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%%EndProlog
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3665 y(ing)g(for)f(function)i(in)n(v)n(ersion)d(and)i(searc)n(h)e(for)h
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3765 y(in)n(tegrated)33 b(languages)e(is)j(usually)f(based)g(on)h
(narro)n(wing,)d(a)j(com)n(bination)e(of)i(v)-5 b(ariable)523
3864 y(instan)n(tiation)25 b(and)h(reduction.)f(The)h(instan)n(tiation)
f(of)h(v)-5 b(ariables)24 b(is)h(often)h(computed)g(b)n(y)523
3964 y(unifying)32 b(a)f(subterm)h(of)f(the)h(goal)e(expression)g(with)
i(the)g(left-hand)g(side)g(of)f(some)g(pro-)523 4063
y(gram)24 b(rule;)h(then)h(narro)n(wing)d(reduces)i(the)h(instan)n
(tiated)f(goal)f(using)h(that)h(rule.)f(Needed)523 4163
y(narro)n(wing)20 b(is)i(curren)n(tly)f(the)i(b)r(est)g(narro)n(wing)c
(strategy)i(for)h(\014rst-order,)e(lazy)i(functional)523
4263 y(logic)30 b(programs)f(due)i(to)g(its)h(optimalit)n(y)e(prop)r
(erties)g([5].)h(Needed)h(narro)n(wing)c(pro)n(vides)523
4362 y(completeness)22 b(in)i(the)f(sense)g(of)g(logic)f(programming)f
(\(computation)h(of)h(all)g(solutions\))g(as)523 4462
y(w)n(ell)f(as)f(functional)i(programming)d(\(computation)i(of)g(v)-5
b(alues\),)22 b(and)g(it)h(can)e(b)r(e)i(e\016cien)n(tly)523
4562 y(implemen)n(ted)28 b(b)n(y)g(pattern)f(matc)n(hing)g(and)g
(uni\014cation.)648 4662 y(The)g(fold/unfold)h(transformation)d
(approac)n(h)h(w)n(as)g(\014rst)i(in)n(tro)r(duced)f(in)h([8])f(to)g
(opti-)523 4762 y(mize)h(functional)h(programs)d(and)i(then)h(used)f
(for)g(logic)f(programs)f([28].)i(This)g(approac)n(h)p
523 4839 473 4 v 538 4893 a Fr(?)600 4924 y Fx(This)21
b(w)n(ork)g(has)g(b)r(een)g(partially)h(supp)r(orted)e(b)n(y)g(CICYT)h
(under)f(gran)n(t)h(TIC)g(98-0445-C03-01.)p eop
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2 1 bop 523 448 a FA(is)26 b(commonly)f(based)h(on)g(the)g
(construction,)f(b)n(y)h(means)f(of)h(a)g Fq(str)l(ate)l(gy)p
FA(,)g(of)g(a)g(sequence)f(of)523 548 y(equiv)-5 b(alen)n(t)32
b(programs)e(eac)n(h)i(obtained)g(b)n(y)h(the)f(preceding)g(ones)g
(using)g(an)g Fq(elementary)523 648 y FA(transformation)c(rule.)g(The)i
(essen)n(tial)e(rules)g(are)h Fq(folding)38 b FA(and)29
b Fq(unfolding)p FA(,)h(i.e.,)g(con)n(trac-)523 747 y(tion)g(and)f
(expansion)f(of)i(sub)r(expressions)e(of)h(a)g(program)f(using)h(the)h
(de\014nitions)f(of)h(this)523 847 y(program)h(\(or)i(of)g(a)f
(preceding)g(one\).)h(Other)g(rules)g(whic)n(h)g(ha)n(v)n(e)e(b)r(een)j
(considered)e(are,)523 946 y(for)26 b(example,)f(instan)n(tiation,)h
(de\014nition)g(in)n(tro)r(duction/elimination,)f(and)h(abstraction.)
648 1046 y(There)33 b(exists)g(a)g(large)g(class)g(of)g(program)f
(optimizations)h(whic)n(h)h(can)f(b)r(e)h(ac)n(hiev)n(ed)523
1146 y(b)n(y)29 b(fold/unfold)g(transformations)f(and)h(are)f(not)i(p)r
(ossible)f(b)n(y)g(using)g(a)g(fully)h(automatic)523
1245 y(metho)r(d)c(\(suc)n(h)g(as,)f(e.g.,)g(partial)g(ev)-5
b(aluation\).)26 b(T)n(ypical)e(instances)i(of)f(this)h(class)f(are)g
(the)523 1345 y(strategies)37 b(that)i(p)r(erform)f Fq(tupling)46
b FA(\(also)37 b(kno)n(wn)h(as)g Fq(p)l(airing)7 b FA(\))41
b([8,)13 b(11],)38 b(whic)n(h)g(merges)523 1445 y(separate)27
b(\(nonnested\))j(function)f(calls)f(with)i(some)e(common)g(argumen)n
(ts)g(in)n(to)g(a)h(single)523 1544 y(call)i(to)h(a)f(\(p)r(ossibly)h
(new\))g(recursiv)n(e)e(function)i(whic)n(h)f(returns)g(a)h(tuple)g(of)
f(the)i(results)523 1644 y(of)j(the)h(separate)e(calls,)h(th)n(us)h(a)n
(v)n(oiding)d(either)j(m)n(ultiple)g(accesses)d(to)j(the)g(same)e(data)
523 1743 y(structures)30 b(or)f(common)h(sub)r(computations,)h
(similarly)e(to)i(the)f(idea)h(of)f Fq(sharing)39 b FA(whic)n(h)523
1843 y(is)25 b(used)h(in)f(graph)g(rewriting)f(to)h(impro)n(v)n(e)f
(the)i(e\016ciency)f(of)h(computations)f(in)g(time)h(and)523
1943 y(space)32 b([6].)h(A)g(closely)e(related)i(strategy)e(is)i
Fq(c)l(omp)l(osition)40 b FA([30])32 b(\(also)g(kno)n(wn)g(as)g
Fq(fusion)p FA(,)523 2042 y Fq(defor)l(estation)p FA(,)g(or)d
Fq(vertic)l(al)34 b(jamming)k FA([12]\),)30 b(whic)n(h)g(essen)n
(tially)g(consists)f(of)h(the)h(merg-)523 2142 y(ing)26
b(of)h(nested)g(function)g(calls,)f(where)g(the)h(inner)g(function)g
(call)f(builds)h(up)g(a)f(comp)r(osite)523 2242 y(ob)5
b(ject)31 b(whic)n(h)g(is)h(used)f(b)n(y)g(the)h(outer)e(call,)h(and)g
(comp)r(osing)g(these)g(t)n(w)n(o)g(calls)f(in)n(to)h(one)523
2341 y(has)i(the)h(e\013ect)g(to)f(a)n(v)n(oid)f(the)i(generation)e(of)
i(the)g(in)n(termediate)f(data)g(structure.)g(The)523
2441 y(comp)r(osition)26 b(can)g(b)r(e)g(made)h(automatically)e([30],)g
(whereas)g(tupling)i(has)f(only)g(b)r(een)h(au-)523 2540
y(tomated)h(to)f(some)g(exten)n(t)h([9,)13 b(10].)648
2640 y(Although)31 b(a)g(lot)h(of)f(literature)g(has)g(b)r(een)h(dev)n
(oted)f(to)g(pro)n(ving)f(the)i(correctness)e(of)523
2740 y(fold/unfold)c(systems)g(w.r.t.)g(the)h(v)-5 b(arious)25
b(seman)n(tics)h(prop)r(osed)f(for)h(logic)f(programs)f([7,)523
2839 y(13,)13 b(20,)g(21,)g(23,)g(28],)28 b(in)i(functional)f
(programming)e(the)j(problem)f(of)g(correctness)e(has)i(re-)523
2939 y(ceiv)n(ed)24 b(surprisingly)e(little)j(atten)n(tion)g([26,)12
b(27].)24 b(Of)g(the)h(v)n(ery)e(few)h(studies)h(of)f(correctness)523
3039 y(of)g(fold/unfold)g(transformations)e(in)i(functional)g
(programming,)e(the)i(most)g(general)e(and)523 3138 y(recen)n(t)32
b(w)n(ork)f(is)h([26],)g(whic)n(h)g(de\014nes)h(a)f(simple)g(\(syn)n
(tactic\))h(condition)f(for)g(restricting)523 3238 y(general)20
b(fold/unfold)h(transformations)f(and)h(whic)n(h)h(can)f(b)r(e)h
(applied)f(to)g(giv)n(e)g(correctness)523 3337 y(pro)r(ofs)h(for)f(sev)
n(eral)g(w)n(ell-kno)n(wn)g(transformation)g(metho)r(ds,)h(suc)n(h)h
(as)e(the)i(deforestation.)648 3437 y(In)35 b([2],)g(w)n(e)g(in)n(v)n
(estigated)f(fold/unfold)h(rules)g(in)g(the)h(con)n(text)f(of)g(a)g
(strict)g(\()p Fq(c)l(al)t(l-by-)523 3537 y(value)6 b
FA(\))28 b(functional)f(logic)f(language)g(based)g(on)h(unrestricted)f
(\(i.e.,)i(not)f(optimized\))g(nar-)523 3636 y(ro)n(wing.)36
b(The)h(use)h(of)f(narro)n(wing)e(emp)r(o)n(w)n(ers)h(the)i
(fold/unfold)f(system)g(b)n(y)g(implicitly)523 3736 y(em)n(b)r(edding)
29 b(the)g(instan)n(tiation)f(rule)h(\(the)g(op)r(eration)f(of)g(the)h
(Burstall)f(and)h(Darlington)523 3836 y(framew)n(ork)23
b([8])h(whic)n(h)h(in)n(tro)r(duces)f(an)g(instance)g(of)h(an)f
(existing)g(equation\))h(in)n(to)f(unfold-)523 3935 y(ing)31
b(b)n(y)h(means)f(of)g(uni\014cation.)h(Ho)n(w)n(ev)n(er,)d([2])i(do)r
(es)h(not)f(consider)f(a)i(general)e(transfor-)523 4035
y(mation)f(system)g(\(only)h(t)n(w)n(o)e(rules:)h(fold)h(and)f
(unfold\),)h(and)g(hence)f(the)h(comp)r(osition)f(or)523
4134 y(tupling)j(transformations)d(cannot)h(b)r(e)i(ac)n(hiev)n(ed.)e
(Also,)h([2])g(refers)f(to)h(a)f(notion)h(of)g(\\re-)523
4234 y(v)n(ersible")c(folding,)i(whic)n(h)g(is)f(strictly)h(w)n(eak)n
(er)e(than)i(the)g(one)f(whic)n(h)h(w)n(e)g(consider)e(here.)523
4334 y(On)f(the)h(other)f(hand,)g(the)h(use)f(of)g(unrestricted)g
(narro)n(wing)e(to)i(p)r(erform)g(unfolding)h(ma)n(y)523
4433 y(pro)r(duce)g(an)h(imp)r(ortan)n(t)f(increase)f(in)i(the)g(n)n
(um)n(b)r(er)f(of)h(program)d(rules.)648 4533 y(In)c(this)h(pap)r(er)g
(w)n(e)f(de\014ne)h(a)f(transformation)f(metho)r(dology)h(for)g(lazy)g
(\()p Fq(c)l(al)t(l-by-name)6 b FA(\))523 4633 y(functional)33
b(logic)f(programs.)f(On)i(the)g(theoretical)g(side,)g(w)n(e)f(extend)i
(the)f(T)-7 b(amaki)32 b(and)523 4732 y(Sato)h(transformation)f(rules)i
([28])f(for)g(logic)g(programs)e(to)j(cop)r(e)f(with)i(lazy)e
(functional)523 4832 y(logic)39 b(programs)e(based)i(on)h(needed)f
(narro)n(wing.)f(The)h(transformation)f(pro)r(cess)h(con-)p
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3 2 bop 523 448 a FA(sists)33 b(of)h(applying)f(an)h(arbitrary)e(n)n
(um)n(b)r(er)h(of)h(times)g(the)g(basic)f(transformation)g(rules,)523
548 y(whic)n(h)24 b(are:)e(de\014nition)i(in)n(tro)r(duction,)g
(de\014nition)g(elimination,)g(unfolding,)g(folding,)f(and)523
648 y(abstraction.)31 b(Needed)h(narro)n(wing)e(is)i(complete)g(for)g
Fq(inductively)j(se)l(quential)41 b FA(programs)523 747
y([4].)32 b(Th)n(us,)f(w)n(e)h(demonstrate)f(that)h(suc)n(h)g(a)f
(program)f(structure)i(is)f(preserv)n(ed)g(through)523
847 y(the)e(transformation)d(sequence)i(\()p Fp(R)1680
859 y Fz(0)1718 847 y Fo(;)14 b(:)g(:)g(:)f(;)h Fp(R)1972
859 y Fn(n)2018 847 y FA(\),)28 b(whic)n(h)h(is)f(a)f(k)n(ey)h(p)r(oin)
n(t)g(for)g(pro)n(ving)f(the)523 946 y(correctness)36
b(of)i(the)h(transformation)d(system)i(as)g(w)n(ell)g(as)f(its)h
(e\013ectiv)n(e)g(applicabilit)n(y)-7 b(.)523 1046 y(F)g(or)37
b(instance,)g(b)n(y)g(using)f(other)h(v)-5 b(arian)n(ts)36
b(of)h(narro)n(wing)e(\(e.g.,)i(lazy)g(narro)n(wing)d([22]\))523
1146 y(the)28 b(structure)e(of)i(the)f(original)f(program)f(is)i(not)g
(preserv)n(ed,)f(th)n(us)h(seriously)f(restricting)523
1245 y(the)k(applicabilit)n(y)f(of)g(the)h(resulting)f(system.)g(The)h
(ma)5 b(jor)28 b(tec)n(hnical)h(result)g(consists)g(of)523
1345 y(pro)n(ving)20 b(strong)g(correctness)f(for)i(the)h
(transformation)d(system,)i(namely)g(that)h(the)f(v)-5
b(alues)523 1445 y(and)25 b(answ)n(ers)f(computed)i(b)n(y)f(needed)g
(narro)n(wing)e(in)j(the)g(initial)g(and)f(the)h(\014nal)f(program)523
1544 y(coincide)33 b(\(for)g(goals)f(constructed)h(using)g(the)g(sym)n
