8/14/2019 ko.pdf

1/46

Ko(c) Robert Jasiek Version 7b; last update 2010-01- 1; first publis!ed "ersion 7 first da# 2010-01-12.

Abstract$or t!e first ti%e& ko-intersection and its t#pes& ko& ko-stone& ko-strin'& and t!e necessar#funda%ental ter%s are defined in 'eneral for all positions so t!at not all stones in all positions areko-stones. a%ples are 'i"en.

Preface*o is one of t!e ke# strate'ic concepts. +!erefore it is e tre%el# i%portant to kno, ,!at it is& i.e.&to be able to distin'uis! ,!at is a ko fro% ,!at is not a ko. +!is paper 'i"es t!e ans,er.

dentif#in' basic-kos is "er# eas# - identif#in' kos related to c#cles of %ore t!an 2 %o"es is "er#&and so%eti%es e tre%el#& difficult. $or t!at reason& traditional o t!eor# could define t!e basic-koscurrentl# on t!e board as kos but !ad no s#ste%atic understandin' of kos in 'eneral. +!e rarit# ofkos related to c#cles ,it! / or %ore pla#s (one of t!e% occurs onl# about once e"er# ca. &000t! to10&000t! 'a%e) contributed to t!e dela#. o, t!e ans,er co%es fro% t!e "ie, of %at!e%aticalabstraction.

o pla#ers !a"in' difficult# ,it! t!e abstract definitions of c#cle-set& left-part& (ans,er-)strate'#and (ans,er-)co%patible %i'!t still tr# to 'et an intuiti"e understandin' nstead of ordinar# ko and'a%e end rules& t!e 3default restriction rules3 are used. +!e# pro!ibit sin'le stone suicide& use t!efi ed-ko-rule& allo, a basic-ko recapture after inter"enin' passes and ot!er,ise end a se4uence

after a c#cle endin' at t!e position at t!e start of a se4uence or end a se4uence b# successi"e passes. +!e fi ed-ko-rule allo,s a recreated position but t!en pro!ibits t!e sa%e ne t pla#& i.e.& ac#cle %a# occur but not recur. - *o is defined "ia its ko-intersections. n infor%al ,ords& a local-ko-intersection re4uires a pla#er to force a c#cle ,!ile t!e opponent pre"ents t!e pla#er fro%i%pro"in' t!e local area on t!e related c#cles5 intersections. 6 'lobal-ko-intersection re4uires a

pla#er to force a c#cle ,!ile t!e opponent pre"ents t!e pla#er5s ,in ,!en considerin' t!e ,!ole board score and t!e ko%i.

oes t!e reader ,onder ,!# t!e rules used for t!e definitions differ fro% ordinar# rules8 +!edefault restriction rules are used to define 3ko3. 9o%e set of ordinar# rules is used to regulate

practical playing. :# distin'uis!in' t!e t,o purposes& t!e definition ,orks re'ardless of ,!ic!

ordinar# rules are used.6 ko-intersection can be of one& t,o or t!ree of t!e t#pes basic-& local- or 'lobal-ko-intersection.9o%e intersections are onl# local-ko-intersections. 9o%e ot!ers are onl# 'lobal-ko-intersections.

!at does t!e abstract5s condition 3so t!at not all stones in all positions are ko-stones3 %ean8 t istri"ial but for practical purposes also %eanin'less to define ko so t!at all intersections of t!e boardand in all positions are ko-intersections. at!er t!an e pectin' t!e pla#ers to cooperate& t!edefinitions rel# on t!e concept 3force3.

Fundamentals

Presuppositions and Basic Definitions I efined else,!ere are in particular pla#& %o"e& current-position& basic-ko& (situational)

8/14/2019 ko.pdf

2/46

c#cle.

i"en t!e basic o rules& eit!er allo,ed or pro!ibited suicide& and a scorin' %et!od.

+!e history-bans is t!e set of ko bans prior to t!e start of anal#sis.

+!e start-position is t!e current-position or a particular position.

Default Restriction Rules +!e 1-play-rule pro!ibits a c#cle consistin' of 1 pla#.

+!e basic-ko-rule pro!ibits i%%ediate recapture in a basic-ko.

+!e fixed-ko-rule pro!ibits a pla# to lea"e position 6 and create position : if an earlier pla#left position 6 and created position :.

+!e 3-pass-rule ends t!e 'a%e in case of successi"e passes.

+!e cycle-end-rule ends t!e 'a%e ,!en a) a situational c#cle startin' at t!e start-positionoccurs and b) a recapture in a basic-ko after e actl# 2 successi"e& inter"enin' passes doesnot occur.

+!e default restriction rules are t!e 1-pla#-rule& t!e basic-ko-rule& t!e fi ed-ko-rule& t!e - pass-rule& t!e c#cle-end-rule.

Basic Definitions II 6 move-sequence is a se4uence of %o"es under t!e default restriction rules& startin' fro%

t!e start-position& lettin' t!e pla#ers alternate %o"es and endin' due to t!e c#cle-end-rule ort!e -pass-rule.

6 left-part of a %o"e-se4uence is eit!er t!e ,!ole %o"e-se4uence or a part t!at consists ofone or %ore t!an one successi"e %o"es of it and starts ,it! its first %o"e.

6 pla#er5s strategy is a set of one or %ore t!an one left-parts of %o"e-se4uences so t!at eac!left-part starts ,it! a %o"e of !is& eac! left-part ends ,it! a %o"e of !is& t!ere are not t,oleft-parts so t!at t!e# ,it!out t!eir last %o"e are e4ual& and t!e afore%entioned conditionsare not true for t!e set to'et!er ,it! an# left-part not in t!e set.

$or a pla#er& an answer-strategy of t!e opponent is a set of one or %ore t!an one left-partsof %o"e-se4uences so t!at eac! left-part starts ,it! a %o"e of t!e pla#er& eac! left-part ends,it! a %o"e of t!e opponent& t!ere are not t,o left-parts so t!at t!e# ,it!out t!eir last %o"eare e4ual& and t!e afore%entioned conditions are not true for t!e set to'et!er ,it! an# left-

part not in t!e set.

6 %o"e-se4uence is compatible ,it! a strate'# of a pla#er if eac! left-part t!at is of t!e%o"e-se4uence and ends ,it! a %o"e of t!e pla#er is in t!e strate'#.

$or a pla#er& a %o"e-se4uence is answer-compatible ,it! an ans,er-strate'# of t!eopponent if eac! left-part t!at is of t!e %o"e-se4uence and ends ,it! a %o"e of t!eopponent is in t!e ans,er-strate'#.

Applied Definitions 6 pla#er can force so%et!in' if t!ere is at least one strate'# of !is so t!at eac! co%patible

%o"e-se4uence fulfils t!at so%et!in'.

6 pla#er does force so%et!in' if !e uses a strate'# of !is so t!at eac! co%patible %o"e-se4uence fulfils t!at so%et!in'.

8/14/2019 ko.pdf

3/46

prevent so%et!in' is force to fulfil not t!e so%et!in'.

6 pla#er can answer-force 6 if t!e opponent %o"in' second uses an ans,er-strate'# t!at -re'ardless of t!e pla#er5s first %o"e - does force : and if t!ere is at least one strate'# of t!e

pla#er so t!at eac! %o"e-se4uence t!at is co%patible ,it! t!e pla#er5s strate'# and ans,er-co%patible ,it! t!e opponent5s ans,er-strate'# fulfils 6.

$or a %o"e-se4uence& t!ere is t!e first %o"in' pla#er5s local-area-improvement on e actl# aset of intersections if its area score before t!e %o"e-se4uence is s%aller t!an its area scoreafter t!e %o"e-se4uence.

6 c#cle5s cycle-set is t!e set of all t!e intersections of t!e c#cle5s pla#s.

Types of Ko Intersections 6 basic -ko-intersection is an intersection of a basic-ko.

=nder default restriction rules ,it!out !istor#-bans& a local -ko-intersection is an intersectionfor ,!ic! a set of c#cles e ists so t!at

eac! of t!e c#cles starts fro% t!e start-position&

eac! of t!e c#cles !as at least one pla# creatin' t!e current-position&

t!e intersection belon's to eac! of t!e c#cles5 c#cle-sets& and

a pla#er can ans,er-force one of t!e c#cles b# %o"in' first in it if t!e opponent%o"in' second does pre"ent local-area-i%pro"e%ent of t!e pla#er on t!e c#cle-set.

i"en t!e ko%i& t!e !istor#-bans and t!e %o"in' pla#er and usin' t!e default restrictionrules& a global -ko-intersection is an intersection for ,!ic! a set of c#cles e ists so t!at

eac! of t!e c#cles starts fro% t!e current-position&

eac! of t!e c#cles !as at least one pla# creatin' t!e current-position&

t!e intersection belon's to eac! of t!e c#cles5 c#cle-sets& and

t!e pla#er can ans,er-force one of t!e c#cles b# %o"in' first in it if t!e opponent%o"in' second does pre"ent t!e pla#er5s ,in.

Ko 6 ko-intersection is an intersection t!at is at least one of basic-ko-intersection& local-ko-

intersection& 'lobal-ko-intersection.

6 ko is a ko-intersection and& recursi"el#& an# adjacent ko-intersection. 6 ko-stone is a stone on a ko-intersection.

6 ko-string is a ko-stone5s strin'.

8/14/2019 ko.pdf

4/46

Reasons for the Non ob!ious "onditions

# play ruleit!out t!e 1-pla#-rule and under rules allo,in' suicide& eac! of t!e %arkedintersections ,ould be a local-ko-intersection because :lack could pre"ent!ite5s local-area-i%pro"e%ent and !ite could force a 1-%o"e-c#cle b#suicidin' on eit!er intersection.

Fi$ed ko rule

ontinuation. +!e %o"es 1 to ? couldrecur fore"er. +!e %o"e-se4uence,ould not end.

9uppose t!ere ,ould be no fi ed-ko-rule. +!e c#cle-end-rule does not endt!e 'a%e ,!en a recapture in a basic-ko after e actl# 2 successi"e&inter"enin' passes occurs. +!e -pass-rule does not end t!e 'a%e after onl#

2 successi"e passes. +!e basic-ko-rule pro!ibits to recapture / i%%ediatel# but does not pro!ibit 7 to recapture /.+!erefore t!e fi ed-ko-rule is alsoneeded to ensure an end of t!e 'a%e.ere it pro!ibits 7 to pla# at 1.

