Table Of ContentHypergames and full completeness for system F
(ROUGH DRAFT)
Dominic J. D. Hughes ∗
StanfordUniversity
August25,2006
8
0
ThispaperreviewsthefullycompletehypergamesmodelofsystemF,presentedadecadeago
0
in the author’s thesis. Instantiating type variables is modelled by allowing “games as moves”.
2
The uniformity of a quantifiedtype variable ∀X is modelled by copycat expansion: X represents
n anunkowngame,akindofblackbox,soalltheplayercandoiscopymovesbetweenapositive
a
occurrenceandanegativeoccurrenceofX.
J
Thispresentationisbasedonslidesforatalkentitled“Hypergamesemantics: tenyearslater”
5
givenatGamesforLogicandProgrammingLanguages,Seattle,August2006.
1
] 1 Introduction
O
L Zwicker’sHypergame[Zwi87]isanalternatingtwo-playergame: oneplayerchoosesanyalternating
. gameGwhichterminates1 (e.g.“O’s&X’s”orChess2),thenplayproceedsinG.3
h
t
a Hypergame
m O’s&X’s Chess
[
rmblkans
opopopop
1 0Z0Z0Z0Z
Z0Z0Z0Z0
v 0Z0Z0Z0Z
Z0Z0Z0Z0
5 POPOPOPO
SNAQJBMR
7
5 left centre Nf3 e4
2
.
1
rmblkans rmblkans
0 opopopop opopopop
0Z0Z0Z0Z 0Z0Z0Z0Z
8 X X Z0Z0Z0Z0 Z0Z0Z0Z0
0 0ZZ00ZZ00ZZN0ZZ0 0ZZ00ZZ0PZZ00ZZ0
v: PSONPAOQPJOBPZOR PSONPAOQ0JOBPMOR
Xi top-left top right c5 Nf6
r
a
O O rompbZlpkoapnosp rompbolpkoap0osp
0Z0Z0Z0Z 0Z0Z0m0Z
X X X O Z0o0Z0Z0 Z0Z0Z0Z0
0Z0ZPZ0Z 0Z0ZPZ0Z
Z0Z0Z0Z0 Z0Z0Z0Z0
POPO0OPO POPO0OPO
SNAQJBMR SNAQJBMR
∗Visitingscholar,ComputerScienceDepartment,StanfordUniversity,CA94305.
1Everylegalsequenceofmovesisfinite.
2Toensuretermination,assumeadrawisforceduponathreefoldrepetitionofaposition(avariantofastandardrule).
3Thequestion“DoesHypergameterminate?”,theHypergameparadox,amountstoahereditaryformofRussell’sparadox,
knownasMirimanoff’sparadox[Mir17]: “Isthesetofwell-foundedsetswell-founded?”.(Each‘paradox’isillusory,being
merelyduetothelackofformaldefinitionof“game”or“set”.)
1
At theImperialCollegeGamesWorkshopin1996, theauthorillustrated howhypergames—games
in which games can be played as moves — can model languages with universal quantification.
Originally implemented in [Hug97] for Girard’s system F [Gir71, GLT89], the idea is quite gen-
eral, and has been successfully applied to affine linear logic [MO01, Mur01] and Curry-style type
isomorphisms[dL06].
1.1 Universallyquantifiedgames
RecallthelittlegirlAnna-Louisewhowinsonepointoutoftwoina“simultaneousdisplay”against
chessworldchampionsSpasskyandFischer[Con76,Theorem51]. ShefacesSpasskyasBlackand
Fischer as White, and copies moves back and forth, indirectly playing one champion against the
other. WhenSpasskyopenswiththeQueen’spawnd4,sheopensd4againstFischer;whenFischer
respondswiththeKing’sknightNf6,sherespondsNf6againstSpassky,andsoon.
Fischer Spassky
rmblka0s RMBJQANS
opopopop OPOPZPOP
0Z0Z0m0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0O0Z0
0Z0O0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0m0Z0Z0
POPZPOPO popopopo
SNAQJBMR s0aklbmr
d4
Nf6
Anna-Louise
We shall write G→G for such a simultaneous display with a game G (so Anna-Louise played the
gameChess→Chessabove,assecondplayer,againsttheFischer-Spasskyteam).4
Observing that her copycat strategy is not specific to chess, Anna-Louise declares that she will
tackletheFischer-Spasskyteaminamoregrandiosespectacle: shewillgivethemanadditionalfirst
move, to decide the game for simultaneous display. For example, the Fischer-Spassky team might
choose Chess, thereby opting for the simultaneous display Chess→Chess, and play continues as
above. OrtheymightchooseO’s&X’s,optingforthesimultaneousdisplayO’s&X’s→O’s&X’s,and
openwithXinthecentreofSpassky’sgrid;Anna-LouisecopiesthatXaccrossasheropeningmove
onFischer’sgrid;FischerrespondswithOin(his)top-left;AnnacopiesthisObacktoSpassky;and
soon:
4Conwaywrites−G+G,orG−G[Con76,Chapter7]. Lateron,weshalladdaformofbacktrackingtoourgames
sothatAnna-LouisemayrestartthegamewithFischerasmanytimesasshelikes,correspondingtotheintuitionismofthe
arrow→ofsystemF,inwhichafunctionmayreaditsargumentanynumberoftimes[Lor60,Fel85,Coq91,HO00]. To
maintainthefocusonuniversalquantification,hereintheintroductionweshallignoretheavailabilitybacktracking.
2
Fischer Spassky
O
X X
O
Xcentre
Otop-left
Anna-Louise
Thekeynoveltyof[Hug97]wastodefinethisasaformalgame,ahypergameoruniversallyquanti-
fiedgame,whichweshallwriteas
∀G.G→G
The tree of ∀G.G → G is illustrated below. Similar in spirit to Zwicker’shypergame5, it differs in
the fact that the first player not only chooses G but also plays an opening move m in G. We call
suchacompoundmove(importingagame,andplayingamoveinagame)ahypermove.
∀G.G→G
Chess,d4 O’s&X’s,left
rmblkans RMBJQANS
opopopop OPOPZPOP
0ZZ00ZZ00ZZ00ZZ0→0ZZ00ZZ00OZ00ZZ0 → X
0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO popopopo
SNAQJBMR snaklbmr
d4 Nf6 left top
rompbolpkoapnosp ROMPBOJPQZAPNOSP rompbolpkoapnosp ROMPBOJPQZAPNOSP O
0ZZ00ZZ00ZZ00ZZ0→0ZZ00ZZ00OZ00ZZ0 0ZZ00ZZ00ZZ00ZZ0→0ZZ00ZZ00OZ00ZZ0 X → X → X
0Z0O0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0m0Z0Z0
POPZPOPO popopopo POPOPOPO popopopo
SNAQJBMR snaklbmr SNAQJBMR s0aklbmr
f5 d5 top-left right
rmblkans RMBJQANS rmblkans RMBJQANS
opopo0op OPOPZPOP opo0opop OPOPZPOP
0ZZ00ZZ00ZZp0ZZ0→0ZZ00ZZ00OZ00ZZ0 0ZZ00ZZp0ZZ00ZZ0→0ZZ00ZZ00OZ00ZZ0 X → X O X → X
0Z0O0Z0Z 0Z0Z0Z0Z 0Z0O0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
POPZPOPO popopopo POPZPOPO popopopo O
SNAQJBMR snaklbmr SNAQJBMR snaklbmr
1.2 Self-reference(withoutparadox)
Inthetreeabove,wehaveshowntwocasesforinstantiatingGinthehypergameH = ∀G.G→G,
eithertoChessortoO’s&X’s. ButitisalsopossibletoinstantiateGtoahypergame,orindeed,to
Hitself. Weconsiderthiscasebelow. Theinitialstateis:
5TheauthorwasunawareofZwicker’sworkwhilepreparing[Hug97], hencethe lackofreferencetoZwickerinthat
paper,andintheauthor’sthesis[Hug00].