(b)r(ols)g(of)g(the)h(initial)f(program\).)f(The)523
1644 y(e\016ciency)c(impro)n(v)n(emen)n(t)g(of)g Fp(R)1546
1656 y Fn(n)1620 1644 y FA(with)h(regard)e(to)h Fp(R)2242
1656 y Fz(0)2308 1644 y FA(is)g(not)h(ensured)f(b)n(y)g(an)g(arbitrary)
523 1743 y(use)22 b(of)h(the)f(elemen)n(tary)g(transformation)f(rules)g
(but)i(it)g(rather)e(dep)r(ends)i(on)f(the)h(heuristic)523
1843 y(whic)n(h)32 b(is)g(emplo)n(y)n(ed.)f(On)h(the)h(practical)e
(side,)h(w)n(e)g(in)n(v)n(estigate)e(ho)n(w)i(the)g(classical)f(and)523
1943 y(p)r(o)n(w)n(erful)c(transformation)f(metho)r(dologies)h(of)h
Fq(tupling)35 b FA(and)28 b Fq(c)l(omp)l(osition)36 b
FA([24])27 b(transfer)523 2042 y(to)c(our)f(framew)n(ork.)f(W)-7
b(e)23 b(sho)n(w)f(the)i(adv)-5 b(an)n(tages)21 b(of)h(using)h(needed)g
(narro)n(wing)e(to)h(ac)n(hiev)n(e)523 2142 y(comp)r(osition)h(and)h
(tupling)g(in)g(an)g(in)n(tegrated)f(setting,)h(and)g(illustrate)f(the)
h(p)r(o)n(w)n(er)f(of)h(our)523 2242 y(transformation)f(system)i(b)n(y)
g(\(automatically\))g(optimizing)g(sev)n(eral)e(signi\014cativ)n(e)h
(exam-)523 2341 y(ples)k(using)f(a)g(protot)n(yp)r(e)g(implemen)n
(tation)g([1].)648 2441 y(The)22 b(structure)g(of)h(the)g(pap)r(er)f
(is)g(as)g(follo)n(ws.)g(After)h(recalling)e(some)h(basic)g
(de\014nitions)523 2540 y(in)34 b(the)f(next)h(section,)f(w)n(e)g(in)n
(tro)r(duce)g(the)g(basic)g(transformation)f(rules)g(and)h(illustrate)
523 2640 y(them)21 b(b)n(y)g(means)f(of)g(sev)n(eral)f(simple)i
(examples)f(in)h(Sec.)g(3.)f(W)-7 b(e)21 b(also)e(state)i(the)g
(correctness)523 2740 y(of)g(the)g(transformation)e(system)i(and)g(sho)
n(w)f(some)g(results)h(ab)r(out)f(the)i(structure)e(of)h(trans-)523
2839 y(formed)29 b(programs.)e(Section)i(4)g(sho)n(ws)g(ho)n(w)f(to)i
(ac)n(hiev)n(e)e(the)h(\(automatic\))h(comp)r(osition)523
2939 y(and)j(tupling)g(strategies)e(in)i(our)f(framew)n(ork)f(as)h(w)n
(ell)g(as)g(an)g(exp)r(erimen)n(tal)h(ev)-5 b(aluation)523
3039 y(of)30 b(the)g(metho)r(d)g(on)f(a)g(small)h(set)f(of)h(b)r(enc)n
(hmarks.)f(Section)g(5)g(concludes.)h(More)e(details)523
3138 y(and)f(pro)r(ofs)g(of)h(all)f(tec)n(hnical)g(results)g(can)h(b)r
(e)g(found)g(in)f([3].)523 3417 y Fs(2)112 b(Preliminaries)523
3629 y FA(W)-7 b(e)35 b(assume)e(familiarit)n(y)g(with)i(basic)f
(notions)f(of)i(term)f(rewriting)f([6])h(and)g(functional)523
3729 y(logic)e(programming)e([14].)h(W)-7 b(e)33 b(consider)e(a)h
Fq(signatur)l(e)39 b Fo(\006)e FA(partitioned)32 b(in)n(to)g(a)g(set)g
Fp(C)37 b FA(of)523 3828 y Fq(c)l(onstructors)c FA(and)26
b(a)g(set)h Fp(F)34 b FA(of)26 b(\(de\014ned\))i Fq(functions)33
b FA(or)26 b Fq(op)l(er)l(ations)p FA(.)h(W)-7 b(e)27
b(write)g Fo(c=n)22 b Fp(2)h(C)523 3928 y FA(and)g Fo(f)9
b(=n)22 b Fp(2)h(F)32 b FA(for)22 b Fo(n)p FA(-ary)g(constructor)f(and)
i(op)r(eration)g(sym)n(b)r(ols,)f(resp)r(ectiv)n(ely)-7
b(.)22 b(There)h(is)523 4028 y(at)g(least)g(one)g(sort)f
Fo(B)t(ool)j FA(con)n(taining)e(the)g(0-ary)f(Bo)r(olean)g
(constructors)f Fo(tr)r(ue)i FA(and)g Fo(f)9 b(al)r(se)p
FA(.)523 4127 y(The)27 b(set)g(of)h Fq(terms)33 b FA(and)27
b Fq(c)l(onstructor)i(terms)34 b FA(with)27 b Fq(variables)37
b FA(\(e.g.,)27 b Fo(x;)14 b(y)s(;)g(z)t FA(\))27 b(from)f
Fp(X)40 b FA(are)523 4227 y(denoted)23 b(b)n(y)f Fp(T)f
FA(\()p Fp(C)13 b([)8 b(F)g Fo(;)14 b Fp(X)e FA(\))25
b(and)d Fp(T)f FA(\()p Fp(C)5 b Fo(;)14 b Fp(X)e FA(\),)23
b(resp)r(ectiv)n(ely)-7 b(.)22 b(The)h(set)g(of)f(v)-5
b(ariables)21 b(o)r(ccurring)523 4327 y(in)26 b(a)f(term)g
Fo(t)h FA(is)f(denoted)g(b)n(y)g Fp(V)7 b Fo(ar)r FA(\()p
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FA(if)26 b(it)g(do)r(es)f(not)h(con)n(tain)e(m)n(ultiple)523
4426 y(o)r(ccurrences)i(of)i(an)n(y)e(v)-5 b(ariable.)27
b(W)-7 b(e)28 b(write)p 1910 4381 86 4 v 27 w Fo(o)1950
4438 y Fn(n)2024 4426 y FA(for)f(the)h Fq(list)35 b FA(of)28
b(ob)5 b(jects)27 b Fo(o)2856 4438 y Fz(1)2894 4426 y
Fo(;)14 b(:)g(:)g(:)f(;)h(o)3118 4438 y Fn(n)3163 4426
y FA(.)648 4526 y(A)39 b Fq(p)l(attern)45 b FA(is)38
b(a)h(term)g(of)f(the)i(form)e Fo(f)9 b FA(\()p 1984
4458 89 4 v Fo(d)2027 4538 y Fn(n)2072 4526 y FA(\))40
b(where)e Fo(f)9 b(=n)41 b Fp(2)h(F)47 b FA(and)38 b
Fo(d)2997 4538 y Fz(1)3035 4526 y Fo(;)14 b(:)g(:)g(:)f(;)h(d)3262
4538 y Fn(n)3350 4526 y Fp(2)523 4625 y(T)21 b FA(\()p
Fp(C)5 b Fo(;)14 b Fp(X)e FA(\).)28 b(Note)g(the)f(di\013erence)g(with)
h(the)g(usual)f(notion)g(of)g(pattern)g(in)h(functional)f(pro-)523
4725 y(gramming:)i(a)h(constructor)g(term.)g(A)h(term)g(is)f
Fq(op)l(er)l(ation-r)l(o)l(ote)l(d)40 b FA(\()p Fq(c)l(onstructor-r)l
(o)l(ote)l(d)9 b FA(\))523 4825 y(if)27 b(it)g(has)f(an)h(op)r(eration)
e(\(constructor\))h(sym)n(b)r(ol)g(at)h(the)g(ro)r(ot.)f(A)g
Fq(p)l(osition)35 b Fo(p)26 b FA(in)h(a)f(term)h Fo(t)523
4924 y FA(is)d(represen)n(ted)f(b)n(y)h(a)g(sequence)f(of)h(natural)g
(n)n(um)n(b)r(ers)f(\()p Fm(\003)i FA(denotes)f(the)g(empt)n(y)g
(sequence,)p eop
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4 3 bop 523 448 a FA(i.e.,)25 b(the)h(ro)r(ot)e(p)r(osition\).)i(P)n
(ositions)d(are)h(ordered)g(b)n(y)h(the)g Fq(pr)l(e\014x)35
b FA(ordering:)24 b Fo(p)f Fp(\024)f Fo(q)s FA(,)j(if)h
Fp(9)p Fo(w)523 548 y FA(suc)n(h)h(that)g Fo(p:w)e FA(=)e
Fo(q)s FA(.)k(P)n(ositions)e Fo(p;)14 b(q)30 b FA(are)c
Fq(disjoint)36 b FA(if)28 b(neither)f Fo(p)22 b Fp(\024)h
Fo(q)30 b FA(nor)c Fo(q)g Fp(\024)d Fo(p)p FA(.)k(Giv)n(en)f(a)523
648 y(term)31 b Fo(t)p FA(,)f(w)n(e)g(let)h Fp(P)7 b
Fo(os)p FA(\()p Fo(t)p FA(\))31 b(and)g Fp(N)12 b(V)7
b(P)g Fo(os)p FA(\()p Fo(t)p FA(\))31 b(denote)f(the)h(set)g(of)f(p)r
(ositions)g(and)h(the)f(set)h(of)523 747 y(non-v)-5 b(ariable)25
b(p)r(ositions)i(of)g Fo(t)p FA(,)g(resp)r(ectiv)n(ely)-7
b(.)26 b Fo(t)p Fp(j)2039 759 y Fn(p)2104 747 y FA(denotes)h(the)g
Fq(subterm)33 b FA(of)26 b Fo(t)h FA(at)g(p)r(osition)523
847 y Fo(p)p FA(,)h(and)f Fo(t)p FA([)p Fo(s)p FA(])892
859 y Fn(p)958 847 y FA(denotes)g(the)h(result)g(of)f
Fq(r)l(eplacing)k(the)f(subterm)k Fo(t)p Fp(j)2591 859
y Fn(p)2657 847 y FA(b)n(y)27 b(the)h(term)g Fo(s)p FA(.)648
946 y(W)-7 b(e)28 b(denote)g(b)n(y)g Fp(f)p Fo(x)1264
958 y Fz(1)1325 946 y Fp(7!)c Fo(t)1462 958 y Fz(1)1500
946 y Fo(;)14 b(:)g(:)g(:)f(;)h(x)1731 958 y Fn(n)1801
946 y Fp(7!)24 b Fo(t)1938 958 y Fn(n)1983 946 y Fp(g)k
FA(the)g Fq(substitution)34 b Fo(\033)d FA(with)e Fo(\033)s
FA(\()p Fo(x)3046 958 y Fn(i)3075 946 y FA(\))24 b(=)g
Fo(t)3250 958 y Fn(i)3306 946 y FA(for)523 1046 y Fo(i)f
FA(=)f(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)22 b FA(\(with)i
Fo(x)1224 1058 y Fn(i)1275 1046 y Fp(6)p FA(=)e Fo(x)1409
1058 y Fn(j)1467 1046 y FA(if)h Fo(i)f Fp(6)p FA(=)h
Fo(j)5 b FA(\),)23 b(and)f Fo(\033)s FA(\()p Fo(x)p FA(\))i(=)f
Fo(x)g FA(for)e(all)h(other)g(v)-5 b(ariables)21 b Fo(x)p
FA(.)i(A)f(sub-)523 1146 y(stitution)k Fo(\033)j FA(is)c
Fq(\(gr)l(ound\))i(c)l(onstructor)p FA(,)e(if)h Fo(\033)s
FA(\()p Fo(x)p FA(\))h(is)e(\(ground\))g(constructor)f(for)g(all)i
Fo(x)f FA(suc)n(h)523 1245 y(that)k Fo(\033)s FA(\()p
Fo(x)p FA(\))d Fp(6)p FA(=)e Fo(x)p FA(.)29 b(The)f(iden)n(tit)n(y)h
(substitution)g(is)f(denoted)h(b)n(y)f Fo(id)p FA(.)g(Giv)n(en)g(a)g
(substitution)523 1345 y Fo(\022)e FA(and)e(a)g(set)g(of)g(v)-5
b(ariables)23 b Fo(V)42 b Fp(\022)22 b(X)12 b FA(,)25
b(w)n(e)e(denote)h(b)n(y)g Fo(\022)2199 1360 y Fl(j)-17
b Fz(\022)h Fn(V)2301 1345 y FA(the)24 b(substitution)h(obtained)e
(from)523 1445 y Fo(\022)k FA(b)n(y)e(restricting)f(its)h(domain)g(to)g
Fo(V)18 b FA(.)26 b(W)-7 b(e)25 b(write)g Fo(\022)g FA(=)e
Fo(\033)28 b FA([)p Fo(V)19 b FA(])25 b(if)h Fo(\022)2545
1460 y Fl(j)-17 b Fz(\022)g Fn(V)2645 1445 y FA(=)23
b Fo(\033)2780 1460 y Fl(j)-17 b Fz(\022)h Fn(V)2858
1445 y FA(,)25 b(and)f Fo(\022)i Fp(\024)c Fo(\033)29
b FA([)p Fo(V)19 b FA(])523 1544 y(denotes)31 b(the)h(existence)f(of)g
(a)g(substitution)h Fo(\015)k FA(suc)n(h)31 b(that)h
Fo(\015)26 b Fp(\016)20 b Fo(\022)32 b FA(=)d Fo(\033)35
b FA([)p Fo(V)19 b FA(].)31 b(A)h Fq(uni\014er)40 b FA(of)523
1644 y(t)n(w)n(o)25 b(terms)g Fo(s)h FA(and)f Fo(t)h
FA(is)f(a)h(substitution)g Fo(\033)j FA(with)d Fo(\033)s
FA(\()p Fo(s)p FA(\))e(=)f Fo(\033)s FA(\()p Fo(t)p FA(\).)j(Tw)n(o)f
(substitutions)h Fo(\033)j FA(and)523 1743 y Fo(\033)573
1713 y Fl(0)631 1743 y FA(are)k Fq(indep)l(endent)43
b FA(\(on)34 b(a)f(set)h(of)g(v)-5 b(ariables)33 b Fo(V)19
b FA(\))34 b(i\013)h(there)f(exists)f(some)h Fo(x)g Fp(2)g
Fo(V)53 b FA(suc)n(h)523 1843 y(that)28 b Fo(\033)s FA(\()p
Fo(x)p FA(\))h(and)e Fo(\033)1104 1813 y Fl(0)1128 1843
y FA(\()p Fo(x)p FA(\))i(are)d(not)i(uni\014able.)648
1943 y(A)i(set)g(of)g(rewrite)f(rules)h Fo(l)e Fp(!)f
Fo(r)33 b FA(suc)n(h)d(that)g Fo(l)f Fp(62)e(X)12 b FA(,)31
b(and)f Fp(V)7 b Fo(ar)r FA(\()p Fo(r)r FA(\))28 b Fp(\022)f(V)7
b Fo(ar)r FA(\()p Fo(l)r FA(\))30 b(is)g(called)523 2042
y(a)e Fq(term)j(r)l(ewriting)g(system)k FA(\(TRS\).)30
b(The)f(terms)f Fo(l)i FA(and)f Fo(r)j FA(are)27 b(called)i(the)g
Fq(left-hand)j(side)523 2142 y FA(\()p Fq(lhs)7 b FA(\))35
b(and)f(the)g Fq(right-hand)j(side)k FA(\()p Fq(rhs)7
b FA(\))35 b(of)e(the)i(rule,)e(resp)r(ectiv)n(ely)-7
b(.)33 b(A)h(TRS)h Fp(R)f FA(is)g(left-)523 2242 y(linear)23
b(if)g Fo(l)i FA(is)f(linear)e(for)h(all)g Fo(l)h Fp(!)g
Fo(r)h Fp(2)f(R)p FA(.)g(A)f(TRS)h(is)f(constructor)f(based)h(\(CB\))h
(if)f(eac)n(h)g(lhs)523 2341 y Fo(l)i FA(is)e(a)g(pattern.)g(In)g(the)h
