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DESCRIBING PAWN STRUCTURES.

TAN S.T,EDINBURGH, DEC. I974.

JO <sLp^Qi2r <"*--.

fT^W^y^^JLO **-, GJU-/W4

|\fo(* to Ye CopJtd. °T AistriG^lkJl -L^capTfc* fr^vo-fe. e^o^ f>u-rpoS*-S .

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i . THE I HEORY OF GAMES .AND PRINCIPLES

3 .PAST. AND PRESENT.4. THE 9XIO --1 BOARD.r> . P A w N TO PAWN RELATIONS.6.T IE ATTACK DEFENCE DIAGRAM./. CONS 1 RUCTION AND USE.h.u^dAT ING AND CAUSAL I TY .PREFERENCES.

ARsTR A C T .

DEsCRIBTIONS OF A CHESS POSITION SHOULD CONTAIN THOSEFEATURES WHICH ARE:(I)AS INVARIANT AS POSS I RL E , I .E .DO NUT CHANGE TOO RAPIDLY FROM MOVE TO MOVE ,( I I ) PER-TINENT TO ANALYSIS AND TO OUR CHESS KNOWLEDGE , A ND(Iit) NUT TOO EASY FOR DIRECT EVALUATION.-MNARY PAWN TO PAWN RELATIONS GIVE RISE TO A 'PAWN-KELAMONS GRAPH' WHOSE CONNECTED COMPONENTS FORM PAWNISLANDS AND FRONTS . REL A T I ONS BETWEEN THESE ISLANDS AND..EHEEN THESE ISLANDS AND THE KINGS ,IN THE FORM OFIHREATS AND DEFENCES TO THREATS, ARE REPRESENTED AS'ATTACK-DEFENCE D I AGRAMS ' . THF UPDATING RU_.ES FOR THE'OAWN-HELATIONS GRAPHS' AND 'ATTACK-DEFENCE DIAGRAMS'SHOW HOW CHANGES CAUSED BY MOVES CAN BE LOCALIZED ASMUCH As POSSIBLE.

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F"Ew PEOPLE WOULD BE SURPRISED BY THE FACT THAT THE BORFL-VONNEiMA.N THEORY OF GAMF3 AS IT IS TOD A Y , AL THOUGH ORIGINALLY INTENDED10 MODEL PARLOUR GAMES, HAS LITTLE BEYOND TERMINOLOGICAL TRIVIAIITIF-.SH COMMON WITH COMPUTER CHESS. THE REDUCTION OF EXTENSIVE FORMS OF/r-KO-SUM, TWO-PERSON GAMES WITH PERFECT INFORMATION TO THEIR NORMAL-I /El) FORMS BY MEANS OF PURE STRATEGIES (SEE E.G. KUHNTBJ) LEA'-S TOA COMBINATORIAL EXPLOSION THAT PUTS EVEN TIC-TAc-TOfc OUT OE REACHOF OaMe THEORY. IN PRACTICE, WHAT PEOPIE DO IS TO ASSUME THE NORMALIZEDFORM I.N THE FIRST PLACE, AND THE AVAILABILITY OF UTIL I T V FUNC T lONS .THIS MAKES THE THEORY INAPPLICABLE TO CASES WHERE WE CAN'T FIND AS--IAI.L s£T OF PURL: STRA TEG I E 3 , WHERE THE UTILITIES CAN'T BF OBTAINED,o. WHERE STATISTICAL SOLUTIONS IN TERMS 0~ MIXED STRATEGIES ARE OFnO INTEREST TO US.■-FVErtTHELESS THE TRANSFORMATION FROM EXTENSIVE TO NORMALIZED FORMSIS CONCEPTUALLY 5 I GN I F IC A NT ; TH I NK I NG IN TERMS OF STRATEGIES INSTEADue *tuVbS IS A STEP 'UP WARDS * , AND THE RESULTING MATRIX FORMAT ISSIMPLER AND SO MORE AMENABLE TO MATHEMATICAL TREATMENT THAN THE TREEFORMAT. NOTE THAT THE NORMALIZED FORM IS COMPLETE IN ITSELF, IN THEsEHSE THAT ONCE IT IS CONSTRUCTED , NO REFERENCE TO THE ORIGINALEX I ENS IVE FORM IS NEEDED.being less ambit i ous, one may then ask, if it is not possible, bydropping the completeness property, to perform similar 'sfmanthascents* which will re more useful in the case of chess and fnngamesh particular. this is attempted here for e md i ngs : at tack-defencedependency relations for groups of pawns and kings are used toexpress some general properties of the end i ngs . ho we ver , before oe goany further into the matter, lest anybody should think that we areequating any 'semantic ascent* with progress itself, we must firsthive reasons why such a step is not only desirable but also necessary.

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THE ADVANTAGES OK MAKING RFOU I REMEN TS , Wi l CH WE WOULD LIKE TO SEEAAi'ISFIED IN I NTF:LL ICENT ChESS PROGRAMS, AMD THE UNDERLYING PRINCIPLESEXPLICIT, ARE: IT FACIEI TATbS THEIR DEHUGG I MG , A MD IT ALLOWSCMM-'ARISON WITH THE ACTUAL RE AL IZATI3N , TH JS SAVING US FROM MANYILLUSIONS.

(I)PLURALITY AND MODULARITY.

H.O.ALiTY IS HERE uSFji AS THE OPPOSITE OF' UNI I FORM IT V , IT ISCHA RACIERIZED HY ITS ABILITY TO ACCOMMODATE MANy SPECIAL METHODSAMU SPECIAL KNOWLEDGE.IT HAS REEN CONjECTJREO THAT INTELLIGENCEIS A QUESTION OF HAVING A 'RIG SWITCH* <SiE E.G. NEWELL 1 10 .1 , 0 . 1 0 )WHICH EFFICIENTLY CONNECTS PARTICULAR DOMAINS WITH SPECIAL METHODS.EACH METHOD REPRESENTING EXPERIISe IN ITS OWN DOMAIN.C JESS IN GENERAL AND ENDGAMES IN PARTICULAR MAY HE VIEWED AS ONEPRO ,LE:i AND ONE MAY DEVISE A UNIFORM METHOD TO DEAL WITH IT,AR ALTERNATIVELY, IT MAY Re CONSIDERED AS A MOSAIC OF" THOUSANDS OFA-HPNOULEMS, EACH OF WHICH REoUIRIMG DIFFERENT WAYS OF HAMDL I MG . JUSTFUNK HOW NICELY STRUCTURED INTO STAGES IS THE MATING PROCESS WITH* SINGLE ROOK, HOW X I NIG »KN I GHT AND SISHOP COORDINATE TO SET MATE,HOW ONE 4QRKS BACKWARD FROM CRITICA.. POSITIONS TO FIND RELATEDAjUAHEb IN BLOCKED OR ALMOST CLUCKED PAWN ENDINGS, HOW ONE TAKESA WANTAGE OF PRFVIOUS EXHAUSTIVE ANALYSIS IM THE FORM OF TROIT7KYLI.mE,HAEHR ZONES, THE ZONES FOR U VS. P,ETC.

THE OBJECTION TO UNIFORMITY IS ITS I MEFF I CI ENCY : TAKE STATICEVALUATION FUNCTIONS; IF THEY ARE TO HE PRECISE, THEY MUST 811B 11 SUFFI-CIENTLY COMPLEX AMD COSTLY, FOR AS EXAMPLES DEMONS 19ATE , M I NORPER TURHA TIONS IN A POSITION MAY EASIIY ALTER THE VALUE OR EVEN THEC lARACTER OF THE" POSITION; IF THEY ARE CHEAP THEM IN GENERALIT IS NECESSARY TO COMPENSATE THEIR IMPRECISION WITH EXTENSIVESEARCH. RESIDES, MANY EVALUATION FUNCTIONS ARE COSTLY AND IMPRECISi,AND URL) I NARY PRUNING AND TERMINATION HEURISTICS ARE MORE SUITABLEMEN SEARCH IS EXTENSIVE, which IS why THE SECOND ALTERNATIVE IS THEA..A favoured one at present, as shown 3Y recent tournament results.UnILY. I \l SPECIAL CASES DO NF HAVE EVALUATION FUNCTIONS WHICH ARESIMPLE AND GOOD AT THE SAME TIME. THE SAME IS TRUE WITH REGARD10 OiHER FORMS OF UNIFORMITY.

