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b(.)g(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)68 b(155)639 2658 y(6.2)84 b(The)28 b(Con\015uence)f (Hierarc)n(h)n(y)49 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.) h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(166)639 2760 y(6.3)84 b(Summary)67 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(178)515 2957 y FN(7)76 b(Conditional)31 b(Rewrite)g(Systems)1199 b(179)639 3059 y F0(7.1)84 b(Basic)27 b(Notions)44 b(.)d(.)h(.)g(.)f(.) h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)68 b(180)639 3161 y(7.2)84 b(E\013ectiv)n(e)27 b(T)-7 b(ermination)82 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(186)830 3264 y(7.2.1)94 b(Systems)27 b(without)h(Extra)f(V)-7 b(ariables)48 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(187)830 3366 y(7.2.2)94 b(T)-7 b(ransforming)26 b(1-CTRSs)82 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(190)830 3468 y(7.2.3)94 b(Left-Linear)27 b(Normal)f(1-CTRSs)57 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(195)830 3571 y(7.2.4)94 b(Summary:)27 b(Metho)r(ds)h(for)f(1-CTRSs)49 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(204)830 3673 y(7.2.5)94 b(Extra)26 b(V)-7 b(ariables)27 b(on)g(Righ)n(t-Hand)g (Sides)i(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(204)830 3775 y(7.2.6)94 b(Quasi-Reductiv)n(e)26 b(Deterministic)i(3-CTRSs)36 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)68 b(206)830 3878 y(7.2.7)94 b(T)-7 b(ransforming)26 b(3-CTRSs)82 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(211)830 3980 y(7.2.8)94 b(Innermost)27 b(T)-7 b(ermination)27 b(of)g FP(U)9 b F0(\()p FO(R)p F0(\))85 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)68 b(215)830 4082 y(7.2.9)94 b(Quasi-Simplifying)27 b(Deterministic)h(3-CTRSs)50 b(.)42 b(.)g(.)f(.)h(.)f(.)68 b(225)830 4185 y(7.2.10)52 b(Summary:)27 b(Metho)r(ds)h(for)f(3-CTRSs) 49 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(229)639 4287 y(7.3)84 b(Con\015uence)28 b(of)f(T)-7 b(erminating)27 b(Systems)34 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)68 b(230)639 4389 y(7.4)84 b(Orthogonal)26 b(Systems)70 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(232)830 4492 y(7.4.1)94 b(Normal)27 b(2-CTRSs)38 b(.)k(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(233)830 4594 y(7.4.2)94 b(Almost)28 b(Normal)e(2-CTRSs)73 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(235)830 4696 y(7.4.3)94 b(Lev)n(el-Con\015uence)26 b(for)h(3-CTRSs)84 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) 68 b(237)830 4799 y(7.4.4)94 b(CTRSs)27 b(with)i(Strict)e(Equalit)n(y) 38 b(.)j(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(237)p eop %%Page: 15 3 15 14 bop 2671 208 a FV(Con)n(ten)n(ts)154 b(xv)515 415 y FN(8)76 b(Mo)s(dularit)m(y)1951 b(243)639 515 y F0(8.1)84 b(Di\013eren)n(t)28 b(Kinds)g(of)f(Com)n(binations)73 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(243)639 614 y(8.2)84 b(A)28 b(Short)f(History)g(of)h(Mo)r(dularit)n (y)f(Results)43 b(.)e(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(248)830 714 y(8.2.1)94 b(The)28 b(Con\015uence)f(Hierarc)n(h)n(y)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(248)830 814 y(8.2.2)94 b(The)28 b(T)-7 b(ermination)27 b(Hierarc)n(h)n(y)57 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)68 b(252)830 913 y(8.2.3)94 b(Com)n(bined)27 b(Prop)r(erties)f(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)68 b(257)639 1013 y(8.3)84 b FO(C)874 1025 y FM(E)919 1013 y F0(-T)-7 b(ermination)26 b(and)i(the)g(T)-7 b(ermination)27 b(Hierarc)n(h)n(y)52 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)68 b(259)639 1112 y(8.4)84 b(Disjoin)n(t)28 b(Unions)44 b(.)e(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(263)830 1212 y(8.4.1)94 b FO(C)1140 1224 y FM(E)1212 1212 y