(remainder)e(of)h(this)g(pap)r(er,)g(a)g(functional)g(logic)f
Fq(pr)l(o)l(gr)l(am)31 b FA(is)23 b(a)g(left-)523 2441
y(linear)h(CB-TRS.)g(Conditions)g(in)g(program)e(rules)i(are)f(treated)
h(b)n(y)g(using)g(the)h(prede\014ned)523 2540 y(functions)33
b Fk(and)o FA(,)f Fk(if)p 1165 2540 25 4 v 29 w(then)p
1370 2540 V 28 w(else)o FA(,)g Fk(case)p 1804 2540 V
29 w(of)f FA(whic)n(h)i(are)e(reduced)h(b)n(y)g(standard)g(de\014ning)
523 2640 y(rules)e([17,)13 b(22].)30 b(Tw)n(o)f(\(p)r(ossibly)i
(renamed\))f(rules)g Fo(l)f Fp(!)f Fo(r)33 b FA(and)e
Fo(l)2573 2610 y Fl(0)2623 2640 y Fp(!)d Fo(r)2773 2610
y Fl(0)2828 2640 y Fq(overlap)p FA(,)k(if)f(there)523
2740 y(is)d(a)f(non-v)-5 b(ariable)26 b(p)r(osition)h
Fo(p)c Fp(2)g(N)12 b(V)7 b(P)g Fo(os)p FA(\()p Fo(l)r
FA(\))28 b(and)f(a)g(most-general)f(uni\014er)h Fo(\033)k
FA(suc)n(h)d(that)523 2839 y Fo(\033)s FA(\()p Fo(l)r
Fp(j)655 2851 y Fn(p)694 2839 y FA(\))e(=)g Fo(\033)s
FA(\()p Fo(l)952 2809 y Fl(0)975 2839 y FA(\).)k(A)g(left-linear)f(TRS)
h(with)g(no)f(o)n(v)n(erlapping)e(rules)i(is)g(called)g
Fq(ortho)l(gonal)p FA(.)523 2939 y(A)37 b Fq(r)l(ewrite)h(step)43
b FA(is)36 b(an)h(application)f(of)g(a)g(rewrite)g(rule)h(to)f(a)g
(term,)h(i.e.,)g Fo(t)h Fp(!)3134 2951 y Fn(p;R)3281
2939 y Fo(s)e FA(if)523 3039 y(there)f(exists)f(a)h(p)r(osition)f
Fo(p)h FA(in)g Fo(t)p FA(,)g(a)f(rewrite)h(rule)f Fo(R)i
FA(=)f Fo(l)h Fp(!)f Fo(r)j FA(and)d(a)f(substitution)i
Fo(\033)523 3138 y FA(with)29 b Fo(t)p Fp(j)766 3150
y Fn(p)830 3138 y FA(=)c Fo(\033)s FA(\()p Fo(l)r FA(\))k(and)g
Fo(s)c FA(=)g Fo(t)p FA([)p Fo(\033)s FA(\()p Fo(r)r
FA(\)])1636 3150 y Fn(p)1705 3138 y FA(\()p Fo(p)30 b
FA(and)e Fo(R)i FA(will)f(often)g(b)r(e)h(omitted)f(in)g(the)h
(notation)523 3238 y(of)h(a)f(computation)h(step\).)g(The)g(instan)n
(tiated)g(lhs)g Fo(\033)s FA(\()p Fo(l)r FA(\))g(is)g(called)g(a)f
Fq(r)l(e)l(dex)p FA(.)h(A)g(term)g Fo(t)g FA(is)523 3337
y(called)24 b(a)f Fq(normal)k(form)k FA(if)25 b(there)f(is)g(no)f(term)
h Fo(s)g FA(with)h Fo(t)e Fp(!)g Fo(s)p FA(.)h Fp(!)2548
3307 y Fz(+)2627 3337 y FA(denotes)g(the)g(transitiv)n(e)523
3437 y(closure)i(of)i Fp(!)g FA(and)f Fp(!)1250 3407
y Fl(\003)1316 3437 y FA(denotes)g(the)h(re\015exiv)n(e)f(and)g
(transitiv)n(e)g(closure)f(of)i Fp(!)p FA(.)648 3537
y(T)-7 b(o)31 b(ev)-5 b(aluate)31 b(terms)h(con)n(taining)e(v)-5
b(ariables,)31 b(narro)n(wing)e(non-deterministically)i(in-)523
3636 y(stan)n(tiates)23 b(the)h(v)-5 b(ariables)23 b(suc)n(h)g(that)h
(a)g(rewrite)f(step)h(is)f(p)r(ossible.)h(F)-7 b(ormally)g(,)23
b Fo(t)g Fj(;)3160 3648 y Fn(p;R;\033)3352 3636 y Fo(t)3382
3606 y Fl(0)523 3736 y FA(is)37 b(a)f Fq(narr)l(owing)j(step)j
FA(if)37 b Fo(p)f FA(is)h(a)f(non-v)-5 b(ariable)35 b(p)r(osition)i(in)
g Fo(t)f FA(and)h Fo(\033)s FA(\()p Fo(t)p FA(\))i Fp(!)3030
3748 y Fn(p;R)3177 3736 y Fo(t)3207 3706 y Fl(0)3230
3736 y FA(.)e(W)-7 b(e)523 3836 y(denote)38 b(b)n(y)g
Fo(t)957 3848 y Fz(0)1035 3836 y Fj(;)1118 3805 y Fl(\003)1118
3856 y Fn(\033)1204 3836 y Fo(t)1234 3848 y Fn(n)1318
3836 y FA(a)g(sequence)f(of)i(narro)n(wing)d(steps)i
Fo(t)2508 3848 y Fz(0)2586 3836 y Fj(;)2669 3848 y Fn(\033)2707
3856 y Fi(1)2785 3836 y Fo(:)14 b(:)g(:)41 b Fj(;)3006
3848 y Fn(\033)3044 3856 y Fh(n)3130 3836 y Fo(t)3160
3848 y Fn(n)3243 3836 y FA(with)523 3935 y Fo(\033)47
b FA(=)d Fo(\033)773 3947 y Fn(n)846 3935 y Fp(\016)26
b(\001)14 b(\001)g(\001)27 b(\016)f Fo(\033)1153 3947
y Fz(1)1191 3935 y FA(.)40 b(Since)g(w)n(e)g(are)f(in)n(terested)h(in)h
(computing)f Fq(values)47 b FA(\(constructor)523 4035
y(terms\))33 b(as)g(w)n(ell)g(as)f Fq(answers)41 b FA
(\(substitutions\))34 b(in)g(functional)f(logic)f(programming,)f(w)n(e)
523 4134 y(sa)n(y)21 b(that)h(the)h(narro)n(wing)d(deriv)-5
b(ation)21 b Fo(t)i Fj(;)1878 4104 y Fl(\003)1878 4155
y Fn(\033)1946 4134 y Fo(c)f Fq(c)l(omputes)j(the)g(r)l(esult)f
Fo(c)g Fq(with)i(answer)f Fo(\033)g FA(if)e Fo(c)523
4234 y FA(is)i(a)f(constructor)f(term.)i(The)g(ev)-5
b(aluation)24 b(to)h(\(ground\))f(constructor)f(terms)i(\(and)f(not)h
(to)523 4334 y(arbitrary)c(expressions\))h(is)i(the)f(in)n(tended)h
(seman)n(tics)f(of)g(functional)g(languages)f(and)h(also)523
4433 y(of)29 b(most)g(functional)g(logic)f(languages.)f(In)i
(particular,)f(the)i(equalit)n(y)e(predicate)g Fp(\031)h
FA(used)523 4533 y(in)j(some)f(examples)g(is)h(de\014ned,)g(lik)n(e)g
(in)g(functional)g(languages,)e(as)h(the)h Fq(strict)h(e)l(quality)523
4633 y FA(on)e(terms,)f(i.e.,)h(the)h(equation)e Fo(t)1578
4645 y Fz(1)1644 4633 y Fp(\031)e Fo(t)1767 4645 y Fz(2)1835
4633 y FA(is)j(satis\014ed)f(if)h Fo(t)2353 4645 y Fz(1)2422
4633 y FA(and)f Fo(t)2616 4645 y Fz(2)2684 4633 y FA(are)g(reducible)h
(to)g(the)523 4732 y(same)25 b(ground)f(constructor)f(term.)i(W)-7
b(e)26 b(sa)n(y)e(that)h Fo(\033)k FA(is)c(a)g Fq(c)l(ompute)l(d)i
(answer)h(substitution)523 4832 y FA(for)f(an)g(equation)g
Fo(e)h FA(if)g(there)f(is)h(a)f(narro)n(wing)e(deriv)-5
b(ation)27 b Fo(e)c Fj(;)2537 4802 y Fl(\003)2537 4852
y Fn(\033)2605 4832 y Fo(tr)r(ue)p FA(.)p eop
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5 4 bop 523 448 a Fg(Needed)28 b(Narro)m(wing.)39 b FA(A)26
b(c)n(hallenge)e(in)i(the)g(design)e(of)i(functional)f(logic)g
(languages)e(is)523 548 y(the)i(de\014nition)f(of)g(a)g(\\go)r(o)r(d")e
(narro)n(wing)g(strategy)-7 b(,)23 b(i.e.,)i(a)e(restriction)g
Fo(\025)i FA(on)f(the)g(narro)n(w-)523 648 y(ing)34 b(steps)g(issuing)f
(from)h Fo(t)p FA(,)g(without)h(losing)e(completeness.)h
Fq(Ne)l(e)l(de)l(d)i(narr)l(owing)42 b FA([5])34 b(is)523
747 y(curren)n(tly)25 b(the)h(b)r(est)g(kno)n(wn)f(narro)n(wing)e
(strategy)i(due)h(to)f(its)h(optimalit)n(y)g(prop)r(erties.)f(It)523
847 y(extends)f(the)h(Huet)g(and)f(L)n(\023)-39 b(evy's)22
b(notion)i(of)g(a)g(needed)g(reduction)g([18].)f(The)h(de\014nition)h
(of)523 946 y(needed)30 b(narro)n(wing)d([5])i(uses)g(the)h(notion)g
(of)f Fq(de\014nitional)k(tr)l(e)l(e)i FA([4].)30 b(Roughly)f(sp)r
(eaking,)523 1046 y(a)24 b(de\014nitional)h(tree)f(for)g(a)g(function)h
(sym)n(b)r(ol)f Fo(f)33 b FA(is)25 b(a)f(tree)g(whose)g(lea)n(v)n(es)f
(con)n(tain)h(all)g(\(and)523 1146 y(only\))31 b(the)h(rules)f(used)g
(to)g(de\014ne)h Fo(f)40 b FA(and)31 b(whose)g(inner)g(no)r(des)g(con)n
(tain)f(information)h(to)523 1245 y(guide)23 b(the)h(\(optimal\))f
(pattern)h(matc)n(hing)e(during)h(the)h(ev)-5 b(aluation)22
b(of)i(expressions.)d(Eac)n(h)523 1345 y(inner)k(no)r(de)f(con)n(tains)
g(a)h(pattern)f(and)h(a)f(v)-5 b(ariable)24 b(p)r(osition)h(in)g(this)g
(pattern)f(\(the)i Fq(induc-)523 1445 y(tive)32 b(p)l(osition)6
b FA(\))32 b(whic)n(h)e(is)g(further)g(re\014ned)g(in)g(the)g(patterns)
g(of)g(its)g(immediate)g(c)n(hildren)523 1544 y(b)n(y)h(using)f
(di\013eren)n(t)h(constructor)e(sym)n(b)r(ols.)i(The)g(pattern)f(of)h
(the)g(ro)r(ot)f(no)r(de)h(is)g(simply)523 1644 y Fo(f)9
b FA(\()p 605 1598 93 4 v Fo(x)652 1656 y Fn(n)698 1644
y FA(\),)30 b(where)p 1025 1598 V 29 w Fo(x)1072 1656
y Fn(n)1147 1644 y FA(are)f(di\013eren)n(t)g(v)-5 b(ariables.)29
b(A)h(de\014ned)g(function)g(is)f(called)h Fq(inductively)523
1743 y(se)l(quential)40 b FA(if)31 b(it)g(has)g(a)f(de\014nitional)h
(tree.)f(A)i(rewrite)e(system)g Fp(R)h FA(is)g(called)g
Fq(inductively)523 1843 y(se)l(quential)36 b FA(if)29
b(all)e(its)h(de\014ned)g(functions)g(are)e(inductiv)n(ely)i(sequen)n
(tial.)648 1943 y(T)-7 b(o)21 b(compute)g(needed)h(narro)n(wing)d
(steps)i(for)g(an)g(op)r(eration-ro)r(oted)e(term)i Fo(t)p
FA(,)h(w)n(e)f(tak)n(e)f(a)523 2042 y(de\014nitional)j(tree)h
Fp(P)30 b FA(for)22 b(the)i(ro)r(ot)f(of)g Fo(t)h FA(and)f(compute)h
Fo(\025)p FA(\()p Fo(t;)14 b Fp(P)7 b FA(\).)24 b(Then,)f(for)g(all)g
(\()p Fo(p;)14 b(R)q(;)g(\033)s FA(\))24 b Fp(2)523 2142
y Fo(\025)p FA(\()p Fo(t;)14 b Fp(P)7 b FA(\),)41 b Fo(t)k
Fj(;)989 2154 y Fn(p;R;\033)1203 2142 y Fo(t)1233 2112
y Fl(0)1297 2142 y FA(is)40 b(a)g Fq(ne)l(e)l(de)l(d)j(narr)l(owing)f
(step)p FA(.)f(Informally)f(sp)r(eaking,)g(needed)523
2242 y(narro)n(wing)35 b(applies)j(a)f(rule,)h(if)g(p)r(ossible,)g(or)f
(c)n(hec)n(ks)f(the)i(subterm)g(corresp)r(onding)e(to)523
2341 y(the)41 b(inductiv)n(e)g(p)r(osition)f(of)h(the)f(branc)n(h:)g
(if)h(it)g(is)g(a)f(v)-5 b(ariable,)39 b(it)i(is)g(instan)n(tiated)f
(to)523 2441 y(the)c(constructor)e(of)i(a)f(c)n(hild;)g(if)i(it)f(is)f
(already)f(a)h(constructor,)f(w)n(e)i(pro)r(ceed)f(with)h(the)523
2540 y(corresp)r(onding)d(c)n(hild;)j(if)g(it)g(is)f(a)g(function,)h(w)
n(e)f(ev)-5 b(aluate)35 b(it)h(b)n(y)f(recursiv)n(ely)f(applying)523
2640 y(needed)28 b(narro)n(wing)d(\(see)i([5])h(for)f(a)g(detailed)h
(de\014nition\).)523 2814 y Fq(Example)j(1.)42 b FA(Consider)27
b(the)h(follo)n(wing)e(set)i(of)f(rules)g(for)h(\\)p
Ff(6)p FA(")e(and)h(\\+":)850 2966 y Fk(0)51 b Ff(6)f
Fk(N)186 b Fp(!)79 b Fk(true)316 b(0)18 b FA(+)g Fk(N)78
b Fp(!)g Fk(N)742 3066 y(s)p FA(\()p Fk(M)p FA(\))51
b Ff(6)f Fk(0)186 b Fp(!)79 b Fk(false)164 b(s)p FA(\()p
Fk(M)p FA(\))18 b(+)g Fk(N)78 b Fp(!)g Fk(s)p FA(\()p
Fk(M)18 b FA(+)g Fk(N)p FA(\))742 3166 y Fk(s)p FA(\()p
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n(w)n(o)h(distinct)h(needed)f(narro)n(wing)e(deriv)-5
b(ations)523 4360 y Fo(e)23 b Fj(;)668 4330 y Fl(\003)668
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6 5 bop 523 448 a FA(v)-5 b(alues)29 b(and)h(answ)n(er)e