WIH THE EXISTENCE OF THOUSANDS OF EXPERT PROGRAMS, THE ROLE OFMA ' 3 1 G SWITCH* IS OF UTMOST I MPORTANCE , I T DIAGNOSES AND DFCIDES-UtN TO CALL WHICH EXPERT. IT CAN BECOME VERY COMPLICATED BECAUSE■HE DIAGNOSIS ITSELF OFTEM DEPENDS ON THE RESULTS OF THE EXPERTPRUGKAMS(LOOKAHEAD) . I T IS THEREFORE DESIRABLE TO HAVE AS MUCHOUJLAfMTY AS POSS I QLE ; MODULES SHOULD ONLY DEPEND ON THE RESULTSOF The SUBMODULES, NOT ON HOW THEY ARE OBTAINED.

Mb- -ME XT REQUIREMENT IS CLOSELY CONNECTED TO THE FIRST, [^ FtrjITb SA I I FACT lON IS A PREREQUISITE FOR THE FIRST.

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A REPRESENTATION IS SAID TO RE EP I S TEMOLO O I CALL Y ADEQUATE FOR APERSuN OR MACHINE (MCCARTHY ci HAyESCQJ) 'IF IT CAN BE USED PRACTICALLYfO ExPf-ESS THE FACTS THAT ONE ACTUALLY HAS ABOUT THE ASPECTS(OT INTEREST TO THE PERSON OR MACHINE) OF THE#ORLD x ...HAT wt WANT IS A CHESS PROGRAM THAT IS Vy \ STEMOL.OG I C ALL V ADEQUATEAOR v.3,50 THAT WE CAN EXPRESS SOME 0~ OUR CHESS KNOWLEDGE < IT IS OURKNOWLEDGE PUT IN THE MACHINE, THE CONTROVERSY ABOUT WHAT THE MAOHINFACiUALLY KNOWS IS IRRELEVANT HERE, AT THE FOMENT AT ANY RATE) AMDCOMMUNICATE IT TO THE PROGRAM.IT IS VERY FASHIONABLE TODAY TO TALK OF 'ADVISING* A PROGRAM, RUT-"-lE*- IT CONSISTS OF ADVICES LIKE:'P_AY PE TROFF , SUHSTRACT r-0 POINTS*,' KAf- LIGHT PIECES FROM THE RACK RANK IN THE OPEN I NG , RE WARDy AH OS * , ETC. THIS IS ADVISING THE PROGRAM ON WHAT WE DON'T KNOW( .MY NOT 26.75 POINTS), IF A CHESS MASTER DOES A SIMILAR JOR OR1-ILiDLES WITH PARAMETERS THEN H£ IS WASTING HIS T ALT- NTS , RES I DES , THE*EIS eNUUGH OF' WHAT WE DO KNOW AROUT CHESS WHICH MAY RE COMMUNICATED

■<V GRADUALLY BUILDING UP THE PROGRAM'S DESCR IPTI ON AL TOOLS.IT IS NOT A QUESTION OF SIMULATING HUMAN THOUGHT: WE DON'T RFQUJRFAJYNHEHE THAT THE MACHINE HAS THE SAME THOUGHT PROCESSES AS OURS.

SAI LSFACTION OF THIS REQUIREMENT WOULD HAVE THE BENEVOLENT SIDE-EFFECT OF MAKING OTHER IMPORTANT COMMUNICATIONS POSSIBLE:

I IE PROGRAM'S INTERNAL COMMUNICATIONS BETWEEN MOVES ('COMTI■M-.IITY v ) , BETWEEN BRANCHES OF THE A MAL YS IS- TREE ( AILOW ING ELIHI NAT lON OF REDUNDANT SUB" ANAL YS I S ) , BE T WE= N THE EXPERTPROGRAMS(SHARING OF DATA ),ETC . THE CONSEQUENCES ARE : POSS I R I L IIV OF A DIRECT CONNECTION BETWEEN BOARD FEATURES AND ACTIONSWITHOUT HAVING TO GO THROUGH T\\E RITUAI 0" REDUCING EVERYi HI NG TO A NUMBER AND THEM PRODUCING MOVES BACK FROMIHE NUMBERS, GREAT lON OF CONDITIONS FOR ANALOGICAL REASONING,LeARNING, ETC. THIS IS WHAT WF BELIEVE WILL IN THE LONG RONleLL ON THE PLAYING PERFORMANCE OF THE PROGRAM.

rf wf- HARDLY NEED TO SAY THAT U\t REQUIREMENTS OF THE LASTa-CTION CAN ONLY BE REALIZED STEP-WISE AND THAT WE ARE STILLVEKY FAR FROM THE FINAL S T A TE . I N I T I AIL Y , W- STUDIED THE FEASI-HILTIY OF THE APPROACH , W I THOUT TOO MJCH CONCERN WITH OPTI-MIZATION, BY ACTUALLY WRITING THE PROGRAMS OR SUBPROGRAMSI J )E TAIL, TO GET THE FEEL OF THeIR BULK.

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IN THE SINGLE PAWN CASE (TANLIID) WE CONCENTRATED ON RULES! J ThE FOR*! OF ASSOCIATIONS BETWEEN PREDICATES AND ACTJON-

s .HEMES, HQ,. T.HeY combine to form DECISION trees, and howAL MADE INDEPENDENT FROM THE ACTUAL EVALUATIONS AND EXECU*liuNs JF THE PREDICATES A'Mj) AC TI 0 N-SCHEMES . RECURS I Vfc-^|-DICATES REPRESENTING LOOKAHEADS ARE NOT DISTINGUISHEDr J")A OTHERS, AND PARTICULAR PREDICATES ASSOCIATED WITH CR I -II GAL SQJARES WERE DEALT WITH.

II The BISHOP VS. PAWN PROGRAM ( T A N [ I2 J ) , PRED I O ATES RFPRES- Jl' ING-' PROPERTIES OF A LONG RANGE PIECE AND THE USE OFCONSTRAINTS AND GOALS WITHIN ACTION-SCHEMES WERE-AUn.Fj.HUT THE MAIN CONCERN TritßE WAS WITH THE DICHOTOMYJF -'EVALUATION VS. LOOK-UP AND UPD AT ING,DEC I D I NG WHICH PREDIGATES 'TO EVALUATE AND WHICH TO LOOK UP, AND WITH THEuSh OF TRANSITION NETWORKS IN THE UPDATING.

IHE CASE OF ROOK VS. BISHOP IS USED (TANCI3]) TO STUDYM.)rt EXAMPLES FROM A ROOK MAY BE USED WHEN SIMILAR SITUATIONSUCC JR OH THE BOARD. IT ASSUMES THAT T^t MOVES IN THE FXAMPLFSa.c INTERPRETED, I .E. NOT ONLY THE MOVES BUI THE ACTION-SCHEMESARE GIVEN.

IHE hAsIC IDEAS ARE:1) OBVIOUS MOVES SHOULD Ht HANDLED SEPARATELY BY RULES

SIMILAR TO THOSE USED IM EARLIER WORKS, SO AS NOTTO BURDEN THE PROPER PROGRAM "OR ANALOGY.