Allo%in& instead of Prohibitin& "ycles+!e fi ed-ko-rule to'et!er ,it! t!e basic-ko-rule allo, c#cles. nstead t!e positional-superko-rule,ould pro!ibit c#cles. +!e definitions of local- and 'lobal-ko-intersection rel# on t!e e istence ofc#cles. +!erefore c#cles !a"e to be allo,ed for t!e default restriction rules used for t!ese

definitions. +!at c#cles are allo,ed for t!e purpose of identif#in' kos on t!e definition le"el doesnot i%pl# at all t!at ordinar# ko and 'a%e end rules ,ould be re4uired to allo, c#cles - rat!er t!e#!a"e t!e freedo% of c!oice ,!et!er to allo, c#cles. +!is coe istence of definition "ersus ordinar#rules can be seen& e.'.& also for basic-ko in so%e real ,orld rulesets +!e# define a Abasic-Bko b#3i%%ediate recapture ,ould recreate t!e position C initial s!ape3 and t!en add t!e stricter basic-ko-rule& ,!ic! pro!ibits i%%ediate recapture. +!is paper uses t!e sa%e kind of approac! but in 'eneralfor all kos instead of onl# basic-kos.

' pass rule+!e -pass-rule instead of a 2-pass-rule enables usa'e of passes as ko t!reats.

"ycle end rule

ontinuation. it!out c#cle-end-rule&10 ,ould not end t!e 'a%e.

ontinuation. $ro% t!e position before 11& :lack !as not pla#ed on t!eintersection of 11 #et. +!erefore 11 isnot pro!ibited under t!e fi ed-ko-rule.

8/14/2019 ko.pdf

5/46

(continuation)

!ite %a# not pla# 1/ at t!eintersection of / because t!is ,ould

be pro!ibited b# t!e fi ed-ko-rule.

9ince !ite could not pre"ent :lack5slocal-area-i%pro"e%ent on t!e D

interestin' intersections& none of t!e%,ould be a local-ko-intersection.ence& to let t!e% be local-ko-intersections& also t!e c#cle-end-ruleis needed.

(tart position "ondition in "ycle end rule

(continuation) ontinuation. to 10 is a situationalt!at does not end at t!e start-position.f t!e c#cle-end-rule ended t!e 'a%ene"ert!eless& :lack could not ans,er-force a c#cle t!at creates t!e start-

position before %o"e 1.

(ituational "ondition in "ycle end rule

:efore 1& it is !ite5s turn. 6fter & itis :lack5s turn. +!erefore t!e c#clefro% 1 to is not situational. f :lack/ passed& t!en t!e c#cle fro% 1 to /,ould be situational.

ontinuation. it!out t!e situationalcondition in t!e c#cle-end-rule&alread# ,ould in"oke t!e 'a%e endand :lack ,ould not 'et a c!ance tore%o"e t!e ,!ite stones.

ontinuation. +!e re%o"al %ust beallo,ed to %odel also t!ose ordinar#rulesets t!at allo, it.

Basic ko Recapture "ondition in "ycle end rule

+!e re%o"al of all t!e ,!ite stones%ust be allo,ed to %odel also t!oseordinar# rulesets t!at allo, it.

(continuation)

6lt!ou'! t!e %o"es 1 to / are asituational c#cle& %o"e / does not end

t!e 'a%e b# t!e c#cle-end-rule because of its e ception 3a recapturein a basic-ko after e actl# 2successi"e& inter"enin' passes3.

Both )ocal And *lobal ko intersection6lt!ou'! 'lobal-ko-intersections t!at are neit!er basic- nor local-ko-intersections are rare (onl# t,os!ape classes are kno,n and onl# one 'a%e !as been reported so far)& t!e# do e ist. !at e ists%ust be e plained re'ardless of rarit#.

"er#bod# calls a s!ape like& e.'.& a round-robin-ko a ko s!ape. +!is is so re'ardless of ,!et!ercurrentl# perfect-pla# s!ould %ake pla#s in it or ,!et!er possibl# so%e pla#s else,!ere on t!e board (like suc! capturin' !u'e strin's or closin' !u'e territories) are %ore ur'ent. 9i%ilarl#

8/14/2019 ko.pdf

6/46

e"er#bod# calls an end'a%e ko a ko re'ardless of ,!et!er it e ists on t!e board alread# since t!eopenin' or %iddle 'a%e. =sa'e of t!e t#pe local-ko-intersection %akes it al,a#s possible toidentif# t!e round-robin-ko5s intersections as ko-intersections - e"en ,!en t!e# are not 'lobal-ko-intersections.

nstead of ,onderin' ,!# t,o ne,l# defined t#pes of ko-intersections are necessar#& t!e readers!ould be astonis!ed t!at only t,o suffice. +!at just t,o ne, t#pes describe all kno,n ko s!apess!o,s !o, ,ell t!ese t#pes !a"e been c!osen. 6 careless t!eor# ,ould use %ore t#pes - a 'oodt!eor# uses as fe, t#pes as necessar#.

+istory bansEocal-ko-intersection i'nores !istor#-bans because a) it is a local concept and b) eit!er pla#er %i'!t

be t!e one ,!o can ans,er-force a c#cle b# %o"in' first in it. lobal-ko-intersection does not need!istor#-bans but ,orks ,ell ,it! or ,it!out t!e%. +!erefore it is defined in t!e %ore 'eneral ,a#.+!ereb# a current-position of a 'a%e can be studied in t,o ,a#s b# considerin' or i'norin' t!e!istor#-bans.

(et of "ycles

Variation 1 (continuation) Variation 2

(continuation)

f !ite pre"ents :lack5s local-area-i%pro"e%ent& t!en :lack can ans,er-force a c#cle of t!e set of c#cles includin' t!ose in Variations 1F2. :lackcannot ans,er-force a particular c#cle t!ou'!. .'.& if :lack tried to ans,er-

force t!e c#cle of Variation 1& t!en !ite %i'!t c!oose to pla# as in Variation2 to pre"ent :lack5s local-area-i%pro"e%ent ne"ert!eless. +!is s!o,s t!at in'eneral it does not suffice to consider onl# one c#cle but one !as to consider aset of c#cles.

(tart position for )ocal ko intersection+!ere are t,o reasons ,!# it does not al,a#s suffice to start fro% t!e current-position 1) +!e

pla#er %i'!t alread# !a"e as %uc! local-area on t!e c#cle-set as !e can 'et (e.'.& !e %i'!t !a"e t!eentire c#cle-set as !is area). +!en it is tri"ial for t!e opponent to pre"ent t!e pla#er5s local-area-i%pro"e%ent. 2) t %i'!t be possible to ans,er-force so%e c#cle but not necessaril# a c#cle of t!ata pla# creates t!e current-position& as is s!o,n in t!e e a%ple

f :lack ,ants to pre"ent !ite5slocal-area-i%pro"e%ent& !e !as toallo, !ite ans,er-forcin' so%ec#cle.

(continuation) ontinuation. :lack does not need to pla# D at 10 t!ou'!. +!erefore !itecannot force creation of t!e current-

position. +!e c#cle fro% 2 to 10 doesnot create it. f local-ko-intersectionrelied on t!e current-position as t!estart-position& t!e D intersections

,ould not be local-ko-intersections.

8/14/2019 ko.pdf

7/46

Relation+!e c#cles of a set of c#cles are related to eac! ot!er +!e# start fro% t!e sa%e position. ac! oft!e c#cles is related to t!e current-position ac! %ust create it. ac! of t!e c#cles is related to t!eintersection tested as a ko-intersection t belon's to eac! of t!e c#cles5 c#cle-sets.

(pecific "onditions for *lobal ko intersectionlobal-ko-intersection refers to t!e ko%i& t!e %o"in' pla#er and t!e current-position as t!e start-

position because it s!all be applicable to t!e current state of a 'a%e in pro'ress.

"ore "onditions3a pla#er can ans,er-force one of t!e c#cles A...B if t!e opponent A...B does pre"ent local-area-i%pro"e%ent of t!e pla#er on t!e c#cle-set3 and 3t!e pla#er can ans,er-force one of t!e c#cles A...Bif t!e opponent A...B does pre"ent t!e pla#er5s ,in3 are t!e core conditions of t!e definitions of local-or 'lobal-ko-intersection& respecti"el#. eference to c#cle does surprise nobod#. Gnl# usin' 3a

pla#er can force a c#cle3 ,ould not ,ork because a) t!e opponent %i'!t al,a#s pass in case of anatte%pted non-suicide c#cle and b) t!e opponent %i'!t %ake si%ple inter"enin' pla#s in case of anatte%pted sucide c#cle. $orcin' a c#cle ,orks onl# as an ans,er (t!erefore ans,er-force) to a dut#for t!e opponent ,it! t!at !e is bein' so bus# forcin' it t!at %ean,!ile t!e pla#er 'ets !is c!ance toconstruct a c#cle. !at re%ains to be told is t!e opponent5s dut# eit!er to pre"ent local-area-i%pro"e%ent of t!e pla#er on t!e c#cle-set or to pre"ent t!e pla#er5s ,in for local- or 'lobal-ko-intersection& respecti"el#.

Remarks +!e aut!or pro"ed t!at eac! strin' in eac! position can recur in a c#cle if t!ere are no ko

rules and t!e pla#ers cooperate. +!in's are "er# different and %uc! %ore difficult underso%e 'i"en restriction rules if one pla#er tries to force ,!ile t!e opponent tries to pre"ent ac#cle.

+!e 'reatest !urdles before disco"er# of t!e ko-related definitions ,ere 1) definition offorce& ,!ic! ,as sol"ed& e.'.& in t!e Hapanese 200 ules& 2) findin' and definin' ,ell t!edefault restriction rules& ) understandin' t!at practicall# useful definitions of ko-relatedter%s depend on force& /) understandin' t!at it is easier to approac! a definition of ko-intersection first re'ardless of ,!et!er an intersection is occupied& ) identif#in' t!e e actconditions of local-C'lobal-ko-intersection. 6lto'et!er t!e aut!or needed 1 . #ears& all !iskno,led'e on ko and rules and so%e of !is earlier in"entions to o"erco%e all t!ese !urdles.

.'.& ,!en !e !ad in"ented t!e fi ed-ko-rule in 1II7& it !ad been pure t!eor# at t!at ti%e; int!e %eanti%e it !as beco%e applied t!eor#. ence in retrospect it is not surprisin' t!at a'eneral definition of t!e see%in'l# "er# basic concept ko !as escaped all pla#ers andresearc!ers for a "er# lon' ti%e.

Jre"ious "ersions of t!e definitions ,orked for all but one or a fe, s!ape classes. +!ecurrent "ersion appears to ,ork for all kno,n s!ape classes. t could !appen t!ou'! t!at ane, s!ape class is disco"ered for t!at t!e current "ersion ,ould not assi'n t!e e pectedc!aracteriKation. +!is possibilit# lets t!e current t!eor# of definitions be a %odel rat!er t!annecessaril# t!e final t!eor#. +!e be!a"iour of kos !as to be understood #et better before onecan be sure to !a"e found t!e final t!eor#.

urin' t!e alternatin'-se4uence& it is d#na%ic ,!et!er an intersection is a ko-intersectionand of ,!ic! t#pes it is.