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3604 y(rule,)h(th)n(us)h(b)r(earing)f(the)h(capabilit)n(y)f(of)g(narro)
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y(;)p eop
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7 6 bop 523 448 a FA(By)39 b(a)g(needed)g(narro)n(wing)e(step)i
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463 y Fl(f)p Fe(X)p Fl(7!)p Fe(0)p Fl(g)2251 448 y Fk(s)p
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548 y(second)28 b(rule)g(of)h Fp(R)p FA(,)g(w)n(e)f(get)h(\(b)n(y)f
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518 y Fl(0)3025 548 y FA(.)i(No)n(w,)h(the)523 648 y(goal)d
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3181 y([23]\))h(is)h(disallo)n(w)n(ed.)e(There)h(are)g(sev)n(eral)f(p)r
(oin)n(ts)h(regarding)f(our)h(de\014nition)h(of)f(the)h(fold-)523
3281 y(ing)j(rule)g(whic)n(h)h(are)e(w)n(orth)h(noticing:)g(\(i\))h(As)
g(a)f(di\013erence)g(w.r.t.)h(the)g(unfolding)f(rule,)523
3380 y(the)e(subterm)g(whic)n(h)g(is)g(selected)g(for)f(the)i(folding)f
(step)g(needs)g(not)f(b)r(e)i(a)e(\(needed\))i(nar-)523
3480 y(ro)n(wing)d(redex.)h(This)g(generalit)n(y)f(is)h(not)g(only)g
(safe)g(but)h(also)e(helpful)i(as)f(it)h(will)f(b)r(ecome)523
3580 y(apparen)n(t)d(in)i(Example)e(3.)h(\(ii\))h(In)g(con)n(trast)d
(to)j([2],)f(the)g(substitution)h Fo(\022)i FA(of)d(Def.)h(4)f(is)g
(not)523 3679 y(a)f(uni\014er)g(but)i(just)f(a)f(matc)n(her.)f(This)i
(is)f(similar)g(to)g(man)n(y)g(other)g(folding)g(rules)g(for)g(logic)
523 3779 y(programs,)i(whic)n(h)j(ha)n(v)n(e)e(b)r(een)i(de\014ned)g
(in)g(a)g(similar)e(\\functional)i(st)n(yle")e(\(see,)i(e.g.,)f([7,)523
3878 y(20,)13 b(24,)g(28]\).)34 b(\(iii\))i(Finally)-7
b(,)35 b(the)g Fq(non-er)l(asing)43 b FA(condition)34
b(in)h(Def.)h(1)f(can)f(no)n(w)g(b)r(e)h(fully)523 3978
y(clari\014ed:)30 b(it)h(a)n(v)n(oids)e(to)i(consider)e(a)h(rule)h
Fo(l)e Fp(!)f Fo(r)r FA(,)k(with)f Fp(V)7 b Fo(ar)r FA(\()p
Fo(r)r FA(\))30 b Fp(\032)e(V)7 b Fo(ar)r FA(\()p Fo(l)r
FA(\),)31 b(as)f(a)g(folding)523 4078 y(rule,)d(since)h(it)g(migh)n(t)f
(in)n(tro)r(duce)h(extra)e(v)-5 b(ariables)27 b(in)g(the)h(rhs)f(of)h
(the)g(resulting)f(rule.)648 4177 y(Man)n(y)34 b(attempts)h(ha)n(v)n(e)
e(b)r(een)i(also)f(made)g(to)h(de\014ne)g(a)f(folding)h(transformation)
e(in)523 4277 y(a)h(\(pure\))h(functional)g(con)n(text)f([8,)14
b(27].)34 b(A)h Fq(marke)l(d)44 b FA(folding)34 b(for)h(a)f(lazy)g
(\(higher-order\))523 4377 y(functional)h(language)d(has)i(b)r(een)h
(presen)n(ted)e(in)i([26],)f(whic)n(h)g(preserv)n(es)e(the)j(seman)n
(tics)523 4476 y(of)30 b(\(ground)g(constructor\))f(v)-5
b(alues)31 b(under)f(applicabilit)n(y)g(conditions)g(whic)n(h)g(are)g
(similar)523 4576 y(to)c(ours.)f(Ho)n(w)n(ev)n(er,)f(our)i(correctness)
e(results)h(are)g(sligh)n(tly)h(stronger,)e(since)i(w)n(e)g(preserv)n
(e)523 4675 y(the)i(\(non-ground\))f(seman)n(tics)f(of)i(computed)g(v)
-5 b(alues)27 b(and)g Fq(answers)p FA(.)p 523 4748 473
4 v 540 4801 a Fy(1)600 4833 y Fx(A)21 b Ft(de\014nition)k(rule)j
Fx(main)n(tains)22 b(its)g(status)g(only)g(as)h(long)f(as)h(it)f
(remains)g(unc)n(hanged,)f(i.e.,)j(once)600 4924 y(a)i(de\014nition)f
(rule)h(is)g(transformed)g(it)g(is)g(not)f(considered)h(a)g
Ft(de\014nition)i(rule)k Fx(an)n(ymore.)p eop
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8 7 bop 648 448 a FA(As)35 b(in)g(our)f(de\014nition)h(of)g(folding,)g
(a)g(large)e(n)n(um)n(b)r(er)i(of)g(prop)r(osals)e(also)h(allo)n(w)g
(the)523 548 y(folded)d(and)f(the)h(folding)g(rule)f(to)h(b)r(elong)f
(to)g(di\013eren)n(t)h(programs)e(\(see,)h(e.g.,)g([7,)14
b(20,)f(23,)523 648 y(24,)g(28]\),)34 b(whic)n(h)g(in)h(general)e(is)h
(crucial)f(to)i(ac)n(hiev)n(e)e(an)h(e\013ectiv)n(e)g(optimization.)g
(Some)523 747 y(other)26 b(w)n(orks)f(in)h(the)h(literature)f(ha)n(v)n
(e)f(adv)n(o)r(cated)g(a)h(di\013eren)n(t)h(st)n(yle)f(of)g(folding)h
(whic)n(h)f(is)523 847 y Fq(r)l(eversible)36 b FA([13],)27
b(i.e.,)h(a)f(kind)h(of)g(folding)g(whic)n(h)f(can)h(alw)n(a)n(ys)e(b)r
(e)i(undone)g(b)n(y)f(an)h(unfold-)523 946 y(ing)k(step.)h(This)f
(greatly)f(simpli\014es)h(the)h(correctness)d(pro)r(ofs)i(|correctness)
e(of)i(folding)523 1046 y(follo)n(ws)24 b(immediately)h(from)f(the)h
(correctness)e(of)h(unfolding|,)h(but)h(usually)e(require)f(to)r(o)523
1146 y(strong)31 b(applicabilit)n(y)g(conditions,)h(suc)n(h)f(as)g
(requiring)g(that)h(b)r(oth)h(the)f(folded)g(and)g(the)523
1245 y(folding)f(rules)g(b)r(elong)g(to)g(the)g(same)g(program,)e(whic)
n(h)i(drastically)f(reduces)h(the)g(p)r(o)n(w)n(er)523
1345 y(of)36 b(the)h(transformation.)d(The)i(folding)g(rule)g(prop)r
(osed)f(in)i([2])f(for)f(a)h(strict)g(functional)523
1445 y(logic)28 b(language)f(is)i(rev)n(ersible)e(and)i(th)n(us)g(its)g
(transformational)d(p)r(o)n(w)n(er)i(is)h(v)n(ery)e(limited.)523
1544 y(The)f(folding)g(rule)g(in)n(tro)r(duced)g(in)h(this)f(pap)r(er)g
(is)g(more)f(p)r(o)n(w)n(erful)h(and)g(the)h(applicabilit)n(y)523
1644 y(conditions)33 b(are)f(less)g(restrictiv)n(e.)1613
1614 y Fz(2)1683 1644 y FA(Therefore,)g(its)h(use)g(within)h(a)e
(transformation)g(sys-)523 1743 y(tem)24 b(|when)g(guided)f(b)n(y)g
(appropriate)f(strategies|)g(is)i(able)f(to)g(pro)r(duce)g(more)g
(e\013ectiv)n(e)523 1843 y(optimizations)k(for)g(\(lazy\))h(functional)
f(logic)g(programs.)648 1943 y(The)e(set)g(of)h(rules)f(presen)n(ted)f
(so)h(far)g(constitutes)g(the)h(k)n(ernel)f(of)g(our)g(transformation)
523 2042 y(system.)i(These)f(rules)h(su\016ce)g(for)f(automatizing)g
(the)h Fq(c)l(omp)l(osition)35 b FA(strategy)-7 b(.)26
b(Ho)n(w)n(ev)n(er,)523 2142 y(the)34 b(transformation)e(system)i(m)n
(ust)g(b)r(e)g(emp)r(o)n(w)n(ered)e(for)h(ac)n(hieving)g(the)h
Fq(tupling)41 b FA(opti-)523 2242 y(mization,)31 b(whic)n(h)g(w)n(e)f
(attain)h(b)n(y)f(extending)h(the)h(transformation)d(system)h(with)i(a)
e(rule)523 2341 y(of)d(abstraction)f([8,)13 b(26])27
b(\(often)g(kno)n(wn)g(as)f Fq(wher)l(e{abstr)l(action)36
b FA(rule)27 b([24]\).)g(It)g(essen)n(tially)523 2441
y(consists)g(of)g(replacing)f(the)i(o)r(ccurrences)d(of)j(some)e
(expression)g Fo(e)h FA(in)h(the)f(rhs)g(of)g(a)g(rule)g
Fo(R)523 2540 y FA(b)n(y)20 b(a)h(fresh)f(v)-5 b(ariable)20
b Fo(z)t FA(,)g(adding)g(the)h(\\lo)r(cal)f(declaration")f
Fo(z)26 b FA(=)d Fo(e)d FA(within)i(a)e Fo(w)r(her)r(e)h
FA(expres-)523 2640 y(sion)29 b(in)i Fo(R)q FA(.)e(F)-7
b(or)30 b(instance,)f(the)i(rule)e Fk(double)p 1995 2640
25 4 v 28 w(sum)p FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\))55
b Fp(!)i Fk(sum)p FA(\()p Fk(sum)p FA(\()p Fk(X)p Fo(;)14
b Fk(Y)p FA(\))p Fo(;)g Fk(sum)p FA(\()p Fk(X)p Fo(;)g
Fk(Y)p FA(\)\))523 2740 y(can)27 b(b)r(e)h(transformed)f(in)n(to)g(the)
h(new)g(rule)731 2901 y Fk(double)p 1000 2901 V 27 w(sum)p
FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\))49 b Fp(!)i Fk(sum)p
FA(\()p Fk(Z)p Fo(;)14 b Fk(Z)p FA(\))26 b Fk(where)g(Z)c
FA(=)h Fk(sum)p FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\))26
b Fo(:)523 3063 y FA(As)j(noted)g(b)n(y)g([24],)g(the)g(use)g(of)g(the)
h(where{abstraction)c(rule)j(has)g(the)g(adv)-5 b(an)n(tage)28
b(that)523 3162 y(in)40 b(the)g(call-b)n(y-v)-5 b(alue)38
b(mo)r(de)i(of)g(execution,)f(the)h(ev)-5 b(aluation)40
b(of)f(the)h(expression)f Fo(e)g FA(is)523 3262 y(p)r(erformed)21
b(only)h(once.)f(This)h(is)g(also)f(true)g(in)i(a)e(lazy)g(con)n(text)h
(under)f(an)h(implemen)n(tation)523 3361 y(based)k(on)f
Fq(sharing)p FA(,)j(whic)n(h)e(allo)n(ws)f(us)h(to)g(k)n(eep)f(trac)n
(k)g(of)h(v)-5 b(ariables)25 b(whic)n(h)h(o)r(ccur)g(sev)n(eral)523
3461 y(times)i(in)g(the)g(expression)e(to)h(b)r(e)h(ev)-5
b(aluated.)648 3561 y(The)19 b(new)g(rules)g(in)n(tro)r(duced)g(b)n(y)f
(the)i(where{abstraction)d(do)i(con)n(tain)f(extra)g(v)-5
b(ariables)523 3660 y(in)24 b(the)h(righ)n(t-hand)e(sides.)g(Ho)n(w)n
(ev)n(er,)g(as)g(noted)h(in)g([26],)g(this)g(can)g(b)r(e)g(easily)f
(amended)h(b)n(y)523 3760 y(using)19 b(standard)f(\\lam)n(b)r(da)g
(lifting")h(tec)n(hniques)g(\(whic)n(h)g(can)g(b)r(e)g(though)n(t)g(of)
g(as)g(an)f(appro-)523 3860 y(priate)h(application)f(of)h(a)g
(de\014nition)g(in)n(tro)r(duction)g(step)g(follo)n(w)n(ed)f(b)n(y)h(a)
f(folding)h(step\).)h(F)-7 b(or)523 3959 y(instance,)24
b(if)h(w)n(e)f(consider)f(again)g(the)i(rule)f Fk(double)p
2168 3959 V 27 w(sum)p FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p
FA(\))46 b Fp(!)i Fk(sum)p FA(\()p Fk(Z)p Fo(;)14 b Fk(Z)p
FA(\))22 b Fk(where)h(Z)f FA(=)523 4059 y Fk(sum)p FA(\()p
Fk(X)p Fo(;)14 b Fk(Y)p FA(\))p Fo(;)25 b FA(w)n(e)h(can)f(transform)g
(it)i(\(b)n(y)f(lam)n(b)r(da)g(lifting)h([19]\))e(in)n(to)h(the)h(new)f
(pair)f(of)i(rules)742 4219 y Fk(double)p 1011 4219 V
28 w(sum)p FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\))21 b
Fp(!)j Fk(ds)p 1576 4219 V 29 w(aux)p FA(\()p Fk(sum)p
FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\)\))997 4319 y Fk(ds)p
1090 4319 V 29 w(aux)p FA(\()p Fk(Z)p FA(\))22 b Fp(!)i
Fk(sum)p FA(\()p Fk(Z)p Fo(;)14 b Fk(Z)p FA(\))523 4477
y(Note)31 b(that)g(these)f(rules)g(can)g(b)r(e)h(directly)g(generated)e
(from)h(the)h(initial)g(de\014nition)g(b)n(y)f(a)523
4576 y(de\014nition)c(in)n(tro)r(duction)g(\()p Fk(ds)p
1489 4576 V 29 w(aux)p FA(\()p Fk(Z)p FA(\))c Fp(!)h
Fk(sum)p FA(\()p Fk(Z)p Fo(;)14 b Fk(Z)p FA(\)\))25 b(and)g(then)h(b)n