3
Fischer Spassky
∀G.G→G
Anna-Louise
FischerandSpasskybeginbyimportingagameforG,inthiscase,H = ∀G.G→Gitself,yielding
asimultaneousdisplayofH:
Fischer Spassky
H → H
Anna-Louise
Inotherwords,wehave:
Fischer Spassky
∀G .G →G → ∀G .G →G
1 1 1 2 2 2
Anna-Louise
The local bound variable G is renamed in each component to clarify the evolution of the game
below.6 AsinthesimultaneousdisplayChess→Chess,whereSpasskyopenedwithamoveonhis
chessboard, here in H → H Spassky must complete the opening hypermoveby playing a move on
his copy of H. Since H = ∀G .G → G is a hypergame, opening H requires importing another
2 2 2
game,instantiatingG . SupposehechoosesChessforG :
2 2
Fischer Spassky
rmblkans RMBJQANS
opopopop OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z
∀G1.G1 →G1 → Z00ZZ00ZZ00ZZ00Z → Z00ZZ00ZZ00ZZ00Z
Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO popopopo
SNAQJBMR snaklbmr
Anna-Louise
6Thescopeof∀G1inthediagramdoesnotextendpastthecentralarrow→.Inotherwords,formallythegameplayed
byAnna-Louiseis(∀G1.G1→G1) → (∀G2.G2→G2).
4
NowSpasskyhashisownlocalsimultaneousdisplayChess→Chess. Tocompletehisopening(hy-
per)moveontheoverallgame,hemustopenthischessdisplay. SupposeheplaysNf3(necessarily
ontherightboard,whereitishisturnsincehehasWhite):
Fischer Spassky
rmblkans RZBJQANS
opopopop OPOPOPOP
0Z0Z0Z0Z 0ZNZ0Z0Z
∀G1.G1 →G1 → Z00ZZ00ZZ00ZZ00Z → Z00ZZ00ZZ00ZZ00Z
Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO popopopo
SNAQJBMR snaklbmr
Anna-Louise
Now it is Anna-Louise’s turn. She has three options: (1) respond to Spassky as Black on the
rightmost chess board, (2) respond to Spassky as White on the other chess board, or (3) play an
opening move against Fischer in ∀G .G → G . We consider the last case, since it is the most
1 1 1
interesting. SupposeAnna-LouisechoosestoimportO’s&X’sforG :
1
Fischer Spassky
rmblkans RZBJQANS
opopopop OPOPOPOP
0Z0Z0Z0Z 0ZNZ0Z0Z
→ Z0Z0Z0Z0 Z0Z0Z0Z0
→ →
0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO popopopo
SNAQJBMR snaklbmr
Anna-Louise
Now Fischer has his own local simultaneous display O’s&X’s → O’s&X’s. For Anna-Louise to
completeherhypermove,shemustplayamoveonO’s&X’s→O’s&X’s(necessarilyintherightof
thetwogrids,theoneinwhichitherturn). SupposesheplaysherXtop-right:
5
Fischer Spassky
rmblkans RZBJQANS
opopopop OPOPOPOP
X
0Z0Z0Z0Z 0ZNZ0Z0Z
→ Z0Z0Z0Z0 Z0Z0Z0Z0
→ →
0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO popopopo
SNAQJBMR snaklbmr
Anna-Louise
FischerrespondseitherwithanOinthesamegrid,orwithanXintheemptygrid,andplaycontinu-
ousinthetwolocalsimultaneousdisplays,O’s&X’s→O’s&X’sagainstFischerandChess→Chess
againstSpassky.
