Independence-friendly logic. A game-theoretic approach

215 Pages·2011·1.029 MB·English

by  Mann, Allen L., Sevenster, Merlijn, Sandu, Gabriel

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240 Stablegroups,F.O.WAGNER
241 Surveysincombinatorics,1997,R.A.BAILEY(ed)
242 GeometricGaloisactionsI,L.SCHNEPS&P.LOCHAK(eds)
243 GeometricGaloisactionsII,L.SCHNEPS&P.LOCHAK(eds)
244 Modeltheoryofgroupsandautomorphismgroups,D.M.EVANS(ed)
245 Geometry,combinatorialdesignsandrelatedstructures,J.W.P.HIRSCHFELDetal(eds)
246 p-Automorphismsoffinitep-groups,E.I.KHUKHRO
247 Analyticnumbertheory,Y.MOTOHASHI(ed)
248 TametopologyandO-minimalstructures,L.VANDENDRIES
249 Theatlasoffinitegroups–Tenyearson,R.T.CURTIS&R.A.WILSON(eds)
250 Charactersandblocksoffinitegroups,G.NAVARRO
251 Gro¨bnerbasesandapplications,B.BUCHBERGER&F.WINKLER(eds)
252 Geometryandcohomologyingrouptheory,P.H.KROPHOLLER,G.A.NIBLO&R.STO¨HR(eds)
253 Theq-Schuralgebra,S.DONKIN
254 Galoisrepresentationsinarithmeticalgebraicgeometry,A.J.SCHOLL&R.L.TAYLOR(eds)
255 Symmetriesandintegrabilityofdifferenceequations,P.A.CLARKSON&F.W.NIJHOFF(eds)
256 AspectsofGaloistheory,H.VO¨LKLEIN,J.G.THOMPSON,D.HARBATER&P.MU¨LLER(eds)
257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications(2ndedition),
J.MADORE
258 Setsandproofs,S.B.COOPER&J.K.TRUSS(eds)
259 Modelsandcomputability,S.B.COOPER&J.TRUSS(eds)
260 GroupsStAndrews1997inBathI,C.M.CAMPBELLetal(eds)
261 GroupsStAndrews1997inBathII,C.M.CAMPBELLetal(eds)
262 Analysisandlogic,C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL
263 Singularitytheory,W.BRUCE&D.MOND(eds)
264 Newtrendsinalgebraicgeometry,K.HULEK,F.CATANESE,C.PETERS&M.REID(eds)
265 Ellipticcurvesincryptography,I.BLAKE,G.SEROUSSI&N.SMART
267 Surveysincombinatorics,1999,J.D.LAMB&D.A.PREECE(eds)
268 Spectralasymptoticsinthesemi-classicallimit,M.DIMASSI&J.SJO¨STRAND
269 Ergodictheoryandtopologicaldynamicsofgroupactionsonhomogeneousspaces,M.B.BEKKA&
M.MAYER
271 Singularperturbationsofdifferentialoperators,S.ALBEVERIO&P.KURASOV
272 Charactertheoryfortheoddordertheorem,T.PETERFALVI.TranslatedbyR.SANDLING
273 Spectraltheoryandgeometry,E.B.DAVIES&Y.SAFAROV(eds)
274 TheMandelbrotset,themeandvariations,T.LEI(ed)
275 Descriptivesettheoryanddynamicalsystems,M.FOREMAN,A.S.KECHRIS,A.LOUVEAU&
B.WEISS(eds)
276 Singularitiesofplanecurves,E.CASAS-ALVERO
277 Computationalandgeometricaspectsofmodernalgebra,M.ATKINSONetal(eds)
278 Globalattractorsinabstractparabolicproblems,J.W.CHOLEWA&T.DLOTKO
279 Topicsinsymbolicdynamicsandapplications,F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds)
280 CharactersandautomorphismgroupsofcompactRiemannsurfaces,T.BREUER
281 Explicitbirationalgeometryof3-folds,A.CORTI&M.REID(eds)
282 Auslander-Buchweitzapproximationsofequivariantmodules,M.HASHIMOTO
283 Nonlinearelasticity,Y.B.FU&R.W.OGDEN(eds)
284 Foundationsofcomputationalmathematics,R.DEVORE,A.ISERLES&E.SU¨LI(eds)
285 Rationalpointsoncurvesoverfinitefields,H.NIEDERREITER&C.XING
286 Cliffordalgebrasandspinors(2ndEdition),P.LOUNESTO
287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCE,A.F.COSTA&E.MART´INEZ(eds)
288 Surveysincombinatorics,2001,J.W.P.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities,L.SALOFF-COSTE
290 QuantumgroupsandLietheory,A.PRESSLEY(ed)
291 Titsbuildingsandthemodeltheoryofgroups,K.TENT(ed)
292 Aquantumgroupsprimer,S.MAJID
293 SecondorderpartialdifferentialequationsinHilbertspaces,G.DAPRATO&J.ZABCZYK
294 Introductiontooperatorspacetheory,G.PISIER
295 Geometryandintegrability,L.MASON&Y.NUTKU(eds)
296 Lecturesoninvarianttheory,I.DOLGACHEV
297 Thehomotopycategoryofsimplyconnected4-manifolds,H.-J.BAUES
298 Higheroperads,highercategories,T.LEINSTER(ed)
