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<COLUMN=351><FONT=0><FONT=11><COLUMN=1182>CONTENTS<SP><SP><SP><SP>
<SP>i<FONT=0><CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=1><COLUMN=609>T<SP>A<SP>B<SP>L<SP>E<SP><SP><SP>O
<SP>F<SP><SP><SP>C<SP>O<SP>N<SP>T<SP>E<SP>N<SP>T<SP>S<CR><LF>
<CR><LF>
<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><FONT=1>PREFACE<COLUMN=1349>i<FONT=0><CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><FONT=1>CHAPTER<COLUMN=1276>PAGE<FONT=0><CR><LF>
<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=387>1<COLUMN=435>SYMBOLIC<SP>EXPRESSIONS
<COLUMN=1344>1<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=483>1.1<COLUMN=555>Introduction
<COLUMN=711><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP>
<SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.
<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP><SP>.<SP>
<SP>.<SP><SP>.<SP><COLUMN=1334><SP>1<CR><LF>
<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><FONT=1>INDEX<COLUMN=1345>7<FONT=0><CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11>1.<COLUMN=1041>Symbolic<SP>expressions
<SP><SP><SP><SP><SP>1<FONT=0><CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=1186><FONT=1>CHAPTER<SP>1<FONT=0><CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><FONT=9>Symbolic<SP>Expressions<FONT=0><CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=744><FONT=1>1.1<SP><SP>Introduction
<FONT=0><CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>This<COL INC 18>book<COL INC 17>is<COL INC 18>a
<COL INC 18>study<COL INC 17>of<COL INC 18>data<COL INC 18>structures
<COL INC 17>and<COL INC 18>programming<COL INC 18>languages;
<COL INC 17>in<CR><LF>
<COLUMN=351><FONT=0>particular<COL INC 21>it<COL INC 20>is
<COL INC 21>a<COL INC 21>study<COL INC 20>of<COL INC 21>data
<COL INC 20>structures<COL INC 21>and<COL INC 21>programming
<COL INC 20>languages<CR><LF>
<COLUMN=351><FONT=0>centered<COL INC 11>around<COL INC 11>the
<COL INC 11>language<COL INC 11>LISP.<FONT=7><SP>1<FONT=0>
<COL INC 11>We<COL INC 11>will<COL INC 11>study<COL INC 11>many
<COL INC 11>of<COL INC 11>the<COL INC 11>formal<COL INC 10>and<CR><LF>
<COLUMN=351><FONT=0>theoretical<COL INC 12>aspects<COL INC 12>of
<COL INC 11>languages<COL INC 12>and<COL INC 12>data<COL INC 12>structures
<COL INC 11>as<COL INC 12>well<COL INC 12>as<COL INC 12>examining
<COL INC 11>the<CR><LF>
<COLUMN=351><FONT=0>practical<COL INC 18>applications<COL INC 18>of
<COL INC 18>data<COL INC 18>structures.<COL INC 19>We<COL INC 18>will
<COL INC 18>show<COL INC 18>that<COL INC 18>this<COL INC 19>area
<COL INC 18>of<CR><LF>
<COLUMN=351><FONT=0>computer<COL INC 12>science<COL INC 12>is
<COL INC 12>a<COL INC 12>discipline<COL INC 12>of<COL INC 12>importance
<COL INC 12>and<COL INC 12>beauty,<COL INC 12>worthy<COL INC 13>of
<COL INC 12>careful<CR><LF>
<COLUMN=351><FONT=0>study.<COL INC 17><SP>How<COL INC 18>are
<COL INC 17>we<COL INC 17>to<COL INC 18>proceed?<FONT=7><SP>2<FONT=0>
<COL INC 17>We<COL INC 17>must<COL INC 18>not<COL INC 17>pursue
<COL INC 17>theory<COL INC 18>and<COL INC 17>rigor<CR><LF>
<COLUMN=351><FONT=0>without<COL INC 25>proper<COL INC 25>regard
<COL INC 25>for<COL INC 25>practice.<COL INC 26>Our<COL INC 25>study
<COL INC 25>is<COL INC 25>not<COL INC 25>that<COL INC 26>of
<COL INC 25>pure<CR><LF>
<COLUMN=351><FONT=0>mathematics;<COL INC 17>our<COL INC 18>results
<COL INC 17>will<COL INC 17>have<COL INC 18>applications<COL INC 17>in
<COL INC 18>everyday<COL INC 17>programming<CR><LF>
<COLUMN=351><FONT=0>practice.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>Number<COL INC 21>theory<COL INC 21>studies