(y)g(folding)f(the)i(original)523 4676 y(rule)33 b(at)h(the)g
(expression)e Fk(sum)p FA(\()p Fk(sum)p FA(\()p Fk(X)p
Fo(;)14 b Fk(Y)p FA(\))p Fo(;)g Fk(sum)p FA(\()p Fk(X)p
Fo(;)g Fk(Y)p FA(\)\))29 b(using)34 b(as)e(folding)i(rule)f(the)h
(newly)p 523 4748 473 4 v 540 4801 a Fy(2)600 4833 y
Fx(It)26 b(w)n(ould)g(b)r(e)g(in)n(teresting)h(to)g(study)e(a)h
(generalization)j(of)e(our)f(folding)i(rule)e(to)h(a)f
Ft(disjunctive)600 4924 y Fx(folding)h(rule,)f(i.e.,)h(allo)n(wing)h
(the)d(folding)i(of)f(m)n(ultiple)f(recursiv)n(e)h(rules)g(\(see)g
([25]\).)p eop
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9 8 bop 523 448 a FA(generated)34 b(de\014nition)h(for)f
Fk(ds)p 1512 448 25 4 v 29 w(aux)p Fo(=)p Fk(1)p FA(.)f(The)i
(inclusion)f(of)h(an)f(abstraction)g(rule)g(is)h(tra-)523
548 y(ditional)c(in)g(functional)g(fold/unfold)g(framew)n(orks)e([8,)14
b(24,)f(26,)g(27].)30 b(In)h(the)h(case)e(of)h(logic)523
648 y(programs,)e(abstraction)g(is)i(only)g(p)r(ossible)g(b)n(y)f
(means)h(of)g(the)g(so)g(called)f Fq(gener)l(alization)523
747 y FA(strategy)23 b([24],)g(whic)n(h)h(generalizes)e(some)i(calls)f
(to)h(eliminate)g(the)h(mismatc)n(hes)e(that)i(pre-)523
847 y(v)n(en)n(t)i(a)g(folding)h(step.)648 947 y(No)n(w,)35
b(w)n(e)h(are)f(ready)g(to)h(formalize)f(our)h(abstraction)f(rule,)g
(whic)n(h)h(is)g(inspired)g(b)n(y)523 1046 y(the)f(standard)e(lam)n(b)r
(da)h(lifting)g(transformation)f(of)h(functional)h(programs.)d(By)h
(means)523 1146 y(of)k(the)h(tuple)f(constructor)f Fp(h)h(i)p
FA(,)h(our)e(de\014nition)i(allo)n(ws)e(the)h(abstraction)f(of)h
(di\013eren)n(t)523 1246 y(expressions)32 b(in)i(one)g(go.)f(F)-7
b(or)33 b(a)h(sequence)f(of)h(\(pairwise)f(disjoin)n(t\))i(p)r
(ositions)e Fo(P)46 b FA(=)p 3295 1200 87 4 v 33 w Fo(p)3337
1258 y Fn(n)3382 1246 y FA(,)523 1345 y(w)n(e)29 b(let)g
Fo(t)p FA([)p 821 1300 85 4 v Fo(s)860 1357 y Fn(n)905
1345 y FA(])928 1357 y Fn(P)1009 1345 y FA(=)24 b(\(\(\()p
Fo(t)p FA([)p Fo(s)1286 1357 y Fz(1)1324 1345 y FA(])1347
1357 y Fn(p)1381 1365 y Fi(1)1418 1345 y FA(\)[)p Fo(s)1512
1357 y Fz(2)1550 1345 y FA(])1573 1357 y Fn(p)1607 1365
y Fi(2)1644 1345 y FA(\))14 b Fo(:)g(:)g(:)f FA([)p Fo(s)1862
1357 y Fn(n)1908 1345 y FA(])1931 1357 y Fn(p)1965 1365
y Fh(n)2010 1345 y FA(\).)29 b(By)g(abuse,)f(w)n(e)h(denote)f
Fo(t)p FA([)p 2926 1300 V Fo(s)2965 1357 y Fn(n)3011
1345 y FA(])3034 1357 y Fn(P)3118 1345 y FA(b)n(y)h Fo(t)p
FA([)p Fo(s)p FA(])3350 1357 y Fn(P)523 1445 y FA(when)f
Fo(s)779 1457 y Fz(1)839 1445 y FA(=)23 b Fo(:)14 b(:)g(:)23
b FA(=)f Fo(s)1173 1457 y Fn(n)1241 1445 y FA(=)h Fo(s)p
FA(,)28 b(as)f(w)n(ell)g(as)g(\(\()p Fo(t)p FA([)p Fo(s)1947
1457 y Fz(1)1985 1445 y FA(])2008 1457 y Fn(P)2050 1465
y Fi(1)2087 1445 y FA(\))14 b Fo(:)g(:)g(:)f FA([)p Fo(s)2305
1457 y Fn(n)2351 1445 y FA(])2374 1457 y Fn(P)2416 1465
y Fh(n)2461 1445 y FA(\))28 b(b)n(y)f Fo(t)p FA([)p 2689
1399 V Fo(s)2728 1457 y Fn(n)2773 1445 y FA(])p 2796
1422 88 3 v 25 x Fn(P)2838 1478 y Fh(n)2883 1445 y FA(.)523
1636 y Fg(De\014nition)k(5)41 b(\(Abstraction\).)34 b
Fq(L)l(et)d Fo(R)c FA(=)e(\()p Fo(f)9 b FA(\()p 2124
1575 76 4 v Fo(t)2154 1648 y Fn(n)2200 1636 y FA(\))26
b Fp(!)g Fo(r)r FA(\))h Fp(2)f(R)2616 1648 y Fn(k)2688
1636 y Fq(b)l(e)32 b(a)f(rule)h(and)f(let)p 3317 1570
89 4 v 32 w Fo(P)3370 1648 y Fn(j)523 1736 y Fq(b)l(e)26
b(se)l(quenc)l(es)g(of)h(disjoint)h(p)l(ositions)f(in)g
Fp(N)12 b(V)7 b(P)g Fo(os)p FA(\()p Fo(r)r FA(\))27 b
Fq(such)f(that)g Fo(r)r Fp(j)2645 1748 y Fn(p)2708 1736
y FA(=)c Fo(e)2834 1748 y Fn(i)2888 1736 y Fq(for)27
b(al)t(l)g Fo(p)f Fq(in)h Fo(P)3352 1748 y Fn(i)3379
1736 y Fq(,)523 1836 y Fo(i)f FA(=)g(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(j)5
b Fq(,)31 b(i.e.,)j Fo(r)29 b FA(=)c Fo(r)r FA([)p 1380
1790 74 4 v Fo(e)1419 1848 y Fn(j)1455 1836 y FA(])p
1478 1813 78 3 v 25 x Fn(P)1520 1869 y Fh(j)1555 1836
y Fq(.)32 b(We)g(may)g(get)f(pr)l(o)l(gr)l(am)h Fp(R)2474
1848 y Fn(k)q Fz(+1)2631 1836 y Fq(fr)l(om)g Fp(R)2899
1848 y Fn(k)2971 1836 y Fq(by)g(r)l(eplacing)523 1953
y Fo(R)37 b Fq(with)g Fp(f)p Fo(f)9 b FA(\()p 934 1892
76 4 v Fo(t)964 1965 y Fn(n)1008 1953 y FA(\))35 b Fp(!)g
Fo(f)p 1248 1953 26 4 v 40 w(aux)p FA(\()p 1445 1907
104 4 v Fo(y)1486 1965 y Fn(m)1548 1953 y Fo(;)14 b Fp(h)p
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1965 y Fn(j)1952 1953 y Fp(i)p FA(\))p Fo(;)87 b(f)p
2180 1953 26 4 v 39 w(aux)p FA(\()p 2377 1907 104 4 v
Fo(y)2418 1965 y Fn(m)2481 1953 y Fo(;)14 b Fp(h)p Fo(z)2589
1965 y Fz(1)2626 1953 y Fo(;)g(:)g(:)g(:)f(;)h(z)2849
1965 y Fn(j)2884 1953 y Fp(i)p FA(\))35 b Fp(!)g Fo(r)r
FA([)p 3163 1907 74 4 v Fo(z)3202 1965 y Fn(j)3237 1953
y FA(])p 3260 1930 78 3 v 25 x Fn(P)3302 1986 y Fh(j)3338
1953 y Fp(g)p Fq(,)523 2071 y(wher)l(e)p 753 2025 74
4 v 26 w Fo(z)792 2083 y Fn(j)852 2071 y Fq(ar)l(e)25
b(fr)l(esh)h(variables)i(not)d(o)l(c)l(curring)g(in)p
2125 2010 76 4 v 26 w Fo(t)2155 2083 y Fn(n)2200 2071
y Fq(,)g Fo(f)p 2305 2071 26 4 v 40 w(aux)g Fq(is)g(a)h(fr)l(esh)g
(function)f(symb)l(ol)523 2170 y(that)30 b(do)l(es)g(not)g(o)l(c)l(cur)
f(in)h FA(\()p Fp(R)1445 2182 y Fz(0)1483 2170 y Fo(;)14
b(:)g(:)g(:)g(;)g Fp(R)1738 2182 y Fn(k)1779 2170 y FA(\))p
Fq(,)30 b(and)g Fp(V)7 b Fo(ar)r FA(\()p Fo(r)r FA([)p
2262 2125 74 4 v Fo(z)2301 2182 y Fn(j)2337 2170 y FA(])p
2360 2147 78 3 v 26 x Fn(P)2402 2204 y Fh(j)2438 2170
y FA(\))23 b(=)g Fp(f)p 2623 2125 104 4 v Fo(y)2664 2182
y Fn(m)2726 2170 y Fo(;)p 2763 2125 74 4 v 14 w(z)2802
2182 y Fn(j)2837 2170 y Fp(g)p Fq(.)523 2346 y FA(Informally)-7
b(,)36 b(the)h(t)n(w)n(o)e(rules)h(generated)g(b)n(y)g(the)h
(abstraction)e(transformation)g(can)h(b)r(e)523 2445
y(understo)r(o)r(d)27 b(as)g(a)g(syn)n(tactic)g(v)-5
b(arian)n(t)27 b(of)h(the)g(follo)n(wing)e(rule:)731
2613 y Fo(f)9 b FA(\()p 813 2552 76 4 v Fo(t)843 2625
y Fn(n)888 2613 y FA(\))51 b Fp(!)f Fo(r)r FA([)p 1166
2567 74 4 v Fo(z)1205 2625 y Fn(j)1241 2613 y FA(])p
1264 2590 78 3 v 25 x Fn(P)1306 2646 y Fh(j)1397 2613
y Fo(w)r(her)r(e)28 b Fp(h)p Fo(z)1722 2625 y Fz(1)1759
2613 y Fo(;)14 b(:)g(:)g(:)g(;)g(z)1983 2625 y Fn(j)2017
2613 y Fp(i)24 b FA(=)e Fp(h)p Fo(e)2231 2625 y Fz(1)2268
2613 y Fo(;)14 b(:)g(:)g(:)g(;)g(e)2492 2625 y Fn(j)2526
2613 y Fp(i)28 b Fo(:)523 2780 y FA(No)n(w)33 b(w)n(e)g(state)g(the)h
(main)f(theoretical)g(results)g(for)g(the)h(basic)f(transformations)e
(in)n(tro-)523 2879 y(duced)26 b(in)g(this)g(section.)f(W)-7
b(e)26 b(state)g(the)g(correctness)d(of)j(transformation)e(sequences)h
(con-)523 2979 y(structed)e(from)g(an)g(inductiv)n(ely)h(sequen)n(tial)
e(program)g(b)n(y)h(applying)g(the)g(follo)n(wing)g(rules:)523
3079 y(de\014nition)35 b(in)n(tro)r(duction,)f(de\014nition)h
(elimination,)g(unfolding,)g(folding,)f(and)h(abstrac-)523
3178 y(tion.)25 b(In)g(pro)n(ving)f(this,)h(w)n(e)g(assume)f(that)i(no)
e(folding)h(step)g(is)g(applied)g(after)g(a)g(de\014nition)523
3278 y(elimination,)g(whic)n(h)f(guaran)n(tees)e(that)j(no)f(function)h
(call)f(to)h(a)f(previously)f(deleted)i(func-)523 3377
y(tion)39 b(is)h(in)n(tro)r(duced)f(along)f(a)h(transformation)e
(sequence)i([23].)f(First,)i(w)n(e)f(state)g(that)523
3477 y(transformations)26 b(preserv)n(e)g(the)h(inductiv)n(ely)h
(sequen)n(tial)f(structure)g(of)h(programs.)523 3653
y Fg(Theorem)i(1.)41 b Fq(L)l(et)c FA(\()p Fp(R)1297
3665 y Fz(0)1335 3653 y Fo(;)14 b(:)g(:)g(:)g(;)g Fp(R)1590
3665 y Fn(n)1635 3653 y FA(\))38 b Fq(b)l(e)g(a)g(tr)l(ansformation)h
(se)l(quenc)l(e.)e(If)i Fp(R)3006 3665 y Fz(0)3081 3653
y Fq(is)f(induc-)523 3753 y(tivlely)31 b(se)l(quential,)f(then)g
Fp(R)1427 3765 y Fn(i)1485 3753 y Fq(is)g(also)g(inductively)h(se)l
(quential,)g(for)f Fo(i)23 b FA(=)g(1)p Fo(;)14 b(:)g(:)g(:)f(;)h(n)p
Fq(.)523 3928 y FA(Sands)38 b(formalizes)f(a)h(syn)n(tactic)g
Fq(impr)l(ovement)47 b FA(theory)37 b([26])h(whic)n(h)g(restricts)f
(general)523 4028 y(fold/unfold)f(transformations)e(and)i(can)g(b)r(e)h
(applied)f(to)g(giv)n(e)f(correctness)f(pro)r(ofs)i(for)523
4127 y(some)25 b(existing)h(transformation)e(metho)r(ds)i(\(suc)n(h)f
(as)h(deforestation)e([30]\).)i(Ho)n(w)n(ev)n(er,)d(w)n(e)523
4227 y(\014nd)31 b(it)g(more)e(con)n(v)n(enien)n(t)h(to)g(stic)n(k)g
(to)g(the)h(logic)f(programming)e(metho)r(ds)j(for)f(pro)n(ving)523
4327 y(correctness)k(b)r(ecause)h(the)h(narro)n(wing)d(mec)n(hanism)i
(can)g(b)r(e)h(prop)r(erly)e(seen)i(as)f(a)g(gen-)523
4426 y(eralization)28 b(of)i(the)g(SLD-resolution)f(metho)r(d)i(whic)n
(h)f(implicitly)g(applies)g(instan)n(tiation)523 4526
y(b)r(efore)21 b(replacing)e(a)i(call)f(b)n(y)h(the)g(corresp)r(onding)
e(instance)i(of)g(the)g(b)r(o)r(dy)-7 b(.)22 b(That)f(is,)f(instan-)523
4625 y(tiation)j(is)g(computed)g(in)h(a)e(systematic)h(w)n(a)n(y)f(b)n
(y)g(the)i(needed)f(narro)n(wing)e(mec)n(hanism)h(\(as)523
4725 y(in)31 b(the)g(unfolding)f(of)h(logic)f(programs\),)e(whereas)h
(it)i(is)g(not)f(restricted)g(in)h(the)g(Burstall)523
4825 y(and)c(Darlington's)e(fold/unfold)i(framew)n(ork)e(considered)g
(in)i([26].)f(Unrestricted)g(instan-)523 4924 y(tiation)j(is)h
(problematic)e(since)i(it)f(do)r(es)g(not)h(ev)n(en)f(preserv)n(e)e(lo)
r(cal)i(equiv)-5 b(alence,)29 b(and)h(for)p eop
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10 9 bop 523 448 a FA(this)26 b(reason)e(the)j(instan)n(tiation)e(rule)
g(is)h(not)g(considered)f(explicitly)h(in)g([26].)f(As)h(a)f(conse-)523
