)g(:)g(:)g(;)g FB([)p Fr(C)2664 2737 y Fm(n)2709 2725 y FB(])483 b(\(18\))374 2896 y Fn(where)27 b Fr(l)639 2908 y Fo(1)676 2896 y Fr(;)14 b(:)g(:)g(:)f(;)h(l)885 2908 y Fm(p)951 2896 y Fn(are)27 b(the)h(p)r(ositiv)n(e)d(head)i(elemen)n(ts)f(of)h(\(10\).)457 3074 y FB(The)g(conjunctiv)n(e)f(terms)g(in)h(\()p Fr(l)1452 3086 y Fo(1)1489 3074 y FB(;)14 b Fs(not)35 b Fr(l)1703 3086 y Fo(1)1740 3074 y FB(\))p Fr(;)14 b(:)g(:)g(:)g(;)g FB(\()p Fr(l)2014 3086 y Fm(p)2052 3074 y FB(;)g Fs(not)35 b Fr(l)2266 3086 y Fm(p)2304 3074 y FB(\))28 b(express,)e(in)n(tuitiv)n (ely)-7 b(,)23 b(that)28 b(w)n(e)374 3182 y(are)g(free)g(to)h(decide)f (ab)r(out)h(ev)n(ery)f(p)r(ositiv)n(e)f(head)h(elemen)n(t)g(of)g(the)i (rule)e(whether)g(or)h(not)g(to)374 3290 y(include)d(it)h(in)f(the)i (answ)n(er)f(set.)457 3401 y(T)-7 b(o)28 b(illustrate)d(this)j (de\014nition,)g(let)g(us)g(apply)g(it)g(\014rst)h(to)f(program)e (\(3\).)j(The)g(translation)374 3509 y(of)e(the)h(cardinalit)n(y)23 b(constrain)n(t)j(0)c Fq(\024)h(f)p Fr(a;)14 b(b)p Fq(g)22 b(\024)g FB(1)27 b(is)1440 3680 y([0)c Fq(\024)g(f)p Fr(a;)14 b(b)p Fq(g)p FB(])p Fr(;)g FB([)p Fq(f)p Fr(a;)g(b)p Fq(g)19 b(\024)k FB(1])p Fr(:)919 b FB(\(19\))374 3850 y(The)27 b(\014rst)h(conjunctiv)n(e)e(term)h(is)1399 4021 y Fq(h)p Fr(a;)14 b(b)p Fq(i)23 b FB(:)g Fq(f;)p Fr(;)14 b Fq(f)p FB(1)p Fq(g)p Fr(;)g Fq(f)p FB(2)p Fq(g)p Fr(;)g Fq(f)p FB(1)p Fr(;)g FB(2)p Fq(gg)374 4192 y FB(whic)n(h)26 b(equals)1650 4334 y Fq(>)p FB(;)14 b Fr(a)p FB(;)g Fr(b)p FB(;)g(\()p Fr(a;)g(b)p FB(\))374 4475 y(and)41 b(is)e(equiv)-5 b(alen)n(t)39 b(to)i Fq(>)p FB(.)g(Similarly)-7 b(,)36 b(the)41 b(second)g(conjunctiv)n(e)f(term)g(is)g(equiv)-5 b(alen)n(t)39 b(to)374 4583 y Fs(not)32 b FB(\()p Fr(a;)14 b(b)p FB(\).)25 b(Consequen)n(tly)-7 b(,)23 b(\(19\))h(can)g(b)r(e)h (written)f(as)g Fs(not)32 b FB(\()p Fr(a;)14 b(b)p FB(\).)25 b(It)g(follo)n(ws)d(that)j(the)g(trans-)374 4691 y(lation)g(of)i (program)e(\(3\))j(can)f(b)r(e)h(written)f(as)1320 4862 y(\()p Fr(a)p FB(;)14 b Fs(not)35 b Fr(a)p FB(\))p Fr(;)14 b FB(\()p Fr(b)p FB(;)g Fs(not)35 b Fr(b)p FB(\))p Fr(;)14 b Fs(not)35 b FB(\()p Fr(a)p FB(;)14 b Fr(b)p FB(\))p Fr(:)799 b FB(\(20\))457 5036 y(Similarly)-7 b(,)22 b(w)n(e)27 b(can)g(c)n(hec)n(k)g(that)h(program)c(\(13\))k(turns)f(in)n(to)1190 5206 y Fr(a)c Fq( )g FB(\()p Fs(not)35 b Fr(a)p FB(;)14 b Fs(not)35 b Fr(b)p FB(\))p Fr(;)14 b Fs(not)36 b FB(\()p Fs(not)f Fr(a;)14 b Fs(not)35 b Fr(b)p FB(\))p Fr(:)457 5380 y FB(The)27 b(translation)e(de\014ned)j(ab)r(o)n(v)n(e)e(is)g (sound:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 374 432 a FB(12)904 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)374 631 y(The)l(or)l(em)g(1)374 753 y FB(F)-7 b(or)27 b(an)n(y)f(program)f (\012)j(with)f(w)n(eigh)n(t)f(constrain)n(ts,)f(\012)i(and)h([\012])g (ha)n(v)n(e)e(the)i(same)e(answ)n(er)g(sets.)457 960 y(W)-7 b(e)40 b(will)d(conclude)h(this)h(section)g(with)g(a)g(few)h (commen)n(ts)e(ab)r(out)i(translating)c(w)n(eigh)n(t)374 1068 y(constrain)n(ts)25 b(of)i(the)h(forms)e Fr(L)d Fq(\024)g Fr(S)32 b FB(and)27 b Fr(S)h Fq(\024)23 b Fr(U)9 b FB(.)457 1179 y(In)31 b(Section)f(3)h(w)n(e)f(ha)n(v)n(e)g(agreed)g (to)h(iden)n(tify)e(an)n(y)i(rule)e(elemen)n(t)h Fr(c)h FB(with)f(the)i(cardinalit)n(y)374 1287 y(constrain)n(t)19 b(1)k Fq(\024)g(f)p Fr(c)p Fq(g)p FB(,)e(and)h(to)g(drop)f(the)h(head)g (of)g(a)g(rule)e(with)i(w)n(eigh)n(t)e(constrain)n(ts)g(when)i(this)374 1395 y(head)29 b(is)f(1)e Fq(\024)g(f)j(g)p FB(.)g(It)h(is)e(easy)h(to) g(c)n(hec)n(k)g(that)g([1)d Fq(\024)g(f)p Fr(c)p Fq(g)p FB(])i(is)h(equiv)-5 b(alen)n(t)27 b(to)i Fr(c)p FB(,)h(and)f([1)d Fq(\024)g(f)j(g)p FB(])374 1503 y(is)d(equiv)-5 b(alen)n(t)25 b(to)j Fq(?)p FB(.)457 1614 y(If)d(the)g(w)n(eigh)n(ts)e Fr(w)1029 1626 y Fo(1)1067 1614 y Fr(;)14 b(:)g(:)g(:)g(;)g(w)1311 1626 y Fm(m)1399 1614 y FB(are)24 b(in)n(tegers)f(then)i(the)h (inequalit)n(y)21 b(in)j(\(17\))h(is)f(equiv)-5 b(alen)n(t)23 b(to)374 1722 y Fq(b)p Fr(U)9 b Fq(c)t FB(+)t(1)21 b Fq(\024)738 1660 y Fi(P)825 1747 y Fm(i)p Ff(2)p Fm(I)946 1722 y Fr(w)1005 1734 y Fm(i)1033 1722 y FB(.)f(Consequen)n(tly)-7 b(,)19 b(in)g(the)i(case)e(of)i(in)n(teger)d(w)n(eigh)n(ts)h(\(in)g (particular,)e(in)j(the)374 1830 y(case)26 b(of)i(cardinalit)n(y)23 b(constrain)n(ts\),)j([)p Fr(S)i Fq(\024)23 b Fr(U)9 b FB(])27 b(can)g(b)r(e)h(written)f(as)g Fs(not)35 b FB([)p Fq(b)p Fr(U)9 b Fq(c)19 b FB(+)f(1)k Fq(\024)h Fr(S)5 b FB(].)27 b(This)374 1938 y(is)k(similar)d(to)33 b(a)f(transformation)d(that)k(is)f(used)h(b)n(y)f(the)h(prepro)r (cessor)e Fp(lp)-6 b(arse)33 b FB(of)g(system)374 2046 y Fp(smodels)p FB(.)457 2157 y(The)c(sign)e Fr(<)i FB(in)f(place)g(of)h Fq(\024)f FB(is)g(not)h(allo)n(w)n(ed)d(in)i(w)n(eigh)n(t)f(constrain)n (ts.)g(But)i(sometimes)d(it)374 2265 y(is)g(con)n(v)n(enien)n(t)g(to)h (write)f(expressions)f(of)j(the)g(form)1321 2437 y([)p Fr(L)23 b(<)f Fq(f)p Fr(c)1589 2449 y Fo(1)1649 2437 y FB(=)h Fr(w)1796 2449 y Fo(1)1834 2437 y Fr(;)14 b(:)g(:)g(:)f(;)h(c) 2054 2449 y Fm(m)2140 2437 y FB(=)23 b Fr(w)2287 2449 y Fm(m)2350 2437 y Fq(g)p FB(])374 2610 y(understo)r(o)r(d)k(as)g (shorthand)g(for)1269 2783 y Fq(h)p Fr(c)1337 2795 y Fo(1)1374 2783 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)1595 2795 y Fm(m)1658 2783 y Fq(i)23 b FB(:)1759 2716 y Fi(\010)1808 2783 y Fr(I)30 b FB(:)23 b Fr(L)f(<)2087 2721 y Fi(P)2175 2808 y Fm(i)p Ff(2)p Fm(I)2295 2783 y Fr(w)2354 2795 y Fm(i)2382 2716 y Fi(\011)2444 2783 y Fr(:)3215 2782 y FB(\(21\))374 