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␈↓␈↓↓␈↓α␈↓
␈↓ ↓H

␈↓ ↓H
WHAT IS A PATTERN?
␈↓ ↓H

␈↓ ↓H

␈↓ ↓H
␈↓ α(In␈α∞many␈α∞areas␈α∞of␈α∞programming␈α∞and␈α∞especially␈α∞in␈α∞artificial␈α∞intelligence,␈α
it␈α
is␈α
effective
␈↓ ↓H
␈↓ ↓Hto␈α⊂create␈α⊂and␈α⊂use␈α⊂patterns.␈α⊂The␈α⊂use␈α⊂involves␈α⊂recognizing␈α⊂the␈α⊂pattern␈α⊂in␈α⊂a␈α⊂situation␈α∂and
␈↓ ↓H
␈↓ ↓Htaking␈α⊃an␈α⊃appropriate␈α⊃action.␈α⊃More␈α⊃generally,␈α⊃the␈α⊃action␈α⊃taken␈α⊃may␈α⊃depend␈α⊃on␈α⊃a␈α⊂set␈α⊂of
␈↓ ↓H
␈↓ ↓Hpatterns␈αand␈αother␈αinformation␈αas␈αwell.
␈↓ ↓H

␈↓ ↓H
␈↓ α(The␈α∂most␈α∂well␈α∂developed␈α∂forms␈α∂of␈α∂pattern␈α∂and␈α∞pattern␈α∞recognition␈α∞occur␈α∞in␈α∞formal
␈↓ ↓H
␈↓ ↓Hsyntax␈α
of␈α
languages␈α
where␈α
the␈α
languages␈αare␈αstrings␈αof␈αsymbols.␈αI␈αwant␈αto␈αdefine␈αpatterns
␈↓ ↓H
␈↓ ↓Habstractly␈α∩quite␈α∩separately␈α∩from␈α∩language␈α∩but␈α∩hopefully␈α∩in␈α∩a␈α∩way␈α∩that␈α⊃is␈α⊃applicable␈α⊃to
␈↓ ↓H
␈↓ ↓Hlanguage␈α∂and␈α∂in␈α∂a␈α∂way␈α∂that␈α∂will␈α∂enable␈α∞some␈α∞of␈α∞the␈α∞properties␈α∞of␈α∞linguistic␈α∞patterns␈α∞that
␈↓ ↓H
␈↓ ↓Hhave␈αbeen␈αstudied␈αto␈αcarry␈αover␈αto␈α␈↓↓abstract␈αpatterns␈↓.
␈↓ ↓H

␈↓ ↓H
␈↓ α(When␈α∪I␈α∪have␈α∪tried␈α∪to␈α∪give␈α∪a␈α∪fully␈α∪general␈α∩definition␈α∩of␈α∩pattern,␈α∩the␈α∩ideas␈α∩have
␈↓ ↓H
␈↓ ↓Hgotten␈α
quite␈α
complex␈α
and␈α
no␈α
␈↓↓most␈α
general␈↓␈α
idea␈α
has␈α
come␈α
out.␈α
Therefore,␈α
I␈α
will␈α
try␈α
to␈α
sneak
␈↓ ↓H
␈↓ ↓Hup␈αon␈αit␈αand␈αstart␈αwith␈αthe␈αsimplest␈αkinds␈αof␈αpatterns.
␈↓ ↓H

␈↓ ↓H


␈↓ ↓HType␈αI␈αpatterns
␈↓ ↓H

␈↓ ↓H
␈↓ α(A␈α⊂type␈α⊂I␈α⊂pattern␈α⊂is␈α⊂specified␈α⊂by␈α⊂giving␈α⊂a␈α∂collection␈α∂of␈α∂predicates␈α∂and␈α∂a␈α∂quantifier
␈↓ ↓H
␈↓ ↓Hfree␈α∩expression␈α∩in␈α∩these␈α∩predicate␈α⊃symbols.␈α⊃An␈α⊃instance␈α⊃of␈α⊃the␈α⊃pattern␈α⊃is␈α⊃a␈α⊃pairing␈α⊃of
␈↓ ↓H
␈↓ ↓Hentities␈α∂with␈α∂the␈α∂free␈α∂individual␈α∂variables␈α∂such␈α∂that␈α∞when␈α∞the␈α∞variables␈α∞are␈α∞assigned␈α∞the
␈↓ ↓H
␈↓ ↓Hcorresponding␈αentities,␈αthe␈αexpression␈αis␈αtrue.
␈↓ ↓H