<COL INC 21>properties<COL INC 21>of<COL INC 21>a<COL INC 21>certain
<COL INC 21>class<COL INC 21>of<COL INC 21>operations<CR><LF>
<COLUMN=351><FONT=0>definable<COL INC 10>over<COL INC 11>the
<COL INC 10>set<COL INC 10><FONT=12>N<FONT=0><COL INC 11>of
<COL INC 10>non-negative<COL INC 10>integers<COL INC 11>also
<COL INC 10>called<COL INC 11>natural<COL INC 10>numbers.<CR><LF>
<COLUMN=351><FONT=0>A<COL INC 12>very<COL INC 12>formal<COL INC 11>presentation
<COL INC 12>might<COL INC 12>begin<COL INC 11>with<COL INC 12>a
<COL INC 12>construction<COL INC 11>of<COL INC 12><FONT=12>N<FONT=0>
<COL INC 12>from<COL INC 11>more<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>________________<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423><FONT=7><SP>1<FONT=0>However,
<COL INC 12>this<COL INC 12>is<COL INC 13>not<COL INC 12>a
<COL INC 12>manual<COL INC 12>to<COL INC 13>help<COL INC 12>you
<COL INC 12>become<COL INC 12>a<COL INC 13>proficient<COL INC 12>LISP
<CR><LF>
<COLUMN=351><FONT=0>coder.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423><FONT=7><SP>2<FONT=0>How<COL INC 10>do
<COL INC 10>we<COL INC 10>introduce<COL INC 10>rigor<COL INC 10>into
<COL INC 10>a<COL INC 10>field<COL INC 10>whose<COL INC 10>countenance
<COL INC 10>is<COL INC 10>as<COL INC 11><FONT=2>ad<COL INC 10>hoc
<FONT=0><CR><LF>
<COLUMN=351><FONT=0>and<COL INC 12>diverse<COL INC 11>as<COL INC 12>that
<COL INC 12>of<COL INC 11>programming?<COL INC 12>We<COL INC 11>must
<COL INC 12>bear<COL INC 12>in<COL INC 11>mind<COL INC 12>that
<COL INC 12>the<COL INC 11>results<CR><LF>
<COLUMN=351><FONT=0>of<SP>our<SP>studies<SP>are<SP>to<SP>have<SP>practical
<SP>applications.<CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11>2<SP><SP>Symbolic<SP>expressions
<COLUMN=1320>1.1<FONT=0><CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>primitive<COL INC 13>notions,<COL INC 13>but
<COL INC 13>it<COL INC 13>is<COL INC 14>usually<COL INC 13>assumed
<COL INC 13>that<COL INC 13>the<COL INC 13>reader<COL INC 13>is
<COL INC 14>familiar<COL INC 13>with<CR><LF>
<COLUMN=351><FONT=0>the<COL INC 14>fundamental<COL INC 14>properties
<COL INC 14>of<COL INC 14><FONT=12>N<FONT=0>.<COL INC 14><SP>In
<COL INC 14>either<COL INC 14>case<COL INC 14>the<COL INC 14>next
<COL INC 14>step<COL INC 14>would<COL INC 15>be<COL INC 14>to<CR><LF>
<COLUMN=351><FONT=0>define<SP>the<SP>class<SP>of<SP>operations<SP>which
<SP>we<SP>would<SP>allow<SP>on<SP>our<SP>domain.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>For<COL INC 12>most<COL INC 12>people
<COL INC 11>and<COL INC 12>most<COL INC 12>purposes,<COL INC 11>the
<COL INC 12>following<COL INC 12>characterization<COL INC 12>of
<COL INC 11>a<CR><LF>
<COLUMN=351><FONT=0>natural<SP>number<SP>is<SP>satisfactory:<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><FONT=1>I<FONT=0><COLUMN=399>A<SP>natural<SP>number
<SP>is<SP>a<SP>sequence<SP>of<SP>decimal<SP>digits.<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>The<COL INC 13>definition<COL INC 13>assumes
<COL INC 13>the<COL INC 13>terminology<COL INC 13>of<COL INC 13>"sequence",
<COL INC 13>"decimal"<COL INC 14>and<COL INC 13>"digit"<CR><LF>
<COLUMN=351><FONT=0>are<COL INC 15>known.<COL INC 16>If<COL INC 15>any
<COL INC 16>of<COL INC 15>these<COL INC 16>terms<COL INC 15>are
<COL INC 16><FONT=5>not<FONT=0><COL INC 15>understood,<COL INC 16>they
<COL INC 15>can<COL INC 16>be<COL INC 15>further<CR><LF>
<COLUMN=351><FONT=0>elaborated.<COL INC 22>However,<COL INC 23>this
<COL INC 22>process<COL INC 22>of<COL INC 23>explanation<COL INC 22>and
<COL INC 23>description<COL INC 22>must<CR><LF>
<COLUMN=351><FONT=0>terminate.<COL INC 28>We<COL INC 27>must