=nder t!e definitions& a sin'le stone suicide does not define a ko-intersection because t!e

8/14/2019 ko.pdf

8/46

default restriction rules pro!ibit 1-pla#-c#cles. $or ot!er purposes& one %i'!t pro"ideanot!er t#pe of ko-intersections to include t!e intersections of sin'le stone suicides.

n t!e definitions of local-C'lobal-ko-intersection& 3does pre"ent3 is supposed to i%pl# 3can pre"ent3.

n so%e s!apes like& e.'.& round-robin-ko or %olasses ko& a pla#er cannot ans,er-force a

particular c#cle but can ans,er-force so%e c#cle of a set of c#cles. +!erefore t!e definitionsneed to rel# on set of c#cles. 6 set %i'!t contain one c#cle onl# t!ou'!.

9ince& in t!e definitions& t!e pla#er can ans,er-force one of t!e c#cles& t!en t!e set of c#clescannot be e%pt# ,!ene"er t!ere s!all be so%e local-C'lobal-ko-intersection. +!erefore it isnot necessar# to specif# a not e%pt# condition e plicitl#.

+!e condition 3eac! of t!e c#cles !as at least one pla# creatin' t!e current-position3 relatest!e c#cles to t!e current-position and t!e condition 3t!e intersection belon's to eac! of t!ec#cles5 c#cle-sets3 relates t!e potential ko-intersection to t!e c#cles.

:esides pro!ibited sin'le stone suicide (1-pla#-c#cle)& so%e assu%ptions are %ade for lon'c#cles. a) Eon' c#cles also t!rou'! basic-kos beco%e interestin' at all due to so%erestriction for pla#in' in a basic-ko t!e basic-ko-rule. :esides basic-ko recapture %i'!t beallo,ed or pro!ibited after inter"enin' passes. $ro% traditional and a%bi'uous rulesets (liket!e Hapanese 1IDI ules)& it is unclear ,!et!er t!is ,as intended. Gt!er rulesets (like t!en' 1II1 ules) allo, passes to ser"e as ko t!reats. 6s a conse4uence of inconclusi"etradition& researc! %ust allo, bot! cases ,it! or ,it!out passes as ko t!reats. +!erefore t!eless restricti"e option of allo,in' passes to ser"e as ko t!reats is specified in t!e c#cle-end-rule and t!ereb# used for t!e default restriction rules and t!eir usa'e of a 'eneral definitionof ko. +!is does not %ean t!ou'! t!at eac! defined ko ,ould be possible under allrestriction rules stricter t!an t!e default restriction rules. b) +o enable passes as ko t!reats&t!e default restriction rules are sufficientl# rela ed t!e c#cle-end-rule allo,s pass as a kot!reat for a basic-ko and t!ree (not just t,o) successi"e passes end t!e 'a%e; a basic-kocapture in a ko& t,o passes and t!en a recapture in t!e sa%e basic-ko constitute a situationalc#cle ,it! e actl# 2 pla#s but still allo, t!e %eanin' of pass as a ko t!reat because t!e'a%e does not end #et. c) ot just an assu%ption but e"en a re4uire%ent is to reject c#clescreated b# t!e pla#ers5 cooperation for its o,n sake. +!e default restriction rules co%bined,it! t!e force-dependent concepts a"oid it t!at ot!er,ise all strin's in all positions ,ould

be ko strin's. d) it! respect to one c#cle& t!e fi ed-ko-rule& ,!ic! is included in t!e defaultrestriction rules& is t!e %ost liberal. - 6s a conse4uence of t!e abo"e %utuall# balancin'assu%ptions and t!eir realiKation in t!e definitions& eac! intersection t!at so%eone !as seenas belon'in' to a ko is defined as a ko-intersection ,!ile eac! intersection t!at nobod# !asseen as belon'in' to a ko is defined not to be a ko-intersection. Jeople ,it! a %ore

restricti"e perception of ko can use ko C restriction rules t!at are %ore restricti"e t!an t!edefault restriction rules and can de"elop t!eir strate'# %ore closel# to perfect-pla# so t!att!e# ,ill %ake fe,er strate'ic %istakes b# %akin' a c#cle5s %o"e at inappropriate ti%in'.+!e# s!ould understand t!ou'! t!at t!e definitions !ere are related to t!e %ost liberal rat!ert!an t!e strictest restriction rules and are allo,in' bot! strate'icall# inacti"e and perfect-

pla# kos.

1) !en currentl# in a 'a%e position fi'!tin' a ko is pre%ature& it does not !a"e 'lobal-ko-intersections. +o identif# suc! kos ne"ert!eless& one needs local-ko-intersections. 2) +!ee istence of 'lobal-ko-intersection a%ple / pro"es t!at t!e t#pe 'lobal-ko-intersection isneeded because& on t!e interestin' intersections& it does not !a"e local-ko-intersections.

6 local-ko-intersection is called 3local3 because onl# t!e local-area-i%pro"e%ent isconsidered. 6 'lobal-ko-intersection is called 3'lobal3 because t!e ,!ole board5s score andt!e ko%i are considered.

8/14/2019 ko.pdf

9/46

+!e default restriction rules are used to define 3ko3 but !ere t!e# are not used to define perfect-pla# under 'i"en rules.

+raditional o t!eor# !as considered ko as so%et!in' on t!e board. e"ert!eless& one %i'!t be te%pted to t!ink of ko as so%et!in' bein' e4ui"alent to a c#cle ,it! pla#s on t!e ko5sintersections. +!is kind of definition ,ould !a"e se"eral proble%s t!ou'! a) n 'eneral& ako !as to be defined b# a set of c#cles rat!er t!an al,a#s onl# one particular c#cle. b) +oac!ie"e une4ui"ocalit# for a set of c#cles& an additional condition of set-%a i%alit# ,ould

be needed +!e ot!er conditions are not true for t!e union of t!e set of c#cles and a c#cle notin t!e set. c) 9uc! a %a i%al set of c#cles ,ould define all t!e ko5s intersections as t!e unionof all t!e c#cle-sets of t!e set5s c#cles. d) 9o%e!o, all up to t,o sets 'i"en due to t!e t,ot#pes local-C'lobal-ko-intersection need to be taken into consideration. e) +!e resultin'concept ,ould not be so %uc! ko-like an# lon'er but %ore n' st#le ko-position-like. -+!ere %a# be reasons for related t!eoretical studies. $or practical purposes t!ou'!& ko asdefined !ere ('i"in' a connected set of intersections) is "er# useful. f one reall# ,ants torefer to t!e c#cles& t!is can be done usin' t!e su''ested definitions for t!e t#pes of ko-intersection. t ,ould be superfluous t!ou'! to al,a#s refer to c#cles if all one ,ants to

consider is t!e location of a ko on t!e board.

Future Research esearc! in ko and its c!aracteristics s!ould be i%pro"ed to decide ,!et!er t!e current

definitions are a te%porar# %odel or t!e final t!eor# for a definition of ko.

e, ko s!ape classes& if an#& s!ould be disco"ered and c!ecked a'ainst t!e currentdefinitions.

:i' kos not consistin' of onl# basic-kos s!ould be classified into t#pes 4uite like basic-kosare classified.

Lore ad"anced ter%s like ko-t!reat s!ould be defined.

6 suicide pla# of a c#cle-set5s c#cle definin' a ko-intersection s!ould be found if so%ee ists.

*o-position s!ould be defined in t!e sense of n' *o ules or t!e e, *o ules. eferenceto ko-couplin' s!ould be replaced b# reference to set of c#cles like in t!e definitions oflocal-C'lobal-ko-intersection. nstead of 3ko-position3& a better na%e s!ould be c!osen

because ko-position is used also in a different %eanin' of 3a position ,it! at least one ko init3.

idden kos s!ould be studied in 'reater detail.

6ll intersections t!at are not ko-intersections s!ould be classified.

,$amples of Ko intersection=nless specified ot!er,ise& t!e e a%ples presu%e area scorin'. Gb"ious rests of %o"e-se4uencesare so%eti%es not s!o,n. $or %ost e a%ples& onl# one t#pe of ko-intersection is "erified; ko-intersections %i'!t also be of ot!er t#pes.

6part fro% a) bi''er !ell kos and b) a c#cle ,it! 7 pla#s& t!e e a%ples s!o, all kno,n s!apeclasses of lon' c#cle s!apes consistin' not onl# of basic-kos. Lultiples do not create ne, proble%sfor local-ko-intersections because it suffices to stud# %o"e-se4uences restricted to eac! s!apeseparatel#. +!e aut!or is 'rateful for ne, disco"eries.

+!e local-area of a c#cle-set is t!e area score on onl# its intersections in t!e position5s conte t.

8/14/2019 ko.pdf

10/46

"ounter e$amplesounter-e a%ples are as i%portant as positi"e e a%ples. 9ince t!ere is no 'eneral classification ofall non-ko-intersections #et& an# counter-e a%ple t!at ,ould be classified as a positi"e e a%ple b#t!e definition of ko-intersection ,ould let t!is definition be a failure.

"ounter e$ample #- n particular&none of t!e %arked intersections is alocal-ko-intersection.

#cle 1 a c#cle. +!e start-position ist!e current-position. Jla# / creates t!ecurrent-position.

(continuation)

(continuation) Variation 1.1. +!e opponent !itecannot pre"ent t!e pla#er :lack5slocal-area-i%pro"e%ent.

:efore Variation 1.1& t!e local-area is0.

6fter Variation 1.1& t!e local-area is 2. #cle 2 start-position. Jla# createst!e current-position.

(continuation) (continuation) Variation 2.1 (continuation fro% first%o"es of #cle 2).

:efore Variation 2.1& t!e local-area is1.

6fter Variation 2.1& t!e local-area is 2. Variation 2.2.

(continuation)

(continuation)

6fter Variation 2.2& t!e local-area is 2.

it! se4uences like #cle 2& Variation2.1 and Variation 2.2& t!e opponent:lack does pre"ent t!e pla#er !ite5slocal-area-i%pro"e%ent.

:lack c!ooses / in Variation 2.2 rat!ert!an / in #cle 2. +!e pla#er !itecannot ans,er-force t!e c#cle in#cle 2.

#cle start-position. Jla# 2 createst!e current-position.

8/14/2019 ko.pdf

11/46

(continuation) (continuation)

Variation .1 :# ans,erin' 1& t!eopponent !ite cannot pre"ent t!e

pla#er :lack5s local-area-i%pro"e%ent.

#cle / start-position of a c#cle. Jla#1 creates t!e current-position.

(continuation)

Variation /.1 +!e opponent :lackcannot pre"ent t!e pla#er !ite5slocal-area-i%pro"e%ent.

:efore Variation /.1& t!e local-area is.

6fter Variation /.1& t!e local-area is 2. #cle start-position of a c#cle. Jla#DI creates t!e current-position.

(continuation)

(continuation) (continuation)

8/14/2019 ko.pdf

12/46

Variation .1 9ee Variation 1.1. +!eopponent !ite cannot pre"ent t!e

pla#er :lack5s local-area-i%pro"e%ent.