␈↓ ↓H
␈↓ α(The␈α
notion␈α
of␈α
pin␈α
in␈α
chess␈α
can␈α
be␈α
given␈α
as␈α
an␈αexample␈αof␈αa␈αtype␈αI␈αpattern␈αprovided
␈↓ ↓H
␈↓ ↓Hit␈αis␈αdescribed␈αin␈αterms␈αof␈αthe␈αfollowing␈αpredicates:
␈↓ ↓H

␈↓ ↓H
␈↓ α(1.␈αiswhite(piece)␈αasserts␈αthat␈α␈↓↓piece␈↓␈αis␈αwhite.
␈↓ ↓H

␈↓ ↓H
␈↓ α(2.␈α⊂sees(piece1,piece2,direction,board)␈α⊂asserts␈α⊂that␈α∂if␈α∂one␈α∂moves␈α∂from␈α∂␈↓↓piece1␈↓␈α∂in␈α∂the
␈↓ ↓H
␈↓ ↓Hdirection␈α∞␈↓↓direction␈↓␈α∞on␈α∞the␈α∞board␈α
situation␈α
␈↓↓board␈↓,␈α
one␈α
encounters␈α
␈↓↓piece␈↓␈α
before␈α
encountering␈α
a
␈↓ ↓H
␈↓ ↓Hnon-blank␈αsquare␈αor␈αthe␈αedge␈αof␈αthe␈αboard.
␈↓ ↓H

␈↓ ↓H
␈↓ α(2.␈α≠moves(piece,direction)␈α≠asserts␈α≠that␈α≠␈↓↓piece␈↓␈α≠moves␈α≠an␈α≠arbitrary␈α~distance,␈α~(it
␈↓ ↓H
␈↓ ↓Htherefore␈αmust␈αbe␈αa␈αqueen,␈αrook,␈αor␈αbishop)␈αin␈α␈↓↓direction␈↓.
␈↓ ↓H

␈↓ ↓H
␈↓ α(3.␈αbetter(piece1,piece2)␈αasserts␈αthat␈α␈↓↓piece1␈↓␈αhas␈αa␈αhigher␈αvalue␈αthan␈α␈↓↓piece2␈↓.
␈↓ ↓H

␈↓ ↓H
␈↓ α(4.␈αdefended(piece,board)␈αasserts␈αthat␈α␈↓↓piece␈↓␈αis␈αdefended␈αin␈αthe␈αposition␈α␈↓↓board␈↓.
␈↓ ↓H

␈↓ ↓H
␈↓ α(Pins␈α∂are␈α∂then␈α∂described␈α∂by␈α∂a␈α∞formula␈α∞involving␈α∞five␈α∞free␈α∞variables,␈α∞namely␈α∞␈↓↓piece1␈↓␈α∞-
␈↓ ↓H
␈↓ ↓Hthe␈α∩pinning␈α∩piece,␈α∩␈↓↓piece2␈↓␈α∩-␈α∩the␈α∩pinned␈α∩piece,␈α∩␈↓↓piece3␈↓␈α∩-␈α⊃the␈α⊃piece␈α⊃against␈α⊃which␈α⊃␈↓↓piece2␈↓␈α⊃is
␈↓ ↓H
␈↓ ↓Hpinned,␈α∩␈↓↓direction␈↓␈α∩-␈α∩the␈α∩direction␈α∩from␈α∩␈↓↓piece1␈↓␈α∩to␈α∩␈↓↓piece2␈↓,␈α∩and␈α∩␈↓↓board␈↓␈α∩-␈α∩the␈α∩board␈α∩position␈α⊃in
␈↓ ↓H
␈↓ ↓Hwhich␈αall␈αthis␈αoccurs.␈αThe␈αformula␈αis
␈↓ ↓H
␈↓ α(␈↓↓iswhite(piece1)␈α
≡␈α
iswhite(piece3)␈α
∧
␈↓ ↓H
␈↓ α_iswhite(piece1)␈α
≡␈α
¬iswhite(piece2)␈α
∧
␈↓ ↓H
␈↓ α_sees(piece1,piece2,direction,board)␈α
∧
␈↓ ↓H
␈↓ α_sees(piece2,piece3,direction,board)␈α
∧
␈↓ ↓H
␈↓ α_moves(piece1,direction)␈α
∧
␈↓ ↓H
␈↓ α_[¬defended(piece3,board)␈α∨␈αbetter(piece3,piece1)].␈↓
␈↓ ↓H