299 Kleiniangroupsandhyperbolic3-manifolds,Y.KOMORI,V.MARKOVIC&C.SERIES(eds)
300 IntroductiontoMo¨biusdifferentialgeometry,U.HERTRICH-JEROMIN
301 StablemodulesandtheD(2)-problem,F.E.A.JOHNSON
302 DiscreteandcontinuousnonlinearSchro¨dingersystems,M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH
303 Numbertheoryandalgebraicgeometry,M.REID&A.SKOROBOGATOV(eds)
304 GroupsStAndrews2001inOxfordI,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
305 GroupsStAndrews2001inOxfordII,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
306 Geometricmechanicsandsymmetry,J.MONTALDI&T.RATIU(eds)
307 Surveysincombinatorics2003,C.D.WENSLEY(ed.)
308 Topology,geometryandquantumfieldtheory,U.L.TILLMANN(ed)
309 Coringsandcomodules,T.BRZEZINSKI&R.WISBAUER
310 Topicsindynamicsandergodictheory,S.BEZUGLYI&S.KOLYADA(eds)
311 Groups:topological,combinatorialandarithmeticaspects,T.W.MU¨LLER(ed)
312 Foundationsofcomputationalmathematics,Minneapolis2002,F.CUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles,S.MU¨LLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs,D.CVETKOVIC´,P.ROWLINSON&S.SIMIC´
315 Structuredringspectra,A.BAKER&B.RICHTER(eds)
316 Linearlogicincomputerscience,T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds)
317 Advancesinellipticcurvecryptography,I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds)
318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations,D.HENRY
319 DoubleaffineHeckealgebras,I.CHEREDNIK
320 L-functionsandGaloisrepresentations,D.BURNS,K.BUZZARD&J.NEKOVA´Rˇ (eds)
321 Surveysinmodernmathematics,V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory,F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds)
324 Singularitiesandcomputeralgebra,C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciflow,P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype,J.E.HUMPHREYS
327 Surveysincombinatorics2005,B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds,R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology,A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005,L.MPARDO,A.PINKUS,E.SU¨LI&
M.J.TODD(eds)
332 Handbookoftiltingtheory,L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition),A.KOCK
334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory,J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory,N.YOUNG(ed)
339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,
F.MEZZADRI&N.C.SNAITH(eds)
342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry,A.NEEMAN
346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,
A.PILLAY&A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,
A.PILLAY&A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis,J.BLATH,P.MO¨RTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis,K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO
357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER&
I.J.LEARY(eds)
359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARC´IA-PRADA&
S.RAMANAN(eds)
360 Zariskigeometries,B.ZILBER
361 Words:Notesonverbalwidthingroups,D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERO´N&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena,G.BLOWER
368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZA´LEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIE´
370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals,F.PRZYTYCKI&M.URBAN´SKI
372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularity,andcompleteintersection,J.MAJADAS&A.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC
375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns,S.LINTON,N.RUSˇKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY
378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 RealandComplexSingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.L.WALLS(eds)
381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA&
P.WINTERNITZ(eds)
382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJ´ICˇEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN&
T.WEISSMAN(eds)
London Mathematical Society Lecture Note Series: 386
Independence-Friendly
Logic
A Game-Theoretic Approach
ALLEN L. MANN
University of Tampere, Finland
GABRIEL SANDU
University of Helsinki, Finland
MERLIJN SEVENSTER
Philips Research Laboratories, The Netherlands
cambridge university press
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(cid:2)c A.L.Mann,G.SanduandM.Sevenster2011
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Contents
1 Introduction page 1
2 Game theory 9
2.1 Extensive games 9
2.2 Strategic games 17
3 First-order logic 28
3.1 Syntax 28
3.2 Models 30
3.3 Game-theoretic semantics 32
3.4 Logical equivalence 39
3.5 Compositional semantics 48
3.6 Satisfiability 55
4 Independence-friendly logic 59
4.1 Syntax 59
4.2 Game-theoretic semantics 62
4.3 Skolem semantics 67
4.4 Compositional semantics 76
4.5 Game-theoretic semantics redux 83
5 Properties of IF logic 87
5.1 Basic properties 87
5.2 Extensions of IF logic 95
5.3 Logical equivalence 99
5.4 Model theory 127
6 Expressive power of IF logic 134
6.1 Definability 134
6.2 Second-order logic 136
6.3 Existential second-order logic 139
6.4 Perfect recall 145
v
vi Contents
7 Probabilistic IF logic 150
7.1 Equilibrium semantics 151
7.2 Monotonicity rules 156
7.3 Behavioral strategies and compositional semantics 170
7.4 Elimination of strategies 171
7.5 Expressing the rationals 181
8 Further topics 185
8.1 Compositionality 185
8.2 IF modal logic 191
References 198
Index 203
1
Introduction
First-order logic meets game theory as soon as one considers sentences
withalternatingquantifiers.Eventhesimplestalternatingpatternillus-
trates this claim:
∀x∃y(x<y). (1.1)
We can convince an imaginary opponent that this sentence is true on
thenaturalnumbersbypointingoutthatforeverynaturalnumbermhe
chooses for x, we can find a natural number n for y that is greater than
m. If, on the other hand, he were somehow able to produce a natural
number for which we could not find a greater one, then the sentence
would be false.
We can make a similar arrangement with our opponent if we play
on any other structure. For example, if we only consider the Boolean
values0and1orderedintheir naturalway, wewould agreeonasimilar
protocolfortestingthesentence,exceptthateachpartywouldpick0or
1 instead of any natural number.
It is natural to think of these protocols as games. Given a first-order
sentencesuchas(1.1),oneplayertriestoverifythesentencebychoosing
a value of the existentially quantified variable y, while the other player
attempts to falsify it by picking the value of the universally quantified
variable x. Throughout this book we will invite Eloise to play the role
of verifier and Abelard to play the role of falsifier.
We can formalize this game by drawing on the classical theory of
extensive games. In this framework, the game between Abelard and
Eloisethatteststhetruthof(1.1)ismodeledasatwo-stagegame.First
Abelard picks an object m. Then Eloise observes which object Abelard
chose, and picks another object n. If m<n, we declare that Eloise has
wonthegame;otherwisewedeclareAbelardthewinner.Wenoticethat
2 Introduction
Eloise’s ability to “see” the object m before she moves gives her an ad-
vantage. The reason we give Eloise this advantage is that the quantifier
∃y lieswithinthescopeof∀x.Inotherwords,thevalueofy dependson
the value of x.