$or all super-lon' c#cles& applicationof t!e definition of local-ko-intersection is si%ilarl# tri"ial.

9u%%ar# 6 c#cle fittin' all t!ere4uire%ents of t!e definition oflocal-ko-intersection does not e ist.+!erefore none of t!e %arkedintersections is a local-ko-intersection.

"ounter e$ample .- n particular& t!e%arked intersection is not a local-ko-intersection.

6 c#cle includin' a pla# on t!e%arked intersection %ust be lon'.9uc! %akes it essentiell# i%possiblefor a pla#er to ans,er-force.

"ounter e$ample '- "en none oft!e %arked intersections is a ko-intersection.

6 c#cle cannot be ans,er-forced b#eit!er pla#er (!ere :lack). - 9ince itis eas# for t!e pla#ers to cooperatee"en in perfect-pla# and create around-robin-ko-like c#cle& t!is is a"er# i%portant e a%ple for s!o,in'n'5s failure ,!en describin' kostones nai"el# as stones t!at can becaptured c#clicall# or repeatedl#.

e"ert!eless& n' deser"es t!e !onour

of !a"in' %oti"ated t!e aut!or of t!iste t to searc! for a careful definition.

"ounter e$ample /- 9uicide allo,ed&ko%i 0. +!e position5s topic ,asdescribed b# enis $eld%an but

%a#be it !ad been in"ented before.

"en none of t!e %arked intersectionsis a local ko intersection . Eocal-areaM 1.

Variation 1 for t!e current-position ast!e start-position.

(continuation)

(continuation)

Eocal-area M 2.

f :lack c!ooses %o"e (and :lack& for t!e purpose of t!e definition& can c!oose it)& t!en !ite cannot pre"ent:lack5s local-area-i%pro"e%ent. +!erefore t!e local-ko-intersection definition5s condition 3if t!e opponent ( !ite)%o"in' second does pre"ent local-area-i%pro"e%ent of t!e pla#er on t!e c#cle-set3 cannot be fulfilled for t!e t,ointersections as t!e supposed c#cle-set. ence neit!er of t!ese intersections is a local-ko-intersection.

8/14/2019 ko.pdf

13/46

Variation 2 for t!e current-position ast!e start-position. ontinuation. +!e %o"es 1 to arenot a situational c#cle. +!erefore t!ec#cle-end-rule does not end t!e 'a%e

just after .

(continuation)

(continuation) (continuation) (continuation)

Lo"e 2 is s!o,n for t!e purpose of s!o,in' t!e c#cle fro% 1 to . o,e"er& %o"e is a strate'ic %istake for t!e purpose of t!e definition5s condition 3if t!e opponent ( !ite) %o"in' second does pre"ent local-area-i%pro"e%ent oft!e pla#er on t!e c#cle-set3. :lack can c!oose better t!an %o"e in Variation 2 b# replacin' it ,it! %o"e ofVariation 1. 6lt!ou'! !ite tries differentl# ,it! %o"e 2& t!e local-area e%er'in' after so%e %ore %o"es follo,in'%o"e 12 of Variation 2 is !ite5s drea% onl#. n realit#& t!e local-area ,ill end up like in Variation 1. $or t!edefinition5s purpose& ,e %a# as ,ell assu%e %o"e 2 of Variation 2 to be do%inated b# %o"e 2 of Variation 1.

:lack to %o"e. "en none of t!e%arked intersections is a &lobal kointersection .

9core after Variation 1 M ? - ? - 0 M0.

6lt!ou'! !ite does pre"ent :lack5s,in& :lack cannot ans,er-force ac#cle. +!erefore neit!er of t!e %arkedintersections is a 'lobal-ko-intersection. ( ote 6lt!ou'!suicidin' 2 stones is t!e easiest

possible suicide task& !ite is notforced to pass in bet,een t!e pla#s oft!ese stones. t is surprisin'l# difficultto construct a ko ,it! suicide. +ri"ialterritor# s!apes don5t ,ork& eit!er.)

"ounter e$ample 0 "en none of t!e %arked intersectionsis a local -ko-intersection.

Variation 1

(continuation)

n Variation 1& !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro% to . +!erefore in particular none oft!e %arked intersections is a local-ko-intersection. +!e sendin'-2-returnin'-1 ko c#cles are i%%aterial for t!e

purpose of t!e application of t!edefinition5s condition 3if t!e opponent( !ite) does pre"ent A...B3. :esidest!e definition does not care for t!e

!idden bent-/-in-t!e-corner kos; itonl# looks for kos currentl# on t!e board.

:lack to %o"e. f t!e ko%i is 2 & t!ene"en none of t!e %arked intersectionsis a &lobal -ko-intersection. - 6fterVariation 1& t!e score is 0.

8/14/2019 ko.pdf

14/46

Variation 2

6fter Variation 2& t!e score is 0. it!Variations 1F2& !ite pre"ents:lack5s ,in. :lack cannot ans,er-force a c#cle t!ou'! because& inVariation 2& !ite does not need tocapture. +!erefore e"en none of t!eintersections is a 'lobal-ko-

intersection.

:lack to %o"e. f t!e ko%i is 2/. &

t!en e"en none of t!e %arkedintersections is a &lobal -ko-intersection. - 6fter Variation 1& t!escore is 0. . !ite cannot pre"ent:lack5s ,in.

"ounter e$ample 1 "en none of t!e %arked intersectionsis a local -ko-intersection.

Variation 1

(continuation)

(continuation)

f :lack does pre"ent !ite5s local-area-i%pro"e%ent& !ite cannotans,er-force a c#cle t!at creates t!ecurrent-position because& in Variation1& :lack can c!oose to approac! t!e,!ite liberties differentl#.

Variation 2

ontinuation. does not co%plete asituational c#cle. +!erefore t!e c#cle-end-rule does not end t!e 'a%e.

(continuation)

(continuation)

+o pre"ent !ite5s local-area-i%pro"e%ent under default restriction rules&:lack could also c!oose Variation 2. o,e"er& for t!e purpose of t!edefinition5s condition 3t!e pla#er ( !ite) can ans,er-force one of t!e c#cles3&:lack does not c!oose Variation 2. f one ,anted to percei"e t!e s!ape as aead *o& t!en one s!ould introduce anot!er t#pe of ko-intersection ,it! all oft!ese conditions a) t is not a 'lobal-ko-intersection. b) 6 3related3 c#clee ists so t!at durin' it t!e pla#ers use perfect-pla#. c) 6 pla#er can c!oose !is

perfect-pla# bet,een a c#cle as in (b) or not creatin' a 3related3 c#cle.

"ounter e$ample 2- one of t!e%arked intesections is a local-ko-intersection.

!ite pre"ents :lack5s local-area-i%pro"e%ent. :lack cannot ans,er-force a c#cle t!at creates t!e current-

position.

Basic ko intersection

ac! basic-ko currentl# on t!e board as suc! ser"es as an e a%ple& re'ardless of bein' a sin'le koor part of a %ultiple or %ulti-sta'e ko etc. =suall# !idden kos do not !a"e an# basic-ko-intersection.

8/14/2019 ko.pdf

15/46

)ocal ko intersection+!e definition of local-ko-intersection does not care ,!et!er a ko is ,ort! fi'!tin' in a 'lobalconte t no,& later or ne"er.

,$ample #- ac! of t!e %arkedintersections is a local-ko-intersection.

6lt!ou'! !ite does pre"ent :lack5slocal-area-i%pro"e%ent& :lack canans,er-force a c#cle on t!e %arkedintersections. Jla# creates t!ecurrent-position.

ontinuation. / in"okes t!e c#cle-end-rule.

:efore or after t!e c#cle (before %o"e1 or after %o"e )& t!e local-area is 2.

,$ample .- like a%ple 1.,$ample '- like a%ple 1 ,it!s,apped colours.

,$ample / ac! of t!e %arked intersections is alocal-ko-intersection.

Variation 1

ontinuation. does not co%plete asituational c#cle. +!erefore t!e c#cle-end-rule does not end t!e 'a%e. +!isis t!e %ost pro%inent e a%ple ,!#t!e rule !as t!e 3situational3 conditionin it.

ontinuation.

Variation 2

ontinuation. / in"okes t!e c#cle-end-rule.

Variation

(continuation)

!ile :lack& usin' an# of t!e Variations 1- & pre"ents !ite5s local-area-i%pro"e%ent& !ite ans,er-forces t!e c#cle%o"e 1 to & ,!ic! occurs in eac! of t!ese "ariations. Lo"e creates t!e current-position. +!erefore eac! of t!e%arked intersections is a local-ko-intersection.

,$ample 0- ac! of t!e %arkedintersections is a local-ko-intersection.

+!e c#cle starts fro% t!is start- position.

Jla# 1 creates t!e current-position.

8/14/2019 ko.pdf

16/46

ontinuation. t suffices if :lack passes ne t because t!at in"okes t!ec#cle-end-rule.

6lt!ou'! :lack does pre"ent !ite5slocal-area-i%pro"e%ent& !ite canans,er-force a c#cle on t!e %arkedintersections.

,$ample 1- Eike a%ple 9ee t!ec#cle and c#cle start-position t!ere.Jla# 2 creates t!e current-position.

$ro% t!e positions of a%ples F?& neit!er pla#er can ans,er-force a c#cle. +!is is i%%aterial for t!e definition oflocal-ko-intersection t!ou'!; it does not re4uire t!e c#cle to start fro% t!e current-position. at!er it re4uires in

particular so%e pla# of t!e c#cle to create t!e current-position.

,$ample 2- ac! of t!e %arkedintersections is a local-ko-intersection.

ontinuation. ? in"okes t!e c#cle-end-rule.

+!e opponent :lack does pre"ent t!e pla#er !ite5s local-area-i%pro"e%ent. !ite can ans,er-force a c#cle like t!eone fro% 1 to ?. +!erefore eac! of t!e %arked intersections is a local-ko-intersection.

,$ample 3- ac! of t!e %arkedintersections is a local-ko-intersection.

9tart-position of t!e c#cles inVariations 1F2. Jla# 1 creates t!ecurrent-position.

Variation 1.

ontinuation. D in"okes t!e c#cle-end-rule.

Variation 2. ontinuation. D in"okes t!e c#cle-end-rule.

$or eac! of t!e c#cles of Variations 1 or 2& t!e pla#er :lack is ,it!out local-area-i%pro"e%ent. +!is %eans t!at t!eopponent !ite does pre"ent :lack5s local-area-i%pro"e%ent. Ver# ot!er !ite c!oices can be dis%issed as notfulfillin' t!is. :lack does not create a so called stable seki because t!e definition asks !i% to ans,er-force a c#cle.:lack 1 is necessar# to fulfil t!e condition to recreate t!e current-position; :lack %a# not refuse to do so; it is outside:lack5s c!oice because it is si%pl# a %atter of e istence of t!is c#cle. :lack cannot ans,er-force a particular c#cle -:lack can ans,er-force one of t!e c#cles of Variations 1 or 2 or ot!er "ariations ,it! si%ilar c#cles. 9ince :lack canans,er-force one of t!ese c#cles& eac! of t!e %arked intersections is a local-ko-intersection.