␈↓ ↓H
␈↓ α(This␈αsimple␈αexample␈αsuggests␈αthe␈αfollowing␈αremarks:
␈↓ ↓H

␈↓ ↓H
␈↓ α(1.␈α⊂The␈α⊂simplest␈α⊂linguistic␈α⊂patterns␈α⊂are␈α∂of␈α∂type␈α∂I.␈α∂These␈α∂give␈α∂a␈α∂special␈α∂role␈α∂to␈α∂the
␈↓ ↓H
␈↓ ↓Hrelation
␈↓ ↓H

␈↓ ↓H
␈↓ α(␈↓↓ispredecessor(string1,string2,string3)␈↓
␈↓ ↓H

␈↓ ↓H
␈↓ ↓Hwhich␈α
asserts␈α
that␈α
␈↓↓string1␈↓␈α
immediately␈α
precedes␈α
␈↓↓string2␈↓␈αin␈αtheir␈αoccurence␈αas␈αsubstrings␈αof
␈↓ ↓H
␈↓ ↓H␈↓↓string3␈↓.
␈↓ ↓H

␈↓ ↓H
␈↓ ↓HThus␈α∞strings␈α∞consisting␈α∞of␈α∞noun␈α∞phrases␈α∞followed␈α
by␈α
verb␈α
phrases␈α
which␈α
give␈α
one␈α
form␈α
of
␈↓ ↓H
␈↓ ↓Hsentence␈α
in␈α
some␈α
simple␈α
grammars␈α
are␈α
examples␈α
of␈α
the␈α
pattern␈↓↓
␈↓ ↓H

␈↓ ↓H
␈↓ α_ispredecessor(string1,string2,string3)␈α
∧
␈↓ ↓H
␈↓ α_nounphrase(string1)␈α
∧
␈↓ ↓H
␈↓ α_verbphrase(string2).␈↓
␈↓ ↓H

␈↓ ↓H
␈↓ α(2.␈α∞Since␈α∞a␈α∞substantial␈α∞part␈α∞of␈α∞the␈α
generalization␈α
we␈α
are␈α
looking␈α
for␈α
consists␈α
merely
␈↓ ↓H
␈↓ ↓Hin␈α∀not␈α∀giving␈α∀a␈α∀special␈α∀role␈α∀to␈α∀the␈α∀relation␈α∀␈↓↓ispredecessor␈↓,␈α∀it␈α∀is␈α∀likely␈α∪that␈α∪some␈α∪of␈α∪the
␈↓ ↓H
␈↓ ↓Hinteresting␈αproperties␈αof␈αstring␈αpatterns␈αcarry␈αover␈αto␈αthe␈αgeneral␈αcase.
␈↓ ↓H