<COL INC 28>assume<COL INC 27>that<COL INC 28>some<COL INC 27>concepts
<COL INC 28>require<COL INC 28>no<COL INC 27>further<CR><LF>
<COLUMN=351><FONT=0>elaboration.<COL INC 25>The<COL INC 25>current
<COL INC 25>definition<COL INC 25>suffers<COL INC 25>from<COL INC 25>a
<COL INC 25>different<COL INC 25>kind<COL INC 25>of<CR><LF>
<COLUMN=351><FONT=0>inadequacy.<COL INC 12>It<COL INC 13>fails
<COL INC 12>to<COL INC 13>illuminate<COL INC 12>the<COL INC 13>relationships
<COL INC 12>between<COL INC 13>natural<COL INC 12>numbers.<CR><LF>
<COLUMN=351><FONT=0>The<COL INC 12>"meaning"<COL INC 13>of
<COL INC 12>the<COL INC 13>natural<COL INC 12>numbers<COL INC 13>is
<COL INC 12>missing.<COL INC 13>It<COL INC 12>is<COL INC 13>like
<COL INC 12>giving<COL INC 13>a<COL INC 12>person<CR><LF>
<COLUMN=351><FONT=0>an<COL INC 22>alphabet<COL INC 21>and<COL INC 22>rules
<COL INC 21>for<COL INC 22>forming<COL INC 21>syntactically
<COL INC 22>correct<COL INC 21>words<COL INC 22>but<COL INC 21>not
<CR><LF>
<COLUMN=351><FONT=0>supplying<SP>a<SP>dictionary<SP>which<SP>relates
<SP>these<SP>words<SP>to<SP>the<SP>person's<SP>vocabulary.<CR>
<LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>If<COL INC 33>pressed<COL INC 33>for
<COL INC 33>details<COL INC 34>we<COL INC 33>might<COL INC 33>attempt
<COL INC 33>a<COL INC 34>more<COL INC 33>elaborate<CR><LF>
<COLUMN=351><FONT=0>characterization<SP>like<SP>the<SP>following:<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=411><FONT=1>1.<FONT=0><SP><FONT=2>zero
<FONT=0><SP>is<SP>an<SP>element<SP>of<SP><FONT=12>N<FONT=0>.<CR>
<LINESPACE 3>
<COLUMN=351><FONT=0><FONT=1>II<FONT=0><SP><SP><SP><FONT=1>2.<FONT=0>
<SP>If<SP><FONT=2>n<FONT=0><SP>is<SP>in<SP><FONT=12>N<FONT=0><SP>then
<SP>the<SP><FONT=2>successor<FONT=0><SP>of<SP><FONT=2>n<FONT=0><SP>is
<SP>in<SP><FONT=12>N<FONT=0>.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=411><FONT=1>3.<FONT=0><COL INC 10>The
<COL INC 10>only<COL INC 10>elements<COL INC 10>of<COL INC 11>
<FONT=12>N<FONT=0><COL INC 10>are<COL INC 10>those<COL INC 10>created
<COL INC 10>by<COL INC 10>finitely<COL INC 11>many<COL INC 10>applications
<CR><LF>
<COLUMN=351><FONT=0><COLUMN=411>of<SP>rules<SP><FONT=1>1<FONT=0><SP>and
<SP><FONT=1>2<FONT=0>.<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>We<COL INC 15>can<COL INC 15>define
<COL INC 15><FONT=2>successor<FONT=0><COL INC 15>as<COL INC 15>a
<COL INC 15>specific<COL INC 15>mapping,<COL INC 15><FONT=1>S<FONT=0>,
<COL INC 15>which<COL INC 15>creates<COL INC 15>new<CR><LF>
<COLUMN=351><FONT=0>elements<COL INC 12>subject<COL INC 11>to
<COL INC 12>the<COL INC 12>rules<COL INC 11>that<COL INC 12>two
<COL INC 11>elements,<COL INC 12><FONT=2>x<FONT=0><COL INC 12>and
<COL INC 11><FONT=2>y<FONT=0><COL INC 12>are<COL INC 11>equal
<COL INC 12>just<COL INC 12>in<COL INC 11>the<CR><LF>
<COLUMN=351><FONT=0>case<COL INC 14>that<COL INC 14><FONT=1>S<FONT=2>(x)
<FONT=0><COL INC 14>equals<COL INC 13><FONT=1>S<FONT=2>(y)<FONT=0>;
<COL INC 14>and<COL INC 14><FONT=1>S<FONT=2>(x)<FONT=0><COL INC 14>is
<COL INC 14>different<COL INC 13>from<COL INC 14><FONT=2>x<FONT=0>,
<COL INC 14>for<COL INC 14>any<COL INC 14>element<COL INC 13><FONT=2>x
<FONT=0>.<CR><LF>
<COLUMN=351><FONT=0>We<COL INC 13>select<COL INC 13>a<COL INC 14>distinguished
<COL INC 13>element,<COL INC 13><FONT=2>0<FONT=0>,<COL INC 13>as
<COL INC 14>a<COL INC 13>notation<COL INC 13>for<COL INC 13><FONT=2>zero
<FONT=0>;<COL INC 14>and<COL INC 13>abbreviate<CR><LF>
<COLUMN=351><FONT=0><FONT=1>S<FONT=2>(0)<FONT=0><SP>as<SP><FONT=2>1
<FONT=0>,<SP>and<SP>abbreviate<SP><FONT=1>S<FONT=2>(<FONT=1>S<FONT=2>(0))
<FONT=0><SP>as<SP><FONT=2>2<FONT=0><SP>etc.<SP>in<SP>the<SP>usual<SP>manner.