,$ample 4- ac! of t!e %arkedintersections is a local-ko-intersection.

Variation 1. !ite does not c!ooset!is "ariation because !e ,ould not

pre"ent :lack5s local-area-i%pro"e%ent.

(continuation)

(continuation) (continuation) ontinuation. +!e fi ed-ko-rule pro!ibits !ite 1D at 10.

8/14/2019 ko.pdf

17/46

Variation 2. 9tart like Variation 1.Lo"e 1? co%pletes a situational c#cle

t!at creates t!e current-position.

Variation . !ite does not c!ooset!is "ariation because !e ,ould not

pre"ent :lack5s local-area-i%pro"e%ent.

(continuation)

(continuation) ontinuation. +!e fi ed-ko-rule pro!ibits !ite 1/ at ?.

Variation /. !ite does not c!ooset!is "ariation because !e ,ould not

pre"ent :lack5s local-area-i%pro"e%ent.

(continuation) ontinuation. +!e fi ed-ko-rule pro!ibits !ite 10 at 2.

Variation . !ite does not c!ooset!is "ariation because !e ,ould not

pre"ent :lack5s local-area-i%pro"e%ent.

(continuation) (continuation) (continuation)

ontinuation. +!e fi ed-ko-rule pro!ibits !ite 1D at 10.

Variation ?. 9tart like Variation ?.Lo"e 1? co%pletes a situational c#clet!at creates t!e current-position.

Variation 7. !ite does not c!ooset!is "ariation because !e ,ould not

pre"ent :lack5s local-area-i%pro"e%ent.

(continuation) (continuation) ontinuation. +!e fi ed-ko-rule pro!ibits !ite 1/ at ?.

Variation D. !ite does not c!ooset!is "ariation because !e ,ould not

pre"ent :lack5s local-area-i%pro"e%ent.

(continuation) +!e fi ed-ko-rule pro!ibits !ite 10at 2.

8/14/2019 ko.pdf

18/46

+o pre"ent :lack5s local-area-i%pro"e%ent& !ite can c!oose oneof t!e Variations 2 or ?. n eac! oft!e%& a situational c#cle creatin' t!ecurrent-position occurs. +!erefore:lack can ans,er-force suc! a c#cle.ence eac! of t!e %arked

intersections is a local-ko-intersection.

,$ample #5- ac! of t!e %arkedintersections is a local-ko-intersection.

Variation 1. !ite s!ould not c!oosecapture in t!e basic-ko.

(continuation) Variation 2. !ite s!ould not c!oosecapture in t!e basic-ko.

ontinuation. +!e opponent can alter!is pla#s a bit. +!erefore !itecannot ans,er-force a particular c#cle.

9i%ilar se4uences are possible. :lack pre"ents !ite5s local-area-i%pro"e%ent. !ite can ans,er-force

so%e suc! c#cle t!at starts at t!ecurrent-position and ends ,it! a pla#creatin' it. +!erefore eac! of t!e%arked intersections is a local-ko-intersection.

,$ample ##- ac! of t!e %arkedintersections is a local-ko-intersection.+!e start-position and se4uences areanalo'ue to a%ple D. !ite doesnot 'et a c!ance to capture t!e basic-ko.

,$ample #.- ac! of t!e %arkedintersections is a local-ko-intersection.+!e se4uences are analo'ue toa%ple I. !ite does not 'et a

c!ance to capture t!e basic-ko.

,$ample #'- ac! of t!e %arkedintersections is a local-ko-intersection.+!e s!ape first appeared in a 'a%e of+. Lark all.

Variation 1 ontinuation. 10 in"okes t!e c#cle-end-rule.

:efore or after t!e c#cle of Variation 1or 2& t!e local-area is 0.

Variation 2 ontinuation. 10 in"okes t!e c#cle-end-rule.

Variation

(continuation)

6fter Variation & t!e local-area is D&,!ic! is local-area-i%pro"e%ent for:lack. +!erefore !ite ? is a strate'ic%istake and !ite does not c!ooseVariation . ote t!at pla# 7 does notend t!e 'a%e b# t!e c#cle-end-rule.

+!e opponent !ite5s onl# c!ance to pre"ent t!e pla#er :lack5s local-area-i%pro"e%ent is a c#cle like in Variation 1or 2. n particular& since !ite can "ar# at %o"e ?& t!ere is %ore t!an one suc! c#cle. :lack can ans,er-force one oft!ese c#cles. +!erefore t!e %arked intersections are local-ko-intersections.

8/14/2019 ko.pdf

19/46

,$ample #/- ac! of t!e %arkedintersections is a local-ko-intersection.+!e c#cle ,as disco"ered b# $redansen.

ontinuation. D in"okes t!e c#cle-end-rule.

+!e opponent !ite5s onl# c!ance to pre"ent t!e pla#er :lack5s local-area-i%pro"e%ent is a c#cle like t!is. 9o :lackcan ans,er-force a c#cle. +!erefore t!e %arked intersections are local-ko-intersections.

,$ample #0- ac! of t!e %arkedintersections is a local-ko-intersection.

f t!e aut!or recalls correctl#& :ill+a#lor disco"ered pin,!eel kos.

1 at for clock-,ise rotation ,ouldalso be possible.

(continuation)

ontinuation. D in"okes t!e c#cle-end-rule.

:efore or after t!e c#cle& t!e local-area is -2. +!e opponent !ite5s onl#c!ance to pre"ent t!e pla#er :lack5s local-area-i%pro"e%ent is a c#cle liket!is& ,!ic! !e c!ooses. 9o :lack can ans,er-force a c#cle. +!erefore t!e%arked intersections are local-ko-intersections.

,$ample #1- ac! of t!e %arkedintersections is a local-ko-intersection.+!e aut!or disco"ered t!e s!ape ofa%ples 1?F17. Variation 1. Jla# 10 creates t!e

current-position.

ontinuation. 7 is allo,ed b# t!efi ed-ko-rule. Lo"es to D are a s!ortc#cle ,it!in t!e lon'er c#cle but Ddoes not in"oke t!e c#cle-end-rule

because D does not create t!e start- position& ,!ic! is t!e current-position.

Variation 2. Jla# I creates t!e current- position.

ontinuation. Lo"es to 7 are a s!ortc#cle ,it!in t!e lon'er c#cle but 7does not in"oke t!e c#cle-end-rule

because 7 does not create t!e start- position& ,!ic! is t!e current-position.10 in"okes t!e c#cle-end-rule.

Variation . Jla# 10 creates t!ecurrent-position.

ontinuation. Lo"es to 7 are a s!ortc#cle ,it!in t!e lon'er c#cle but 7does not in"oke t!e c#cle-end-rule

because 7 does not create t!e start- position& ,!ic! is t!e current-position.

10 in"okes t!e c#cle-end-rule.

!ite pre"ents :lack5s local-area-i%pro"e%ent. :lack can ans,er-forcea c#cle t!at creates t!e current-

position. +!erefore eac! of t!e%arked intersections is a local-ko-

intersection.

8/14/2019 ko.pdf

20/46

,$ample #2- ac! of t!e %arkedintersections is a local-ko-intersection

(and a basic-ko-intersection).

+!is can be s!o,n separatel# for t!ese/ and t!e ot!er / intersections.

(6lternati"e =se a%ple 1? as t!estart-position.) Variation 1

ontinuation. does not let t!e c#cle-end-rule end t!e 'a%e. ? in"okes t!ec#cle-end-rule and creates t!e current-

position. :lack is ,it!out local-area-i%pro"e%ent. 1 to ? are a c#cle.

Variation 2 ontinuation. creates t!e current- position. ? in"okes t!e c#cle-end-rule.:lack is ,it!out local-area-i%pro"e%ent; t!erefore it suffices t!at

!ite ? passes alt!ou'! !ite could!a"e %ade a pla# instead. 1 to are ac#cle.

:# Variation 1 or 2& !ite pre"ents :lack5s local-area-i%pro"e%ent. :lack can ans,er-force a c#cle t!e c#cle ofVariation 1 or t!e c#cle of Variation 2. +!erefore eac! of t!e / outer& %arked intersections is a local-ko-intersection.

o, it is s!o,n ,!# t!e ot!er /intersections are local-ko-

intersections.Variation

ontinuation. does not let t!e c#cle-end-rule end t!e 'a%e. ? in"okes t!e

c#cle-end-rule and creates t!e current- position. :lack is ,it!out local-area-i%pro"e%ent. 1 to ? are a c#cle.

Variation / ontinuation. creates t!e current- position. ? in"okes t!e c#cle-end-rule.:lack is ,it!out local-area-i%pro"e%ent; t!erefore it suffices t!at!ite ? passes alt!ou'! !ite could!a"e %ade a pla# instead. 1 to are ac#cle.

:# Variation or /& !ite pre"ents:lack5s local-area-i%pro"e%ent.:lack can ans,er-force a c#cle t!ec#cle of Variation or t!e c#cle ofVariation /. +!erefore eac! of t!e /inner& %arked intersections is a local-ko-intersection.

,$ample #3- ac! of t!e %arkedintersections is a local-ko-intersection.

Variation 1 fro% t!e current-positionas t!e start-position. :lack 'ets local-area-i%pro"e%ent.

9tart-position of Variations 2-/.

8/14/2019 ko.pdf

21/46

Variation 2

ontinuation. ? in"okes t!e c#cle-end-rule. creates t!e current-

position. 1 to are a c#cle. :lack is,it!out local-area-i%pro"e%ent.

Variation

ontinuation. !ite does not c!ooseVariation because :lack ,ouldre%o"e all ,!ite stones& i.e.& !ite,ould not pre"ent :lack5s local-area-i%pro"e%ent.

Variation /

ontinuation. !ite does not c!ooseVariation / because :lack ,ouldre%o"e all ,!ite stones& i.e.& !ite,ould not pre"ent :lack5s local-area-i%pro"e%ent.

+!e start-position of Variations 2-/ allo,s a suitable c#cle for t!e definition of local-ko-intersection. :# c!oosin'Variation 2& t!e opponent !ite pre"ents t!e pla#er :lack5s local-area-i%pro"e%ent. :lack can ans,er-force t!e c#clein Variation 2. +!erefore eac! of t!e %arked intersections is a local-ko-intersection.

,$ample #4- ac! of t!e %arked

intersections is a local-ko-intersection.:ill +a#lor disco"ered t!e dia%ond koin a%ples 1IF20.

+!e se4uence is one of t!e possible

c#cles.