2955 y(Using)k(this)h(notation,)f(w)n(e)h(can)g(write)g ([)p Fr(S)h Fq(\024)22 b Fr(U)9 b FB(])28 b(as)f Fs(not)35 b FB([)p Fr(U)d(<)22 b(S)5 b FB(].)457 3066 y(Finally)-7 b(,)26 b(note)j(that)h(eac)n(h)e(of)i(the)f(sets)g Fr(X)36 b FB(used)29 b(in)g(the)h(expressions)c Fq(h)p Fr(c)2760 3078 y Fo(1)2798 3066 y Fr(;)14 b(:)g(:)g(:)f(;)h(c)3018 3078 y Fm(m)3081 3066 y Fq(i)27 b FB(:)f Fr(X)35 b FB(in)374 3174 y(form)n(ulas)21 b(\(15\),)i(\(17\))h(and)g(\(21\))g(satis\014es)e (the)j(assumption)c(of)j(Prop)r(osition)d(1)j(\(Section)f(2.3\),)374 3282 y(b)r(ecause)k(the)h(w)n(eigh)n(ts)d Fr(w)1178 3294 y Fm(i)1234 3282 y FB(are)i(nonnegativ)n(e.)1211 3589 y Fj(4.2)49 b(Nondisjunctive)34 b(T)-7 b(r)i(anslation)374 3754 y FB(A)30 b(rule)f(with)h(nested)g(expressions)e(\(Section)h(2\))h (is)g Fn(nondisjunctiv)n(e)e FB(if)i(its)f(head)h(is)f(a)h(literal)374 3862 y(or)e Fq(?)p FB(.)g(A)h Fn(nondisjunctiv)n(e)e(program)f FB(is)i(a)g(program)e(with)j(nested)g(expressions)d(whose)i(rules)374 3970 y(are)e(nondisjunctiv)n(e.)457 4081 y(F)-7 b(or)27 b(an)n(y)h(program)d(\012)k(with)f(w)n(eigh)n(t)e(constrain)n(ts,)g (its)i 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b(example,)e(if)j(\005)g(is)f(\(3\))h (then)g([\005],)h(as)e(w)n(e)g(ha)n(v)n(e)g(seen,)h(is)f(\(20\);)g(the) h(nondisjunctiv)n(e)374 4898 y(translation)24 b([\005])902 4868 y Fm(nd)1010 4898 y FB(of)j(the)h(same)e(program)f(is)1517 5067 y Fr(a)e Fq( )g Fs(not)35 b(not)g Fr(a;)1517 5175 y(b)23 b Fq( )g Fs(not)35 b(not)g Fr(b;)1517 5283 y Fq(?)23 b( )g Fs(not)35 b(not)g FB(\()p Fr(a)p FB(;)14 b Fr(b)p FB(\))p Fr(:)3215 5174 y FB(\(23\))p eop end %%Page: 13 13 TeXDict begin 13 12 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)648 b FB(13)374 631 y Fs(Pr)l(op)l(osition) 31 b(2)374 740 y FB(F)-7 b(or)27 b(an)n(y)f(program)f(\012)j(with)f(w)n (eigh)n(t)f(constrain)n(ts,)f([\012])2104 710 y Fm(nd)2211 740 y FB(is)i(strongly)e(equiv)-5 b(alen)n(t)25 b(to)j([\012].)457 906 y(In)e(com)n(bination)c(with)j(Theorem)f(1,)i(this)f(fact)h(sho)n (ws)f(that)h(the)g(nondisjunctiv)n(e)e(transla-)374 1014 y(tion)i(is)h(sound:)g(\012)h(and)f([\012])1246 984 y Fm(nd)1354 1014 y FB(ha)n(v)n(e)f(the)i(same)e(answ)n(er)g(sets.)457 1122 y(Its)h(pro)r(of)g(is)g(based)g(on)g(the)h(follo)n(wing)c(w)n (ell-kno)n(wn)g(fact)k(ab)r(out)f(in)n(tuitionistic)c(logic:)374 1288 y Fs(F)-6 b(act)29 b(1)374 1397 y FB(If)34 b Fr(F)46 b FB(is)33 b(a)g(prop)r(ositional)d(com)n(bination)g(of)k(form)n(ulas)d Fr(F)2212 1409 y Fo(1)2249 1397 y Fr(;)14 b(:)g(:)g(:)g(;)g(F)2487 1409 y Fm(m)2584 1397 y FB(then)35 b Fr(F)f Fq(_)23 b(:)p Fr(F)47 b FB(is)32 b(in)n(tu-)374 1505 y(itionisticall)o(y)22 b(deriv)-5 b(able)25 b(from)h Fr(F)1449 1517 y Fo(1)1505 1505 y Fq(_)19 b(:)p Fr(F)1687 1517 y Fo(1)1725 1505 y FB(,)p Fr(:)14 b(:)g(:)g FB(,)p Fr(F)1935 1517 y Fm(m)2017 1505 y Fq(_)19 b(:)p Fr(F)2199 1517 y Fm(m)2262 1505 y FB(.)374 1728 y Fs(Pr)l(o)l(of)30 b(of)h(Pr)l(op)l(osition)g(2)374 1838 y FB(W)-7 b(e)23 b(will)c(sho)n(w)j(that)h(form)n(ula)c(\(18\))j (is)g(equiv)-5 b(alen)n(t)20 b(to)j(the)f(conjunction)g(of)h(the)f (form)n(ulas)e(\(22\))374 1946 y(in)26 b(the)i(logic)d(of)j (here-and-there.)e(By)h(F)-7 b(act)28 b(1,)f(the)h(form)n(ula)1651 2108 y([)p Fr(C)1733 2120 y Fo(0)1771 2108 y FB(])19 b Fq(_)f(:)p FB([)p Fr(C)2023 2120 y Fo(0)2061 2108 y FB(])1131 b(\(24\))374 2271 y(is)33 b(en)n(tailed)f(b)n(y)h(the)i(form) n(ulas)c Fr(c)23 b Fq(_)g(:)p Fr(c)35 b FB(for)e(all)f(head)i(elemen)n (ts)e Fr(c)i FB(of)g(rule)f(\(10\).)h(F)-7 b(or)34 b(ev)n(ery)374 2379 y(negativ)n(e)f Fr(c)p FB(,)j Fr(c)24 b Fq(_)g(:)p Fr(c)36 b FB(is)f(pro)n(v)-5 b(able)33 b(in)i(the)h(logic)d(of)j (here-and-there.)e(It)i(follo)n(ws)d(that)j(\(24\))374 2487 y(is)28 b(deriv)-5 b(able)26 b(in)j(the)g(logic)d(of)j (here-and-there)f(from)g Fr(l)2126 2499 y Fo(1)2182 2487 y Fq(_)20 b(:)p Fr(l)2337 2499 y Fo(1)2374 2487 y FB(,)p Fr(:)14 b(:)g(:)g FB(,)p Fr(l)2556 2499 y Fm(p)2614 2487 y Fq(_)19 b(:)p Fr(l)2768 2499 y Fm(p)2807 2487 y FB(.)29 b(Consequen)n(tly)-7 b(,)374 2595 y Fq(::)p FB([)p Fr(C)566 2607 y Fo(0)604 2595 y FB(])24 b Fq(\021)g FB([)p Fr(C)822 2607 y Fo(0)859 2595 y FB(])29 b(is)e(deriv)-5 b(able)25 b(from)i(these)h(form)n(ulas)d(as)j(w)n(ell.)e(Hence)i(\(18\))g(is)f (equiv)-5 b(alen)n(t)26 b(in)374 2703 y(the)i(logic)c(of)k (here-and-there)e(to)h(the)h(rule)851 2866 y(\()p Fr(l)908 2878 y Fo(1)946 2866 y FB(;)14 b Fs(not)35 b Fr(l)1160 2878 y Fo(1)1197 2866 y FB(\))p Fr(;)14 b(:)g(:)g(:)g(;)g FB(\()p Fr(l)1471 2878 y Fm(p)1509 2866 y FB(;)g Fs(not)35 b Fr(l)1723 2878 y Fm(p)1761 2866 y FB(\))p Fr(;)14 b Fs(not)36 b(not)f FB([)p Fr(C)2217 2878 y Fo(0)2255 2866 y FB(])23 b Fq( )g FB([)p Fr(C)2489 2878 y Fo(1)2526 2866 y FB(])p Fr(;)14 b(:)g(:)g(:)g(;)g FB([)p Fr(C)2816 2878 y Fm(n)2862 2866 y FB(])374 3028 y(whic)n(h)26 b(can)h(b)r(e)h (brok)n(en)f(in)n(to)f(the)i(rules)1112 3167 y Fr(l)1137 3179 y Fm(j)1171 3167 y FB(;)14 b Fs(not)35 b Fr(l)1385 3179 y Fm(j)1443 3167 y Fq( )23 b FB([)p Fr(C)1631 3179 y Fo(1)1669 3167 y FB(])p Fr(;)14 b(:)g(:)g(:)g(;)g FB([)p Fr(C)1959 3179 y Fm(n)2004 3167 y FB(])166 b(\(1)23 b Fq(\024)g Fr(j)28 b Fq(\024)22 b Fr(p)p FB(\))p Fr(;)1112 3275 y Fs(not)35 b(not)g FB([)p Fr(C)1498 3287 y Fo(0)1536 3275 y FB(])23 b Fq( )g FB([)p Fr(C)1770 3287 y Fo(1)1808 3275 y FB(])p Fr(;)14 b(:)g(:)g(:)f(;)h FB([)p Fr(C)2097 3287 y Fm(n)2143 3275 y FB(])p Fr(:)374 3433 y FB(The)33 b(\014rst)h(line)e(is)h(equiv)-5 b(alen)n(t)31 b(to)j(the)g(\014rst)f (line)f(of)i(\(22\))f(in)g(the)h(logic)d(of)j(here-and-there.)374 3541 y(The)27 b(second)g(line)f(is)h(in)n(tuitionistical)o(ly)21 b(equiv)-5 b(alen)n(t)25 b(to)j(the)g(second)f(line)f(of)h(\(22\).)p 3107 3487 61 4 v 3107 3541 4 54 v 3165 3541 V 3107 3544 61 4 v 1115 3814 a Fj(4.3)49 b(Eliminating)33 b(Neste)-5 b(d)36 b(Expr)-5 b(essions)374 3976 y FB(A)39 b(nondisjunctiv)n(e)e (rule)h(is)g Fn(nonnested)i FB(if)e(its)h(b)r(o)r(dy)g(is)f(a)h (conjunction)g(of)g(literals,)c(eac)n(h)374 4084 y(p)r(ossibly)h (pre\014xed)i(with)h Fs(not)7 b FB(.)39 b(A)g Fn(nonnested)g(program)c FB(is)j(a)g(program)e(whose)i(rules)f(are)374 4192 y(nonnested.)c(Th)n 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(dropping)h(the)h(new)g(atoms)f(from)f(the)j(answ)n(er)d(sets)i(of)g (the)h(translation)374 5056 y(giv)n(es)h(the)j(answ)n(er)e(sets)h(for)h (the)g(original)23 b(program.)457 5164 y(Eac)n(h)40 b(of)h(the)h(new)f (atoms)f(in)n(tro)r(duced)g(in)g(the)i(nonnested)f(translation)d ([\012])3027 5134 y Fm(nn)3155 5164 y FB(b)r(elo)n(w)374 5272 y(is,)f(in)n(tuitiv)n(ely)-7 b(,)34 b(an)j(\\abbreviation")d(for)j (some)g(form)n(ula)e(related)h(to)i(the)g(nondisjunctiv)n(e)374 5380 y(translation)24 b([\012])900 5350 y Fm(nd)980 5380 y FB(.)j(F)-7 b(or)27 b(instance,)g(to)g(eliminate)d(the)k(nesting)e (of)h(negations)f(from)g(the)i(\014rst)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 374 432 a FB(14)904 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)374 631 y FB(line)j(of)j(the)g(nondisjunctiv)n(e)e(translation)e(\(22\),)k(w)n (e)f(will)e(in)n(tro)r(duce,)h(for)h(ev)n(ery)f Fr(j)5 b FB(,)36 b(a)f(new)374 739 y(atom)26 b Fr(q)623 751 y Fd(not)34 b Fm(l)774 759 y Fc(j)809 739 y FB(,)27 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b(\(23\))j(of)h(program)c(\(3\))k(turns)g(after)f (this)g(transformation)d(in)n(to)1517 1660 y Fr(q)1554 1672 y Fd(not)34 b Fm(a)1747 1660 y Fq( )23 b Fs(not)35 b Fr(a;)1517 1768 y(a)23 b Fq( )g Fs(not)7 b Fr(q)1851 1780 y Fd(not)34 b Fm(a)2021 1768 y Fr(;)1517 1876 y(q)1554 1888 y Fd(not)g Fm(b)1740 1876 y Fq( )23 b Fs(not)35 b Fr(b;)1517 1984 y(b)23 b Fq( )g Fs(not)35 b Fr(q)1871 1996 y Fd(not)f Fm(b)2034 1984 y Fr(;)1517 2092 y Fq(?)23 b( )g Fs(not)35 b(not)g FB(\()p Fr(a)p FB(;)14 b Fr(b)p FB(\))p Fr(:)3215 1875 y FB(\(25\))457 2248 y(In)n(tro)r(ducing)39 b(the)i(atoms)e Fr(q)1371 2260 y Fd(not)34 b Fm(l)1522 2268 y Fc(j)1598 2248 y FB(brings)39 b(us)i(v)n(ery)f(close)f(to)i(the) g(goal)e(of)i(eliminating)374 2356 y(nesting)28 b(altogether,)f(b)r (ecause)i(ev)n(ery)g(rule)f(of)h(the)h(program)d(obtained)h(from)g ([\012])3001 2325 y Fm(nd)3111 2356 y FB(b)n(y)h(this)374 2464 y(transformation)23 b(is)i(strongly)g(equiv)-5 b(alen)n(t)24 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y(The)j(de\014nition)f(of)h ([\012])1208 3772 y Fm(nn)1326 3802 y FB(b)r(elo)n(w)e(follo)n(ws)g(a)i (di\013eren)n(t)f(approac)n(h)f(to)j(the)f(elimination)c(of)374 3910 y(the)d(remaining)c(nested)k(expressions.)e(Besides)g(the)j (\\negation)c(atoms")h(of)i(the)h(form)d Fr(q)3153 3922 y Fd(not)34 b Fm(l)3304 3930 y Fc(j)3339 3910 y FB(,)374 4018 y(it)27 b(in)n(tro)r(duces)g(other)g(new)h(atoms,)f(to)h(mak)n(e)e (the)i(translation)d(of)j(w)n(eigh)n(t)f(constrain)n(ts)e(more)374 4126 y(compact)18 b(in)h(some)g(cases.)f(These)i(\\w)n(eigh)n(t)e (atoms")g(ha)n(v)n(e)g(the)j(forms)d Fr(q)2597 4138 y Fm(w)r Ff(\024)p Fm(S)2767 4126 y FB(and)h Fr(q)2957 4138 y Fm(w)r( )g FB(0,)h(stands)g(for)374 4666 y Fq(f)p Fr(c)452 4678 y Fo(1)515 4666 y FB(=)26 b Fr(w)665 4678 y Fo(1)703 4666 y Fr(;)14 b(:)g(:)g(:)g(;)g(c)924 4678 y Fm(m)p Ff(\000)p Fo(1)1098 4666 y FB(=)26 b Fr(w)1248 4678 y Fm(m)p Ff(\000)p Fo(1)1397 4666 y Fq(g)p FB(.)j(Consider)f(a)h 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FB(of)32 b(literals)c(is)j(an)h(answ)n(er)f(set)h(for)g(\005) 2721 5227 y Fo(1)2780 5215 y Fq([)21 b FB(\005)2918 5227 y Fo(2)2988 5215 y FB(i\013)32 b Fr(Z)38 b FB(is)31 b(an)374 5323 y(answ)n(er)26 b(set)h(for)g(\005)967 5335 y Fo(1)1033 5323 y FB(and)g Fr(Z)i Fq(j)-14 b FB(=)23 b(\005)1439 5335 y Fo(2)1476 5323 y FB(.)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)648 b FB(21)374 631 y Fs(L)l(emma)30 b(5)374 742 y FB(F)-7 b(or)27 b(an)n(y)f(consisten)n(t)g(set)i Fr(Z)33 b FB(of)28 b(literals,)c Fr(Z)k Fq(j)-14 b FB(=)23 b(\012)28 b(i\013)f Fr(Z)i Fq(j)-14 b FB(=)23 b(\005)2290 754 y Fo(2)2327 742 y FB(.)374 974 y Fs(Pr)l(o)l(of)374 1084 y FB(It)35 b(is)e(su\016cien)n(t)h(to)h(consider)e(the)i(case)f (when)h(\012)g(consists)e(of)h(a)h(single)d(rule)i(\(10\).)g(In)h(this) 374 1192 y(case,)26 b Fr(Z)j Fq(j)-14 b FB(=)23 b(\012)28 b(i\013)1063 1356 y Fr(Z)h Fq(j)-14 b FB(=)23 b Fr(C)1305 1368 y Fo(0)1370 1356 y 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Fm(i)498 4759 y Fq(\024)23 b Fr(U)643 4771 y Fm(i)670 4759 y FB(])693 4729 y Fm(Z)773 4759 y FB(is)j Fq(?)p FB(,)g(so)g(that)h(eac)n(h)e(rule)h(\(41\))g(is)f(trivially)d (satis\014ed)j(b)n(y)h(an)n(y)g Fr(Z)2821 4729 y Ff(0)2844 4759 y FB(.)h(On)f(the)h(other)374 4867 y(hand,)g(in)g(this)g(case)g (\012)1099 4837 y Fm(Z)1180 4867 y FB(is)f(empt)n(y)-7 b(.)p 1613 4813 V 1613 4867 4 54 v 1670 4867 V 1613 4870 61 4 v 374 5098 a Fs(L)l(emma)30 b(7)374 5209 y FB(If)d(set)h Fs(cl)9 b FB(\(\012)746 5179 y Fm(Z)800 5209 y FB(\))27 b(is)g(consisten)n(t)f(then)i(it)e(is)h(the)h(only)e(answ)n(er)g(set)h (for)g(\005)2602 5179 y Fm(Z)2602 5230 y Fo(1)2656 5209 y FB(;)g(otherwise,)f(\005)3161 5179 y Fm(Z)3161 5230 y Fo(1)3242 5209 y FB(has)374 5317 y(no)h(answ)n(er)f(sets.)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 374 432 a FB(22)904 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)374 631 y(Pr)l(o)l(of)374 736 y FB(Recall)21 b(that)j Fs(cl)9 b FB(\(\012)956 706 y Fm(Z)1010 736 y FB(\))25 b(is)d(de\014ned)j(as)e (the)h(unique)g(minimal)19 b(set)24 b(satisfying)d(\012)2804 706 y Fm(Z)2882 736 y FB(\(Section)i(3.2\).)374 844 y(The)i(answ)n(er)f (sets)h(for)g(a)f(program)f(with)h(nested)i(expressions)c(that)k(do)r (es)f(not)g(con)n(tain)f(nega-)374 952 y(tion)32 b(as)g(failure)f(are)h (de\014ned)h(as)f(the)i(minimal)28 b(consisten)n(t)j(sets)i(satisfying) d(that)k(program)374 1060 y(\(Section)c(2.2\).)g(It)h(remains)d(to)i (notice)g(that)h(\012)1893 1030 y Fm(Z)1977 1060 y FB(and)g(\005)2204 1030 y Fm(Z)2204 1081 y Fo(1)2288 1060 y FB(are)f(satis\014ed)f(b)n(y)h (the)i(same)d(sets)374 1168 y(of)e(literals)d(\(Lemma)i(6\).)p 1274 1114 61 4 v 1274 1168 4 54 v 1331 1168 V 1274 1171 61 4 v 374 1368 a Fs(The)l(or)l(em)k(1)374 1473 y FB(F)-7 b(or)27 b(an)n(y)f(program)f(\012)j(with)f(w)n(eigh)n(t)f(constrain)n (ts,)f(\012)i(and)h([\012])g(ha)n(v)n(e)e(the)i(same)e(answ)n(er)g (sets.)374 1674 y Fs(Pr)l(o)l(of)374 1779 y FB(By)d(the)g(de\014nition) 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FB(.)c(By)f(Prop)r(osition)d(2,)k([\012])1386 2710 y Fm(nd)1493 2740 y FB(has)f(the)h(same)e(answ)n(er)g(sets)i(as)f ([\012].)p 2768 2686 V 2768 2740 4 54 v 2825 2740 V 2768 2743 61 4 v 693 3001 a Fj(6.2)48 b(Two)35 b(L)-5 b(emmas)36 b(on)e(Pr)-5 b(o)g(gr)g(ams)36 b(with)d(Neste)-5 b(d)36 b(Expr)-5 b(essions)374 3161 y FB(The)28 b(idea)e(of)i(program)d (completion)g(\(Clark)h(1978\))h(is)g(that)h(the)g(set)g(of)g(rules)f (of)h(a)g(program)374 3268 y(with)e(the)g(same)f(atom)g Fr(q)30 b FB(in)25 b(the)i(head)f(is)g(the)g(\\if)6 b(")26 b(part)g(of)g(a)g(de\014nition)f(of)h Fr(q)s FB(;)h(the)f(\\only)f(if)6 b(")374 3376 y(half)26 b(of)i(that)g(de\014nition)e(is)g(left)i (implicit.)23 b(If,)28 b(for)f(instance,)g(the)h(rule)1751 3536 y Fr(q)e Fq( )d Fr(F)374 3695 y FB(is)31 b(the)j(only)d(rule)h(in) g(the)i(program)c(whose)i(head)h(is)e Fr(q)37 b FB(then)c(that)g(rule)f (is)g(an)h(abbreviated)374 3803 y(form)26 b(of)h(the)h(assertion)d (that)j Fr(q)j FB(is)c(equiv)-5 b(alen)n(t)25 b(to)i 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FB(stands)k(for)g(the)h(pair)e (of)h(rules)1731 5215 y Fr(F)35 b Fq( )23 b Fr(G)1731 5323 y(G)h Fq( )f Fr(F)r(:)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)648 b FB(23)374 631 y Fs(Completion)31 b(L)l(emma)374 738 y FB(Let)f(\005)g(b)r(e)g(a)f(program)e(with)j (nested)g(expressions,)d(and)i(let)h Fr(Q)f FB(b)r(e)h(a)g(set)g(of)f (atoms)g(that)h(do)374 846 y(not)i(ha)n(v)n(e)f(regular)f(o)r (ccurrences)h(in)g(the)i(heads)f(of)h(the)f(rules)f(of)i(\005.)f(F)-7 b(or)32 b(ev)n(ery)f Fr(q)j Fq(2)e Fr(Q)p FB(,)g(let)374 954 y Fs(Def)18 b FB(\()p Fr(q)s FB(\))28 b(b)r(e)g(a)f(form)n(ula.)e (Then)i(the)h(program)1379 1116 y(\005)19 b Fq([)g(f)p Fr(q)25 b Fq( )e Fs(Def)18 b FB(\()p Fr(q)s FB(\))47 b(:)f Fr(q)26 b Fq(2)e Fr(Q)p Fq(g)374 1277 y FB(has)j(the)h(same)e (answ)n(er)g(sets)h(as)g(the)h(program)1368 1439 y(\005)18 b Fq([)h(f)p Fr(q)26 b Fq($)d Fs(Def)18 b FB(\()p Fr(q)s FB(\))46 b(:)h Fr(q)26 b Fq(2)d Fr(Q)p Fq(g)p Fr(:)457 1600 y FB(In)g(the)g(sp)r(ecial)e(case)h(when)h Fr(Q)f FB(is)g(a)h(singleton)d(this)i(fact)h(w)n(as)f(\014rst)h(pro)n(v)n(ed)e (b)n(y)i(Esra)e(Erdem)374 1708 y(\(p)r(ersonal)k(comm)n(unication\).) 457 1816 y(In)34 b(the)h(statemen)n(t)f(of)g(the)h(completion)c(lemma,) h(if)i(the)h(atoms)d(from)h Fr(Q)i FB(o)r(ccur)e(neither)374 1924 y(in)23 b(\005)h(nor)f(in)g(the)h(form)n(ulas)d Fs(Def)d FB(\()p Fr(q)s FB(\))24 b(then)g(adding)f(the)h(rules)e Fr(q)k Fq( )d Fs(Def)18 b FB(\()p Fr(q)s FB(\))25 b(to)e(\005)h (extends)g(the)374 2032 y(program)h(b)n(y)j(\\explicit)e (de\014nitions")g(of)i(\\new")g(atoms.)f(According)f(to)i(the)h(lemma)d (b)r(elo)n(w,)374 2140 y(suc)n(h)20 b(an)g(extension)f(is)h(conserv)-5 b(ativ)n(e:)17 b(the)k(answ)n(er)e(sets)i(for)f(\005)h(can)f(b)r(e)h (obtained)e(b)n(y)h(dropping)374 2248 y(the)28 b(new)f(atoms)f(from)g (the)i(answ)n(er)f(sets)g(for)g(the)h(extended)g(program.)374 2409 y Fs(L)l(emma)i(on)f(Explicit)i(De\014nitions)374 2516 y 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FB(a)g(subset)g(of)f(a)h(consisten)n(t)e(set)374 3485 y Fr(Z)35 b FB(of)29 b(literals.)d(If)k(the)g(literals)c(in)j Fr(Z)c Fq(n)20 b Fr(Z)1671 3454 y Ff(0)1723 3485 y FB(do)29 b(not)h(ha)n(v)n(e)e(regular)f(o)r(ccurrences)h(in)h(the)h(heads)374 3593 y(of)d(the)h(rules)e(of)i(\005)g(and)f Fr(Z)i Fq(j)-14 b FB(=)23 b(\005)28 b(then)g Fr(Z)1681 3562 y Ff(0)1727 3593 y Fq(j)-14 b FB(=)22 b(\005.)457 3753 y(The)f(pro)r(of)g(of)h (this)f(lemma)e(uses)i(the)h(follo)n(wing)17 b(fact)22 b(that)g(is)f(easy)f(to)i(v)n(erify)e(b)n(y)h(structural)374 3861 y(induction:)374 4022 y Fs(F)-6 b(act)29 b(3)374 4129 y FB(Let)i Fr(F)44 b FB(b)r(e)31 b(a)g(form)n(ula)e(without)i (negation)e(as)i(failure,)e Fr(Z)37 b FB(a)31 b(consisten)n(t)f(set)i (of)f(literals)d(and)374 4237 y Fr(Z)437 4207 y Ff(0)487 4237 y FB(a)f(subset)h(of)g Fr(Z)6 b FB(.)27 b(If)h Fr(Z)1165 4207 y Ff(0)1211 4237 y Fq(j)-14 b FB(=)23 b Fr(F)39 b FB(then)28 b Fr(Z)h Fq(j)-14 b FB(=)23 b Fr(F)12 b FB(.)374 4451 y Fs(Pr)l(o)l(of)30 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2226 y(Since)26 b Fr(Z)34 b FB(is)26 b(an)h(answ)n(er)g(set)g(for)g(\005)19 b Fq([)g FB(\001,)806 2386 y(\()p Fr(Z)25 b Fq(\\)18 b Fr(Q)p FB(\))h Fq([)g Fr(Z)1247 2352 y Ff(0)1293 2386 y Fq(j)-14 b FB(=)22 b(\005)d Fq([)g FB(\001)28 b(i\013)f(\()p Fr(Z)d Fq(\\)19 b Fr(Q)p FB(\))g Fq([)f Fr(Z)2180 2352 y Ff(0)2226 2386 y FB(=)23 b Fr(Z)33 b FB(i\013)27 b Fr(Z)2566 2352 y Ff(0)2612 2386 y FB(=)c Fr(Z)h Fq(n)18 b Fr(Q:)374 2547 y FB(Using)26 b(\(44\),)h(w)n(e)h(conclude:)1269 2707 y(\()p Fr(Z)c Fq(\\)19 b Fr(Q)p FB(\))f Fq([)h Fr(Z)1709 2673 y Ff(0)1755 2707 y Fq(j)-14 b FB(=)23 b(\005)28 b(i\013)f Fr(Z)2104 2673 y Ff(0)2150 2707 y FB(=)c Fr(Z)h Fq(n)18 b Fr(Q:)374 2868 y FB(Since)26 b(no)i(elemen)n(t)d(of)j Fr(Q)f FB(o)r(ccurs)g(in)g(\005,)h(w)n(e)f(can)g(rewrite)f(this)h(as) 1457 3028 y Fr(Z)1520 2994 y Ff(0)1566 3028 y Fq(j)-14 b FB(=)23 b(\005)28 b(i\013)f Fr(Z)1915 2994 y Ff(0)1961 3028 y FB(=)c Fr(Z)h Fq(n)18 b Fr(Q:)374 3189 y FB(Since)24 b Fr(Z)651 3159 y Ff(0)699 3189 y FB(here)g(is)g(an)h(arbitrary)d (subset)j(of)g Fr(Z)18 b Fq(n)13 b Fr(Q)p FB(,)25 b(w)n(e)f(pro)n(v)n (ed)g(that)h Fr(Z)19 b Fq(n)13 b Fr(Q)24 b FB(is)g(an)h(answ)n(er)e (set)374 3297 y(for)k(\005.)457 3405 y(T)-7 b(o)23 b(pro)n(v)n(e)g (Claim)d(1)k(in)f(the)i(general)c(case,)j(consider)e(an)h(answ)n(er)g (set)h Fr(Z)30 b FB(for)23 b(\005)11 b Fq([)g FB(\001.)26 b(It)e(is)f(an)374 3512 y(answ)n(er)h(set)i(for)g(\005)963 3482 y Fm(Z)1032 3512 y Fq([)15 b FB(\001)1171 3482 y Fm(Z)1251 3512 y FB(also.)24 b(By)i(the)g(sp)r(ecial)e(case)h(of)h (Claim)d(1)i(pro)n(v)n(ed)g(ab)r(o)n(v)n(e,)f Fr(Z)d Fq(n)15 b Fr(Q)26 b FB(is)374 3620 y(an)h(answ)n(er)f(set)i(for)f(\005) 1083 3590 y Fm(Z)1137 3620 y FB(.)g(Since)g(no)g(elemen)n(t)f(of)i Fr(Q)f FB(o)r(ccurs)g(in)g(\005,)h(\005)2543 3590 y Fm(Z)t Ff(n)p Fm(Q)2706 3620 y FB(=)22 b(\005)2855 3590 y Fm(Z)2937 3620 y FB(\(Lemma)k(9\).)374 3728 y(It)i(follo)n(ws)c(that)k Fr(Z)c Fq(n)18 b Fr(Q)27 b FB(is)g(an)g(answ)n(er)f(set)i(for)f(\005) 1943 3698 y Fm(Z)t Ff(n)p Fm(Q)2083 3728 y FB(,)h(and)f(consequen)n (tly)f(for)h(\005.)374 3883 y Fn(Claim)g(2:)k(If)g Fr(Z)866 3853 y Ff(\003)934 3883 y Fn(is)f(an)g(answ)n(er)f(set)i(for)f FB(\005)h Fn(then)h(there)e(exists)f(a)i(unique)f(answ)n(er)f(set)i Fr(Z)36 b Fn(for)374 3991 y FB(\005)18 b Fq([)h FB(\001)28 b Fn(suc)n(h)f(that)h Fr(Z)c Fq(n)18 b Fr(Q)23 b FB(=)f Fr(Z)1372 3961 y Ff(\003)1410 3991 y Fn(.)457 4099 y FB(Consider)31 b(\014rst)i(the)g(case)f(when)i(neither)e(\005)h(nor)f (\001)i(con)n(tains)d(negation)g(as)i(failure.)d(Let)374 4207 y Fr(Z)437 4177 y Ff(\003)502 4207 y FB(b)r(e)e(an)f(answ)n(er)f (set)i(for)f(\005.)h(De\014ne)1216 4367 y Fr(Z)1273 4379 y Fo(0)1333 4367 y FB(=)22 b Fr(Z)1483 4333 y Ff(\003)1540 4367 y Fq([)c(f)p Fr(q)26 b Fq(2)d Fr(Q)51 b FB(:)g Fr(Z)2050 4333 y Ff(\003)2110 4367 y Fq(j)-14 b FB(=)23 b Fs(Def)18 b FB(\()p Fr(q)s FB(\))p Fq(g)p Fr(:)374 4528 y FB(W)-7 b(e)35 b(will)c(sho)n(w)j(that)h Fr(Z)1144 4540 y Fo(0)1216 4528 y FB(is)e(the)i(only)e(consisten)n(t)g(set)i Fr(Z)40 b FB(of)35 b(literals)30 b(with)k(the)h(prop)r(erties)374 4636 y(from)26 b(Claim)e(2.)k(Clearly)c Fr(Z)1252 4648 y Fo(0)1307 4636 y Fq(n)18 b Fr(Q)23 b FB(=)g Fr(Z)1607 4606 y Ff(\003)1644 4636 y FB(.)28 b(W)-7 b(e)28 b(will)d(c)n(hec)n(k)i (no)n(w)g(that)420 4793 y(\(i\))41 b Fr(Z)606 4805 y Fo(0)670 4793 y FB(satis\014es)26 b(\005)19 b Fq([)g FB(\001,)397 4900 y(\(ii\))40 b(no)27 b(prop)r(er)g(subset)g(of)h Fr(Z)1336 4912 y Fo(0)1401 4900 y FB(satis\014es)d(\005)19 b Fq([)g FB(\001,)28 b(and)374 5006 y(\(iii\))39 b(ev)n(ery)c (consisten)n(t)g(set)i Fr(Z)42 b FB(of)37 b(literals)32 b(that)37 b(satis\014es)e(\005)25 b Fq([)f FB(\001)37 b(and)g(has)f(the)h(prop)r(ert)n(y)549 5114 y Fr(Z)24 b Fq(n)18 b Fr(Q)23 b FB(=)f Fr(Z)929 5084 y Ff(\003)995 5114 y FB(is)k(a)h(sup)r(erset)h(of)f Fr(Z)1625 5126 y Fo(0)1662 5114 y FB(.)457 5272 y(T)-7 b(o)18 b(sho)n(w)g(that)h Fr(Z)995 5284 y Fo(0)1050 5272 y FB(satis\014es)e(\005,)i(observ)n(e)e (that)i Fr(Z)1972 5242 y Ff(\003)2029 5272 y FB(satis\014es)e(\005)i (and)f(no)h(elemen)n(t)d(of)j Fr(Q)f FB(o)r(ccurs)374 5380 y(in)29 b(\005.)h(T)-7 b(o)29 b(sho)n(w)g(that)h Fr(Z)1159 5392 y Fo(0)1226 5380 y FB(satis\014es)e(\001,)i(assume)e (that)i Fr(Z)2184 5392 y Fo(0)2248 5380 y Fq(j)-14 b FB(=)27 b Fs(Def)18 b FB(\()p Fr(q)s FB(\).)30 b(Since)f(no)h(elemen)n (t)e(of)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 374 432 a FB(26)904 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)374 631 y Fr(Q)h FB(o)r(ccurs)g(in)g Fs(Def)18 b FB(\()p Fr(q)s FB(\),)32 b(it)f(follo)n(ws)e(that)j Fr(Z)1747 601 y Ff(\003)1814 631 y Fq(j)-14 b FB(=)30 b Fs(Def)18 b FB(\()p Fr(q)s FB(\),)32 b(so)f(that)h Fr(q)h Fq(2)e Fr(Z)2724 643 y Fo(0)2760 631 y FB(.)h(Assertion)e(\(i\))i(is)374 739 y(pro)n(v)n(ed.)457 869 y(It)g(is)f(con)n(v)n(enien)n(t)f(to)i(pro) n(v)n(e)e(assertion)g(\(iii\))g(next.)i(T)-7 b(ak)n(e)31 b(a)h(consisten)n(t)e(set)i Fr(Z)38 b FB(of)32 b(literals)374 977 y(that)f(satis\014es)f(\005)21 b Fq([)g FB(\001)32 b(and)f(has)f(the)i(prop)r(ert)n(y)e Fr(Z)d Fq(n)20 b Fr(Q)29 b FB(=)g Fr(Z)2331 947 y Ff(\003)2369 977 y FB(.)i(Since)f Fr(Z)2706 947 y Ff(\003)2776 977 y FB(is)g(an)h(answ)n(er)e(set)374 1085 y(for)e(\005,)h(and)f(\005)h(do)r(es)f(not)h(con)n(tain)e(elemen)n (ts)f(of)j Fr(Q)p FB(,)f Fr(Z)2106 1055 y Ff(\003)2172 1085 y FB(is)f(disjoin)n(t)g(from)g Fr(Q)p FB(,)h(so)g(that)1211 1321 y Fr(Z)1268 1333 y Fo(0)1323 1321 y Fq(n)18 b Fr(Q)23 b FB(=)g Fr(Z)1623 1287 y Ff(\003)1679 1321 y Fq(n)18 b Fr(Q)23 b FB(=)f Fr(Z)1978 1287 y Ff(\003)2039 1321 