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>Given<COL INC 19>a<COL INC 19>choice
<COL INC 19>between<COL INC 19>the<COL INC 19>two<COL INC 19>previous
<COL INC 19>definitions,<COL INC 19><FONT=1>I<FONT=0><COL INC 19>and
<COL INC 19><FONT=1>II<FONT=0>,<COL INC 18>it<CR><LF>
<COLUMN=351><FONT=0>appears<COL INC 13>that<COL INC 13><FONT=1>II
<FONT=0><COL INC 13>is<COL INC 13>more<COL INC 13>precise.
<COL INC 13><SP>Much<COL INC 13>less<COL INC 13>is<COL INC 13>left
<COL INC 13>to<COL INC 13>the<COL INC 13>imagination;<COL INC 13>given
<CR><LF>
<COLUMN=351><FONT=0><FONT=2>zero<FONT=0><COL INC 18>and<COL INC 17>a
<COL INC 18>definition<COL INC 18>of<COL INC 17><FONT=2>successor
<FONT=0><COL INC 18>the<COL INC 17>definition<COL INC 18>will
<COL INC 18>act<COL INC 17>as<COL INC 18>a<COL INC 18>recipe
<COL INC 17>for<CR><LF>
<COLUMN=351><FONT=0>producing<COL INC 18>elements<COL INC 19>of
<COL INC 18><FONT=12>N<FONT=0>.<COL INC 19>This<COL INC 18>style
<COL INC 19>of<COL INC 18>definition<COL INC 19>is<COL INC 18>called
<COL INC 19>an<COL INC 18>inductive<CR><LF>
<COLUMN=351><FONT=0>definition<SP>or<SP>generative<SP>definition.<CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11>1.1<COLUMN=1148>Introduction<SP><SP><SP>
<SP><SP>3<FONT=0><CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>The<COL INC 12>basic<COL INC 11>content
<COL INC 12>of<COL INC 11>an<COL INC 12>inductive<COL INC 12>definition
<COL INC 11>of<COL INC 12>a<COL INC 11>set<COL INC 12>of<COL INC 12>objects
<COL INC 11>consists<CR><LF>
<COLUMN=351><FONT=0>of<SP>three<SP>parts:<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=471>(1)<COL INC 11>A<COL INC 12>description
<COL INC 11>of<COL INC 12>an<COL INC 11>initial<COL INC 12>set
<COL INC 11>of<COL INC 12>objects;<COL INC 11>the<COL INC 12>elements
<COL INC 11>of<COL INC 12>this<COL INC 11>set<CR><LF>
<COLUMN=351><FONT=0><COLUMN=471>are<COL INC 11>the<COL INC 11>initial
<COL INC 11>elements<COL INC 11>of<COL INC 11>the<COL INC 11>set
<COL INC 11>we<COL INC 11>are<COL INC 11>describing<COL INC 11>in
<COL INC 12>the<COL INC 11>inductive<CR><LF>
<COLUMN=351><FONT=0><COLUMN=471>definition.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><FONT=1>IND<FONT=0><CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=471>(2)<COL INC 14>Given<COL INC 15>the
<COL INC 14>description<COL INC 14>of<COL INC 15>some<COL INC 14>existing
<COL INC 14>elements<COL INC 15>in<COL INC 14>the<COL INC 15>set,
<COL INC 14>we<CR><LF>
<COLUMN=351><FONT=0><COLUMN=471>are<SP>given<SP>a<SP>means<SP>of<SP>constructing
<SP>more<SP>elements.<CR><LINESPACE 9>
<COLUMN=351><FONT=0><COLUMN=471>(3)<COL INC 11>A<COL INC 10>termination
<COL INC 11>clause,<COL INC 10>saying<COL INC 11>that<COL INC 11>the
<COL INC 10>only<COL INC 11>elements<COL INC 10>in<COL INC 11>the
<COL INC 11>set<COL INC 10>are<CR><LF>
<COLUMN=351><FONT=0><COLUMN=471>those<SP>which<SP>gained<SP>admittance
<SP>by<SP>either<SP>(1)<SP>or<SP>(2).<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>Notice<COL INC 12>that<COL INC 11>our<COL INC 12>definition
<COL INC 11>of<COL INC 12><FONT=12>N<FONT=0>,<COL INC 11>in
<COL INC 12>terms<COL INC 11>of<COL INC 12><FONT=2>zero<FONT=0>
<COL INC 11>and<COL INC 12><FONT=2>successor<FONT=0>,<COL INC 11>is
<COL INC 12>an<COL INC 11>instance<CR><LF>
<COLUMN=351><FONT=0>of<COL INC 11><FONT=1>IND<FONT=0>:<COL INC 12>we
<COL INC 11>are<COL INC 12>defining<COL INC 11>the<COL INC 12>set