(continuation)

(continuation)

(continuation)

ontinuation. 10 in"okes t!e c#cle-end-rule.

!ite5s onl# c!ance to pre"ent :lack5s local-area-i%pro"e%ent is to follo, ac#cle. :lack can ans,er-force one of t!e c#cles. +!erefore eac! of t!e %arkedintersections is a local-ko-intersection. ,$ample .5- ac! of t!e %arked

intersections is a local-ko-intersection.

8/14/2019 ko.pdf

22/46

,$ample .#- +!e position is fro% t!en' 1II1 ules booklet. 6ccordin' toru%ours& Latti 9ii"ola disco"eredtriple ko stones.

+!e %arked area5s score is 0. +!ere isan e"en nu%ber of da%e . +!ereforet!e e cite%ent is confined to t!e dut#of li"in' ,it! eit!er pla#er5s bi' centerstrin' and co%petin' for t!ere%ainin' intersections.

+!ese are t!e interestin' intersections.ac! of t!e% is a local-ko-intersection. +!e upper part consistsof ? intersections. +!e lo,er partconsists of onl# / intersections butdecides about t!e connection in caseof %issin' t,o e#es in t!e upper part.

Variation 1 a representati"e c#cle. (continuation) ontinuation. Lo"es 1 to 10 are ac#cle. Lo"e 10 creates t!e current-

position and in"okes t!e c#cle-end-rule.

:efore or after t!e c#cle of Variation1& t!e score& as indicated b# t!e local-area& is -2. +!is is ,it!out :lack5slocal-area-i%pro"e%ent.

Variation 2

6fter Variation 2& t!e local-area is 2.:# c!oosin' t!is "ariation& !ite,ould not pre"ent :lack5s local-area-i%pro"e%ent.

8/14/2019 ko.pdf

23/46

Variation departin' fro% t!e c#cle.

6fter Variation & t!e local-area is 2.:# c!oosin' t!is "ariation& !ite,ould not pre"ent :lack5s local-area-i%pro"e%ent.

ence& to pre"ent :lack5s local-area-i%pro"e%ent& !ite !as to follo, ac#cle. :lack can ans,er-force one oft!e c#cles like t!e one of Variation 1.+!erefore eac! of t!e %arkedintersections is a local-ko-intersection.+!is is re%arkable because it %eans

t!at one does not need to calculate t!e,!ole board score or perfect-pla# toidentif# a triple ko stones c#cle c#cle-set5s intersections as ko-intersections.n ot!er ,ords& t!e triple ko stones!ere do not re4uire a different t#pe ofko-intersection.

,$ample ..- ac! of t!e %arkedintersections is a local-ko-intersection.

Variation 1 (continuation)

ontinuation. / in"okes t!e c#cle-end-rule.

:efore or after t!e c#cle& t!e local-area is 0. !ite is ,it!out local-area-i%pro"e%ent.

Variation 2

(continuation) (continuation)

(continuation)

(continuation)

(continuation)

6fter t!e se4uence& t!e local-area is /.6lso Variation 2 is ,it!out !ite5sloca-area-i%pro"e%ent on t!eVariation 1 c#cle5s c#cle-set.

6lt!ou'! t!e c#cle is not pla#ed in Variation 2& for t!e purpose of t!e definition& it suffices t!at t!e c#cle fulfillin' t!econditions e ists. t suffices t!at& since& to fulfil t!e definition of local-ko-intersection& !ite !as to ans,er-force ac#cle& !ite c!ooses Variation 1 instead of Variation 2. - !ite ans,er-forces a c#cle& t!e c#cle of Variation 1& on t!e%arked intersections. Jla# / of t!e c#cle in Variation 1 creates t!e current-position. ac! of t!e %arked intersections

belon's to t!e c#cle5s c#cle-set. ence eac! is a local-ko-intersection. - t suffices t!at a c#cle startin' ,it! !ite%o"in' first and fulfillin' t!e definition e ists. Gne does not need to stud# :lack %o"in' first. 6t least one pla#er

%ust ans,er-force a c#cle. t is not re4uired t!at bot! pla#ers can because t!e definition asks onl# for 3a pla#er3.

8/14/2019 ko.pdf

24/46

,$ample .'- +!e position is b#er%an idde%a& ,!o t!ereb#redisco"ered so%e c#cle ,it! pla#s.

#cle 1 6 c#cle fro% %o"e 1 to .+!e start-position is t!e current-

position. ontinuation. creates t!e current- position. ? in"okes t!e c#cle-end-rule.

Variation 1.1

(continuation)

Variation 1.2

n #cle 1 and Variations 1.1 and 1.2&t!e opponent !ite pre"ents t!e

pla#er :lack5s local-area-i%pro"e%ent. !ite can c!oose& e.'.&Variation 1.2. :lack cannot ans,er-force t!e c#cle in #cle 1. +!ereforet!e c#cle in #cle 1 does not fit t!edefinition of local-ko-intersection. #cle 2 9tart-position of a c#cle. Jla#

creates t!e current-position.

ontinuation. Lo"es 1 to ? are ac#cle. ? creates t!e current-positionand in"okes t!e c#cle-end-rule. :lackis ,it!out local-area-i%pro"e%ent.

Variation 2.1 !ite %a# not c!oose 2 because t!en !e does not pre"ent:lack5s local-area-i%pro"e%ent.

Variation 2.2 f :lack !ad to c!ooset!is "ariation& t!en t!e opponent!ite could pre"ent t!e pla#er:lack5s local-area-i%pro"e%ent.

8/14/2019 ko.pdf

25/46

(continuation)

+o ans,er-force a c#cle& :lackc!ooses #cle 2 rat!er t!an Variation2.2. +!e #cle 2 stud# s!o,s t!at t!e%arked intersections are local-ko-intersections.

,$ample ./ ac! of t!e %arked basic-ko-intersections is a local-ko-intersection.

ontinuation. Lo"e / does not in"okea 'a%e end b# t!e c#cle-end-rule.

!ile :lack pre"ents !ite5s local-area-i%pro"e%ent& !ite ans,er-forces t!e c#cle 1 to /. +!erefore eac!of t!e %arked intersections is a local-ko-intersection.

"ounter e$ample .0

"en none of t!e %arked basic-ko-intersections is a local-ko-intersection.

!ile :lack pre"ents !ite5s local-area-i%pro"e%ent& !ite cannotans,er-force a c#cle.

,$ample .1

ac! of t!e %arked basic-ko-

intersections is a local-ko-intersection.ontinuation. Lo"e / does not in"okea 'a%e end b# t!e c#cle-end-rule.!ile :lack pre"ents !ite5s local-area-i%pro"e%ent& !ite ans,er-forces t!e c#cle 1 to /. +!erefore eac!of t!e %arked intersections is a local-ko-intersection.

8/14/2019 ko.pdf

26/46

,$ample .2- nstable 4uadruple ko.ac! of t!e %arked intersections is a

local-ko-intersection. 9ince t!e t,ori'!t-%ost %arked intersections arenot basic-ko-intersections& it isinterestin' to kno, t!at also t!e# arelocal-ko-intersections.

Jro"in' local-ko-intersections can bedone& e.'.& b# anal#sin' t!ree pairs of

t,o adjacent intersections at a ti%e&t!en doin' an analo'ue anal#sis forot!er t!ree pairs includin' t!e ri'!t-%ost.

ontinuation. ? in"okes t!e c#cle-end-rule. +o pre"ent :lack5s local-area-i%pro"e%ent& !ite needs to%ake t!e captures. :lack can ans,er-force t!e c#cle 1 to ?. +!erefore eac!of t!e %arked intersections is a local-ko-intersection.

,$ample .3- nstable n-tuple-ko(n>/). ac! of t!e crossedintersections is a local-ko-intersection.9ince t!e t,o ri'!t-%ost crossedintersections are not basic-ko-intersections& it is interestin' to kno,t!at also t!e# are local-ko-intersections. ( n stable n-tuple-kos&t!ere are 2n basic-ko-intersectionsan#,a#.)

+o a"oid effecti"e !ite stoneconnections& pro"in' local-ko-intersections %ust be done for all t!e2n crossed intersections to'et!er. ueto Jropositions 1-/& :lack can ans,er-force a c#cle of Jroposition 1 if !ite

pre"ents :lack5s local-area-i%pro"e%ent. ence eac! of t!ecrossed intersections is a local-ko-intersection.

Propositions for ,$ample .3

Proposition #- f !ite %akes onl# basic-ko captures and pre"ents :lack5s local-area-i%pro"e%ent& t!en :lack can ans,er-force a c#cle t!at starts fro% t!e current-position and !as all

crossed intersections as its c#cle-set.Proof- n t!e current-position& t!e local-area is / - 2n. !ene"er& fro% a position ,it! t!at local-area& :lack %akes a basic-ko capture& t!is i%pro"es t!e local-area for !i% to D - 2n. +o pre"ent:lack5s local-area-i%pro"e%ent& !ite %ust repl# b# a basic-ko capture (if !ite passes& t!en a):lack !a"in' open basic-kos connects a stone or else b) :lack i%pro"es t!e local-area b# anot!er

basic-ko capture #et furt!er in !is fa"our). ue to t!e basic-ko-rule& !ite %a# not recapture t!esa%e basic-ko i%%ediatel#& i.e.& !ite !as to capture anot!er basic-ko. n particular& !ite5s first

basic-ko capture captures t!e ri'!t-%ost basic-ko. :lack c!ooses to capture t!e left-%ost basic-kot!at !e !as not captured #et. !ite follo,s suit necessaril#. +!ereb# on all t!e crossed intersections

pla#s ,ill be %ade until :lack5s ri'!t-%ost capture creates t!e current-position. +!is is t!e first

occurrence of a c#cle& ,!ic! is situational& in"okes t!e c#cle-end-rule and !as ne"er "iolated t!efi ed-ko-rule. N .

Proposition .- f alread# n - / stones !a"e been connected& 2 basic-kos are open for :lack and 2 basic-kos are open for !ite& t!en after !ite5s connection of anot!er stone !ite cannot pre"ent:lack5s local-area-i%pro"e%ent.

Proof- e t :lack captures all ,!ite stones. N .

Proposition '- f at so%e ti%e !ite connects a stone instead of %akin' a basic-ko capture& fe,ert!an n - / stones !a"e alread# been connected and after t!e ne,l# connected stone !ite does notconnect #et anot!er stone& t!en !ite cannot pre"ent :lack5s local-area-i%pro"e%ent.

Proof- f :lack - accordin' to t!e proof of Jroposition 1 - !as open basic-kos& t!en :lackconnects a stone; else :lack passes. +!is leads to a situation ,it! e actl# 2 basic-kos open for:lack& %>2 basic-kos open for !ite and !ite to %o"e. 6ssu%in' ,it!out loss of 'eneralit# t!eattacked :lack t!en al,a#s captures t!e left-%ost basic-ko t!at currentl# !e %a# capture& t!en ?