y FB(=)h Fr(Z)h Fq(n)18 b Fr(Q)k Fq(\022)h Fr(Z)q(:)690 b FB(\(45\))374 1557 y(T)-7 b(ak)n(e)22 b(an)n(y)g Fr(q)k Fq(2)d Fr(Z)920 1569 y Fo(0)966 1557 y Fq(\\)9 b Fr(Q)p FB(.)23 b(Since)f Fr(Z)1417 1527 y Ff(\003)1478 1557 y FB(is)g(disjoin)n(t)f(from)g Fr(Q)p FB(,)i Fr(q)j FB(b)r(elongs)21 b(to)i(the)g(second)f(of)h(the)g(t)n(w)n(o)374 1665 y(sets)30 b(whose)g(union)f(w)n(e)h(denoted)h(b)n(y)f Fr(Z)1637 1677 y Fo(0)1674 1665 y FB(,)g(so)g(that)h Fr(Z)2078 1634 y Ff(\003)2143 1665 y Fq(j)-14 b FB(=)28 b Fs(Def)18 b FB(\()p Fr(q)s FB(\).)31 b(Since)f Fr(Z)2830 1634 y Ff(\003)2895 1665 y FB(=)e Fr(Z)e Fq(n)19 b Fr(Q)31 b FB(and)374 1773 y(the)g(elemen)n(ts)d(of)j Fr(Q)f FB(do)h(not)f(o)r (ccur)g(in)g Fs(Def)18 b FB(\()p Fr(q)s FB(\),)31 b(it)g(follo)n(ws)c (that)k Fr(Z)j Fq(j)-14 b FB(=)28 b Fs(Def)18 b FB(\()p Fr(q)s FB(\).)31 b(In)g(view)f(of)374 1880 y(the)24 b(fact)h(that)f Fr(Z)30 b FB(satis\014es)23 b(\001,)h(w)n(e)g(can)g(conclude)f(that)h Fr(q)i Fq(2)e Fr(Z)6 b FB(.)24 b(Since)f Fr(q)28 b FB(here)23 b(is)h(an)g(arbitrary)374 1988 y(elemen)n(t)g(of)i Fr(Z)829 2000 y Fo(0)882 1988 y Fq(\\)17 b Fr(Q)p FB(,)26 b(w)n(e)g(pro)n(v)n (ed)e(that)j Fr(Z)1694 2000 y Fo(0)1747 1988 y Fq(\\)16 b Fr(Q)23 b Fq(\022)g Fr(Z)6 b FB(.)26 b(In)g(com)n(bination)d(with)j (\(45\),)g(this)f(fact)374 2096 y(sho)n(ws)h(that)i Fr(Z)33 b FB(is)27 b(a)g(sup)r(erset)g(of)h Fr(Z)1513 2108 y Fo(0)1550 2096 y FB(.)457 2226 y(T)-7 b(o)22 b(pro)n(v)n(e)g(assertion) e(\(ii\),)i(assume)g(that)h(a)g(prop)r(er)f(subset)h Fr(Z)29 b FB(of)23 b Fr(Z)2560 2238 y Fo(0)2620 2226 y FB(satis\014es)e(\005)9 b Fq([)g FB(\001.)25 b(Since)374 2334 y(the)h(elemen)n(ts)d(of)j Fr(Q)g FB(do)f(not)h(o)r(ccur)f(in)g (\005,)i(it)e(follo)n(ws)e(that)j Fr(Z)20 b Fq(n)15 b Fr(Q)25 b FB(satis\014es)f(\005.)i(On)g(the)g(other)374 2442 y(hand,)f Fr(Z)19 b Fq(n)14 b Fr(Q)25 b FB(is)f(a)h(subset)g(of)g Fr(Z)1380 2412 y Ff(\003)1418 2442 y FB(.)g(Since)g Fr(Z)1744 2412 y Ff(\003)1807 2442 y FB(is)f(an)h(answ)n(er)f(set)h(for)g(\005,)h (it)e(follo)n(ws)e(that)k Fr(Z)19 b Fq(n)14 b Fr(Q)374 2550 y FB(cannot)22 b(b)r(e)h(a)g(prop)r(er)f(subset)h(of)g Fr(Z)1478 2520 y Ff(\003)1516 2550 y FB(.)g(Consequen)n(tly)e Fr(Z)14 b Fq(n)9 b Fr(Q)23 b FB(=)f Fr(Z)2433 2520 y Ff(\003)2471 2550 y FB(.)h(Then,)g(b)n(y)g(assertion)e(\(iii\),)374 2658 y Fr(Z)33 b FB(is)26 b(a)i(sup)r(erset)f(of)g Fr(Z)1094 2670 y Fo(0)1131 2658 y FB(,)h(whic)n(h)f(is)f(imp)r(ossible,)e(b)n(y)j (the)h(c)n(hoice)e(of)h Fr(Z)6 b FB(.)457 2788 y(T)-7 b(o)30 b(pro)n(v)n(e)e(Claim)g(2)i(in)f(the)i(general)d(case,)i (consider)e(an)i(answ)n(er)f(set)i Fr(Z)2782 2758 y Ff(\003)2850 2788 y FB(for)f(\005.)h(It)f(is)g(an)374 2896 y(answ)n(er)23 b(set)h(for)g(\005)958 2866 y Fm(Z)1007 2841 y Fb(\003)1071 2896 y FB(also.)f(By)h(the)h(sp)r(ecial)d(case)h(of)i(Claim)c(2)j(pro)n (v)n(ed)f(ab)r(o)n(v)n(e,)g(there)i(exists)e(a)374 3004 y(unique)i(answ)n(er)g(set)i Fr(Z)32 b FB(for)26 b(\005)1322 2974 y Fm(Z)1371 2949 y Fb(\003)1427 3004 y Fq([)17 b FB(\001)1568 2974 y Fm(Z)1617 2949 y Fb(\003)1683 3004 y FB(suc)n(h)26 b(that)h Fr(Z)22 b Fq(n)16 b Fr(Q)22 b FB(=)h Fr(Z)2424 2974 y Ff(\003)2462 3004 y FB(.)k(No)f(elemen)n(t)f (of)h Fr(Q)g FB(o)r(ccurs)374 3112 y(in)c(\005)h(or)f(\001)h(in)f(the)h (scop)r(e)g(of)f(negation)f(as)h(failure.)f(By)h(Lemma)f(9)h(it)h (follo)n(ws)c(that)24 b(\005)3048 3082 y Fm(Z)3097 3057 y Fb(\003)3159 3112 y FB(=)f(\005)3309 3082 y Fm(Z)374 3220 y FB(and)33 b(\001)610 3190 y Fm(Z)659 3165 y Fb(\003)733 3220 y FB(=)g(\001)900 3190 y Fm(Z)988 3220 y FB(for)g(ev)n(ery)g Fr(Z)40 b FB(suc)n(h)33 b(that)i Fr(Z)28 b Fq(n)22 b Fr(Q)34 b FB(=)f Fr(Z)2234 3190 y Ff(\003)2272 3220 y FB(.)h(Consequen)n(tly)-7 b(,)32 b(there)i(exists)f(a)374 3328 y(unique)24 b(answ)n(er)f(set)i Fr(Z)31 b FB(for)25 b(\005)1315 3298 y Fm(Z)1382 3328 y Fq([)13 b FB(\001)1519 3298 y Fm(Z)1598 3328 y FB(suc)n(h)25 b(that)g Fr(Z)19 b Fq(n)13 b Fr(Q)23 b FB(=)f Fr(Z)2330 3298 y Ff(\003)2368 3328 y FB(.)j(It)h(follo)n(ws)c(that)j(there)g(exists)374 3436 y(a)i(unique)g(answ)n(er)f(set)h Fr(Z)34 b FB(for)27 b(\005)18 b Fq([)h FB(\001)28 b(suc)n(h)f(that)h Fr(Z)c Fq(n)18 b Fr(Q)23 b FB(=)g Fr(Z)2334 3406 y Ff(\003)2371 3436 y FB(.)p 2502 3382 61 4 v 2502 3436 4 54 v 2559 3436 V 2502 3439 61 4 v 1367 3968 a Fj(6.3)48 b(Pr)-5 b(o)g(of)35 b(of)g(The)-5 b(or)g(em)35 b(2)374 4153 y FB(Let)22 b(\012)g(b)r(e)h(a)e(program)e(with)j(w)n(eigh)n(t)e (constrain)n(ts.)g(Consider)g(the)j(subset)f(\001)g(of)g(its)g (nonnested)374 4260 y(translation)f([\012])897 4230 y Fm(nn)1008 4260 y FB(consisting)h(of)j(the)g(rules)e(whose)h(heads)g (are)g(atoms)f(from)g Fr(Q)2924 4272 y Fo(\012)2975 4260 y FB(.)i(The)g(rules)374 4368 y(included)d(in)i(\001)g(ha)n(v)n(e)g (the)g(forms)f(\(26\){\(30\);)g(they)h(\\de\014ne")g(the)h(atoms)d(in)i Fr(Q)2871 4380 y Fo(\012)2922 4368 y FB(.)g(The)h(rest)f(of)374 4476 y([\012])480 4446 y Fm(nn)591 4476 y FB(will)f(b)r(e)i(denoted)g (b)n(y)g(\005;)h(the)f(rules)f(of)h(\005)g(ha)n(v)n(e)f(the)i(forms)e (\(31\))g(and)h(\(32\).)g(The)g(union)374 4584 y(of)i(these)h(t)n(w)n (o)f(programs)d(is)i([\012])1392 4554 y Fm(nn)1479 4584 y FB(:)1593 4820 y([\012])1699 4786 y Fm(nn)1808 4820 y FB(=)d(\005)c Fq([)g FB(\001)p Fr(:)1072 b FB(\(46\))374 5056 y(The)35 b(idea)f(of)h(the)h(pro)r(of)e(of)h(Theorem)f(2)h(is)f (to)h(transform)e(\005)24 b Fq([)g FB(\001)35 b(in)n(to)f(a)h(program)d (with)374 5164 y(the)g(same)e(answ)n(er)g(sets)h(so)g(that)h(\005)g (will)c(turn)k(in)n(to)e([\012])2189 5134 y Fm(nd)2301 5164 y FB(and)h(\001)h(will)d(turn)i(in)n(to)g(a)g(set)h(of)374 5272 y(explicit)26 b(de\014nitions)g(in)i(the)h(sense)f(of)h(Section)e (6.2,)h(and)h(then)g(use)f(the)h(lemma)d(on)i(explicit)374 5380 y(de\014nitions.)