IS BASED ON MATCHING OF DESCRIPTIONS OF POSIHUNS, BUT THIS ALONE (CALLED 'SURFACE MATCHING v ) IS NEVERSUFFICIENTLY RELIABLE WITHOUT THE ACCOMPANYING VERIFICA-HAN OR REFUTATION( ' BEEP MATCHING*) IN T'^'zl ANALYSIS.HAICHING AND ANALYSIS THUS BECOME RECURSIVELY INTERTWINED.PARfICULAR HEURISTICS RELYING ON THE PARTIAL ORDERIMG OF MOVES OF AN ACTION-SCHEME WHICH ARISES NATURALLY/HeN THEY ARE GENERATED VIA GOALS AND CONSTR A I N T S , ARESUGGESTED FOR THIS VERIFICATION/REFUTATION PROCESS.

TIE OR. JECT OF THE PRESENT WORK IS THE DESCRIPTION OF GENERAL"*[) I NGS. PAWN STRUCTURES DECOMPOSE NATURALLY INTO FRONTS

'AJiJ i SIA IDS, EACH FRONT HAVING NO DIRECT InFIUeNCF" ON ANOTHERAMD SO CAN -HE ANALYSED SEPARA TeL Y . X I NGS ARE MORE

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■OHI I. E,NOT BEING TIED TO ANY PARTICU.AR LOCAL I T V » 1 HEY CANhe IsEii TO ATTACK OR BLOCK ENEMY PAWNS, TO SUPPORT ONE'SuWN PAWNS OP ELSE BE USED TO MANOEUVRE OR OUT-MANOEUVRE THEOPPONENT'S KING. THEY WILL NOT IN GENERAL BE ABLE TO FULFILALL ■ THeIR POSSIBLE TASKS 5 I MULTANEOUSL Y » SO THE PROBLEM OF UISTAIIUN OF EFFORT WILL ARISE, AS ALWAYS WHEN THERE IS SCARCITYor ACTIVE RFSOURCES.

PARTICULAR 'A.OBLEMS,FOR EXAMPLE (WITH RES-»FCT TD A PARTICULARFRoNi):CAN ONE BREAK THROUGH , SMfAK THROUGH , PRE VENT BREAKTHROUGH, JHO CAN LOSE TEMPO, OR CAN THE KING STOP ONE, TWO, OR THREEPAWNS ETC., ARE REFERRED TO AS LOCAL °ROBL EMS . T HESE PROi-LEMS -lAY BE SOLVED BY MEANS OF ESTABLISHED RULES, OR IF THESE>.RL NOT A VAIL ABLE, BY A MAL YSES ,WHI CH WILL 3E CALLED* J Ai.TIAL ANALYSES SINCF THEY DON'T INCLUDE ALL THE PIECES ONn)ARD. UNLIKE OUR PREVIOUS WORKS, WE WON'T 30THER WITH THESEL HJ AL PROBLEMS, WE WILL ASSUME THAT THEIR ANSWERS ARE AVAILABLE, E.G. BY WRITING THE PROGRAM INTERACTIVELY AMD PROVIDING IT-ITU 'ORACLES*.--IH THIS ASSUMPTION, THE PROGRAM'S TASK IS TO MAKE A GENERALASSESSMENT OF A GENERAL PAWN-END I MG , I . E . TO REPRESENT LOCALHREATS,HOW STRONG THEY ARE, POSSIBLE DEFENCES, IF A KING IS"YE .LOADED, ETC. ; SUCH THAT MORE GLOBAL PROPERTIES MAY BECOMMUTED FROM THEM:IF IT IS LIKELY TO BE A RACE WITH APHuru-F I NISH( IN WHICH CASE Q VS. Q,j VS . P , S T AIE MA TE ,oUbEi. ING WITH CHECK ETC. MAY BE REL.E VANT) , I F IT IS A POSi TION WHERE CAREFUL MANOEUVRING WITH THE KINGS IS NECESSARY(In WHICH CASE USE THE METHOD OF RELATED 3QUARFS,SEE

.eRBACH &MAISELISLI] , ) , ETC. NOTE THAT THE METHOD OF RELO-e) SQUARES BEI ONGS TO ANALYSIS PROPER AMD IS OUTSIDEiJR sCOPE, WE MUST ONLY ESTABLISH WHEN, NOI HOW TO USF IT

8..EnT AHO ANALYSI3(LOCAL AND GLOBAi_) IS NOT ENTIRELY UNREIATED10 D --ACTICE,KOTOV[6] EOR E X AMPLE , F I NDS IT IDEAL TO MAKE GENeRAL CONSIDERATIONS WHEN ThE OPPONENT'S COCK IS GOING, AND10 GO THROUGH CONCRETE ANALYSIS WHEN HIS OWN CLOCK TICKS,

EXCLUDING OF COURSE TIME TROUBLE AND EXTREMELY TACTICALS I I UA TIONS."

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El A COMPLETE PLAYING PROGRAM SUCH A GENERAL ASSESSMENT■J ILL BE PRECEDED BY AM ORIENTATION PHASE .WHERE PKELIMIVARY CLASSIFICATION, USUALLY RASeD ON MATERIAL COUNT, IS MADEAID ON WHICH BASIS IT IS DECIDED WHETHER IT IS NECESSARYU MAKE A GENERAL ASSESSMENT AT ALL(E.G. SUPERFLUOUSI J TmE SINGLE PAWN CASE). THE GENERAL ASSESSMENT MAY ALSO BECALLED AN EXPLORATION PHASE, BEFORE THE ACTUAL INVESTIGATIONAND VERIFICATION PHASES TAKE PLACE(FQR THESE PHASES/SEE0E GROOTC 6I ) ; AND IT IS NOT WRONG TO SAY THAT ONE OF THF PURROsES OF HAVING DESCRIPTIONS IS TO ENABLE RESULTS OF EXPLORAI 10-NS(=PARTIAL ANALYSES) TO BF EXPRESSED IN A CONVENIENT WAY

'Se IN FURTHER ANAL YS I S , THUS MAKING STEPWISE ANALYSIS POSSIBLEIT FOLLOWS THAT THESE DESCRIPTIONS MUST REFLECT THE PARTICULARITI.-S OF THE PAWN STRUCTURE , I. E . ITS DECOMPOSITION INTO FRONTSaNu ISLANDS, THEIR RELATIONS TO ONE ANOTHER, AND THE RELATIONS

BETWEEN THE KINGS AND THESE GROUPS OF PAWNS. THIS LEADS USI) ANOTHER USE OF THE DESCRIPTIONS: IT LOCALIZES THE EFFECTSOF A MOVE, A PAWN MOVE BELONGING TO FRONT NUMBER 1 MAY AFFECTITS RELATIONS TO ENEMY PAWNS IN THE SAME FRONT AND THE KINGS»BuT CERTAINLY NOT ITS RELATIONS TO OTHER FRONTS. IT WILL THENHE RELATIVELY EASY TO DETERMINE WHICH PREVIOUS PARTIALANALYSES ARE STILL VALID AND WHICH ARE NO LONGER SO. (COMPAREHe 'METHOD OF ANALOGIES* OF KAISSA,SEE E . G . T ANT 14 3 , W H ICHDEPENDS ON COMPLICATED CALCULATIONS OF SETS OF INFLUENCE TODETERMINE IF PREVIOUS REFUTATIONS ARE APPLICABLE IN SIMILARb I I UATIQNS) .