<COL INC 11>of<COL INC 12>natural<COL INC 11>numbers:<COL INC 12>
<FONT=2>zero<FONT=0><COL INC 11>is<COL INC 12>initially<COL INC 11>included
<CR><LF>
<COLUMN=351><FONT=0>in<COL INC 12>the<COL INC 11>set;<COL INC 12>then
<COL INC 11>applying<COL INC 12>the<COL INC 12>second<COL INC 11>phrase
<COL INC 12>of<COL INC 11>the<COL INC 12>definition<COL INC 11>we
<COL INC 12>can<COL INC 12>say<COL INC 11>that<CR><LF>
<COLUMN=351><FONT=0><FONT=2>one<FONT=0><SP>is<SP>in<SP>the<SP>set<SP>since
<SP><FONT=2>one<FONT=0><SP>is<SP>the<SP>successor<SP>of<SP><FONT=2>zero
<FONT=0>.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>We<COL INC 21>can<COL INC 21>recast
<COL INC 20>the<COL INC 21>positional<COL INC 21>notation<COL INC 20>description
<COL INC 21>as<COL INC 21>an<COL INC 20>inductive<CR><LF>
<COLUMN=351><FONT=0>definition.<CR><LF>
<CR><LINESPACE 8>
<COLUMN=351><FONT=0><FONT=1>1.<FONT=0><SP><SP>A<SP>digit<SP>is<SP>a
<SP>numeral.<CR><LINESPACE 9>
<COLUMN=351><FONT=0><FONT=1>2.<FONT=0><SP><SP>If<SP><FONT=2>n<FONT=0>
<SP>is<SP>a<SP>numeral<SP>then<SP><FONT=2>n<FONT=0><SP>followed<SP>by
<SP>a<SP>digit<SP>is<SP>a<SP>numeral.<CR><LINESPACE 9>
<COLUMN=351><FONT=0><FONT=1>3.<FONT=0><SP><SP>The<COL INC 15>only
<COL INC 15>numerals<COL INC 14>are<COL INC 15>those<COL INC 15>created
<COL INC 15>by<COL INC 14>finitely<COL INC 15>many<COL INC 15>applications
<COL INC 15>of<COL INC 14><FONT=1>1<FONT=0><CR><LF>
<COLUMN=351><FONT=0><COLUMN=399>and<SP><FONT=1>2<FONT=0>.<CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>In<SP>words,<SP>"a<SP>numeral<SP>is<SP>a<SP>digit,
<SP>or<SP>a<SP>numeral<SP>followed<SP>by<SP>a<SP>digit".<CR>
<LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>In<COL INC 14>this<COL INC 13>application
<COL INC 14>of<COL INC 13><FONT=1>IND<FONT=0>,<COL INC 14>the
<COL INC 14>initial<COL INC 13>set<COL INC 14>has<COL INC 13>more
<COL INC 14>than<COL INC 14>one<COL INC 13>element;<CR><LF>
<COLUMN=351><FONT=0>namely<COL INC 13>the<COL INC 12>ten<COL INC 13>decimal
<COL INC 13>digits.<COL INC 12><SP>Again,<COL INC 13>we<COL INC 13>assume
<COL INC 12>that<COL INC 13>the<COL INC 13>questioner<COL INC 12>knows
<CR><LF>
<COLUMN=351><FONT=0>what<COL INC 13>"digit"<COL INC 13>means.
<COL INC 13><SP>This<COL INC 13>is<COL INC 13>a<COL INC 13>characteristic
<COL INC 13>of<COL INC 13>all<COL INC 13>definitions:<COL INC 13>we
<COL INC 14>must<COL INC 13>stop<CR><LF>
<COLUMN=351><FONT=0><FONT=5>somewhere<FONT=0><COL INC 14>in
<COL INC 14>our<COL INC 14>explication.<COL INC 14>Notice<COL INC 14>too
<COL INC 14>that<COL INC 14>we<COL INC 14>assume<COL INC 14>that
<COL INC 14>"followed<COL INC 13>by"<CR><LF>
<COLUMN=351><FONT=0>means<SP>juxtaposition.<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>Inductive<COL INC 12>definitions
<COL INC 12>have<COL INC 12>been<COL INC 12>the<COL INC 12>province
<COL INC 12>of<COL INC 12>mathematics<COL INC 13>for<COL INC 12>many
<CR><LF>
<COLUMN=351><FONT=0>years;<COL INC 10>however,<COL INC 11>computer
<COL INC 10>science<COL INC 11>has<COL INC 10>developed<COL INC 10>a
<COL INC 11>style<COL INC 10>of<COL INC 11>syntax<COL INC 10>specification
<CR><LF>
<COLUMN=351><FONT=0>called<COL INC 10>BNF<COL INC 11>(Backus-Naur
<COL INC 10>Form)<COL INC 11>equations<COL INC 10>which<COL INC 11>has
<COL INC 10>the<COL INC 11>same<COL INC 10>intent<COL INC 11>as
<COL INC 10>that<CR><LF>
<COLUMN=351><FONT=0>of<COL INC 23>inductive<COL INC 22>definitions.