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 1104 432 a Fs(Weight)31 b(Constr)l(aints)e(as)i (Neste)l(d)e(Expr)l(essions)648 b FB(27)457 631 y(F)-7 b(or)27 b(ev)n(ery)f(atom)g Fr(q)g Fq(2)e Fr(Q)1245 643 y Fo(\012)1296 631 y FB(,)k(de\014ne)f(the)h(form)n(ula)d Fs(Def)18 b FB(\()p Fr(q)s FB(\))28 b(as)f(follo)n(ws:)643 792 y Fs(Def)18 b FB(\()p Fr(q)856 804 y Fd(not)34 b Fm(l)1012 792 y FB(\))23 b(=)g Fs(not)35 b Fr(l)643 1044 y Fs(Def)18 b FB(\()p Fr(q)856 1056 y Fm(w)r Ff(\024)p Fm(S)1006 1044 y FB(\))24 b(=)1149 824 y Fi(8)1149 899 y(>)1149 924 y(>)1149 949 y(<)1149 1098 y(>)1149 1123 y(>)1149 1148 y(:)1223 920 y Fq(>)p Fr(;)829 b FB(if)27 b Fr(w)f Fq(\024)c FB(0,)1223 1049 y Fr(q)1260 1061 y Fm(w)r Ff(\024)p Fm(S)1406 1045 y Fb(0)1432 1049 y FB(;)14 b(\()p Fr(c)1537 1061 y Fm(m)1600 1049 y Fr(;)g(q)1674 1061 y Fm(w)r Ff(\000)p Fm(w)1824 1069 y Fc(m)1879 1061 y Ff(\024)p Fm(S)1975 1045 y Fb(0)2001 1049 y FB(\))p Fr(;)84 b FB(if)27 b(0)22 b Fr(<)h(w)j Fq(\024)c Fr(w)2599 1061 y Fo(1)2656 1049 y FB(+)c Fq(\001)c(\001)g(\001)k FB(+)g Fr(w)2996 1061 y Fm(m)3059 1049 y FB(,)1223 1179 y Fq(?)p Fr(;)829 b FB(otherwise)643 1443 y Fs(Def)18 b FB(\()p Fr(q)856 1455 y Fm(w)r( )1149 1322 y(>)1149 1347 y(<)1149 1497 y(>)1149 1522 y(>)1149 1547 y(:)1223 1318 y Fq(>)p Fr(;)829 b FB(if)27 b Fr(w)f(<)c FB(0,)1223 1448 y Fr(q)1260 1460 y Fm(w)r( )975 4495 y(>)975 4520 y(<)975 4670 y(>)975 4694 y(>)975 4719 y(:)1049 4491 y Fq(>)p Fr(;)1074 b FB(if)28 b Fr(w)d Fq(\024)e FB(0)o Fr(;)1049 4621 y FB([)p Fr(w)j Fq(\024)c Fr(S)1300 4591 y Ff(0)1323 4621 y FB(];)14 b(\()p Fr(c)1451 4633 y Fm(m)1515 4621 y Fr(;)g FB([)p Fr(w)21 b Fq(\000)d Fr(w)1797 4633 y Fm(m)1883 4621 y Fq(\024)23 b Fr(S)2027 4591 y Ff(0)2050 4621 y FB(]\))p Fr(;)83 b FB(if)28 b(0)22 b Fr(<)h(w)i Fq(\024)e Fr(w)2671 4633 y Fo(1)2727 4621 y FB(+)18 b Fq(\001)c(\001)g(\001)19 b FB(+)f Fr(w)3068 4633 y Fm(m)3131 4621 y FB(,)1049 4750 y Fq(?)p Fr(;)1074 b FB(otherwise)n Fr(:)572 5144 y FB([)p Fr(w)26 b(<)c(S)5 b FB(])23 b Fq($)975 4924 y Fi(8)975 4999 y(>)975 5024 y(>)975 5049 y(<)975 5198 y(>)975 5223 y(>)975 5248 y(:)1049 5020 y Fq(>)p Fr(;)1074 b FB(if)28 b Fr(w)d(<)e FB(0)o Fr(;)1049 5149 y FB([)p Fr(w)j(<)c(S)1300 5119 y Ff(0)1323 5149 y FB(];)14 b(\()p Fr(c)1451 5161 y Fm(m)1515 5149 y Fr(;)g FB([)p Fr(w)21 b Fq(\000)d Fr(w)1797 5161 y Fm(m)1883 5149 y Fr(<)23 b(S)2027 5119 y Ff(0)2050 5149 y FB(]\))p Fr(;)83 b FB(if)28 b(0)22 b Fq(\024)h Fr(w)i(<)e(w)2671 5161 y Fo(1)2727 5149 y FB(+)18 b Fq(\001)c(\001)g(\001)19 b FB(+)f Fr(w)3068 5161 y Fm(m)3131 5149 y FB(,)1049 5279 y Fq(?)p Fr(;)1074 b FB(otherwise)n Fr(:)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 374 432 a FB(28)904 b Fs(P.)30 b(F)-6 b(err)l(aris)31 b(and)f(V.)g(Lifschitz)374 631 y(Pr)l(o)l(of)374 741 y FB(Recall)20 b(that)j([)p Fr(w)j Fq(\024)d Fr(S)5 b FB(])22 b(is)g(an)h(expression)d(of)j(the)g(form)e (\(15\),)i(whic)n(h)f(stands)g(for)h(a)f(disjunction)374 849 y(of)f(conjunctions)f(\(8\).)i(If)f Fr(w)26 b Fq(\024)d FB(0)e(then)g(the)h(set)f(after)g(the)h(:)g(sign)d(in)i(\(15\))g(has)g (the)g(empt)n(y)g(set)g(as)374 957 y(one)h(of)i(its)e(elemen)n(ts,)f (so)i(that)g(one)g(of)g(the)h(disjunctiv)n(e)d(terms)h(of)h(this)g (form)n(ula)d(is)i(the)h(empt)n(y)374 1065 y(conjunction)d Fq(>)p FB(.)i(If)g Fr(w)j Fq(\025)e Fr(w)1235 1077 y Fo(1)1279 1065 y FB(+)6 b Fq(\001)14 b(\001)g(\001)7 b FB(+)f Fr(w)1584 1077 y Fm(m)1669 1065 y FB(then)22 b(set)g(after)f(the)h(:)g(sign)e(in)h(\(15\))g(is)f(empt)n(y)-7 b(,)21 b(so)g(that)374 1173 y(the)27 b(form)n(ula)d(is)i(the)i(empt)n (y)e(disjunction)g Fq(?)p FB(.)g(Assume)h(no)n(w)f(that)i(0)22 b Fr(<)h(w)2711 1185 y Fo(1)2766 1173 y FB(+)17 b Fq(\001)d(\001)g (\001)j FB(+)g Fr(w)3103 1185 y Fm(m)3190 1173 y Fq(\024)23 b Fr(w)r FB(.)374 1281 y(Let)29 b Fr(I)36 b FB(stand)29 b(for)f Fq(f)p FB(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(m)p Fq(g)28 b FB(and)h(let)f Fr(I)1690 1251 y Ff(0)1743 1281 y FB(b)r(e)h Fq(f)p FB(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(m)19 b Fq(\000)g FB(1)p Fq(g)p FB(.)28 b(F)-7 b(or)28 b(an)n(y)h(subset)g Fr(J)37 b FB(of)29 b Fr(I)7 b FB(,)29 b(b)n(y)374 1389 y(\006)p Fr(J)35 b FB(w)n(e)27 b(denote)h(the)g(sum)1224 1327 y Fi(P)1311 1414 y Fm(i)p Ff(2)p Fm(J)1440 1389 y Fr(w)1499 1401 y Fm(i)1527 1389 y FB(.)g(Then)813 1558 y([)p Fr(w)d Fq(\024)e Fr(S)5 b FB(])92 b(=)267 b Fg(;)1336 1645 y Fm(J)5 b Ff(\022)p Fm(I)33 b Fo(:)27 b(\006)p Fm(J)5 b Ff(\025)p Fm(w)1743 1491 y Fi(\000)1829 1558 y Fg(,)1795 1645 y Fm(i)p Ff(2)p Fm(J)1919 1558 y Fr(c)1955 1570 y Fm(i)1983 1491 y Fi(\001)1170 1848 y Fq($)269 b Fg(;)1336 1935 y Fm(J)5 b Ff(\022)p Fm(I)1464 1918 y Fb(0)1514 1935 y Fo(:)27 b(\006)p Fm(J)5 b Ff(\025)p Fm(w)1765 1781 y Fi(\000)1851 1848 y Fg(,)1817 1935 y Fm(i)p Ff(2)p Fm(J)1942 1848 y Fr(c)1978 1860 y Fm(i)2005 1781 y Fi(\001)2043 1848 y FB(;)268 b Fg(;)2080 1935 y Fm(J)5 b Ff(\022)p Fm(I)33 b Fo(:)27 b Fm(m)p Ff(2)p Fm(J)n(;)p Fo(\006)p Fm(J)5 b Ff(\025)p Fm(w)2646 1781 y Fi(\000)2731 1848 y Fg(,)2698 1935 y Fm(i)p Ff(2)p Fm(J)2822 1848 y Fr(c)2858 1860 y Fm(i)2885 1781 y Fi(\001)1179 2138 y FB(=)92 b([)p Fr(w)26 b Fq(\024)c Fr(S)1587 2104 y Ff(0)1610 2138 y FB(];)278 b Fg(;)1670 2225 y Fm(J)5 b Ff(\022)p Fm(I)1798 2209 