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A COMPLETE REPRESENTATION OF A SITUATION! IN CHfSS CAN REHADE VERY SIMpLY:ALL THE POSITIONS 0" THE PIECES, WHO IS TO-DVE,AND A LITTLE BIT OF HISTORY(EN p ASSA N T , CASTL I NG RIGHTS,>-EHE 1 I I' l ON AND THE 50 MOVE RULE). SUCH A 'FIRST-ORDER DESCRIPllu.-r PRESENTS NO PROBLEMS , NEVERTHELESS IT SHOULD BE MADE ASEFFICIENT AS POSSIBLE, BECAUSE MANY COMPUTATIONS DEPEND ONIT. WHEN THERE ARE ONLY. A HH PIECES ON BOARD » 2-D I MENS l ONALGOURD HATES MAY HE USED; FOR GENERAL POSITIONS THE 12X10CYLINDER REPRESENTATION (SEE E.G. KOZDROWICKI SCOOPERC 7 3 )13 POPULAR, IT HAS THF VIRTUE OF COMBINING THE TwO TESTS'Is SQUARE X OUTSIDE BOARD * AND 'IS SQUARE X OCCUPIED* INTOUt: WHEN GENERATING MOVES. AN ALTERNATIVE IS TO KEEP THE"AC .BOARD AND TO USE-TABLES FOR GENERATING MOVES (SEE-v E L L L 2 J ) .>\t CYLINDER REPRESENTATION IS ADOPTED HERE, WITH ONLYPA* OS AND KINGS IT BECOMES A 9XlO+l BOARD (THE LAST SQUAREis NEEDED BECAUSE OF THE SLIGHT TWIST OF THE CYLINDER):

F I 0.1

81 82 8 3 84 85 8 6 87 8 8 89 9Q

72 ; 73 74 75 76

630405 6 6 b 7

54 b 5 5 6 57 5 8

4? 46 47 48 49

363738 39 4 0

27 28 29 30 31

18 19 2 0 21 2 2

9 10 11 12 13

0 12 3 4 5 6 7 8

UO ; MVEKS i JUS 10 2-U 1 . UUUKIM iM AI ES AMU VILE-V E !< S A 'lAT HE EASILToOiA. BY : 10

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... D:RE DIV IS ARITHMETIC DIVISION, AND MOD GIVES THE REMAINDER. TU GENERATE ALL KING MOVES, ADO ( ANT I -CLOCKW I SE ) -10,-9,-8,+1, --10, +9, +8, AND -1.+9 AND +18 ARE THE ONE AND TWO STEP PAWN■OVAS,+B, +IO ,-8 AND -10 REPRESENT CAPTURES.

a POSIIION IS GIVEN BY THE POSITIONS OF WK.BKHBMBFR OF WHITEhah Ms, LI ST OF POSITIONS OF WHITE PAWNS, NUMBER Of BLACK PAWNS,*UU A LIST OF POSITIONS QF BLACK D AWNS.i- OR EXAMPLE ,

3 >5 L6. (473949)4 (64 56 41 34)

REPRESENTS THE POSITION:

t I G . 2 .

EXAMPLE 1 .KERES-ALEKHI NE, DRESDEN 1936 A361.MOTATIO J: WK = WHTTE KING> R< = 31_ ACK KING>Wp= wH I TE pAWN,RP =h:_ACK PAWN, A M REFFRS TO POSITION NUMBER Nl OF A WERR ACH4M A I SEL

He MATERIAL COUNT IS USED FOR PRELIMINARY CLASSIFICATION,aHI:h jETERMINeS WHAT FURTHER REFINEMENT iAY BE NEEDED. INAOERhi ACHHMAISEL IS[l] FOR INSTANCE, THE PRELIMINARY CLASSIFICATION CONSISTS OF:i V5 . 0 , 1 V5 . 1 , 2 V5 . 1 , 2 V5 . ? , 3 V5 . 2 OR 3, OTHERSAS EXPLAINED IN THE LAST SECTION, OUR CONCERN HERE IS MOSTLY. I I H THF- ' OTHERS * CASE.

F 11. E( x )=X MOD 9RANK(X)=X DIV 9

AN J X = 9*RANK(X)+FILE(X),

4- 4- -- 4- -- + 4 +

BPBP] 4- 4- -- 4- -- 4-

us + 4- -- 4- -- 4--- 4 + 4- © 4- 4- m 4- 4--- 4- © 4- 4 m + 4- -- 4--- 4 4- + © B+ 4- -- 4 + 4- -- +

4- + 4- 4 + 4- -- 4-

L h>r\NIHe IERMINO L 0 G V USED I N TH F FOLLOWING WILL H I T H .StMF""lin OK HqoIF ICATIONS,RF THAT OF kMJCHCB] . Wz_iW-STRICT OURSELVES

I) BINARY PAWN TO PAWN REI AT I ONS , 5 IJVLS-t-Tff=SE ARE SUFFICIENTHiKU :JR^i^*i|2USi^j^^ V R E fdU L T" T N A COLOURED GRAPH, FOR- - \i C H (UTILITY FUNCTION^A-RE " MORE EASILY DEFINED THAN IF I T■"ERE A GENERA! ~ALGEBRA OR A RELATIONAL STRUCTURE.

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AN ENEMY PAWN AHEAD ON THE SAME FHE IS A COUNTERPA WN , I TIS A SENTRY WHEN IT IS ON -A NEIGHBOURING "ILE.THE DEFINITIONSa.E iNOT VALID WHEN THE PAWNS HAVE .MASSED EACH OTHER, WE WILLOUT EN PASSANT TO SIMPLIFY M A T TERS . COUNTERP AWNS AND MUTUALsENTKIfcS OF DISTANCE 1 ARE CALLED RAMS AND LEVERS RESP.Ml'-NDLY RELATIONS GIVE RISE TO A&OUO WHEN THE PAWNS AREABREAST ON TWO NEIGHBOURING F I LES , T W I N ( DO JBLEPA WN ) WHENTUL-Y ARE OH THE SAME FILE;A BACKWARD NEIGHBOUR IS AA,UTECTOR(DISTANCE 1) OR A POTENTIAL PROTECTOR (DISTANCE >1).all THE ABOVE RELATIONS ARE I RREFLEX I VE , I NTR A NS IT I VE ; ALLEXCEPT THOSE CORRESPONDING TO THE LAST THREE ARE SYMMETRIC.I W 0 PAwNS . AR E SAID TO HAVE NO RELATIONS IF NONE OF THE ABO V EHOLD.SUMMARY OF ALL RELATIONS:

NAME GRAPHICAL NOTATION CODE

hDST1 l e COUNTERPAWM■ EL AT l ONS: RAM

SENTRYLEVER

MUENDLY DUORELATIONS: TWIN

POTENTIALPROTECTORPROTECTOR

(HERE X =X V, X=> V, X =/=> V IF V IS AHEAD OF X)

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nL'UL i: ■ v s - f\fir|| W i U L_ :jr: Kj ■* l_ L. _T !■■ H rHHIVI^CLH I i7l 'V ... t - -x m r -i

( >R-GRAPH) .FOR THE THREE ASYMMETRIC REL A T I ONS , T HE I R INVERSER-IATIUNS WILL ALWAYS .E AVAILABLE ALTHOUGH NOT EXPLICITLYORAWN IN THE GRAPH(IN THE ACTUAL MACHINE REPRESENTATION THEaRkOwS ARE NOT PHYSICAL PO I NTE RS , AND THEY MAY B e TRAVFRSFD1 J BOTH DIRECTIONS)FOX EXAMPLE. THE GRAPH OF THt FOLLOWING D OSITIUN:

r I o .\. .

4- 4-

Up)4 4-

Q$ --4- 4 4- 4 4- +

XAMF> L E ?.EUWE-ALEKHI N E , 1 9 3 5 , A 3 82 .

+ +__

+ + -- I $6]__

+ +

B -- 0 -- ♦♦ -- @ --BP ♦ + Ibpj ++ -- + +

4-4 - - 4-+ -- -f -f

4-4 4-4

+4- -_ +H__

++ LpJ Qjp\ __++ --. +4 _- (^p) .__. + +

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F 1G. 4.

,64 66 y 70

-- i < -5 6

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19>

( =w H IT E PAWN, = B L A CK PA W N ) .H THE INTERNAL REPRESENTATION ALL RELATIONS OF NODE 33ARE STORED WITHIN THE NODE, IT BECOMES:

+ 33 (3,59) (3,70) (5,34) (7,24)(+ r OR WHITE, 3=CODE FOR SEnTrY,ETC.) AND THE WHOLE GRAPH CANHEN RE EMBEDDED INTO THE REPRESENTATION OF THE BOARD.

EACH CONNECTED COMPONENT OF THE PR-GRAPH IS CALLED A FRONT,EACH CONNECTED COMPONENT OF THE SUBGRAPH FORMED BY FRIEND-LY RELATIONS IS AN ISLAND. IN THE ABOVE EX AMPLE , THE RE ARE TWOF-.uMT3.THE KING-SIDE FRONT CONSISTS OF ONE WHITE AND TWOH...ACK ISLANDS.APART FROM THE DECOMPOSITION INIO FRONTS AND ISLANDS, THElAIN UsE OF THE PR-GRA=»H IS TO FACILITATE DETECTION OF OTHFR

I IPORTANT PROPERTIES OF PAWN STRUCTURES.a PAWN WITH COUNTERPAWN(S) IS UNFREE(OR BLOCKED ), OTHFRW I SE ITFREE IF IT HAS SENTRY(IES) OF OiSTk^Zt >1 OR ELSE FREE(APASSER). NOTE THAT A PASSER, AS DEFINED HERE, MAY HAVE A SENTRYw I I H WHICH IT FORMS A LEVER.AND MAY BE CAPTURED IF THEOPPONENT HAS THF MO VE , OTHER W I SE IT MAY BECOME A REAL PASSER

■V CAPIURING lIS SENTRY OR BYPASSING IT. IN SUCH CASES, THEL-VER REPRESENTS A CRITICAL TRANSITIONAL J HASE.

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33' 34

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AS INDICATED AT ThiE ENj) OF THE LAST SEC T I ON , L E VERS MAYRajUlße SPECIAL ATTENTION BECAUSE OF THEIR UNSTABLE NATURE.GENERALLY, THERE ARE POINTS OF CONTACT WITH THE ENEMY E.G.A CHECK OR A KING THREATEN IMG TO CAPTURE A PAWN lIN ONE MOVE,WHICH NEED MORE OR LESS URGENT ME A SURE S . THUS A CHECK MUST BELIFTED IMMEDIATELY AND THIS HAS THE POWER TO INTERRUPTOR OVERRULE ANY OTHER RULE.ON THE OTHER HAND, WE WOULD LIKE TO SEE IN A HIGHER ORDERDESCRIPTION OF A POSITION THOSE FEATURES THAT HAVE A MORFPERMANENT CHARACTER, SINCE THIS IS THE PROPERTY viHICHFORMS THE BASIS OF THE RATIONALE OF COMMUNICATING INFORMATIONFRol ONE NODE TO ANOTHER IN THE A MAL YS I S- TREE . WE DECIDEDHeREFORE TO EXCLUDE IMMEDIATE POINTS OF CONTACT FROM THEDESCRIPTIONS CONSIDERED BELOW.

a PASSER REPRESENTS A THREAT, THE NEARER TO PROMOTION THEsTRONGtR THE THREAT. AN UNFREF OR A HALF-FREE PAWN, IF ITLAO \^t CONVERTED TO A PASSER, IS ALSO A THREAT. THECONVERSION MAY TAKE PLACE SIMPLY BY PUSHING FORwARD WITHA FRIENDLY NEIGHBOUR HELPING A C AND I D A TE , THE NEIGHBOUR INHRN SUPPORTED BY THE HEL P ER'S HELPER , E TC .. BY DEFLECTIONOF A SENTRY OR A COUNTERP AWN ( SOME T I MES USING THE FORKLEVER TRICK) OR BY A COMRINATION OF BOTH,IHeSE METHODS OF FORCING PASSERS APP^Y TO A FRONT , I NDFPENDFNT

OF THe NUMBER OF THE ISLANDS AND NEGLECTING THE KINGS.SOMET IMES, BOTH SIDES CAN FORCE PASSERS (MJTUAI BREAKTHROUGH),SOMI--UMES ONLY ONE SIDE CAN FORCE A PASSER, AND IT CAN DO SOWITHOUT GIVING THE OTHER SIDE PASSERS (PURE BREAKTHROUGH),OR IT CAN DO SO ONLY BY LETTING ENEMY PAWNS PASS (WEH^t- ER TO THIS LAST CASE AS SNEAK IMG, THE TERMINOLOGY IS OPEN10 SUGGESTIONS HERE) .THREATS REPRESENTED 3Y PASSERS AND

PURE BREAKTHROUGHS ARE DEMOTED BY => (NOTATION NOT TO BECONFUSED WITH THOSE FOR THE PR-GRAPH). A MJTUAL RREAK-HKOUGH CONSISTS OF TWO THREATS IN OPPOSITE DIRECTIONS (<=>),AMD SNEAKING IS A THREAT WITH A CONDITIONAL COUNTER-THREAT(<: :=>) .

PAWNS OF THE INFERIOR SIDE, E.G, BY 3 L OCKING THE POS I T I ON . EL SEHEY MAY BE MET BY THE OPPONENT'S X I MG . SEIF-DEFENCE BY PAWNSWILL NUT BE REPRESENTED EXPLICITLY, ' DEFENCE x (DENOTEDBY ->) WILL BE USED TO RE r ER TO RELATIONS OF KINGS AND THREATS.OTHER FORMS OF THREATS(ALsO REPRESENTED BY =>) ARE

TMfA'i15

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CAPTURE OF PAWN(S) BY A KING, SUPPORT OF AM ADVANCE OF PAWNSBY A KING, AND THE JOINT ATTACK OF KING AMD PAWNS AGAINST THEOTKEN SIDE'S PAWNS. E.G. DISPERSION VIA EXCHANGES OR SACRIFICESHe.N CAPTURE)

A SYNTAX OF THE A T T A CK - OEFE MCE RELATIONS CAN RE GIV-N IN- -IF AS FOLLOWS** AND % APPLIED TO A SYNTAX CLASS INDICATF THEmUMreR OF OCCURRENCES THAT THE SYNTAX CLASS MAY HAVE, FOR* IT IS ONE OR MORE, FOR % IT IS ONE OR ZERO):

(2) <REL>::=<A-REL>/<D-REL>/<B-REL>(3) <A-REL>::=<ACTOR><ACTOR%XA-LIN<><ACTOR>(4 ) <D- <ELK : XAC TORXD-u I NkX A- REL>(5) <U-REL>: :=<GROUPXB-LINKXGROUP>(6) <ACTOR>: :=<KING>/<GROUP>(7) <A- L INK>: : ~<A-L I NKTYPEXP ARXMOD%XD^%>(8) <D-LINK>: :=<D-LINKTY?EXPARXMOD%XOL%>(9) <B-LINK>: :=<B~LINKTY c> EXPAR^XMOD%XDL^>(IU) <KING>::=K <COLOURXLOCXDL%>

n

- 1 ) <ADD>: :=<REL*>

A.'"'AA,

(11) <GROUP>::=P <COLOURXDL>/NIL(12) <A-LINKTYPF>: := => / <= > / <::=>( 13) <D-LINKTYPE>: : = ->

" (14) <B-IINKTYPE>: := [=]/ = / L =(15) <COLOUR>: : =WH I TE/BL ACK(16) <LOC>: : =10/11 . ./80

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HI mi lUA~L. Et EOOE Ul'UjKHil (H|l(j) ib A SEI UF INSIAiNMJtNOF- RELATIONS* 1 > . (THE OPERATION QF CONCATENATION IN THESYNTAX DOES NOT COMMIT OURSELVES TO ANY PARTICULAR MACHINEREPRESENTATION, IN (1) IT MAY BE CONSIDERED AS SET-THEORETICUNION) .kel is an instance of a relation, it may be of attacking, defensive or balanced ty pe<2) .attacking and offensiverelations were considered above, the first may have up totwo active actors and one passive actor(3).wucn no breakthroughs or sneakings can occur, the pawns aresaid tv oe balanced, they may be completely blocked, sfml rlockfor blocked on one side (l4).

is a group of islands of pawns of thf same colourheIONGING TO ONF FRONT, THE SPECIAL CASE WITH NOPAW. MS AT ALI (ALL THE PAWNS OF THE OTHER SIDE AR- PASSERS) ISDENOTED -iV THE DUMMY NIL.IT IS SOMETIMES CONVENIENT TO THINK OF AN ADD AS A GENERALIZE.GRAPFHSOME A-LINKS HAVE BINARY INPUT), WITH ACTORS AS NODESAMU LINKS AND EDGES (SEE FIG. S BELOW).EACH NODE AND LINK MAY HAVE ADDITIONAL INFORMATION ATTACHEDH THEMiCOLOUR, LOCUTION). PAR(AMETER) INDICATING STRENGTHSOF THREATS, ETC. ,MOD( IFIER) INDICATING MEG AT I ON , MODAL I T lES(POSSIBLE, IMPOSSIBLE, ETC. , THIS FACILITY 15 NOT USED AT THEFOMENT), AND DL ( DESCR IPT I ON LIST) WHICH MAY CONTAIN ALL SORTSUE THINGS, THUS A DL OF AN A-LINK MAY CONTAIN THEVARIATIONS THAT LEAD TO BRE AK HROUGHS , OR IN CAsi OF AN ATTACKUF A KING AGAINST A PAWN GROUP, THE PATH. 3) OF THE KING TOi Ts TARGET, A DL OF A D-LINK CAN GIVE THE AREA IN WHICH AKING IS CONFINED TO (REFERRED TO AS BINDING) IF IT IS TOC IECK THE ADVANCE OF A PARTICULAR GROUP, A Dl OF A KING NODECAiN he USED TO COMBINE THE DIFFERENT BINDINGS THAT THE KINGh JFFL ? b I,NI MEFr ING DIFFERENT THREATS, SO ALLOWING DETECTIONOh OVERLOADING, A DL OF A GROUP INDICATES THE NUMBER OF PAWNISLANDS IN THE GROUP, THEIR LOCATION AND NUMERICAL STRENGTH,PAKTICULAR INTERESTING P A WN3 ( P ASSERS , POTE NT l AL TARGETS , F TC .) .

NOT ALL ADDS CONSTRUCTED ACCORDING TO HE ABOVE SYNTAX AREPERM i SSI RLE, THERE ARF ADDITIONAL SEMANTIC RULES, FOR ExAMRLE, IF A THREAT OF A BREAKTHROUGH BY A WHITE GROUP WGIAGAINST A BLACK GROUP BGI CANNOT BE MET BY THE BLACK KING,HEN THERE CAN RE NO QUESTION OF A JOINT ATTACK OF BK AND BGI ,AGAINSI WGI.AI.SCIF THE WK IS IMMOBILIZED BY BLACK THREATS, /<IT WILL NOT THINK OF ATTACKING FAR AWAY B_ACK PAWN GROUPS,IN JOt CASE OF SUPPORT OF PAWNS BY THE KING. THE PAWNSlisT ALREADY BE ABLE TO BREAK OR SNEAK THROUGH RY THEMSELVESI M ORDER TO GET A SUPPORT.

w-E NeEO NOT, HOWEVER, BOTHER WITH A„L THESE RULES , S YNT ACT I 0OR SEMANTIC;WE ARE NOT CONSIDERING WHEN SOME ARBITRARYSTRUCTURE IS A PERMISSIBLE WELL-FORMED ADD, WE ARE ONI VINTERESTED IN ADDS CONSTRUCTED FROM _EGAL PAWN ENDINGS.

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LET US NOW CONSIDER THE CONSTRUCTION OF ADDS OF POsi l IUNS IN GREATER DETAIL-s jppuse that example i <fig.2> is the input position.FROM The PR-GRAPH, the algorithm consadd receives DATA on theDIFFERENT COMPONENTS OF THE GBAPH. THESE DEFINE THE NODES OF THTA DO:

BK(51) Wk(32)8G1(64,56) WGK47 39 49)8G2(39)8G3(49)

(WE OMIT MOST OF THE INFORMATION IN THE DESCR I P T I ON L I S TS ) .GO.SADD NOW STARTS ASKING QUESTIONS TO TOU LOCA L SUBPROGRAMS(OR IMb ORACLE IN OUR CASE) ABOUT DIFFERENT RELATIONS BETWEENHeSE NODES. BECAUSE OF THE SFMANTIC RULES, THE SEQUENCE OFQUESTIONS ASKED MATTERS ; CONS ADD USES THE FOLLOWING SEQUENCE

(I)RELATIOMS WITHIN FRONTS,(2)DEFENCES TO THREATS OF (1),(3)POSSIBLE ATTACKS OF KINGS AGAINST PAWNS,(4)DEFENCES TO (3),(S)SUPPORT POSSIBILITIES,(6) JO INT ATTACKS.

.1) produces :wgl=>6gl,bg2=>n i l,bg3=>n i l, each with the assoclaied estimates of the minimal number of moves(par)necessary for the successful execution of the threat ( alwaysassuming that the superior side has the move).

(P)BK CAN MEET THE THREAT WGI = >BGI,BUT ONIY JUST, IF WHITEAOVES FIRST THEN BK MUST IMMEDIATELY REACT TO IT, IN OTHER" OROS,«LACK CAN AFFORD TO PASS JUST ONCE , I . E . G I V I NG WHITE

I IE MOVE .HENCE PAR =1 .IN GENERAL, PAR=N>O MEANS:CAM PASS N TIMES AND STILL MEET THE

threat,PAR=O MEANS: MUST MOVE IMMEDIATELY TO MEET THREAT,PAR=-N<o MEANS: MUST MOVE FIRST, AND THEN WOULD STILL

NEED N MOVES TO MEET THE THREAT;

H THE EXAMPLE, IF BK HAD BEEN AT G4,PAR WOULD BE O.ANDAT H2, PAR=-2.wBE J THE KING IS DIRECTLY IN FRONT OF A PASSER OR A CANDIDATE, WE MAKE EXCEPTIONS TO THE AROvE,

PAR=> ip the KING IS IMMEDIATELY IN FRONT OF IT,AND ELSE PAR=M(LARGE NUMBER).

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■■> i■'it_r\ i \ l. I <nr\ <. ■ m ;-m il_ t i mc i ir\c M i _> r. b -/■ !> - 1_

AY U "( 00-;> IM t L "EACH OF THESE DEFENCES IMPLIES BINDINGS ON IHE KING, E.G.nK->(BG3 = >NID CONFINES WK TO H1 , H3 , G1 , G2 , F1 , F 3 , E1 , E 2 , ANDe3(f3 IS INCLUDED BECAUSE wE HAD ONLY TAKEN G3 AND WK INTOACCOON I ) .HFFQKE CONTINUING WITH (3),CONSADD LOOKS ~OR COMBINED THREATSAND THEIR ASSOCIATED B I ND I NGS . TH I S IS USUALLY UUST FINDINGINTERSECTIONS OF THE OLD BINDINGS, BUT SOMETIMES IT NEEDS ALITTLE ANALYSIS, FOR EXAMPLE WITH WK ON Fl AND BG2 ON E3INSTEAD OF F4,WK MUST STAY ON Fl AND CANNOT MOVE AT ALL.

(S) A I TACKS OF WK ON BGI AMD BG2 ARE RULED OUT BECAUSE OF THE■HOI NGS THE WK HAS ALREADY ■ RECb I VED (ACTUALLY THIS IS DONE

MY COMPARING THE REWARD IT GETS IF THE ATTACK ON THE84.V.S IS SUCCESSFUL AND THE PENALTY RECEIVED BY IGNORING THEh I j) I NGS ) .ATTACK OF BK ON WGI IS ALSO RULED OUT, SO THERE IS ONLY THEATTACK Wk = >BG3,WITH PAR-* 4 ( NUMBER OF MOVES UP TO CAPTURE)AND- HE PATH E2,F1,G?,G3 IS STORED IN THE DESCRIPTION LIST.

(4)THh PATH E2,F1,G2,G3 DOES NOT CROSS ANY SQUARE CONTROLLEDhY UK,3O NO DEFFNCE OFFERED.

(S) TO SUPPORT THE ADVANCE OF PAWNS THE KING MUST FIND AsOUARE IMMEDIATELY IN FRqNT OF A PASSER OR A C A MD I D ATE . THEONLY POSSIBLE SUPPORTING ACTION HERE IS B<=>BG3,WITH PAR=2(NUMBER OF MOVES UP TO CONTROL OF G2 ) , DL : J A TH=G4 , H3 .(S)THE ONLY JOINT ATTACK POSSIBLE IS THAT OF BK 4BGI =>WGI.iT TAKES >10 MOVES(BK GOES TO B7,THEN A 6ETC.).

He ADD OF THE POSITION IS NOW COMPLETE ( FI G . s>.5 > .

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9 APRIL 1975. ST T *14.51HR3.

CSI74]IT IS THEN PASSED TO THE PROGRAM DIA3ADD AHICH DIAGNOSESHE "POSIT lON "AND PROVIDES I MXiiiFACE WITH THE PROPER ANALYSIS.DIAGAOJ WILL TRY TO SEE If/he POSITION IS: ~ -^

A)BALANCED OR NOT, OR IF IT IS GREATLY I MB AL ANCED , I N WHICHCASE IT MAY SUGGEST TO ANALYSIS THAT IT IS AN EASYPROBLEM NOT REQUIRING VERY DEEP ANALYSIS,

B)DYNA-IICALLY OR STATICALLY BAL ANCED ; THERE ARE OF COURSEDIFFERENT DEGREES OF DYNAMIC BALANCE, THE STRONGEST IS

THREAT AND COUN TERTHRE A T ARE OF ROUGHLY THESAME STRENGTH WITH INSUFFICIENT DEFENCES, AND THE MOSTSTATIC ONE IS WHERE THE POSITION IS COMPLETELY ORALMOST COMPLETELY BLOCKED ; WHEN THE THREATS ANDCOUNTERTHREATS ARE. ABOUT EQUAL, IT MAY SUGGEST TO ANALYSIS

FOR CHECKS, TO BE PRE^ARzD FOR QUEEN ENDINGS, ETC.

i T ALSO ANALYSES COMPATIBILITY OF THE DIFFERENT OPTIONS OPENFOR THE KINGS. IN DOING SO, IT MAY ESTABLISH THAT APPAAA-TLY DIFFERENT TASKS MAY ACTUALLY 3E RECONC I LED , ANDa USEFUL MULTIPURPOSE MOVE WILL THEN BE SJGGESTEO TO ANALYSIS(SeE E.G. RE'TI'S STUDY, I92I, A7I) .H THE EXAMPLE.IT WOULD FIND THAT 3K->( WGIr >BGI ) IS COMPATULE Tv BK£RGI=>WOI (THE PATH E5,D6,C7,37 IS WITHIN THE BINHUG OF THE FIRST), BUT THAT BOTH ARE INCOMPATIBLE WITHX=>BG2.IT COULD THUS PRESENT ANALYSIS WITH AN ALTERNATIVE,eIHER BK GOES TO THE KING-SIDE OR THE QUEEN -SIDE(BOTHINCIDENTALLY WINS, BLACK TO MOVE, THE FIRST IS THE SOLUTIONCHOSeN BY ALEKHtNE IN THE GAME, THE SECOND DISCOVERED BYAWERBACH) .EACH ALTERNATIVE HERE DECOMPOSES THE ADD INTO TWOCONNECTED COMPONENTS, ENABLING FURTHER SIMPLIFICATION.

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IT SHOULD OBVIOUS FRO>! TOe PREVIOUS SECTION, THATAOOoAQJ IS AN EXPENSIVE OPERATION WHICH CANNOI RE APPLIED10 EVERY POSITION DURING TOE AN AL YS I S . TH I s IS TRUE EVEN FORHe RELATIVELY SIMPLER CONSTRUCTION OF TH- PR-GRAPHH.)M-VER,THEY WERE NEVER INTENDED TO SO USED, THE PR-GRAPHAJu THE ADD OF A POSITION ARE MEANT TO CONTAIN INFORMATIONWHICH ARE NOT EXPECTED TO CHANGE RAPIDLY from movf Tf) MOvFfSO THAT UPDATING THEM WILL GENERALLY BE A LOT EASIER THANRECONSTRUCTION.*__ ReFeR to the collection of information, including ADDanu He pr-graph, which are carried Along fROM movE to move,hh some updating, as the frame, in more or less the samfsen^e AS THE 'FRAME* OF THE 'FRAME PROBLEM* (MCCARTHY RHAYi-SI.9j,HAYESL4J) .THE ORIGINAL FRAME PROBLEM RFFERS10- THE UPDATING OF AN INTERNAL MODEL OF AN EXTERNALENVIRONMENT, BUT IT MAKES NO DIFFERENCE IF WE USE [T FO RA MODEL OF A MODEL. THE PROBLEM AND ITS POSSIBLF SOLUTIONSHAVE BEEN DISCUSSED AT LE-MG TH (E.G . HA YE3C 43 ) , OUR CONCERNheRE Is NOT WITH THE GENERAL PROBLEM, BUT WITH THEPROBLEM OF CHOOSING A REPRESENTATION FOR OUR PARTICULARDOMUN,FOR WHICH UPDATING CAN BE DONE AS SIMPLY AS POSSIBLE,I . PARTICULAR WE WOULD LIKE THE BIANkET ASSUMPT I ON ( NO TH I NGCHANGES UNLESS EXPLICITLY STATED) TO 3E VAIID.

FOR HE PR-GRAPH, IT IS OBVIOUS THAT IF X ADVANCES,HEN X MAY CHANGE ITS RELATIONS TO OTHER RELATIONSMAY DISAPPEAR, NEW RELATIONS wILL NEVER APP£ AR ) , RU T ALL OTHERRELATIONS WILL REMAIN THE SAME. WHEN X CAPTURES Y,THE CHANGESARE SLIGTHLY MORE COMPL I C ATED ( NEW RELATIONS MAY APPEAR).BUT SI ILL LOCALIZED.HE COLLECTION OF ALL POSSIBLE CHANGES FOR E ACH TYPE OF MOVECAN tiE RFPRESENTFD BY A TRANSITION DIAGRAM(SEE TANCI3J FORHE USE OF THE SAME FOR BISHOP TO PAWN RELATION)

FOR EXAMPLE, THE TRANSITION DIAGRAM FOR PUSH X (ONE STEP)LOOKS LIKEUT FALLS APART INTO 5 COMPONENTS):

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EACH TRANSITION HAS A CONDITION WHICH MUST BE SATISFIED INORDER THAT THE TRANSITION TAKES PLACE, THESE CONDITIONS REFERH THE STATE BEFORE THE TR ANS ITI ON ( 10= DI S TANCE ).' NO CONDITION*MEANS TRANSITION UNDER ANY COND I T I ON ,' ILLEGAL * MEANS ILLEGALMOVE UNDER THE COND I T I ONS » ' N IL % INDICATES DISAPPEARANCE OFI lE -RELATION.3YPASSING OF PAWNS).He TRANSITION DIAGRAMS FOR PUSH X(TWO STEPS) ANDCAPTURE X V MAY BE GIVEN SIMILARLY.