␈↓ ↓H
␈↓ α(3.␈α
Given␈α
the␈α
small␈α
number␈α
of␈α
pieces␈αin␈αa␈αchess␈αposition,␈αit␈αis␈αclearly␈αfeasible␈αto␈αfind
␈↓ ↓H
␈↓ ↓Hpins␈α∩in␈α∩a␈α∩position␈α∩given␈α⊃programs␈α⊃for␈α⊃calculating␈α⊃the␈α⊃predicates␈α⊃involved.␈α⊃This␈α⊃involves
␈↓ ↓H
␈↓ ↓Hspecifying␈α∞one␈α∞of␈α
the␈α
elements␈α
of␈α
the␈α
pattern,␈α
namely␈α
␈↓↓board␈↓␈α
which␈α
thereby␈α
plays␈α
a␈α
special
␈↓ ↓H
␈↓ ↓Hrole␈α∂and␈α∂scanning␈α∂the␈α∂others.␈α∂An␈α∂ad␈α∂hoc␈α∂pin␈α∞recognizer␈α∞is␈α∞clearly␈α∞easy␈α∞to␈α∞write,␈α∞and␈α∞it␈α∞is
␈↓ ↓H
␈↓ ↓Heven␈α∩easy␈α∩to␈α∩write␈α∩a␈α∩general␈α∩pattern␈α∩recognizer␈α∩for␈α∩patterns␈α⊃involving␈α⊃a␈α⊃single␈α⊃board
␈↓ ↓H
␈↓ ↓Hvariable,␈α⊂several␈α⊂piece␈α∂variables,␈α∂and␈α∂several␈α∂direction␈α∂variables.␈α∂However,␈α∂a␈α∂recognizer
␈↓ ↓H
␈↓ ↓Hthat␈α
had␈α
to␈α
scan␈α
all␈α
possible␈α
positions␈α
is␈α
infeasible␈α
and␈αnot␈αwanted␈αfor␈αchess␈αplay␈αanyway.
␈↓ ↓H
␈↓ ↓HMoreover,␈α∪the␈α∪different␈α∪roles␈α∪played␈α∪by␈α∪pieces␈α∪and␈α∪directions␈α∪(e.g.␈α∪it␈α∪if␈α∪we␈α∪scan␈α∩the
␈↓ ↓H
␈↓ ↓Hpotential␈α∞pinning␈α∞pieces␈α∞and␈α∞the␈α∞directions␈α∞leading␈α∞from␈α∞them␈α∞first,␈α∞we␈α∞need␈α∞only␈α
look␈α
for
␈↓ ↓H
␈↓ ↓Hpotential␈α⊃pinned␈α⊃pieces␈α⊃in␈α⊃the␈α⊃given␈α⊂direction),␈α⊂makes␈α⊂it␈α⊂unlikely␈α⊂that␈α⊂the␈α⊂techniques␈α⊂of
␈↓ ↓H
␈↓ ↓Hgrammatical␈αpattern␈αrecognition␈αhave␈αmuch␈αcarryover␈αto␈αthe␈αgeneral␈αcase.
␈↓ ↓H

␈↓ ↓H
␈↓ α(4.␈α
One␈α
is␈α
tempted␈αto␈αuse␈αexisting␈αpatterns␈αin␈αdefining␈αnew␈αpatterns,␈αand␈αin␈αorder␈αto
␈↓ ↓H
␈↓ ↓Hdo␈α∩this,␈α⊃we␈α⊃need␈α⊃to␈α⊃allow␈α⊃binding␈α⊃some␈α⊃variables␈α⊃with␈α⊃existential␈α⊃quantifiers.␈α⊃Thus␈α⊃we
␈↓ ↓H
␈↓ ↓Hobviously␈αwant␈αto␈αdefine
␈↓ ↓H

␈↓ ↓H
␈↓ α(pinned(piece,board)␈α≡␈α∃␈αpiece1␈αpiece3␈αdirection.
␈↓ ↓H
␈↓ βλpins(piece1,piece,piece3,direction,board),
␈↓ ↓H

␈↓ ↓H
␈↓ ↓Hand␈α∞the␈α∞predicate␈α∞␈↓↓defended␈↓␈α∞used␈α∞in␈α∞the␈α∞definition␈α
of␈α
␈↓↓pins␈↓␈α
obviously␈α
wants␈α
to␈α
be␈α
defined␈α
in
␈↓ ↓H
␈↓ ↓Hterms␈αof␈αa␈αpattern.
␈↓ ↓H

␈↓ ↓H
␈↓ α(In␈α⊃the␈α⊃study␈α⊃of␈α⊃grammatical␈α⊃patterns,␈α⊃the␈α⊃highly␈α⊃recursive␈α⊃cases␈α⊃are␈α⊃emphasized,
␈↓ ↓H
␈↓ ↓Hbecause␈α⊂they␈α⊂are␈α⊂necessary␈α⊂for␈α⊂any␈α∂kind␈α∂of␈α∂generality,␈α∂because␈α∂Chomsky␈α∂fascinated␈α∂the
␈↓ ↓H
␈↓ ↓Hlinguists␈α&with␈α%the␈α%competence-performance␈α%distinction,␈α%and␈α%because␈α%they␈α%are
␈↓ ↓H
␈↓ ↓Hmathematically␈α⊂the␈α⊂most␈α⊂interesting.␈α⊂Nevertheless,␈α⊂non-recursive␈α⊂patterns␈α∂occur␈α∂in␈α∂many
␈↓ ↓H
␈↓ ↓Hpractical␈α↔cases␈α↔and␈α↔probably␈α↔usually␈α↔admit␈α↔simpler␈α↔methods␈α↔of␈α↔recognition␈α⊗and␈α⊗use.
␈↓ ↓H
␈↓ ↓HTherefore␈α∞we␈α∞shall␈α
take␈α
type␈α
I␈α
patterns␈α
to␈α
be␈α
non-recursive,␈α
and␈α
reserve␈α
recursion␈α
for␈α
a
␈↓ ↓H
␈↓ ↓Hlater␈αtype.
␈↓ ↓H