548 y(quence,)f(the)h(impro)n(v)n(emen)n(t)f(theorem)g(of)g([26])g(do)r
(es)g(not)h(directly)f(apply)g(to)h(our)e(con)n(text.)648
648 y(Our)c(demonstration)f(tec)n(hnique)i(for)f(the)i(correctness)c
(result)j(is)g(inspired)f(b)n(y)h(the)g(orig-)523 747
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b(and)i(Sato)e([28])h(concerning)f(the)i(least)f(Herbrand)g(mo)r(del)
523 847 y(seman)n(tics)35 b(of)g(logic)f(programs)f(\(and)i(the)h
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946 y(Kanamori)21 b([20])i(for)f(the)i(seman)n(tics)e(of)h(computed)g
(answ)n(ers\).)f(In)n(tuitiv)n(ely)-7 b(,)23 b(a)g(fold/unfold)523
1046 y(transformation)h(system)g(is)i(correct)d(if)j(there)f(are)f
(\\at)h(least)f(as)h(man)n(y)f(folds)i(as)e(there)h(are)523
1146 y(unfolds")19 b(or,)g(equiv)-5 b(alen)n(tly)e(,)20
b(if)g(\\going)e(bac)n(kw)n(ard)g(in)i(the)g(computation)g(\(as)f
(folding)h(do)r(es\))523 1245 y(do)r(es)k(not)g(prev)-5
b(ail)23 b(o)n(v)n(er)g(going)g(forw)n(ard)f(in)i(the)h(computation)e
(\(as)h(unfolding)g(do)r(es\)")g([23,)523 1345 y(26].)31
b(This)g(essen)n(tially)g(means)g(that)h(there)f(m)n(ust)h(b)r(e)f(a)h
(kind)f(of)h(\\computational)e(cost")523 1445 y(measure)f(whic)n(h)h
(is)g(not)g(increased)f(either)h(b)n(y)g(folding)g(or)f(b)n(y)h
(unfolding)g(steps.)g(Sev)n(eral)523 1544 y(de\014nitions)d(for)g(this)
h(measure)e(can)h(b)r(e)h(found)f(in)h(the)g(literature:)e(the)i
Fq(r)l(ank)i(of)g(a)g(go)l(al)37 b FA(in)523 1644 y([28],)29
b(the)h Fq(weight)i(of)h(a)f(pr)l(o)l(of)h(tr)l(e)l(e)i
FA(in)30 b([20],)f(or)f(the)i(notion)f(of)h Fq(impr)l(ovement)38
b FA(in)30 b([26].)f(In)523 1743 y(our)21 b(con)n(text,)h(w)n(e)g(ha)n
(v)n(e)f(in)n(tro)r(duced)h(the)g(notion)g(of)g Fq(r)l(ank)j(of)h(a)f
(term)j FA(in)22 b(order)f(to)h(measure)523 1843 y(the)j(computational)
f(cost)g(of)h(a)f(giv)n(en)f(term.)i(The)g(detailed)f(pro)r(of)g(sc)n
(heme)g(can)h(b)r(e)g(found)523 1943 y(in)j([3].)f(The)h(strong)e
(correctness)g(of)i(the)g(transformation)d(is)j(stated)f(as)g(follo)n
(ws.)523 2102 y Fg(Theorem)j(2.)41 b Fq(L)l(et)22 b FA(\()p
Fp(R)1282 2114 y Fz(0)1320 2102 y Fo(;)14 b(:)g(:)g(:)g(;)g
Fp(R)1575 2114 y Fn(n)1620 2102 y FA(\))p Fq(,)23 b Fo(n)g(>)g
FA(0)p Fq(,)f(b)l(e)h(a)g(tr)l(ansformation)g(se)l(quenc)l(e.)f(L)l(et)
g Fo(e)g Fq(b)l(e)h(an)523 2201 y(e)l(quation)31 b(with)h(no)f(new)g
(function)g(symb)l(ol)g(and)h Fo(V)44 b Fp(\023)25 b(V)7
b Fo(ar)r FA(\()p Fo(e)p FA(\))31 b Fq(a)g(\014nite)g(set)g(of)g
(variables.)523 2301 y(Then,)e Fo(e)22 b Fj(;)907 2271
y Fl(\003)907 2321 y Fn(\033)975 2301 y Fo(tr)r(ue)27
b Fq(in)h Fp(R)1328 2313 y Fz(0)1393 2301 y Fq(i\013)g
Fo(e)22 b Fj(;)1641 2271 y Fl(\003)1641 2324 y Fn(\033)1681
2307 y Fd(0)1732 2301 y Fo(tr)r(ue)27 b Fq(in)g Fp(R)2084
2313 y Fn(n)2130 2301 y Fq(,)h(with)g Fo(\033)2411 2271
y Fl(0)2458 2301 y FA(=)22 b Fo(\033)31 b FA([)p Fo(V)19
b FA(])28 b Fq(\(up)f(to)g(r)l(enaming\).)523 2554 y
Fs(4)112 b(Some)37 b(Exp)s(erimen)m(ts)523 2741 y FA(The)29
b(building)g(blo)r(c)n(ks)f(of)h(strategic)f(program)f(optimizations)h
(are)g(transformation)g(tac-)523 2840 y(tics)h(\()p Fq(str)l(ate)l
(gies)7 b FA(\),)29 b(whic)n(h)f(are)f(used)i(to)f(guide)g(the)h(pro)r
(cess)e(and)i(e\013ect)f(some)g(particular)523 2940 y(kind)g(of)f(c)n
(hange)g(to)g(the)h(program)e(undergoing)g(transformation)g([12,)13
b(24].)648 3040 y(One)26 b(of)g(the)g(most)g(relev)-5
b(an)n(t)26 b(quests)g(in)g(applying)g(a)g(transformation)f(strategy)f
(is)j(the)523 3139 y(in)n(tro)r(duction)32 b(of)g(new)h(functions,)f
(often)h(called)f(in)g(the)h(literature)f Fq(eur)l(eka)39
b FA(de\014nitions.)523 3239 y(Although)g(there)f(is)g(no)h(general)e
(theory)g(of)i(strategies)e(whic)n(h)h(ensures)g(that)h(deriv)n(ed)523
3339 y(programs)25 b(are)i(more)f(e\016cien)n(t)i(than)f(the)h(initial)
g(ones,)f(some)g(partial)f(results)h(exist.)h(F)-7 b(or)523
3438 y(instance,)29 b(in)f(the)i(setting)e(of)h(higher-order)d
(\(non-strict\))j(functional)f(languages,)f(Sands)523
3538 y([26])32 b(has)h(recen)n(tly)f(in)n(tro)r(duced)h(the)g(theory)f
(of)h Fq(impr)l(ovement)41 b FA(to)33 b(pro)n(vide)f(a)h(syn)n(tactic)
523 3637 y(metho)r(d)27 b(for)e(guiding)h(and)g(constraining)e(the)j
(unfold/fold)f(metho)r(d)g(in)h([8])f(so)f(that)h(total)523
3737 y(correctness)g(and)h(p)r(erformance)g(impro)n(v)n(emen)n(t)f(are)
h(alw)n(a)n(ys)e(guaran)n(teed.)648 3837 y(In)37 b(the)h(follo)n(wing,)
e(w)n(e)h(illustrate)g(the)h(p)r(o)n(w)n(er)e(of)h(our)g
(transformation)e(system)i(b)n(y)523 3936 y(tac)n(kling)30
b(some)g(represen)n(tativ)n(e)e(examples)i(regarding)e(the)j
(optimizations)f(of)h(comp)r(osi-)523 4036 y(tion)d([30])f(and)g
(tupling)h([8,)13 b(11].)523 4273 y Fg(4.1)95 b(T)-8
b(ransformation)31 b(Strategies)523 4426 y FA(The)h(comp)r(osition)f
(strategy)f(w)n(as)h(originally)f(in)n(tro)r(duced)h(in)h([8,)14
b(11])30 b(for)i(the)g(optimiza-)523 4526 y(tion)e(of)f(pure)h
(functional)g(programs.)d(V)-7 b(arian)n(ts)29 b(of)g(this)h(comp)r
(osition)g(strategy)e(are)h(the)523 4625 y Fq(internal)i(sp)l(e)l
(cialization)37 b FA(tec)n(hnique)28 b([27])f(and)h(the)h
Fq(defor)l(estation)36 b FA(metho)r(d)29 b([30].)e(By)h(us-)523
4725 y(ing)k(the)h(comp)r(osition)f(strategy)f(\(or)h(its)h(v)-5
b(arian)n(ts\),)31 b(one)h(ma)n(y)g(a)n(v)n(oid)f(the)i(construction)
523 4825 y(of)c(in)n(termediate)f(data)g(structures)g(that)h(are)f(pro)
r(duced)g(b)n(y)h(some)f(function)h Fk(g)f FA(and)h(con-)523
4924 y(sumed)37 b(as)e(inputs)i(b)n(y)f(another)f(function)i
Fk(f)p FA(.)f(In)h(some)f(cases,)f(most)h(of)g(the)h(e\016ciency)p
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11 10 bop 523 448 a FA(impro)n(v)n(emen)n(t)25 b(of)h(the)h(comp)r
(osition)f(strategy)f(can)h(b)r(e)g(simply)h(obtained)f(b)n(y)g(lazy)f
(ev)-5 b(alu-)523 548 y(ation)28 b([12].)f(Nev)n(ertheless,)g(the)h
(comp)r(osition)f(strategy)g(often)h(allo)n(ws)e(the)j(deriv)-5
b(ation)27 b(of)523 648 y(programs)c(with)i(impro)n(v)n(ed)e(p)r
(erformance)h(also)g(in)h(the)g(con)n(text)g(of)f(lazy)h(ev)-5
b(aluation)24 b([29].)523 747 y(Laziness)i(is)h(decisiv)n(e)f(when,)i
(giv)n(en)e(a)g(nested)i(function)f(call)g Fk(f)p FA(\()p
Fk(g)p FA(\()p Fk(X)p FA(\)\),)g(the)g(in)n(termediate)523
847 y(data)g(structure)g(pro)r(duced)g(b)n(y)h Fk(g)f
FA(is)g(in\014nite)i(but)f(the)g(function)g Fk(f)f FA(can)g(pro)r(duce)
h(its)f(out-)523 946 y(come)d(b)n(y)g(kno)n(wing)f(only)h(a)g(\014nite)
g(p)r(ortion)g(of)g(the)h(output)g(of)f Fk(g)p FA(.)g(The)g(follo)n
(wing)f(example)523 1046 y(illustrates)k(the)h(adv)-5
b(an)n(tages)26 b(of)h(our)g(transformation)f(rules)h(w.r.t.)h(those)f
(of)h([2].)523 1214 y Fq(Example)j(3.)42 b FA(The)26
b(function)g Fk(sum)p 1598 1214 25 4 v 29 w(prefix)p
FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\))23 b(de\014ned)j(in)g(the)g
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FA(:)600 1461 y Fk(sum)p 737 1461 V 29 w(prefix)p FA(\()p
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b(\()p Fo(R)2061 1473 y Fz(1)2099 1461 y FA(\))125 b
Fk(from)p FA(\()p Fk(X)p FA(\))22 b Fp(!)h FA([)p Fk(X)p
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Fo(R)3290 1473 y Fz(4)3327 1461 y FA(\))848 1561 y Fk(suml)p
FA(\()p Fk(L)p Fo(;)14 b Fk(0)p FA(\))22 b Fp(!)h Fk(0)581
b FA(\()p Fo(R)2061 1573 y Fz(2)2099 1561 y FA(\))219
b Fk(0)18 b FA(+)g Fk(X)23 b Fp(!)g Fk(X)483 b FA(\()p
Fo(R)3290 1573 y Fz(5)3327 1561 y FA(\))627 1660 y Fk(suml)p
FA(\([)p Fk(H)p Fp(j)p Fk(T)p FA(])p Fo(;)14 b Fk(s)p
FA(\()p Fk(X)p FA(\)\))22 b Fp(!)h Fk(H)18 b FA(+)g Fk(suml)p
FA(\()p Fk(T)p Fo(;)c Fk(X)p FA(\))115 b(\()p Fo(R)2061
1672 y Fz(3)2099 1660 y FA(\))c Fk(s)p FA(\()p Fk(X)p
FA(\))18 b(+)g Fk(Y)23 b Fp(!)g Fk(s)p FA(\()p Fk(X)18
b FA(+)g Fk(Y)p FA(\))230 b(\()p Fo(R)3290 1672 y Fz(6)3327
1660 y FA(\))523 1809 y(W)-7 b(e)34 b(can)g(impro)n(v)n(e)e(the)i
(e\016ciency)g(of)g Fp(R)1837 1821 y Fz(0)1908 1809 y
FA(b)n(y)g(a)n(v)n(oiding)e(the)i(creation)f(and)g(subsequen)n(t)523
1909 y(use)27 b(of)h(the)g(in)n(termediate,)f(partial)g(list)h
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Fk(from)o FA(:)558 2068 y(1.)41 b(De\014nition)28 b(in)n(tro)r
(duction:)883 2226 y Fk(aux)p FA(\()p Fk(X)p Fo(;)14
b Fk(Y)p FA(\))22 b Fp(!)h Fk(suml)p FA(\()p Fk(from)p
FA(\()p Fk(X)p FA(\))p Fo(;)14 b Fk(Y)p FA(\))103 b(\()p
Fo(R)2135 2238 y Fz(7)2173 2226 y FA(\))558 2386 y(2.)41
b(Unfolding)28 b(of)g(rule)f Fo(R)1371 2398 y Fz(7)1436
2386 y FA(\(note)h(that)f(instan)n(tiation)h(is)f(automatic\):)992
2544 y Fk(aux)p FA(\()p Fk(X)p Fo(;)14 b Fk(0)p FA(\))21
b Fp(!)i Fk(0)994 b FA(\()p Fo(R)2573 2556 y Fz(8)2610
2544 y FA(\))883 2644 y Fk(aux)p FA(\()p Fk(X)p Fo(;)14
b Fk(s)p FA(\()p Fk(Y)p FA(\)\))22 b Fp(!)h Fk(suml)p
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Fk(X)p FA(\)\)])p Fo(;)14 b Fk(s)p FA(\()p Fk(Y)p FA(\)\))104
b(\()p Fo(R)2573 2656 y Fz(9)2610 2644 y FA(\))558 2804
y(3.)41 b(Unfolding)28 b(of)g(rule)f Fo(R)1371 2816 y
Fz(9)1436 2804 y FA(\(note)h(that)f(this)h(is)g(infeasible)f(with)h(an)
g(eager)e(strategy\):)883 2961 y Fk(aux)p FA(\()p Fk(X)p
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b FA(+)g Fk(suml)p FA(\()p Fk(from)p FA(\()p Fk(s)p FA(\()p
Fk(X)p FA(\)\))p Fo(;)c Fk(Y)p FA(\))104 b(\()p Fo(R)2497
2973 y Fz(10)2567 2961 y FA(\))558 3122 y(4.)41 b(F)-7
b(olding)28 b(of)f Fk(suml)p FA(\()p Fk(from)p FA(\()p