8/14/2019 ko.pdf

27/46

pla#s are like a tiple ko c#cle& after,ards 2 successi"e passes enable !ite a basic-ko recapture andafter,ards ? pla#s for t!e ot!er triple-ko-like c#cle on t!e sa%e ? intersections let !ite finall# runout of options (ot!er t!an passin') due to t!e fi ed-ko-rule (,!ile t!e %o"es before ,ere not

pro!ibited b# t!e fi ed-ko-rule and did not in"oke a 'a%e end due to t!e c#cle-end-rule). +!is ,illlead to so%e position ,it! t!e local-area of at least D - 2n and !ite does not pre"ent :lack5s local-area-i%pro"e%ent co%pared to t!e local-area / - 2n of t!e current-position. N .

Proposition /- f %ore and %ore stones are connected as in Jropositions 1 or & t!en at so%e ti%et!e assu%ptions of Jroposition 2 ,ill be in"oked.

Proof- +ri"ial.

*lobal ko intersection+erritor# scorin' is assu%ed to be defined as in t!e Hapanese 200 ules C "ersion a. +!e"ariations are selected so t!at t!e effect of da%e lettin' independentl#-ali"e 'roups be in-seki isirrele"ant in practice.

"ounter e$ample #

"ounter e$ample # Area (corin&6 Komi 7 #36 Black to 8o!e

n particular& none of t!e %arkedintersections is a 'lobal-ko-intersection.

9core M 27 - I - 1D M 0.!ile !ite pre"ents :lack5s ,in&:lack cannot ans,er-force a c#cle.+!erefore none of t!e %arkedintersections is a 'lobal-ko-intersection.

8/14/2019 ko.pdf

28/46

"ounter e$ample . Area (corin&6 Komi 7 #2906 Black to 8o!e

n particular& none of t!e %arkedintersections is a 'lobal-ko-intersection.

9core M 27 - I - 17. M 0. .!ite cannot pre"ent :lack5s ,in.+!erefore none of t!e %arkedintersections is a 'lobal-ko-intersection.

8/14/2019 ko.pdf

29/46

"ounter e$ample ' Territory (corin&6 Komi 7 #06 Black to 8o!e

Variation. Jrisoner-difference M 2.

9core M 1 - 2 F 2 - 1 M 0.!ite does pre"ent :lack5s ,in. :lackcannot ans,er-force a c#cle.

+!erefore in particular none of t!e%arked intersections is a 'lobal-ko-intersection.

,$ample / Territory (corin&6 Komi 7 #/906 Black to 8o!e

,$ample / Rele!ant (corin& Positions

9corin' Josition 1. :lack ,ants toa"oid t!is scorin' position because&e.'.& in particular t!e stone is dead.

+!is representati"e !#pot!etical-se4uence indicates ,!# is dead.

ontinuation. +!e %arked stone is notcapturable-2 eit!er because& in t!e%o"e-se4uence& !ite can re%o"e allt!e black stones.

8/14/2019 ko.pdf

30/46

9corin' Josition 2. +!e prisoner-difference is 2.

9core M 1 - 2 F 2 - 1/. M 0. .!ite ,ants to a"oid t!is scorin'

position because !e does not pre"ent:lack5s ,in.

9corin' Josition . +!e prisoner-difference is -/. Gb"iousl# :lack,ants to a"oid t!is scorin' position.

9corin' Josition /. +!e prisoner-difference is 7.

9core M 10 - 2 F 7 - 1/. M 0. .!ite ,ants to a"oid t!is scorin'

position because !e does not pre"ent:lack5s ,in.

Variation /.1 creatin' 9corin'Josition /.

9corin' Josition . +!e prisoner-difference is 7.

9core M 10 - 2 F 7 - 1/. M 0. .!ite ,ants to a"oid t!is scorin'

position because !e does not pre"ent:lack5s ,in.

Variation .1 creatin' 9corin'Josition .

(continuation)

(continuation)

9corin' Josition ?. +!e prisoner-difference is 7.

9core M I - 2 F 7 - 1/. M -0. . Variation ?.1 creatin' 9corin'Josition ?.

8/14/2019 ko.pdf

31/46

(continuation)

ontinuation. :lack !as to eli%inateall t!e kos so t!at none of !is strin'sis dead.

9corin' Josition 7. +!e prisoner-difference is 7.

9core M 10 - 2 F 7 - 1/. M 0. .!ite ,ants to a"oid t!is scorin'

position because !e does not pre"ent:lack5s ,in.

Variation 7.1 creatin' 9corin'Josition 7.

(continuation)

(continuation)

9corin' Josition D. +!e prisoner-difference is 7.

9core M I - 2 F 7 - 1/. M -0. .:lack ,ants to a"oid t!is scorin'

position.

Variation D.1 creatin' 9corin'Josition D.

(continuation)

(continuation)

8/14/2019 ko.pdf

32/46

,$ample / Decisions "reatin& the "ycle

Lo"e 1

!ite does pre"ent :lack5s ,in 1) f:lack 1 passes& t!en 9corin' Josition1 can occur. 2) f :lack c!ooses %o"e1& t!en see belo,.

Lo"e 2

!ite does not fill t!e da%e +!is,ould lead to 9corin' Josition 2.!ite ,ould not pre"ent :lack5s ,in.

Lo"e

!ite does pre"ent :lack5s ,in 1) f:lack 1 captures t!e lo,er left cornerstone& t!en 9corin' Josition canoccur. 2) f :lack c!ooses %o"e &t!en see belo,.

Lo"e /

!ite does not fill t!e da%e +!is,ould lead to 9corin' Josition /.!ite ,ould not pre"ent :lack5s ,in.

Lo"e

!ite does pre"ent :lack5s ,in 1) f:lack passes& t!en 9corin' Josition? can occur. ( !ite does not c!ooseVariation .1.) 2) f :lack connects&t!en 9corin' Josition D can occur. )f :lack c!ooses %o"e & t!en see

belo,.

Lo"e ?

1) !ite does not fill t!e da%e +!is,ould lead to 9corin' Josition 7.!ite ,ould not pre"ent :lack5s ,in.2) :# c!oosin' %o"e ?& !ite does

pre"ent :lack5s ,in& as 9corin'Josition 1 s!o,s. Lo"e ? in"okes t!ec#cle-end-rule.

9u%%ariKin' t!e i%plication of all decisions !ite does pre"ent :lack5s ,in.:lack ans,er-forces a c#cle& ,!ic! occurs at %o"e ?. +!e c#cle starts fro%t!e current-position and& in t!e c#cle& %o"e ? creates t!e current-position.+!erefore& in t!e follo,in' dia'ra%& eac! of t!e %arked intersections is a'lobal-ko-intersection.

8/14/2019 ko.pdf

33/46

,$ample / None of the 8arked Intersections is a )ocal ko intersection

n particular none of t!e %arkedintersections is a local-ko-intersection.n t!e follo,in' dia'ra%s& t!e local-area-i%pro"e%ent is calculated ont!ese intersections.

Variation 1. !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro%- to -2.

9tartin' position of Variation 2.

Variation 2. !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro%-1 to 0.

9tartin' position of Variation .

Variation . !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro%-2 to 0.

9tartin' position of Variation /.

Variation /. !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro% to ?.

9tartin' position of Variation .

8/14/2019 ko.pdf

34/46

Variation . !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro%1 to ?.

9tartin' position of Variation ?. Variation ?. !ite cannot pre"ent:lack5s local-area-i%pro"e%ent fro%2 to ?.

(continuation)

(continuation)

9u%%ar# of Variations 1-? andsi%ilar "ariations one of t!e %arkedintersections is a local-ko-intersection.

,$ample 0 Territory (corin&6 Komi 7 #/6 Black to 8o!e

,$ample 0 Rele!ant (corin& Positions

9corin' Josition 1. :lack ,ants toa"oid t!is scorin' position because&e.'.& in particular t!e stone is dead.

+!is representati"e !#pot!etical-se4uence indicates ,!# is dead.

ontinuation. +!e %arked stone is notcapturable-2 eit!er because& in t!e%o"e-se4uence& !ite can re%o"e allt!e black stones.

8/14/2019 ko.pdf

35/46

9corin' Josition 2. +!e prisoner-difference is 2.

9core M 1 - 2 F 2 - 1/ M 1. 9corin' Josition . +!e prisoner-difference is -/. Gb"iousl# :lack,ants to a"oid t!is scorin' position.

9corin' Josition /. +!e prisoner-difference is 7.

9core M 10 - 2 F 7 - 1/ M 1.

Variation /.1 creatin' 9corin'Josition /.

9corin' Josition . +!e prisoner-difference is 7.

9core M 10 - 2 F 7 - 1/ M 1.

Variation .1 creatin' 9corin'Josition .

(continuation)

(continuation)

9corin' Josition ?. +!e prisoner-difference is 7.

9core M I - 2 F 7 - 1/ M 0. Variation ?.1 creatin' 9corin'Josition ?.

8/14/2019 ko.pdf

36/46

(continuation)

ontinuation. :lack !as to eli%inateall t!e kos so t!at none of !is strin'sis dead.

9corin' Josition 7. +!e prisoner-difference is 7.

9core M 10 - 2 F 7 - 1/ M 1.!ite ,ants to a"oid t!is scorin'

position because !e does not pre"ent:lack5s ,in.

Variation 7.1 creatin' 9corin'Josition 7.

(continuation)

(continuation)

9corin' Josition D. +!e prisoner-difference is 7.

9core M I - 2 F 7 - 1/ M 0.

Variation D.1 creatin' 9corin'Josition D.

(continuation)

(continuation)

8/14/2019 ko.pdf

37/46

,$ample 0 Decisions "reatin& the "ycle

Lo"e 1

!ite does pre"ent :lack5s ,in 1) f:lack 1 passes& t!en 9corin' Josition1 can occur. 2) f :lack c!ooses %o"e1& t!en see belo,.

Lo"e 2

nstead of %o"e 2& !ite does not fillt!e da%e +!is ,ould lead to 9corin'Josition 2. !ite does pre"ent :lack5s,in see belo,.

Lo"e

!ite does pre"ent :lack5s ,in 1) f:lack captures t!e lo,er left cornerstone& t!en 9corin' Josition canoccur. 2) f :lack c!ooses %o"e &t!en see belo,.

Lo"e /

nstead of %o"e /& !ite cannot fillt!e da%e +!is ,ould lead to 9corin'Josition /. !ite ,ould not pre"ent:lack5s ,in. +!erefore !ite c!oose%o"e / to pre"ent :lack5s ,in& see

belo,.

Lo"e

!ite does pre"ent :lack5s ,in 1) f:lack passes& t!en 9corin' Josition? can occur. ( !ite does not c!ooseVariation .1.) 2) f :lack connects&t!en 9corin' Josition D can occur. )f :lack c!ooses %o"e & t!en see

belo,.

Lo"e ?