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lOE ij j. r l COL I V OF UPUANNa ARisrS -ROM IRE PROPAGATIONhe .CHANGES, FIRST THeRE IS TBI PRIMARY CHANGE WHICH CAUSES- :-LO!,OARY CHANGES WHICH H TijRM PRODUCES TERTIARY CHANGES,'AID sO ON. THE PR-GRAPH AND THe ADO ARE SO CONSIRUCTFD THATHEY SATISFY A 'NO PROPAGATION LAW V .IN THE FIRST CASE, ITIS SIMPLY BECAUSE NOME OF THE PAWN TO PAWN RELATIONS AREI -ANSI lI VE.H THE CASE OF THE ADD, IT CAN OE SEEN AS FOLLOWS.

EXCLUDING PROMOTIONS AND CAPTURES ( SOME T IMES CAUSING DISAPPeARANCe OF A WHOLE PAWN GROUP) WHICH NEED SPECIAL TREATMENT,a ; ARE LEFT WITH 'ORDINARY* PAWN AND KING MOVES.

A PANN MOVE MAY REINFORCE OR REDUCE (IN CASE OF SELF-dEFENCe) A THREAT, IN EITHER CASE DEFENCES TO THE THREAT ARECIeARLY affected, advancing a pawn may make its support moreOIF HCULT, ITS LOGISTIC SUPPLY BECOMES STRAINED, SU TO SPEAK,'HO IT MAY ALSO BECOME MORE nR LESS VULNERABLE AS A TARGETf or< IHE ENEMY X I NG.

f OK A KING, MOVEMENT 10 ONF DIRECTION WILL IN GENERALEFFECT ITS INFLUENCE IN OTHER D I RECT I ONS . I N ADDITION, IF IT"As a CHANGE IN AN ATTACK VS. PAWNS, IT MAY CHANGE THEDEF ENCE HY THE OTHER KING.

MeSE TWO CAUSALITY LAWS FOR PAWN AND KING MOvES MUST NOTA: SUPERIMPOSED, IF FOR EXAMPLE, A PAW N ( BELONG I NG TO BG1)AOVC STRENGTHENS A JOINT ATTACK SY BGI * RK AGAINST WGI ,IHeN THIS CHANGE IN THE JOINT ATTACK WILL CERTAINLY-OT AFFECT ANY OTHER LINK EMANATING erqm RK.

u.NL'Y !<■ THE JOINT ATTACK IS CHANGED A BK MOVE, THF SECONDLAW MAY RE APPLIED. THIS IS TYPICAL 0-~ THE I NTR ANSITI V I T VOF CHANGES OF THF ADD.IT .WOULD SEEM THEN THAT IT CONTRADICTS RAYES'[4] TRANSITIVITYAXIOM FOR HIS CAUSALITY ReL AT I ON , WH I CH WAS DEFINED AS:a ARROW ( READ: CAUSALLY RELATED TO) R IFF

OF- Sb>4t PROPERTY OF" R CHANGES THEN SOME PROPERTY OF A ISCu_AßLi^3r 0 CHANGE.H FACT THE AXIOM CAN BE RECONCILED WITH IRE ABOVE BY CARE-FULLY DEFINING WHAT THE A AND R'S ARE IN OUR CASEHOT THELINKS OF THF ADD, BUT THE PARTS OF THE LINKS CONNECTED TOI H DIFFERENT ACTORS ARE THE UNITS FOR WHICH THE CAUSALITY■ELAT I UN SHOULD 'OF DEFINED. THUS THE ABOVE PAWN -IUVE IS NOWSAID Tv CHANGE THE PAWN'S PART, BUT NOT THE RK'S PART OFHe JOINT ATTACK. THE TWO PARTS ARE OBVIOUSLY NOT CAUSALLYAOUNEC TED, ALTHOUGH EACH OF THEM MAY CHANGE SHAR-0 COMMONDATA. (E.G. THE PAR ASSOCIATED TO THE JOINT ATTACK), AND ACHANGE IN ONE PART LOOKS EXACTLY LIKE A CHANGE IN THEUTUHR .IF- Wb EXPRESS THE CAUSALITY RELATIONS OF THE ADD INa MOTHER (OF STILL HIGHER ORDER) DIAGRAM, WE SEE THAT THISDIAGRAM DECOMPOSES INTO SEVERAL DISCONNECTED COMPONENTS.AD,WHILE THE TRANSITIVITY AXIOM HOLDS, WE CAN KEEP THE' TU PROPAGATION LAW*, AND EVERYTHING COMES TO A HAPPY : .

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LLJAWERRACH J.tfMAISELIS I.:LFHRBUCH DER E NDSP I EIE , VOL . 1OAUERNENDSPIFLE,BFRLIN,I96 0.

L2J-FLL A.G.:GAMES PLAYING WITH COMPUTERS , LONDON 1972.LSJDE GROUT A.:THOUGHT AND CHOICE IN CHESS, THE HAGUE, I96S.L4JMYBS P.J.:A LOGIC OF ACTIONS, IN MELTZ = R B.SMIC'HIE D.EDS.

MACHINE INTELLIGENCE 6>EDI NbURGH, I 97I , 3 P . 495-520 .IHJKMOCH H..-PAWN POWER IN CHESS, NEW YORK, 1964.I.sJKUTUV A.: THINK LIKE A GRANDMASTER , LONDON , 197? .I HKOZDROWICKI E.W. SCOQPER D.W.:COKO 111 : THE COOPER-KOZ CHESS

PROGRAM, CACM VOL. 16, 19 73, (7) ,po. 411.-426.L-.JKUHN H.W. : EXTENSIVE GAMES AND THE PROBLEM OF INFORMATION,

!N KUHN H.W.STJCKER A . W . EDS , CONTR I BUT l ONS TO -THE THEORY3F GAMES, PRINCE T0N, 195 3, Pp. 193 -21 6.

IHJMCCARTHY J.*HAYf.:S P.J.:SOME PHILOSOPHICAL PROBLEMS FROMTHE STANDPOINT OF ARTIFICIAL INTELLIGENCE,^ MELTZER B.

D. MACHINE INTELLIGENCE 4 , ED 1 N8URGH , 1969 , PP . 463-50? .LIuoHEWELL A. ARTIFICIAL INTELLIGENCE AND THE CONCEPT OF

UNO, IN SHANK R.C.^COLBY X . M . EDS . COMPU TER MODELS OFTHOUGHT AND LANGUAGE, SAN FRANS I 5C0 , 1 973 , PP . 1-60 .

LUITAN S. T. REPRESENTATION OF KNOWLEDGE for VERY SIMPLEJAWN ENDINGS IN CHESS , ED I NBURGH , MEMO M IP-R-98 , 1972 .

I. 12] TAN S. T. :KINGS,PAWN AND B I SHO^ ,ED I NBURGH , MEMO MIP-R-108,19LI3ITAN S.T. HOOK VS, BISHOP VIA BOOK EXAMPLES (IN PROGRESS).LIOITAN S.T.iTHF: WINNING PROGRAM ( SUMMARY OF SOME IDEAS OF

KAISSA) ,FIRBUSH NFW5, 5.1974, PP. 38-40.

ACKNOWLEDGEMENTS.

HIS PAPER IS PART OF A WORK ON ENDGAMES SUPPORTED BYa uRANT FROM THE SCIENTIFIC RESEARCH COUNCIL.F )R VARIOUS SUGGESTIONS, CRITICISMS AND CORRECTIONSWE ARE INDEBTED TO MR . A . J . RO YCROFT .