Fk(s)p FA(\()p Fk(X)p FA(\)\))p Fo(;)14 b Fk(Y)p FA(\))25
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3292 y Fz(11)2285 3280 y FA(\))558 3440 y(5.)41 b(F)-7
b(olding)28 b(of)f(the)h(rhs)f(of)h(rule)f Fo(R)1661
3452 y Fz(1)1726 3440 y FA(using)g Fo(R)2006 3452 y Fz(7)2043
3440 y FA(:)883 3598 y Fk(sum)p 1020 3598 V 29 w(prefix)p
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FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p FA(\))104 b(\()p Fo(R)2144
3610 y Fz(12)2215 3598 y FA(\))523 3758 y(Then,)28 b(the)g(transformed)
e(program)g Fp(R)1773 3770 y Fz(5)1838 3758 y FA(is)i(formed)f(b)n(y)g
(the)h(follo)n(wing)f(rules:)742 3906 y Fk(sum)p 879
3906 V 29 w(prefix)p FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p
FA(\))20 b Fp(!)k Fk(aux)p FA(\()p Fk(X)p Fo(;)14 b Fk(Y)p
FA(\))357 b(\()p Fo(R)2256 3918 y Fz(12)2327 3906 y FA(\))1034
4005 y Fk(aux)p FA(\()p Fk(X)p Fo(;)14 b Fk(0)p FA(\))21
b Fp(!)j Fk(0)634 b FA(\()p Fo(R)2256 4017 y Fz(8)2294
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b Fk(s)p FA(\()p Fk(Y)p FA(\)\))22 b Fp(!)i Fk(X)18 b
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(essen)n(tial)523 4526 y(in)h(the)g(ab)r(o)n(v)n(e)f(example.)g(It)h
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(transformation)e(ev)n(en)h(in)h(the)g(presence)e(of)i(non)n
(terminating)523 4725 y(functions)j(\(as)f(opp)r(osed)g(to)h([2]\).)648
4825 y(The)37 b(tupling)h(strategy)d(w)n(as)i(in)n(tro)r(duced)g(in)g
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4924 y(grams.)27 b(The)h(tupling)h(strategy)e(is)h(v)n(ery)f
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eop
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12 11 bop 523 448 a FA(computation)31 b(of)g(the)h(same)e(sub)r
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b(4.)42 b FA(The)28 b(\014b)r(onacci)f(n)n(um)n(b)r(ers)g(can)g(b)r(e)h
(computed)g(b)n(y)f(the)h(program)e Fp(R)3132 1008 y
Fz(0)3170 996 y FA(:)959 1129 y Fk(fib)p FA(\()p Fk(0)p
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Fo(R)2216 1141 y Fz(1)2253 1129 y FA(\))850 1229 y Fk(fib)p
FA(\()p Fk(s)p FA(\()p Fk(0)p FA(\)\))23 b Fp(!)g Fk(s)p
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y Fz(8)2910 2868 y FA(\))883 2968 y Fk(new)p 1020 2968
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1020 3614 V 29 w(aux)p FA(\()p Fp(h)p Fk(Z)1284 3626
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b(.)p eop
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13 12 bop 1474 417 a Fu(T)-7 b(able)29 b(1.)c Fx(Benc)n(hmark)g
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139 w Fx(7)p 1694 887 V 80 w(1.61)p 1854 887 V 59 w(0.17)p
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V 191 w(6\045)p 2871 887 V 1030 890 1843 4 v 1028 982
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2253 1076 V 60 w(2.26)p 2413 1076 V 178 w(-3\045)p 2871
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3131 y(rules)35 b(\()p Fo(R)q(w)885 3143 y Fz(2)923 3131
y FA(\),)i(the)f(absolute)f(run)n(times)h(\()p Fo(R)q(T)2000
3143 y Fz(2)2037 3131 y FA(\),)g(and)g(the)g(sp)r(eedups)g(ac)n(hiev)n
(ed)f(for)h(the)523 3230 y(transformed)g(programs.)f(All)j(the)g
(programs)d(ha)n(v)n(e)h(b)r(een)i(executed)f(b)n(y)g(using)g(T)-7
b(aste-)523 3330 y(Curry)g(,)25 b(whic)n(h)g(is)g(a)h(publicly)f(a)n(v)
-5 b(ailable)25 b(in)n(terpreter)f(for)h(a)g(subset)g(of)h(Curry)e
([16].)h(Times)523 3430 y(are)g(expressed)f(in)i(seconds)f(and)g(are)g
(the)h(a)n(v)n(erage)d(of)i(10)g(executions.)g(W)-7 b(e)26
b(note)g(that)g(our)523 3529 y(\(automatic\))d(comp)r(osition)g
(strategy)f(p)r(erforms)g(w)n(ell)h(w.r.t.)h(the)f(\014rst)g(four)g(b)r
(enc)n(hmarks.)523 3629 y(They)34 b(are)f(classical)g(examples)g(in)h
(whic)n(h)g(comp)r(osition)f(is)h(able)g(to)g(p)r(erform)f(an)h
(e\013ec-)523 3729 y(tiv)n(e)e(optimization)f(\()p Fk(sumprefix)d
FA(is)k(describ)r(ed)f(in)h(Example)f(3,)g(while)h Fk(doubleappend)p
FA(,)523 3828 y Fk(lengthapp)p FA(,)h(and)k Fk(doubleflip)c
FA(are)j(t)n(ypical)h(functional)g(programs)e(to)i(illustrate)f(de-)523
3928 y(forestation)29 b([30]\).)g(Regarding)f(the)i(last)f(b)r(enc)n
(hmark)g Fk(fibprefix)p FA(,)d(whic)n(h)k(is)f(similar)g(to)523
4027 y Fk(sumprefix)d FA(but)k(it)g(sums)g(Fib)r(onacci)f(n)n(um)n(b)r
(ers)g(instead)h(of)f(natural)g(n)n(um)n(b)r(ers,)g(a)g(slo)n(w-)523
4127 y(do)n(wn)18 b(has)h(b)r(een)g(pro)r(duced)g(\(due)g(to)g(an)g
(incorrect)e(folding,)i(whic)n(h)g(added)g(a)f(new)h(function)523
4227 y(call)31 b(to)h(the)g(recursion\).)e(In)i(this)g(case)f(a)g
(tupling)h(strategy)f(is)g(mandatory)g(to)g(succeed,)523
4326 y(as)c(exp)r(ected.)648 4426 y(In)c(general,)f(the)h(transformed)f
(programs)f(cannot)h(b)r(e)i(guaran)n(teed)d(to)i(b)r(e)g(faster)g
(than)523 4526 y(the)e(original)f(ones,)g(since)h(there)g(is)f(a)h
(trade-o\013)f(b)r(et)n(w)n(een)h(the)g(smaller)f(amoun)n(t)h(of)f
(compu-)523 4625 y(tation)28 b(needed)f(after)h(the)g(transformation)e
(\(when)i(guided)g(b)n(y)f(appropriate)f(strategies\))523
4725 y(and)31 b(the)h(larger)d(n)n(um)n(b)r(er)i(of)h(deriv)n(ed)e
(rules.)h(Nev)n(ertheless,)f(our)h(exp)r(erimen)n(ts)g(seem)g(to)523
4825 y(substan)n(tiate)21 b(that)g(the)h(smaller)e(computations)h(mak)n
(e)f(up)i(for)e(the)i(o)n(v)n(erhead)d(of)i(c)n(hec)n(king)523
4924 y(the)28 b(applicabilit)n(y)f(of)h(the)g(larger)d(n)n(um)n(b)r(er)
j(of)f(rules)g(in)h(the)g(deriv)n(ed)f(programs.)p eop
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14 13 bop 523 448 a Fs(5)112 b(Conclusions)523 640 y
FA(The)40 b(de\014nition)h(of)f(a)g(fold/unfold)g(framew)n(ork)e(for)h
(the)i(optimization)f(of)g(functional)523 740 y(logic)35
b(programs)e(w)n(as)i(an)g(op)r(en)h(problem)f(mark)n(ed)g(in)h([24])f
(as)g(p)r(ending)h(researc)n(h.)e(W)-7 b(e)523 840 y(ha)n(v)n(e)23
b(presen)n(ted)g(a)h(transformation)f(metho)r(dology)g(for)g(lazy)h
(functional)g(logic)f(programs)523 939 y(preserving)32
b(the)h(seman)n(tics)g(of)g(b)r(oth)h(v)-5 b(alues)33
b(and)g(answ)n(ers)e(computed)j(b)n(y)f(an)g(e\016cien)n(t)523
1039 y(\(curren)n(tly)28 b(the)h(b)r(est\))h(op)r(erational)d(mec)n
(hanism.)h(F)-7 b(or)29 b(pro)n(ving)e(correctness,)g(w)n(e)h(exten-)
523 1138 y(siv)n(ely)d(exploit)h(the)h(existing)f(results)f(from)h
(Huet)h(and)f(Levy's)g(theory)f(of)h(needed)h(reduc-)523
1238 y(tions)34 b([18])g(and)g(the)g(wide)h(literature)e(ab)r(out)h
(completeness)g(of)g(needed)g(narro)n(wing)e([5])523
1338 y(\(rather)26 b(than)h(striving)f(an)h(ad-ho)r(c)f(pro)r(of)6
b(\).)27 b(W)-7 b(e)27 b(ha)n(v)n(e)f(sho)n(wn)g(that)h(the)h
(transformation)523 1437 y(pro)r(cess)37 b(k)n(eeps)f(the)j(inductiv)n
(ely)e(sequen)n(tial)g(structure)g(of)h(programs.)e(W)-7
b(e)38 b(ha)n(v)n(e)e(also)523 1537 y(illustrated)26
b(with)g(sev)n(eral)e(examples)h(that)h(the)g(transformation)e(pro)r
(cess)h(can)g(b)r(e)h(guided)523 1637 y(b)n(y)c(appropriate)f
(strategies)f(whic)n(h)j(lead)f(to)g(e\013ectiv)n(e)g(impro)n(v)n(emen)
n(ts.)f(Our)g(exp)r(erimen)n(ts)523 1736 y(sho)n(w)26
b(that)h(our)f(transformation)e(framew)n(ork)h(com)n(bines)h(in)h(a)f
(useful)h(and)g(e\013ectiv)n(e)f(w)n(a)n(y)523 1836 y(the)d(systematic)
f(instan)n(tiation)f(of)i(calls)e(during)h(unfolding)h(\(b)n(y)f
(virtue)g(of)g(the)h(logic)f(com-)523 1936 y(p)r(onen)n(t)30
b(of)f(the)i(needed)e(narro)n(wing)f(mec)n(hanism\))h(with)h(the)g(p)r
(o)n(w)n(er)f(of)g(the)h(abstraction)523 2035 y(transformations)i
(\(thanks)h(to)h(the)f(functional)h(dimension\).)g(W)-7
b(e)34 b(ha)n(v)n(e)e(also)h(presen)n(ted)523 2135 y(an)h(implemen)n
(tation)h(whic)n(h)g(allo)n(ws)e(us)i(to)g(p)r(erform)f(automatically)f
(the)j(comp)r(osition)523 2234 y(strategy)31 b(as)g(w)n(ell)h(as)f(to)h
(p)r(erform)f(all)h(basic)f(transformations)f(in)i(a)g
(semi-automatized)523 2334 y(w)n(a)n(y)-7 b(.)23 b(The)h(m)n
(ulti-paradigm)f(language)g(Curry)g([15,)13 b(17])23
b(is)h(an)g(extension)g(of)g(Hask)n(ell)f(with)523 2434
y(features)i(for)g(logic)g(and)g(concurren)n(t)f(programming.)f(The)j
(results)f(in)h(this)f(pap)r(er)g(can)h(b)r(e)523 2533
y(applied)d(to)g(optimize)h(a)e(large)g(class)h(of)g(k)n(ernel)f
(\(i.e.,)i(non)f(concurren)n(t\))f(Curry)g(programs.)523
2775 y Fg(Ac)m(kno)m(wledgemen)m(ts)523 2934 y FA(W)-7
b(e)26 b(thank)f(Iv)-5 b(an)26 b(Ziliotto)f(and)g(Cesar)f(F)-7
b(erri)25 b(for)g(ha)n(ving)g(implemen)n(ted)h(the)g(transformer)523
3034 y Fb(Synth)h FA(under)h(SICStus)g(Prolog)d(and)i(for)h(the)g(ev)-5
b(aluation)27 b(of)g(the)h(b)r(enc)n(hmarks.)523 3292
y Fs(References)561 3474 y Fx(1.)43 b(M.)18 b(Alpuen)n(te,)g(M.)h(F)-6
b(alasc)n(hi,)19 b(C.)g(F)-6 b(erri,)19 b(G.)f(Moreno,)i(G.)e(Vidal,)h
(and)f(I.)g(Ziliotto.)24 b(The)18 b(T)-6 b(rans-)663
3565 y(formation)23 b(System)d Fa(Synth)p Fx(.)29 b(T)-6
b(ec)n(hnical)23 b(rep)r(ort)f(DSIC-I)r(I/16/99,)i(DSIC,)e(1999.)30
b(Av)l(ailable)663 3657 y(from)25 b(URL:)g Fv
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Fx(.)561 3746 y(2.)43 b(M.)25 b(Alpuen)n(te,)f(M.)h(F)-6
b(alasc)n(hi,)26 b(G.)f(Moreno,)h(and)e(G.)h(Vidal.)33
b(Safe)25 b(F)-6 b(olding/Unfolding)26 b(with)663 3837
y(Conditional)h(Narro)n(wing.)35 b(In)25 b(H.)h(Heering)f(M.)i(Han)n
(us)e(and)g(K.)h(Meink)n(e,)g(editors,)g Ft(Pr)l(o)l(c.)i(of)663
3929 y(ALP'97)p Fx(,)e(Springer)f(LNCS)h(1298,)h(pages)g(1{15,)g(1997.)