<COL INC 23><SP>Here<COL INC 22>is<COL INC 23>the<COL INC 22>previous
<COL INC 23>inductive<COL INC 23>definition<COL INC 22>of<CR><LF>
<COLUMN=351><FONT=0>"numeral"<SP>as<SP>a<SP>set<SP>of<SP>BNF<SP>equations:
<CR><LINESPACE 20>
<COLUMN=351><FONT=0><LB>numeral<RB><COLUMN=519>::=<SP><LB>digit<RB><CR>
<LINESPACE 9>
<COLUMN=351><FONT=0><LB>numeral<RB><COLUMN=519>::=<SP><LB>numeral<RB><LB>digit<RB><CR>
<LINESPACE 3>
<COLUMN=351><FONT=0>As<SP>an<SP>abbreviation,<SP>the<SP><SP>two<SP>BNF
<SP>equations<SP>may<SP>also<SP>be<SP>written:<CR><LINESPACE 3>
<COLUMN=351><FONT=0><LB>numeral<RB><COLUMN=519>::=<SP><LB>digit<RB><SP>|<SP><LB>numeral<RB><LB>digit<RB>.
<CR><LINESPACE 20>
<COLUMN=351><FONT=0>A<COL INC 12>comparison<COL INC 11>between
<COL INC 12>the<COL INC 11>BNF<COL INC 12>and<COL INC 11>the
<COL INC 12>inductive<COL INC 11>descriptions<COL INC 12>of
<COL INC 11>"numeral"<CR><LF>
<COLUMN=351><FONT=0>should<COL INC 20>clarify<COL INC 19>much
<COL INC 20>of<COL INC 19>the<COL INC 20>notation,<COL INC 20>but
<COL INC 19>we<COL INC 20>will<COL INC 19>give<COL INC 20>a
<COL INC 20>more<COL INC 19>detailed<CR><LF>
<COLUMN=351><FONT=0>analysis.<COL INC 14><SP>The<COL INC 14>symbol
<COL INC 14>"::="<COL INC 15>may<COL INC 14>be<COL INC 14>read
<COL INC 14>"is<COL INC 14>a",<COL INC 15>the<COL INC 14>symbol
<COL INC 14>"|"<COL INC 14>may<COL INC 15>be<COL INC 14>read<CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11>4<SP><SP>Symbolic<SP>expressions
<COLUMN=1320>1.1<FONT=0><CR><LF>
<CR><LINESPACE 3>
<COLUMN=351><FONT=0>"or".<COL INC 23><SP>The<COL INC 23>character
<COL INC 23>strings<COL INC 23>beginning<COL INC 23>with<COL INC 23>"<LB>"
<COL INC 23>and<COL INC 23>ending<COL INC 24>with<COL INC 23>"<RB>"<CR><LF>
<COLUMN=351><FONT=0>correspond<COL INC 10>to<COL INC 11>"numeral"
<COL INC 10>and<COL INC 10>"digit"<COL INC 11>in<COL INC 10><FONT=1>1
<FONT=0><COL INC 10>and<COL INC 11><FONT=1>2<FONT=0>;<COL INC 10>by
<COL INC 10>convention,<COL INC 11>components<COL INC 10>of<CR><LF>
<COLUMN=351><FONT=0>BNF<COL INC 17>equations<COL INC 17>which
<COL INC 17><FONT=5>describe<FONT=0><COL INC 17>elements<COL INC 17>are
<COL INC 17>enclosed<COL INC 17>in<COL INC 17>"<LB>"<COL INC 17>and
<COL INC 17>"<RB>";<COL INC 17>and<CR><LF>
<COLUMN=351><FONT=0>elements<COL INC 10>which<COL INC 10>are
<COL INC 11>given<COL INC 10><FONT=5>explicitly<FONT=0><COL INC 10>are
<COL INC 10>written<COL INC 11>without<COL INC 10>the<COL INC 10>"<LB>
<COL INC 10><RB>"<COL INC 11>fence.<COL INC 10><SP>Thus<CR><LF>
<COLUMN=351><FONT=0>"<LB>digit<RB>"<COL INC 16>is<COL INC 16>not
<COL INC 15>a<COL INC 16>numeral<COL INC 16>but<COL INC 15>is
<COL INC 16>a<COL INC 16>description;<COL INC 15>to<COL INC 16>make
<COL INC 16>the<COL INC 16>definition<COL INC 15>of<CR><LF>
<COLUMN=351><FONT=0><LB>numeral<RB><SP>complete<SP>we<SP>should<SP>include
<SP>an<SP>equation<SP>like:<CR><LINESPACE 20>
<COLUMN=351><FONT=0><LB>digit<RB><COLUMN=519>::<SP>=<SP><FONT=2>0<SP>|<SP>1
<SP>|<SP>2<SP>|<SP>3<SP>|<SP>4<SP>|<SP>5<SP>|<SP>6<SP>|<SP>7<SP>|<SP>8
<SP>|<SP>9<FONT=0><CR><LINESPACE 20>
<COLUMN=351><FONT=0>Juxtaposition<COL INC 13>of<COL INC 13>objects
<COL INC 13>implies<COL INC 13>concatenation<COL INC 13>of
<COL INC 13>the<COL INC 13>syntactic<COL INC 13>objects.<COL INC 12>Thus
<CR><LF>
<COLUMN=351><FONT=0>"89"<SP>is<SP>an<SP>instance<SP>of<SP>"<LB>numeral<RB><LB>digit<RB>".