␈↓ ↓H
␈↓ α(5.␈α↔In␈α↔one␈α↔respect,␈α↔type␈α↔I␈α↔patterns␈α↔go␈α↔beyond␈α↔context␈α↔free␈α⊗phrase␈α⊗structure
␈↓ ↓H
␈↓ ↓Hgrammars␈α∞in␈α∞the␈α∞string␈α
case.␈α
Namely,␈α
the␈α
same␈α
variable␈α
can␈α
occur␈α
in␈α
several␈α
places␈α
in␈α
the
␈↓ ↓H
␈↓ ↓Hdefining␈α
expression,␈α
and␈α
this␈α
allows␈α
the␈α
imposition␈α
of␈α
requirements␈α
of␈α
agreement␈α
from␈αthe
␈↓ ↓H
␈↓ ↓Hbeginning.␈α⊃I␈α⊃doubt␈α⊃that␈α⊃the␈α⊃requirement␈α⊃that␈α⊂all␈α⊂occurences␈α⊂of␈α⊂variables␈α⊂be␈α⊂of␈α⊂different
␈↓ ↓H
␈↓ ↓Hvariables␈α∂leads␈α∂to␈α∂an␈α∂interesting␈α∞class␈α∞of␈α∞patterns.␈α∞Moreover,␈α∞it␈α∞doesn't␈α∞give␈α∞the␈α∞context
␈↓ ↓H
␈↓ ↓Hfree␈α
patterns␈α
in␈α
the␈α
grammar␈α
case␈α
unless␈α
we␈α
distinguish␈αthe␈α␈↓↓ispredecessor␈↓␈αfunction,␈αbecause
␈↓ ↓H
␈↓ ↓Ha␈α∂variable␈α∂has␈α∂to␈α∂occur␈α∂in␈α∂the␈α∂␈↓↓ispredecessor␈↓␈α∂relation␈α∂and␈α∂also␈α∞in␈α∞the␈α∞other␈α∞predicates.␈α∞This
␈↓ ↓H
␈↓ ↓Hsimply␈αconfirms␈αmy␈αprejudice␈αagainst␈αcontext␈αfreedom.
␈↓ ↓H

␈↓ ↓H
␈↓ α(6.␈α⊃In␈α⊃writing␈α⊃the␈α⊃␈↓↓pin␈↓␈α⊃pattern,␈α⊃it␈α⊃was␈α⊂an␈α⊂inconvenience␈α⊂not␈α⊂to␈α⊂be␈α⊂able␈α⊂to␈α⊂use␈α⊂the
␈↓ ↓H
␈↓ ↓Hfunction␈α↔␈↓↓color(piece)␈↓,␈α↔but␈α⊗we␈α⊗got␈α⊗around␈α⊗it␈α⊗with␈α⊗the␈α⊗predicate␈α⊗␈↓↓iswhite␈↓␈α⊗at␈α⊗the␈α⊗cost␈α⊗of
␈↓ ↓H
␈↓ ↓Hawkwardness.␈α∞It␈α∞would␈α∞seem␈α∞that␈α∞any␈α∞pattern␈α∞that␈α∞can␈α
be␈α
written␈α
with␈α
functions␈α
can␈α
also
␈↓ ↓H
␈↓ ↓Hbe␈α≤written␈α≤using␈α≤only␈α≤the␈α≤corresponding␈α≤predicates␈α≤provided␈α≠we␈α≠use␈α≠existential
␈↓ ↓H
␈↓ ↓Hquantifiers␈α
to␈α
get␈α
rid␈α
of␈α
unwanted␈α
variables␈α
that␈α
have␈α
to␈α
be␈α
introduced.␈α
Thus␈α
the␈α
relation␈↓↓
␈↓ ↓H

␈↓ ↓H
␈↓ α_y␈α=␈αf(g(h(x)))␈↓
␈↓ ↓H

␈↓ ↓H
␈↓ ↓Hwhich␈αmight␈αoccur␈αin␈αsome␈αpattern␈αwould␈αhave␈αto␈αbe␈αreplaced␈αby
␈↓ ↓H