561 4018 y(3.)43 b(M.)22 b(Alpuen)n(te,)f(M.)h(F)-6 b(alasc)n(hi,)23
b(G.)g(Moreno,)g(and)e(G.)h(Vidal.)29 b(A)21 b(Transformation)i(System)
d(for)663 4110 y(Lazy)27 b(Functional)h(Logic)h(Programs.)40
b(T)-6 b(ec)n(hnical)29 b(rep)r(ort,)f(DSIC,)f(UPV,)g(1999.)41
b(Av)l(ailable)663 4201 y(from)25 b(URL:)g Fv
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Fx(.)561 4290 y(4.)43 b(S.)37 b(An)n(to)n(y)-6 b(.)67
b(De\014nitional)38 b(T)-6 b(rees.)70 b(In)36 b Ft(Pr)l(o)l(c.)j(of)f
(ALP'92)p Fx(,)f(Springer)g(LNCS)g(632,)i(pages)663 4382
y(143{157,)28 b(1992.)561 4471 y(5.)43 b(S.)21 b(An)n(to)n(y)-6
b(,)21 b(R.)g(Ec)n(hahed,)h(and)g(M.)g(Han)n(us.)27 b(A)22
b(Needed)f(Narro)n(wing)i(Strategy)-6 b(.)27 b(In)21
b Ft(Pr)l(o)l(c.)k(21st)663 4563 y(A)n(CM)i(Symp.)g(on)h(Principles)g
(of)f(Pr)l(o)l(gr)l(amming)h(L)l(anguages)p Fx(,)g(pages)f(268{279,)i
(1994.)561 4652 y(6.)43 b(F.)29 b(Baader)g(and)g(T.)g(Nipk)n(o)n(w.)43
b Ft(T)-6 b(erm)30 b(R)l(ewriting)h(and)g(A)n(l)t(l)e(That)p
Fx(.)44 b(Cam)n(bridge)29 b(Univ)n(ersit)n(y)663 4744
y(Press,)e(1998.)561 4833 y(7.)43 b(A.)32 b(Bossi)i(and)e(N.)g(Co)r
(cco.)56 b(Basic)34 b(T)-6 b(ransformation)34 b(Op)r(erations)f(whic)n
(h)f(preserv)n(e)g(Com-)663 4924 y(puted)24 b(Answ)n(er)i
(Substitutions)f(of)i(Logic)f(Programs.)36 b Ft(JLP)p
Fx(,)26 b(16:47{87,)j(1993.)p eop
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15 14 bop 561 448 a Fx(8.)43 b(R.M.)23 b(Burstall)h(and)f(J.)h
(Darlington.)30 b(A)23 b(T)-6 b(ransformation)24 b(System)d(for)j(Dev)n
(eloping)f(Recur-)663 540 y(siv)n(e)i(Programs.)36 b
Ft(Journal)28 b(of)f(the)i(A)n(CM)p Fx(,)c(24\(1\):44{67,)k(1977.)561
624 y(9.)43 b(W.)22 b(Chin.)28 b(T)-6 b(o)n(w)n(ards)23
b(an)f(Automated)f(Tupling)h(Strategy)-6 b(.)28 b(In)21
b Ft(Pr)l(o)l(c.)k(of)f(Partial)g(Evaluation)663 715
y(and)k(Semantics-Base)l(d)i(Pr)l(o)l(gr)l(am)f(Manipulation)p
Fx(,)d(pages)g(119{132.)j(A)n(CM,)d(1993.)523 800 y(10.)43
b(W.)28 b(Chin,)g(A.)f(Goh,)h(and)g(S.)f(Kho)r(o.)41
b(E\013ectiv)n(e)27 b(Optimisation)h(of)g(Multiple)h(Tra)n(v)n(ersals)g
(in)663 891 y(Lazy)24 b(Languages.)34 b(In)24 b Ft(Pr)l(o)l(c.)j(of)f
(PEPM'99)g(\(T)-6 b(e)l(chnic)l(al)27 b(R)l(ep)l(ort)h(BRICS-NS-99-1\))
p Fx(,)f(pages)663 982 y(119{130.)h(Univ)n(ersit)n(y)d(of)i(Aarh)n(us,)
e(DK,)g(1999.)523 1067 y(11.)43 b(J.)29 b(Darlington.)44
b(Program)29 b(transformation.)44 b(In)28 b Ft(F)-6 b(unctional)31
b(Pr)l(o)l(gr)l(amming)g(and)g(its)f(Ap-)663 1158 y(plic)l(ations)p
Fx(,)c(pages)h(193{215.)h(Cam)n(bridge)e(Univ)n(ersit)n(y)f(Press,)i
(1982.)523 1242 y(12.)43 b(M.)38 b(S.)h(F)-6 b(eather.)71
b(A)38 b(Surv)n(ey)e(and)i(Classi\014cation)j(of)e(some)f(Program)h
(Transformation)663 1334 y(Approac)n(hes)25 b(and)g(Tec)n(hniques.)35
b(In)25 b Ft(IFIP'87)p Fx(,)g(pages)h(165{195,)j(1987.)523
1418 y(13.)43 b(P)-6 b(.)28 b(A.)f(Gardner)h(and)f(J.)i(C.)f
(Shepherdson.)40 b(Unfold/fold)29 b(T)-6 b(ransformation)28
b(of)h(Logic)g(Pro-)663 1509 y(grams.)65 b(In)35 b(J.L)i(Lassez)g(and)e
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(in)663 1601 y(Honor)28 b(of)f(A)n(lan)h(R)l(obinson)p
Fx(,)e(pages)h(565{583.)i(The)d(MIT)g(Press,)h(Cam)n(bridge,)f(MA,)f
(1991.)523 1685 y(14.)43 b(M.)30 b(Han)n(us.)45 b(The)29
b(In)n(tegration)h(of)g(Functions)f(in)n(to)h(Logic)g(Programming:)g
(From)f(Theory)663 1776 y(to)d(Practice.)35 b Ft(Journal)29
b(of)e(L)l(o)l(gic)h(Pr)l(o)l(gr)l(amming)p Fx(,)f(19&20:583{628,)j
(1994.)523 1861 y(15.)43 b(M.)24 b(Han)n(us.)29 b(A)23
b(Uni\014ed)f(Computation)h(Mo)r(del)h(for)g(F)-6 b(unctional)23
b(and)g(Logic)i(Programming.)663 1952 y(In)g Ft(Pr)l(o)l(c.)j(of)f(the)
h(24th)h(A)n(CM)e(POPL)p Fx(,)f(pages)g(80{93.)i(A)n(CM,)e(New)g(Y)-6
b(ork,)25 b(1997.)523 2037 y(16.)43 b(M.)22 b(Han)n(us,)g(H.)g(Kuc)n
(hen,)g(and)g(J.J.)h(Moreno-Na)n(v)l(arro.)29 b(Curry:)23
b(A)e(Truly)h(F)-6 b(unctional)23 b(Logic)663 2128 y(Language.)55
b(In)32 b Ft(Pr)l(o)l(c.)i(ILPS'95)f(Workshop)i(on)f(Visions)g(for)f
(the)i(F)-6 b(utur)l(e)35 b(of)f(L)l(o)l(gic)g(Pr)l(o-)663
2219 y(gr)l(amming)p Fx(,)26 b(pages)g(95{107,)j(1995.)523
2304 y(17.)43 b(M.)30 b(Han)n(us)f(\(ed.\).)47 b(Curry:)30
b(An)f(In)n(tegrated)h(Functional)g(Logic)h(Language.)48
b(Av)l(ailable)31 b(at)663 2395 y Fv(http://www-i2.informatik.rwth-a)q
(ache)q(n.de)q(/~han)q(us/c)q(urry)q Fx(,)g(1999.)523
2479 y(18.)43 b(G.)31 b(Huet)f(and)g(J.J.)j(L)n(\023)-36
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(Systems,)f(P)n(art)h(I)663 2571 y(+)d(I)r(I.)43 b(In)28
b(J.L.)i(Lassez)g(and)e(G.D.)h(Plotkin,)g(editors,)h
Ft(Computational)h(L)l(o)l(gic)g({)f(Essays)i(in)663
2662 y(Honor)c(of)f(A)n(lan)h(R)l(obinson)p Fx(,)e(pages)h(395{443,)i
(1992.)523 2746 y(19.)43 b(T.)24 b(Johnsson.)31 b(Lam)n(b)r(da)23
b(Lifting:)h(Transforming)h(Programs)f(to)f(Recursiv)n(e)h(Equations.)
31 b(In)663 2838 y Ft(FPLCA'85)p Fx(,)26 b(pages)g(190{203.)j(Springer)
c(LNCS)h(201,)h(1985.)523 2922 y(20.)43 b(T.)37 b(Ka)n(w)n(am)n(ura)f
(and)h(T.)g(Kanamori.)67 b(Preserv)l(ation)38 b(of)f(Stronger)g(Equiv)l
(alence)f(in)h(Un-)663 3013 y(fold/Fold)27 b(Logic)g(Program)f(T)-6
b(ransformation.)35 b Ft(TCS)p Fx(,)26 b(75:139{156,)k(1990.)523
3098 y(21.)43 b(M.J.)38 b(Maher.)68 b(A)36 b(Transformation)i(System)d
(for)j(Deductiv)n(e)d(Database)j(Mo)r(dules)g(with)663
3189 y(Perfect)26 b(Mo)r(del)h(Seman)n(tics.)34 b Ft(The)l(or)l(etic)l
(al)29 b(Computer)g(Scienc)l(e)p Fx(,)d(110\(2\):377{403,)31
b(1993.)523 3273 y(22.)43 b(J.J.)28 b(Moreno-Na)n(v)l(arro)f(and)f(M.)h
(Ro)r(dr)-9 b(\023)-30 b(\020guez-Artalejo.)39 b(Logic)28
b(Programming)e(with)h(F)-6 b(unc-)663 3365 y(tions)26
b(and)f(Predicates:)i(The)f(language)h(Bab)r(el.)36 b
Ft(JLP)p Fx(,)26 b(12\(3\):191{224,)k(1992.)523 3449
y(23.)43 b(A.)32 b(P)n(ettorossi)j(and)d(M.)i(Proietti.)56
b(T)-6 b(ransformation)34 b(of)g(Logic)g(Programs:)f(Foundations)663
3540 y(and)25 b(Tec)n(hniques.)34 b Ft(Journal)29 b(of)e(L)l(o)l(gic)h
(Pr)l(o)l(gr)l(amming)p Fx(,)f(19&20:261{320,)j(1994.)523
3625 y(24.)43 b(A.)27 b(P)n(ettorossi)j(and)d(M.)i(Proietti.)42
b(Rules)27 b(and)h(Strategies)h(for)f(Transforming)h(Functional)663
3716 y(and)c(Logic)i(Programs.)35 b Ft(A)n(CM)27 b(Computing)h(Surveys)
p Fx(,)g(28\(2\):360{414,)i(1996.)523 3800 y(25.)43 b(A.)27
b(Ro)n(yc)n(houdh)n(ury)-6 b(,)24 b(K.)j(Nara)n(y)n(an)g(Kumar,)f(C.R.)
h(Ramakrishnan,)g(and)f(I.V.)h(Ramakrish-)663 3892 y(nan.)35
b(A)25 b(P)n(arameterized)h(Unfold/Fold)h(T)-6 b(ransformation)27
b(F)-6 b(ramew)n(ork)26 b(for)g(De\014nite)g(Logic)663
3983 y(Programs.)35 b(In)25 b Ft(Pr)l(o)l(c.)j(of)f(PPDP'99)p
Fx(,)f(Springer)f(LNCS,)h(1999.)i(T)-6 b(o)26 b(app)r(ear.)523
4067 y(26.)43 b(D.)38 b(Sands.)72 b(T)-6 b(otal)39 b(Correctness)h(b)n
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(of)663 4159 y(Functional)26 b(Programs.)35 b Ft(A)n(CM)27
b(T)-6 b(oPL)l(aS)p Fx(,)27 b(18\(2\):175{234,)j(Marc)n(h)c(1996.)523
4243 y(27.)43 b(W.L.)29 b(Sc)n(herlis.)45 b(Program)29
b(Impro)n(v)n(emen)n(t)d(b)n(y)i(In)n(ternal)g(Sp)r(ecialization.)47
b(In)28 b Ft(Pr)l(o)l(c.)j(of)f(8th)663 4334 y(A)n(CM)d(POPL)p
Fx(,)f(pages)g(41{49.)i(A)n(CM)e(Press,)h(1981.)523 4419
y(28.)43 b(H.)35 b(T)-6 b(amaki)36 b(and)f(T.)i(Sato.)65
b(Unfold/Fold)36 b(Transformations)i(of)e(Logic)h(Programs.)66
b(In)663 4510 y(S.)25 b(T\177)-38 b(arnlund,)26 b(editor,)g
Ft(Pr)l(o)l(c.)i(of)g(2nd)g(ICLP)p Fx(,)d(pages)i(127{139,)h(1984.)523
4595 y(29.)43 b(P)-6 b(.L.)26 b(W)-6 b(adler.)37 b Ft(Listlessness)30
b(is)e(b)l(etter)i(than)f(Laziness)p Fx(.)37 b(Computer)26
b(Science)g(Departmen)n(t,)663 4686 y(CMU-CS-85-171,)19
b(Carnegie)h(Mellon)f(Univ)n(ertsit)n(y)-6 b(,)16 b(Pittsburgh,)i(P)-6
b(A,)18 b(1985.)23 b(Ph.D.)18 b(Thesis.)523 4770 y(30.)43
b(P)-6 b(.L.)22 b(W)-6 b(adler.)28 b(Deforestation:)23
b(Transforming)g(programs)f(to)f(eliminate)h(trees.)28
b Ft(The)l(or)l(etic)l(al)663 4862 y(Computer)g(Scienc)l(e)p
Fx(,)f(73:231{248,)j(1990.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF

Event Timeline

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