Buttoremainconsistentwithhercopycatstrategy,Anna-LouisemustmimicSpassky. Insteadof
importing O’s&X’s for G against Fischer, she must import Chess and open with the White move
1
Nf3,exactlyasSpasskydid:
Fischer Spassky
RMBJQANS rmblkans rmblkans RZBJQANS
OPOPOPOP opopopop opopopop OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0ZNZ0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 → Z0Z0Z0Z0 Z0Z0Z0Z0
→ →
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0ZNZ0 Z0Z0Z0Z0 Z0Z0Z0Z0
popopopo POPOPOPO POPOPOPO popopopo
snaklbmr SNAQJBZR SNAQJBMR snaklbmr
Chess,Nf3
Anna-Louise
Fischermightnowopenhisotherboardwithe4,whichAnna-Louisewouldcopybacktothecorre-
spondingboardagainstSpassky:
6
Fischer Spassky
RMBJQANS rmblkans rmblkans RZBJQANS
OPO0OPOP opopopop opopopop OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0ZNZ0Z0Z
Z0ZPZ0Z0 Z0Z0Z0Z0 → Z0Z0Z0Z0 Z0Z0Z0Z0
→ →
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0ZPZ0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0ZNZ0 Z0Z0Z0Z0 Z0Z0Z0Z0
popopopo POPOPOPO POPO0OPO popopopo
snaklbmr SNAQJBZR SNAQJBMR snaklbmr
e4
Anna-Louise
Or perhaps Fischer responds with Black in the rightmost of his pair of boards, with d5, which
Anna-LouisecopiestoSpassky:
Fischer Spassky
RMBJQANS rmblkans rmblkans RZBJQANS
OPOPOPOP opo0opop opopopop OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0ZNZ0Z0Z
Z0Z0Z0Z0 Z0ZpZ0Z0 → Z0Z0Z0Z0 Z0Z0Z0Z0
→ →
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0ZpZ0Z
Z0Z0Z0Z0 Z0Z0ZNZ0 Z0Z0Z0Z0 Z0Z0Z0Z0
popopopo POPOPOPO POPOPOPO popo0opo
snaklbmr SNAQJBZR SNAQJBMR snaklbmr
d5
Anna-Louise
Eitherway,shecontinuestocopymovesbetweenthefourboardsaccordingtothefollowinggeom-
etryofcopycatlinks:
Fischer Spassky
RMBJQANS rmblkans rmblkans RZBJQANS
OPOPOPOP opopopop opopopop OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0ZNZ0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 → Z0Z0Z0Z0 Z0Z0Z0Z0
→ →
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0ZNZ0 Z0Z0Z0Z0 Z0Z0Z0Z0
popopopo POPOPOPO POPOPOPO popopopo
snaklbmr SNAQJBZR SNAQJBMR snaklbmr
Anna-Louise
7
ThiscopycatstrategycorrespondstothepolymorphicidentitysystemFterm
ΛG.λgG.g
oftype∀G.G→G.
1.3 Uniformity
Consideragain theoriginal Fischer-Spasskysimultaneous display,withchess. AddKasparovtothe
team,playingBlack.
Kasparov Fischer Spassky
rmblkans rmblkans RMBJQANS
opopopop opopopop OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO POPOPOPO popopopo
SNAQJBMR SNAQJBMR snaklbmr
Anna-Louise
Anna-Louisehastwodistinctwaystoguaranteepickingupapoint. Eithershecopiesmovesbetween
SpasskyandFischer,asbefore,whileignoringKasparov(neverplayingamoveagainsthim),
Kasparov Fischer Spassky
rmblkans rmblka0s RMBJQANS
opopopop opopopop OPOPZPOP
0Z0Z0Z0Z 0Z0Z0m0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0O0Z0
0Z0Z0Z0Z 0Z0O0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0m0Z0Z0
POPOPOPO POPZPOPO popopopo
SNAQJBMR SNAQJBMR s0aklbmr
d4
Nf6
Anna-Louise
orshecopiesmovesbetweenSpasskyandKasparov,ignoringFischer:
8
Kasparov Fischer Spassky
rmblkans rmblkans RMBJQANS
opZpopop opopopop OPO0OPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0o0Z0Z0 Z0Z0Z0Z0 Z0ZPZ0Z0
0Z0ZPZ0Z 0Z0Z0Z0Z 0Z0Z0o0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
POPO0OPO POPOPOPO popopZpo
SNAQJBMR SNAQJBMR snaklbmr
e4
c5
Anna-Louise
We shall write this triple simultaneous display as Chess → Chess → Chess, and more generally,
foranygameG,asG→G→G.7
Nowconsidertheuniversallyquantifiedformofthisgame,thehypergame
∀G.G→G→G.