Hintikka used the game-theoretic interpretation of first-order logic to
emphasize the distinction between constitutive rules and strategic prin-
ciples [28,29]. The former apply to individual moves, and determine
whether a particular move is correct or incorrect. In other words, con-
stitutive rules determine the set of all possible plays, i.e., the possible
sequencesofmovesthatmightariseduringthegame.Incontrast,strate-
gic principles pertain to the observed behavior of the players over many
plays of the game. Choosing blindly is one thing, following a strategy is
another. A strategy is a rule that tells a particular player how to move
in every position where it is that player’s turn. A winning strategy is
one that ensures a win for its owner, regardless of the behavior of the
other player(s). Put another way, constitutive rules tell us how to play
the game, while strategic principles tell us how to play the game well.
When working with extensive games, it is essential to distinguish be-
tweenwinningasingleplay,andhavingawinningstrategyforthegame.
If we are trying to show that (1.1) holds, it is not enough to exhibit one
single play in which m = 4 and n = 7. Rather, to show (1.1) is true,
Eloise must have a strategy that produces an appropriate n for each
value of m her opponent might choose. For instance, to verify (1.1) is
true in the natural numbers, Eloise might use the winning strategy: if
Abelard picks m, choose n = m+1. If we restrict the choice to only
Boolean values, however, Abelard has a winning strategy: he simply
picks the value 1. Thus (1.1) is true in the natural numbers, but false if
we restrict the choice to Boolean values.
To take an example from calculus, recall that a function f is contin-
uous if for every x in its domain, and every ε > 0, there exists a δ > 0
such that for all y,
|x−y|<δ implies |f(x)−f(y)|<ε.
This definition can be expressed using the quantifier pattern
∀x∀ε∃δ∀y(...), (1.2)
where the dots stand for an appropriate first-order formula. Using the
game-theoretic interpretation, (1.2) is true if for every x and ε chosen
by Abelard, Eloise can pick a value for δ such that for every y chosen
by Abelard it is the case that...
Introduction 3
Thekeyfeatureofgame-theoreticsemanticsisthatitrelatesacentral
concept of logic (truth) to a central concept of game theory (winning
strategy).Oncetheconnectionbetweenlogicandgameshasbeenmade,
logical principles such as bivalence and the law of excluded middle can
be explained using results from game theory. To give one example, the
principle of bivalence is an immediate consequence of the Gale-Stewart
theorem,whichsaysthatineverygameofacertainkindthereisaplayer
with a winning strategy.
Mathematicallogicianshavebeenusinggame-theoreticsemanticsim-
plicitlyforalmostacentury.TheSkolemform ofafirst-ordersentenceis
obtained by eliminating each existential quantifier, and substituting for
the existentially quantified variable a Skolem term f(y1,...,yn), where
f isafreshfunctionsymbol andy1,...,yn arethevariables uponwhich
the choice of the existentially quantified variable depends. A first-order
formulaistrueinastructureifandonlyiftherearefunctionssatisfying
its Skolem form. (cid:2) (cid:3)
For instance the Skolem form of (1.1) is ∀x x<f(x) . In the natural
numbers, we can take f to be defined by f(x) = x+1, which shows
that (1.1) is true. Thus we see that Skolem functions encode Eloise’s
strategies.
Logic with imperfect information
Thegame-theoreticperspectiveallowsonetoconsiderextensionsoffirst-
order logic that are not obvious otherwise. Independence-friendly logic,
the subject of the present volume, is one such extension.
Anextensivegamewithimperfectinformationisoneinwhichaplayer
maynot“see”(“know”)allthemovesleadinguptothecurrentposition.
Imperfect information is a common phenomenon in card games such as
bridgeandpoker,inwhicheachplayerknowsonlythecardsonthetable
and the cards she is holding in her hand.
In order to specify semantic games with imperfect information, the
syntax of first-order logic can be extended with slashed sets of variables
that indicate which past moves are unknown to the active player. For
example, in the independence-friendly sentence
(cid:2) (cid:3)
∀x∀y ∃z/{y} R(x,y,z), (1.3)
the notation /{y}indicates that Eloise is not allowed to seethe value of
y when choosing the value of z.