<CR><LINESPACE 3>
<COLUMN=351><FONT=0><COLUMN=423>We<COL INC 11>have<COL INC 11>also
<COL INC 12>introduced<COL INC 11>the<COL INC 11>difference
<COL INC 12>between<COL INC 11>an<COL INC 11>abstract<COL INC 12>object
<COL INC 11>and<CR><LF>
<COLUMN=351><FONT=0>a<COL INC 15>representation<COL INC 14>for
<COL INC 15>that<COL INC 15>object.<COL INC 14><SP>This<COL INC 15>distinction
<COL INC 15>has<COL INC 14>been<COL INC 15>well<COL INC 15>studied
<COL INC 14>in<CR><LF>
<COLUMN=351><FONT=0>philosophy<COL INC 19>and<COL INC 19>mathematics,
<COL INC 19>and<COL INC 19>we<COL INC 20>will<COL INC 19>see
<COL INC 19>that<COL INC 19>this<COL INC 19>idea<COL INC 20>has
<COL INC 19>strong<CR><LF>
<COLUMN=351><FONT=0>consequences<COL INC 14>for<COL INC 14>the
<COL INC 13>field<COL INC 14>of<COL INC 14>programming<COL INC 13>and
<COL INC 14>computer<COL INC 14>science.<COL INC 13><SP>Abstract<CR><LF>
<COLUMN=351><FONT=0>objects<SP>and<SP>their<SP>representations<SP>will
<SP>play<SP>crucial<SP>roles<SP>in<SP>this<SP>text.<CR><LINESPACE 3>
<COLUMN=351><FONT=0>With<SP>this<SP>introduction,<SP>here<SP>is<SP>
<FONT=2>complis<FONT=0><SP>and<SP>friends:<CR><LINESPACE 20>
<COLUMN=351><FONT=0><FONT=2>complis<SP><LB>=<SP>λ[[u;off;vpl]<SP>complis
<FONT=8>'<FONT=2>[u;off;off;vpl;();();();1]<FONT=0><CR><LINESPACE 20>
<COLUMN=351><FONT=2>complis<FONT=8>'<FONT=2><SP><LB>=<SP>λ[[u;org;off;vpl;triv;cmplx;pop;ac]
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531>[null[u]<SP>→<COLUMN=663>[null[cmplx]
<SP>→<SP>triv;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><SP><FONT=13>t<FONT=2>
<SP>→<SP>append<COLUMN=819>[rest[cmplx];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=819><SP>list[mkmove[mem[first[pop]];1]];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=819><SP>rest[pop];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=819><SP>triv]];
<CR><LINESPACE 9>
<COLUMN=351><FONT=2><COLUMN=531><SP>isconst[first[u]]<SP>→<SP>complis
<FONT=8>'<FONT=2>[<COLUMN=891>rest[u];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>org;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>off;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>vpl;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>concat[mkconst[ac;first[u]];triv];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>cmplx;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>pop;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=891>add1[ac]];
<CR><LINESPACE 9>
<COLUMN=351><FONT=2><COLUMN=531><SP>isvar[first[u]]<SP>→<SP>complis
<FONT=8>'<FONT=2>[<COLUMN=867>rest[u];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>org;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>off;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>vpl;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>concat[
<COLUMN=951>mkvar[<COLUMN=1035>ac;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867><COLUMN=951>
<COLUMN=1035>loc[first[u];off;vpl]];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867><COLUMN=951>triv];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>cmplx;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>pop;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=867>add1[ac]];
<CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11>1.1<COLUMN=1148>Introduction<SP><SP><SP>
<SP><SP>5<FONT=0><CR><LF>
<CR><LINESPACE 9>
<COLUMN=351><FONT=2><COLUMN=531><SP>iscarcdr[first[u]]<SP>→<SP>complis
<FONT=8>'<FONT=2>[<COLUMN=903>rest[u];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>org;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>off;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>vpl;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>append[
<COLUMN=1011>reverse[compcarcdr[<COLUMN=1251>ac;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903><COLUMN=1011>
<COLUMN=1251><SP>first[u]<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903><COLUMN=1011>
<COLUMN=1251><SP>off;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903><COLUMN=1011>
<COLUMN=1251><SP>vpl]];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903><COLUMN=1011>triv];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>cmplx;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>pop;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=663><COLUMN=903>add1[ac]];
<CR><LINESPACE 9>
<COLUMN=351><FONT=2><COLUMN=531><SP>iscond[first[u]<SP>→<SP>complis