Aswith∀G.G→Gdiscussedabove,theKasparov-Fischer-Spasskyteam,KFS,nowhastherightto
choosethegameofthetriplesimultaneousdisplay,aspartoftheiropening(hyper)move. Weshall
saythatAnna-Louise’sstrategyisuniforminthissettingif
• irrespective of the game chosen by KFS, she always ignores the same player, Kasparov or
Fischer.
Otherwiseherstrategyisad hoc. Forexample,herstrategywouldbeadhocif,whenKFSchooses
Chess, she ignores Kasparov and copies chess moves betweenFischer and Spassky, but when KFS
chooses O’s & X’s, she ignores Fischer and copies X and O moves between Kasparov and Spassky.
In this case the geometry of her move copying depends on the game imported by FKS: she is not
treatingGasa“blackbox”.
There are only two uniform strategies for Anna-Louise: either she always copies between Kas-
parov and Spassky, ignoring Fischer, or she always copies between Fischer and Spassky, ignoring
Kasparov. ThesecorrespondtothesystemFterms
ΛG.λkG.λfG.k
ΛG.λkG.λfG.f
respectively,oftype
∀G.G→G→G,
wherethevariablekcorrespondstoKasparovandfcorrespondstoFischer.
Moregenerally,withmultiple bound∀variablesand morecomplicatedgame imports,weshall
take uniformity to mean that the links Anna-Louise sets up between components (such as the
Kasparov↔SpasskyorFischer↔Spasskylinksabove)mustbeindependentofthegamesimported
bytheopposingteam: theseimportedgamesareimpenetrable“blackboxes”.
7Againwiththebacktrackingcaveat:seefootnote4.
9
Fixed links. Uniformity as independence from the particular games imported by the opposing
team will include independence from the not only the identity of those games, but also from their
state. ThiswillensurethatthegeometryofAnna-Louise’scopycatplayremainsconstantovertime:
once she has committed to linking one component to another, she must stick with that link for
the rest of the hypergame. To illustrate this aspect of uniformity, consider the quadruple chess
simultaneous display with Kasparov and Fischer playing Black, and Karpov and Spassky playing
White:
Kasparov Fischer Karpov Spassky
rmblkans rmblkans RMBJQANS RMBJQANS
opopopop opopopop OPOPOPOP OPOPOPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO POPOPOPO popopopo popopopo
SNAQJBMR SNAQJBMR snaklbmr snaklbmr
Anna-Louise
We shall write Chess×Chess → Chess×Chess for this simultaneous display.8 Suppose Spassky
begins with e4. Anna-Louise, playing copycat, has a choice between copying this move to Fischer
ortoKasparov. SupposeshecopiesittoFischer,whorespondswithc5,whichshedulycopiesback
toSpassky:
Kasparov Fischer Karpov Spassky
rmblkans rmblkans RMBJQANS RMBJQANS
opopopop opZpopop OPOPOPOP OPO0OPOP
0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z 0Z0Z0Z0Z
Z0Z0Z0Z0 Z0o0Z0Z0 Z0Z0Z0Z0 Z0ZPZ0Z0
0Z0Z0Z0Z 0Z0ZPZ0Z 0Z0Z0Z0Z 0Z0Z0o0Z
Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0 Z0Z0Z0Z0
POPOPOPO POPO0OPO popopopo popopZpo
SNAQJBMR SNAQJBMR snaklbmr snaklbmr
e4
c5
Anna-Louise
SupposeKarpovopenshisgamewiththeverysamemoveasSpassky,e4,whichAnna-Louisecopies
accrosstoKasparov(theonlydestinationwherethismovemakessense):
8Withthebacktrackingcaveat:seefootnote4.
10