1) !ite does not fill t!e da%e +!is,ould lead to 9corin' Josition 7.!ite ,ould not pre"ent :lack5s ,in.2) :# c!oosin' %o"e ?& !ite does

pre"ent :lack5s ,in& as 9corin'Josition 1 s!o,s. Lo"e ? in"okes t!ec#cle-end-rule.

9u%%ariKin' t!e i%plication of all decisions !ite does pre"ent :lack5s ,in.:lack ans,er-forces a c#cle& ,!ic! occurs at %o"e ?. +!e c#cle starts fro%t!e current-position and& in t!e c#cle& %o"e ? creates t!e current-position.+!erefore& in t!e follo,in' dia'ra%& eac! of t!e %arked intersections is a'lobal-ko-intersection.

"ounter e$ample 1 Territory (corin&6 Komi 7 #'906 Black to 8o!e

+!e anal#sis is like in a%ple & e cept t!at t!e score of e"er# scorin' position is 0. 'reater. 6t%o"e & t!is 'i"es :lack anot!er option to c!oose 9corin' Josition D. +!erefore t!e conclusiondiffers& too.

8/14/2019 ko.pdf

38/46

9corin' Josition D. +!e prisoner-difference is 7.9core M I - 2 F 7 - 1 . M 0. .

Variation D.1 creatin' 9corin'Josition D.

(continuation)

(continuation)

!ite cannot pre"ent :lack5s ,in.+!erefore in particular none of t!e%arked intersections is a 'lobal-ko-intersection.

"ounter e$ample 2

"ounter e$ample 2 Area (corin&6 Komi 7 #0906 Black to 8o!e

n particular& none of t!e %arkedintersections is a 'lobal-ko-intersection.

Variation 1

9corin' position after Variation 1.9core M 2? - 10 - 1 . M 0. .:# Variation 1& !ite does not

pre"ent :lack5s ,in. +!erefore!ite5s onl# c!ance is to create ac#cle.

8/14/2019 ko.pdf

39/46

Variation 2 (continuation)

(continuation)

9corin' position after Variation 2.9core M 2? - 10 - 1 . M 0. .:# Variation 2& !ite does not

pre"ent :lack5s ,in& eit!er.

9ince !ite cannot pre"ent :lack5s,in& none of t!e %arked intersectionsis a 'lobal-ko-intersection.

"ounter e$ample 3

"ounter e$ample 3 Area (corin&6 Komi 7 #16 Black to 8o!e

!ile& for t!is position an so%e appropriate ko%i& under territo# scorin' 'lobal-ko-intersectionscan e ist& under area scorin' t!e# do not e ist for t!is position.

n particular& none of t!e %arkedintersections is a 'lobal-ko-intersection.

epresentati"e se4uence.

9core M 2? - 10 - 1? M 0.!ite does pre"ent :lack5s ,in. :lackcannot ans,er-force a c#cle.+!erefore none of t!e %arkedintersections is a 'lobal-ko-intersection.

8/14/2019 ko.pdf

40/46

,$ample 4 Komi 7 5906 Black to 8o!e

f t!e ko%i ,ere 0 or an# ot!er "alue& t!ere ,ould be t!e sa%e 'lobal-ko-intersections. ependin'on t!e ko%i& t!e pla#ers5 roles %i'!t be s,apped. f t!e ko%i is 0& t!en eit!er pla#er can assu%e t!erole of ans,er-forcin' a c#cle.

ac! of t!e %arked intersections is a'lobal-ko-intersection. ( ote ac! isalso a basic-ko-intersection and alocal-ko-intersection.) Variation 1

ontinuation. is allo,ed b# t!efi ed-ko-rule and does not end t!e'a%e b# t!e c#cle-end-rule. ? in"okest!e c#cle-end-rule. +!e score is -0. .!ite pre"ents :lack5s ,in. +!einterestin' c#cle is fro% 1 to ?. :lack%o"es first in it fro% t!e current-

position. Lo"e ? creates t!e current- position. ac! of t!e %arked

intersections is in t!e c#cle-set.

Variation 2 ontinuation. ? in"okes t!e c#cle-end-rule. +!e score is -0. .

!ite pre"ents :lack5s ,in. +!einterestin' c#cle is fro% 1 to . :lack%o"es first in it fro% t!e current-

position. Lo"e creates t!e current- position. ac! of t!e %arkedintersections is in t!e c#cle-set.

!ite pre"ents :lack5s ,in. :lack can ans,er-force one of t!e interestin' c#cles in Variations 1 or 2. ence eac! oft!e %arked intersections is a 'lobal-ko-intersection.

,$ample #5 Komi 7 5906 Black to 8o!e

Nuadruple ko stones and t!e position5s topic ,ere disco"ered b# Latti 9ii"ola.

+!e area on t!e uninterestin' rest of t!e board is -1.

8/14/2019 ko.pdf

41/46

,$ample #5 (corin& Positions

Variation 1. 6fter ?& pla#in' in t!e triple ko isuninterestin' under t!e default restriction rules.

9corin' position after Variation 1.9core M 1 F 1D F 7 F 0. - 17 - 22 - 1 M /0. - /0 M 0. .

Variation 2.

(continuation)

9corin' position after Variation 2.9core M 2 F 17 F 0. - 2 - 7 - 1 M /I. - 1 M 1D. .

Variation . 6fter D& pla#in' in t!e triple ko isuninterestin' under t!e default restriction rules.

8/14/2019 ko.pdf

42/46

(continuation)

9corin' position after Variation .9core M 17 F 17 F 0. - 1 - 2 - 7 - 1 M /. - /? M -11. .

Variation /. Lo"e I capturin' D ,ould be ans,ered b# 12.

9corin' position after Variation /.9core M 2 F 7 F 0. - /0 - 1 M I. - /1 M -1. .

Variation . 6fter 1 & pla#in' in t!e triple ko is

uninterestin' under t!e default restriction rules. (continuation)

9corin' position after Variation .9core M 17 F 22 F 0. - 1 - 1D - 7 - 1 M I. - /1 M -1. .

Variation ?.

8/14/2019 ko.pdf

43/46

(continuation)

(continuation)

9corin' position after Variation ?.9core M 2 F 7 F 0. - 2 - 17 - 1 M 0. - 0 M -1I. .

Variation 7. 6fter 1 & pla#in' in t!e triple ko isuninterestin' under t!e default restriction rules.

(continuation)

(continuation)

9corin' position after Variation 7.9core M 1 F 2 F 7 F 0. - 17 - 17 - 1 M / . - M 10. .

Variation D.

8/14/2019 ko.pdf

44/46

ontinuation. Lo"e 1? capturin' 1 ,ould be ans,ered b# 1I.

9corin' position after Variation D.9core M /0 F 0. - 2 - 7 - 1 M /0. - /0 M 0. .

,$ample #5 The :uadruple Ko (tones "ycle

+!e c#cle starts fro% t!e current-position. Lo"e 1/ creates t!e current-position.

ac! of t!e %arked intersections is a 'lobal-ko-intersection.

otes 1) Jrobabl# none of t!e %arked intersections is alocal-ko-intersection. +!is s!ould be supported b# carefulanal#sis t!ou'!. 2) $or t!e ko%i ran'e -0. to -2& eac! oft!e %arked intersections is a 'lobal-ko-intersection. ) ft!e ko%i is -2. or s%aller& t!en !ite cannot pre"ent:lack5s ,in and none of t!e %arked intersections is a'lobal-ko-intersection. /) f t!e ko%i is 0 or 'reater& t!en!ite can pre"ent :lack5s ,in b# c!oosin' %o"e 2 inVariation 1 ,!ile :lack cannot ans,er-force t!e c#cle;none of t!e %arked intersections is a 'lobal-ko-intersection.

,$ample #5 Decisions "reatin& the "ycle

Lo"e 1 :lack cannot force !is ,in (i.e.& !ite does pre"ent it) but !e can contribute to t!e c#cle.

Lo"e 2 !ite does pre"ent :lack5s ,in 1) !ite does not c!oose %o"e 2 in Variation 1 becauset!en :lack ,ould ,in. 2) !ite does not c!oose %o"e 2 in Variation 2 because t!en :lack ,ould,in. ) !ite c!ooses %o"e 2 of t!e c#cle& see belo,.

Lo"e :lack cannot force !is ,in but !e can contribute to t!e c#cle.

Lo"es /F?FDF10 !ite does pre"ent :lack5s ,in 1) !ite does not c!oose ot!er %o"es ob"ious.

8/14/2019 ko.pdf

45/46

2) !ite c!ooses %o"es /F?FDF10 of t!e c#cle& see belo,.

Lo"e :lack in Variation does not force :lack5s ,in. +!erefore :lack contributes to t!e c#cle.

Lo"e 7 :lack 7 in Variation / does not force :lack5s ,in. +!erefore :lack contributes to t!e c#cle.

Lo"e I 1) :lack I in Variation does not force :lack5s ,in. 2) :lack I in Variation ? does notforce :lack5s ,in. ) +!erefore :lack contributes to t!e c#cle.

Lo"es 11F1 :lack cannot force !is ,in but !e can contribute to t!e c#cle.

Lo"e 12 !ite does pre"ent :lack5s ,in 1) !ite does not c!oose %o"e 12 in Variation 7 because t!en :lack ,ould ,in. 2) !ite c!ooses %o"e 12 of t!e c#cle& see later %o"es.

Lo"e 1/ !ite does pre"ent :lack5s ,in 1) !ite does not c!oose %o"e 1/ in Variation D because t!en :lack ,ould ,in. 2) !ite c!ooses %o"e 1/ of t!e c#cle. +!is in"okes t!e c#cle-end-rule. :lack !as no, ans,er-forced t!e c#cle.

,$amples of Kon eac! e a%ple& all kos in t!e current-position are denoted. 6ll intersections of t!e sa%e ko carr#t!e sa%e letter. 9o different letters indicate different kos.

6lt!ou'! a triple ko !as onl# t,o basic-kos at an# ti%e& t!e na%e ,asc!osen ,ell +!ere are t!ree kos.

8/14/2019 ko.pdf

46/46

*o%i M 1/. & :lack to %o"e& +erritor#9corin'

*o%i M 1?& :lack to %o"e& 6rea9corin'

*o%i M -0. & :lack to %o"e *o%i M 0& :lack to %o"e

"han&es )o&Version 7b

- correct t#pos in t!ree dia'ra%s

- clarif# ,ordin' of local-& 'lobal-ko-intersection ,!ile not c!an'in' t!e intended contents

- clarif# ,ordin' else,!ere

Version 7a

- correct t#pos and ot!er tin# %istakes

- replace declaration of !istor#-bans b# definition

- correct a%ples 27F2D on local-ko-intersection- clarif# ,ordin' of (ans,er-)strate'#& ans,er-force ,!ile not c!an'in' t!e intended contents

Version 7

- first publis!ed "ersion