␈↓ ↓H
␈↓ ↓H␈↓↓␈αF(y,u)␈α∧␈αG(u,v)␈α∧␈αH(v,x)␈α␈↓
␈↓ ↓H

␈↓ ↓H
␈↓ ↓Hwhere␈α␈↓↓f␈↓␈αand␈α␈↓↓F␈↓␈αare␈αrelated␈αby
␈↓ ↓H

␈↓ ↓H
␈↓ ↓H␈↓↓␈α∀␈αx␈αy.(y␈α=␈αf(x)␈α≡␈αF(y,x)).␈α␈↓
␈↓ ↓H

␈↓ ↓H
␈↓ ↓HThus␈α∞we␈α∞would␈α∞get␈α∞a␈α∞pattern␈α∞with␈α∞more␈α∞variables,␈α∞and␈α∞the␈α
extras␈α
can␈α
be␈α
eliminated␈α
with
␈↓ ↓H
␈↓ ↓Hexistential␈α∩quantifiers.␈α∩Suppose␈α∩we␈α∩call␈α∩the␈α∩patterns␈α∩we␈α∩get␈α∩allowing␈α∩function␈α⊃symbols
␈↓ ↓H
␈↓ ↓Htype␈α⊂I-a.␈α⊂It␈α∂is␈α∂an␈α∂open␈α∂question␈α∂whether␈α∂we␈α∂should␈α∂do␈α∂this␈α∂or␈α∂whether␈α∂we␈α∂should␈α∂admit
␈↓ ↓H
␈↓ ↓Hthe␈αfunction␈αsymbols␈αdirectly␈αinto␈αtype␈αI.
␈↓ ↓H

␈↓ ↓H
␈↓ α(6.␈α∂We␈α∂shall␈α∂not␈α∂try␈α∂to␈α∂define␈α∂more␈α∂types␈α∂of␈α∂patterns␈α∂in␈α∂this␈α∂draft.␈α∂However,␈α∞note
␈↓ ↓H
␈↓ ↓Hthe␈αfollowing␈αdirections␈αof␈αgeneralization:
␈↓ ↓H

␈↓ ↓H
␈↓ βλa.␈α≥The␈α≥computation␈α≥of␈α≥the␈α≥predicate␈α≥expression␈α≤could␈α≤involve␈α≤more
␈↓ ↓H
␈↓ ↓Hcomplicated␈α∂calculations␈α∂such␈α∂as␈α∂recursion.␈α∂Thus␈α∂a␈α∂pin␈α∂is␈α∞not␈α∞a␈α∞type␈α∞I␈α∞pattern␈α∞if␈α∞we␈α∞can't
␈↓ ↓H
␈↓ ↓Huse␈α+␈↓↓sees(piece1,piece2,direction,board)␈↓,␈α+but␈α+have␈α+to␈α+build␈α+it␈α+from␈α+the␈α+relation
␈↓ ↓H
␈↓ ↓H␈↓↓adjacent(square1,square2,direction)␈↓.
␈↓ ↓H

␈↓ ↓H
␈↓ βλb.␈αVariable␈αnumbers␈αof␈αentities␈αmay␈αappear␈αin␈αpatterns.
␈↓ ↓H

␈↓ ↓H
␈↓ βλc.␈α∩The␈α∩conditions␈α∩for␈α∩one␈α⊃pattern␈α⊃may␈α⊃require␈α⊃the␈α⊃non-existence␈α⊃of␈α⊃other
␈↓ ↓H
␈↓ ↓Hpatterns.
␈↓ ↓H

␈↓ ↓H
␈↓ α(7.␈α⊂The␈α∂example␈α∂of␈α∂pinning␈α∂shows␈α∂that␈α∂patterns␈α∂may␈α∂be␈α∂used␈α∂in␈α∂much␈α∂more␈α∂varied
␈↓ ↓H
␈↓ ↓Hways␈α∂than␈α∂in␈α∂linguistics␈α∂whether␈α∂of␈α∂natural␈α∂or␈α∂computer␈α∂languages.␈α∞Some␈α∞typology␈α∞of␈α∞the
␈↓ ↓H
␈↓ ↓Hways␈αin␈αwhich␈αpatterns␈αare␈αused␈αafter␈αbeing␈αrecognized␈αis␈αprobably␈αworthwhile.
␈↓ ↓H