y Fb(0)1848 2225 y Fo(:)28 b(\006)p Fm(J)5 b Fo(+)p Fm(w)2083 2233 y Fc(m)2139 2225 y Ff(\025)p Fm(w)2254 2071 y Fi(\000)2425 2138 y Fg(,)2306 2229 y Fm(i)p Ff(2)p Fm(J)g Ff([f)p Fm(m)p Ff(g)2602 2138 y Fr(c)2638 2150 y Fm(i)2665 2071 y Fi(\001)1170 2435 y Fq($)83 b FB([)p Fr(w)26 b Fq(\024)c Fr(S)1587 2401 y Ff(0)1610 2435 y FB(];)1670 2368 y Fi(\000)1708 2435 y Fr(c)1744 2447 y Fm(m)1807 2435 y Fr(;)278 b Fg(;)1844 2522 y Fm(J)5 b Ff(\022)p Fm(I)1972 2506 y Fb(0)2022 2522 y Fo(:)28 b(\006)p Fm(J)5 b Ff(\025)p Fm(w)r Ff(\000)p Fm(w)2360 2530 y Fc(m)2429 2368 y Fi(\000)2514 2435 y Fg(,)2481 2522 y Fm(i)p Ff(2)p Fm(J)2605 2435 y Fr(c)2641 2447 y Fm(i)2668 2368 y Fi(\001)q(\001)1179 2725 y FB(=)92 b([)p Fr(w)26 b Fq(\024)c Fr(S)1587 2691 y Ff(0)1610 2725 y FB(];)14 b(\()p Fr(c)1738 2737 y Fm(m)1801 2725 y Fr(;)g FB([\()p Fr(w)22 b Fq(\000)c Fr(w)2116 2737 y Fm(m)2179 2725 y FB(\))24 b Fq(\024)e Fr(S)2378 2691 y Ff(0)2401 2725 y FB(]\))q Fr(:)374 2987 y FB(The)27 b(pro)r(of)g(of)h(the)g(second)f(equiv)-5 b(alence)25 b(is)h(similar.)p 2164 2933 61 4 v 2164 2987 4 54 v 2221 2987 V 2164 2990 61 4 v 374 3218 a Fs(L)l(emma)k(12)374 3328 y FB(Program)1431 3463 y Fq(f)p Fr(q)25 b Fq($)e Fs(Def)18 b FB(\()p Fr(q)s FB(\))47 b(:)f Fr(q)26 b Fq(2)e Fr(Q)2213 3475 y Fo(\012)2264 3463 y Fq(g)909 b FB(\(47\))374 3598 y(is)26 b(strongly)f(equiv)-5 b(alen)n(t)26 b(to)1239 3754 y Fq(f)p Fr(q)1318 3766 y Fd(not)33 b Fm(l)1496 3754 y Fq($)23 b Fs(not)35 b Fr(l)52 b FB(:)f Fr(q)1942 3766 y Fd(not)34 b Fm(l)2120 3754 y Fq(2)24 b Fr(Q)2265 3766 y Fo(\012)2316 3754 y Fq(g[)1239 3862 y(f)p Fr(q)1318 3874 y Fm(w)r Ff(\024)p Fm(S)1490 3862 y Fq($)f FB([)p Fr(w)j Fq(\024)d Fr(S)5 b FB(])50 b(:)h Fr(q)2032 3874 y Fm(w)r Ff(\024)p Fm(S)2205 3862 y Fq(2)23 b Fr(Q)2349 3874 y Fo(\012)2400 3862 y Fq(g[)1239 3970 y(f)p Fr(q)1318 3982 y Fm(w)r( )c(w)765 967 y Fo(1)811 955 y FB(+)8 b Fq(\001)14 b(\001)g(\001)7 b FB(+)h Fr(w)1120 967 y Fm(m)1205 955 y FB(then,)22 b(b)n(y)g(the)h(de\014nition)e(of)h Fs(Def)c FB(\()p Fr(q)2325 967 y Fm(w)r Ff(\024)p Fm(S)2475 955 y FB(\))k(and)g(b)n(y)g (Lemma)e(11,)i(\(49\))374 1063 y(is)32 b(pro)n(v)-5 b(able)32 b(in)h(the)h(logic)d(of)j(here-and-there.)e(Assume)h(that)h(0)f Fr(<)g(w)j Fq(\024)d Fr(w)2863 1075 y Fo(1)2923 1063 y FB(+)22 b Fq(\001)14 b(\001)g(\001)23 b FB(+)f Fr(w)3276 1075 y Fm(m)3339 1063 y FB(.)374 1171 y(Then)35 b Fr(q)635 1183 y Fm(w)r Ff(\024)p Fm(S)781 1166 y Fb(0)842 1171 y FB(and)g Fr(q)1048 1183 y Fm(w)r Ff(\000)p Fm(w)1198 1191 y Fc(m)1252 1183 y Ff(\024)p Fm(S)1348 1166 y Fb(0)1410 1171 y FB(b)r(elong)e(to)i Fr(Q)1857 1183 y Fo(\012)1908 1171 y FB(,)g(and,)g(b)n(y)g(the)g(induction)f(h)n(yp)r(othesis,)f(the) 374 1279 y(equiv)-5 b(alences)1446 1411 y Fs(Def)18 b FB(\()p Fr(q)1659 1423 y Fm(w)r Ff(\024)p Fm(S)1805 1407 y Fb(0)1831 1411 y FB(\))24 b Fq($)f FB([)p Fr(w)i Fq(\024)e Fr(S)2244 1377 y Ff(0)2267 1411 y FB(])374 1544 y(and)1256 1677 y Fs(Def)18 b FB(\()p Fr(q)1469 1689 y Fm(w)r Ff(\000)p Fm(w)1619 1697 y Fc(m)1674 1689 y Ff(\024)p Fm(S)1770 1673 y Fb(0)1797 1677 y FB(\))23 b Fq($)g FB([)p Fr(w)e Fq(\000)d Fr(w)2203 1689 y Fm(m)2290 1677 y Fq(\024)23 b Fr(S)2434 1643 y Ff(0)2457 1677 y FB(])374 1810 y(are)j(deriv)-5 b(able)25 b(from)i(\(47\).)g(Consequen)n(tly)-7 b(,)26 b(the)i(equiv)-5 b(alences)1550 1964 y Fr(q)1587 1976 y Fm(w)r Ff(\024)p Fm(S)1733 1959 y Fb(0)1782 1964 y Fq($)23 b FB([)p Fr(w)j Fq(\024)d Fr(S)2140 1929 y Ff(0)2163 1964 y FB(])374 2117 y(and)1361 2250 y Fr(q)1398 2262 y Fm(w)r Ff(\000)p Fm(w)1548 2270 y Fc(m)1602 2262 y Ff(\024)p Fm(S)1698 2246 y Fb(0)1748 2250 y Fq($)g FB([)p Fr(w)e Fq(\000)d Fr(w)2099 2262 y Fm(m)2186 2250 y Fq(\024)k Fr(S)2329 2216 y Ff(0)2352 2250 y FB(])374 2383 y(are)j(deriv)-5 b(able)24 b(from)h(\(47\))h(as)g(w)n(ell.)e(By)i(Lemma)f(11,)g(this)h (implies)d(the)j(deriv)-5 b(abilit)n(y)22 b(of)27 b(\(49\).)p 453 2437 61 4 v 453 2491 4 54 v 511 2491 V 453 2494 61 4 v 374 2657 a Fs(The)l(or)l(em)j(2)374 2757 y FB(F)-7 b(or)28 b(an)n(y)h(program)e(\012)i(with)g(w)n(eigh)n(t)f(constrain)n (ts,)f Fr(Z)32 b Fq(7!)26 b Fr(Z)g Fq(n)19 b Fr(Q)2419 2769 y Fo(\012)2500 2757 y FB(is)28 b(a)h(1{1)f(corresp)r(ondence)374 2865 y(b)r(et)n(w)n(een)f(the)h(answ)n(er)e(sets)i(for)f([\012])1508 2835 y Fm(nn)1622 2865 y FB(and)g(the)h(answ)n(er)e(sets)i(for)f(\012.) 374 3031 y Fs(Pr)l(o)l(of)374 3131 y FB(F)-7 b(rom)27 b(Lemmas)f(10)i(and)g(12)g(w)n(e)h(see)f(that)h([\012])1855 3101 y Fm(nn)1970 3131 y FB(has)f(the)h(same)f(answ)n(er)f(sets)h(as)g (the)h(union)374 3239 y(of)e(\005)g(and)g(\(48\).)g(F)-7 b(urthermore,)25 b(this)h(union)g(is)g(strongly)f(equiv)-5 b(alen)n(t)25 b(to)i(the)g(union)g(of)g([\012])3283 3209 y Fm(nd)374 3347 y FB(and)g(\(48\).)g(Indeed,)h(\005)g(consists)e(of)i (the)g(rules)1218 3497 y Fr(l)c Fq( )f Fs(not)36 b Fr(q)1563 3509 y Fd(not)d Fm(l)1718 3497 y Fr(;)14 b FB([)p Fr(C)1837 3509 y Fo(1)1874 3497 y FB(])1897 3467 y Fm(nn)1984 3497 y Fr(;)g(:)g(:)g(:)f(;)h FB([)p Fr(C)2250 3509 y Fm(n)2296 3497 y FB(])2319 3467 y Fm(nn)2405 3497 y Fr(;)1218 3605 y Fq(?)22 b( )h Fs(not)36 b Fr(q)1601 3617 y Fm(L)1647 3625 y Fa(0)1679 3617 y Ff(\024)p Fm(S)1772 3625 y Fa(0)1808 3605 y Fr(;)14 b FB([)p Fr(C)1927 3617 y Fo(1)1964 3605 y FB(])1987 3575 y Fm(nn)2074 3605 y Fr(;)g(:)g(:)g(:)f(;)h FB([)p Fr(C)2340 3617 y Fm(n)2386 3605 y FB(])2409 3575 y Fm(nn)2495 3605 y Fr(;)1218 3713 y Fq(?)22 b( )h Fr(q)1448 3725 y Fm(U)1493 3733 y Fa(0)1526 3725 y Fm(
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