<FONT=8>'<FONT=2>[<COLUMN=879>rest[u];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>org;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>sub1[off];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>vpl;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>triv;<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>append[
<COLUMN=987>cmplx;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879><COLUMN=987>concat[
<COLUMN=1071>mkpush[1];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879><COLUMN=987>
<COLUMN=1071>comcond[<COLUMN=1179>args<FONT=3>c<FONT=2>[<COLUMN=1251>first[u]]
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879><COLUMN=987>
<COLUMN=1071><COLUMN=1179>gensym[];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879><COLUMN=987>
<COLUMN=1071><COLUMN=1179>off;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879><COLUMN=987>
<COLUMN=1071><COLUMN=1179>vpl]]];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>concat[mkpop[ac];pop];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=675><COLUMN=879>add1[ac]];
<CR><LINESPACE 9>
<COLUMN=351><FONT=2><COLUMN=531><SP><FONT=13>t<FONT=2><SP>→<SP>complis
<FONT=8>'<FONT=2>[<COLUMN=711>rest[u];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>org;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>sub1[off];<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>vpl;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>triv;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>append[<COLUMN=819>cmplx;
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819>concat[
<COLUMN=903>mkpush[1];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819><COLUMN=903>λ[[z]
<SP>compapply[<COLUMN=1119>func[first[u]];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819><COLUMN=903>
<COLUMN=1119>complis[<COLUMN=1227>z;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819><COLUMN=903>
<COLUMN=1119><COLUMN=1227>off;<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819><COLUMN=903>
<COLUMN=1119><COLUMN=1227>vpl];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819><COLUMN=903>
<COLUMN=1119>length[z]]]<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711><COLUMN=819><COLUMN=903>
<SP>[arglist[first[u]]<SP>]];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>concat[mkpop[ac];pop];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=531><COLUMN=711>add1[ac]]]<CR>
<LINESPACE 25>
<COLUMN=351><FONT=2>mkmove<SP><LB>=<SP>λ[[ac;loc][eq[ac;loc]<SP>→<SP>();
<SP><FONT=13>t<FONT=2><SP>→<SP>list[MOVE;ac;loc]]]<CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11>6<SP><SP>Symbolic<SP>expressions
<COLUMN=1320>1.1<FONT=0><CR><LF>
<CR><LINESPACE 20>
<COLUMN=351><FONT=2>compcarcdr<SP><LB>=<SP>λ[[ac;exp;off;vpl]<CR>
<LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567>[isvar[arg[exp]]<SP>→<SP>list[mkcarcdr[
<COLUMN=987>func[exp];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><COLUMN=711><COLUMN=987>ac;
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><COLUMN=711><COLUMN=987>loc[arg[exp];off;vpl]]]
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><FONT=13>t<FONT=2><SP>→
<SP>concat[<COLUMN=711>mkcarcdr_ac[func[exp];ac;ac];<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><COLUMN=711>compcarcdr[ac;arg[exp];off;vpl]]]]
<CR><LINESPACE 20>
<COLUMN=351><FONT=2>iscarcdr<SP><LB>=λ[[u]<COLUMN=567>[iscar[u]<SP>→iscarcdr[arg[u]]
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><SP>iscdr[u]<SP>→iscarcdr[arg[u]]
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><SP>atom[u]<SP>→<SP>or[isvar[u];isconst[u]];
<CR><LINESPACE 3>
<COLUMN=351><FONT=2><COLUMN=459><COLUMN=567><SP><FONT=13>t<FONT=2>
<SP>→<SP><FONT=13>f<FONT=2><SP>]]<CR><LINESPACE 20>
<COLUMN=351><FONT=2>iscar<SP><LB>=<SP>λ[[x]<SP>eq[func[x];CAR]]<CR>
<LINESPACE 9>
<COLUMN=351><FONT=2>iscdr<SP><LB>=<SP>λ[[x]<SP>eq[func[x];CDR]]<CR>
<LINESPACE 9>
<COLUMN=351><FONT=2>mkcarcdr<SP><LB>=λ[[carcdr;ac;loc]<SP>list[carcdr;ac;loc]]
<CR><LF>
<FF>
<COLUMN=351><FONT=0><FONT=11><COLUMN=1227>INDEX<SP><SP><SP><SP><SP>7
<FONT=0><CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<COLUMN=351><FONT=0><FONT=9>Index<FONT=0><CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<CR><LF>
<COLUMN=351><FONT=0>generative<SP>definition<SP><SP><SP>2<CR><LF>
<COLUMN=351><FONT=0>inductive<SP>definition<SP><SP><SP>2<CR><LF>
<FF>