F0(and)28 b(Simple)g(T)-7 b(ermination)58 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) f(.)68 b(265)830 1312 y(8.4.2)94 b(DP)27 b(Quasi-Simple)g(T)-7 b(ermination)82 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(275)830 1411 y(8.4.3)94 b(P)n(ersistence)26 b(vs.)h(Mo)r(dularit)n(y) f(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(281)639 1511 y(8.5)84 b(Constructor-Sharing)25 b(Systems)65 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) 68 b(282)830 1611 y(8.5.1)94 b(Con\015uence)76 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)68 b(285)830 1710 y(8.5.2)94 b(Unique)28 b(Normalization)58 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)68 b(291)639 1810 y(8.6)84 b(Hierarc)n(hical)26 b(Com)n(binations)33 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(294)830 1910 y(8.6.1)94 b FO(C)1140 1922 y FM(E)1212 1910 y F0(and)28 b(Simple)g(T)-7 b(ermination)58 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) f(.)68 b(294)830 2009 y(8.6.2)94 b(Innermost)27 b(T)-7 b(ermination)84 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)68 b(309)830 2109 y(8.6.3)94 b(Con\015uence)76 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.) f(.)h(.)g(.)f(.)h(.)f(.)68 b(313)639 2208 y(8.7)84 b(Conditional)27 b(Systems)57 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(317)515 2391 y FN(9)76 b(Graph)33 b(Rewriting)1707 b(325)639 2491 y F0(9.1)84 b(A)28 b(Uniform)g(F)-7 b(ramew)n(ork)25 b(for)j(T)-7 b(erm,)27 b(Graph,)830 2590 y(and)h(Noncop)n(ying)e (Rewriting)e(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h (.)g(.)f(.)h(.)f(.)68 b(327)639 2690 y(9.2)84 b(An)28 b(Application)g(of)g(the)g(F)-7 b(ramew)n(ork)26 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(333)639 2790 y(9.3)84 b(F)-7 b(urther)28 b(T)-7 b(opics)73 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.) f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(337)639 2889 y(9.4)84 b(Conditional)27 b(Graph)g(Rewriting)58 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(338)830 2989 y(9.4.1)94 b(Adequacy)50 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(340)830 3088 y(9.4.2)94 b(Con\015uence)76 b(.)41 b(.)h(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(343)515 3271 y FN(10)28 b(Pro)m(ving)k(T)-8 b(ermination)31 b(of)h(Logic)f(Programs)753 b(351)639 3371 y F0(10.1)42 b(Logic)27 b(Programs)35 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f (.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(354)639 3470 y(10.2)42 b(The)28 b(T)-7 b(ransformation)71 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.) h(.)g(.)f(.)h(.)f(.)68 b(355)639 3570 y(10.3)42 b(Comparison)26 b(of)i(the)g(Di\013eren)n(t)g(Metho)r(ds)d(.)42 b(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(362)639 3670 y(10.4)42 b(Completeness)50 b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h (.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(364)639 3769 y(10.5)42 b(Pro)n(ving)26 b(T)-7 b(ermination)27 b(Automatically)83 b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)68 b(369)639 3869 y(10.6)42 b(Empirical)27 b(Results)i(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)68 b(371)639 3968 y(10.7)42 b(Summary)27 b(and)h(Related)f(W)-7 b(ork)21 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.) g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)68 b(376)515 4151 y FN(A)52 b(Krusk)-5 b(al's)31 b(Theorem)1628 b(379)639 4251 y F0(A.1)64 b(P)n(artial)26 b(W)-7 b(ell-Orderings)45 b(.)c(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g (.)f(.)h(.)f(.)68 b(379)639 4350 y(A.2)c(A)28 b(Pro)r(of)f(of)g(Krusk) -5 b(al's)26 b(Theorem)h(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f (.)h(.)g(.)f(.)h(.)f(.)68 b(381)639 4533 y FN(References)1973 b(385)639 4716 y(Index)2186 b(407)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF