Synchrony and Composition: Toward a Cogni - an der Ruhr ...

BenediktLowe, WolfgangMalzkorn, Thoralf Rasch(eds.)Foundationsof the Formal SciencesIIApplicationsof MathematicalLogic in PhilosophyandLinguisticsBonn,November10-13,2000,pp.255–272.

Synchrony and Composition: Toward a Cogni-tive Ar chitectureBetweenClassicismand Con-nectionismMarkus WerningâLehrstuhlfur WissenschaftsphilosophieUniversitat ErfurtNordhauserStraße6399089ErfurtGermany

E-mail: [email protected]

Abstract. Using the toolsof universalalgebra,it is shown thatoscillatorynet-worksrealizesystematiccognitive representations.It is argued(i) thatanalgebraof propositionsandconceptsfor objectsandpropertiesis isomorphicto analge-braof brainstates,neuronaloscillationsandsetsof oscillationsrelatedto clustersof neurons,(ii) thattheisomorphism,in a strongsense,preservestheconstituentrelationsof the conceptualalgebra,and(iii) that the isomorphismtransfersse-mantic compositionality. Oscillatory networks are neurobiologicallyplausible.They combinethe virtuesandavoid the vicesof classicalandconnectionistar-chitectures.

Received: April 1st,2001;In revisedversion: July 24th,2001;Acceptedby the editors: August17th,2001.2000MathematicsSubjectClassification.92B2008A7091F2091E1068T99.ãResearchfor this paperwassponsoredby the NationalGermanScholarshipFoundation.Itwasenabledby a one-yearresearchscholarshipat RutgersUniversityandtheRutgersCenterof CognitiveScience.I owemany of thepresentedinsightsto discussionswith AndreasEngel,Wolf Singer, JerryFodor, ErnieLePore,Brian McLaughlin,BruceTesar, ThomasMetzinger,andGerhardSchurz.I have benefittedfrom commentsby a numberof colleagueson variousoccasionswherepartsor earlierstagesof this materialwerepresented.I amgrateful,in par-ticular, to participantsat theBerlin ColloquiumPhilosophyMeetsCognitive Science, FotFSII, CogSci2001,andESPP2001.I would, also,like to thankan anonymousrefereefor veryhelpful remarks.

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1 Intr oduction

Minds have the capacityto composecontents.Otherwise,they wouldnot show a systematiccorrelationbetweenrepresentationalcapacities:If a mind is capableof certainintentionalstatesin a certainintentionalmode,it mostprobablyis alsocapableof other intentionalstateswithrelatedcontentsin the samemode.The capacityto seesomethingasaredsquarein agreencircle,e.g., is statisticallyhighly correlatedwith thecapacityto seesomethingasa redcircle in a greensquare.Thecapacityto understandtheEnglishsentence“JohnlovesMary” is correlatedwiththe capacityto understand“Mary lovesJohn”. To explain this correla-tion, compositionaloperationsarepostulated(Throughoutthetext oper-ationsareconceived of as functions).They enablethe systemto buildcomplex representationsfrom primitive onesso that thesemanticvalueof the complex representationis determinedby its structureandby thesemanticvaluesof its components.Severalcognitive theorieshave beendevelopedto meetthe requirementof compositionality. The proposedtheories,though,suffer from severedeficits.

FodorandPylyshyn[FodPyl88] for onetake recourseto a languageof thought,which they link to theclaim that thebrain canbe modelledby a Turing-stylecomputer. A subject’s having anintentionalstate,theybelieve, consistsin the subject’s bearinga computationalrelation to amentalsentence;it is a relationanalogousto the relationa Turing ma-chine’s control headbearsto the tape.A subject’s belief that thereis aredsquarein agreencircle,thus,is conceivedof asacomputationalrela-tion betweenthesubjectandthementalsentence:Thereis a redsquareina greencircle. Likewise,whena subjectunderstandstheutterance“JohnlovesMary”, this utterancereliably causesthesubjectto beara compu-tationalrelationto thementalsentence:John lovesMary. Accordingtothis paradigm,themind composescomplex representationsfrom primi-tiveonesjust thewayacomputercombinesphrasesfrom words:by con-catenation.Thementalsentence–or thought–JohnlovesMary is hencenothingbut a concatenationof thementalwords– or concepts– Mary,John, andloves. Givenacertainsyntacticstructure,thesemanticproper-tiesof thethoughtarecompletelydeterminedby thesemanticpropertiesof theconcepts.

Thetroublewith classicalcomputermodelsiswell knownandreachesfrom the frameproblem,the problemof gracefuldegradation,and the

Synchrony andComposition 257

problemof learningfrom examples(cf. [Hor ä Tie�96]) to problemsthat

arisefrom thecontentsensitivity of logical reasoning(cf. [GigHugä 92]).To avoid the pitfalls of classicism,connectionistmodelshave beende-veloped.In Smolensky’s integratedconnectionist/symbolicarchitecture[Smo91]the termsandthesyntaxof a languagearemappedhomomor-phically onto an algebraof vectorsandtensoroperations.1 Eachprimi-tive term of the languageis assignedto a vector. Every vector rendersa certaindistribution of activity within the connectionistnetwork. Thesyntacticoperationsof the languagehave tensoroperationsascounter-parts.As far assyntaxis concerned,languageswith rich combinatorialpotentialcan,indeed,beimplementedby aconnectionistnetwork.

Thekind of combinationthatis necessaryfor systematicity, however,focusesnot only on syntactic,but alsoon semanticfeatures.Thecapac-ity to think thata child with a redcoatis distractedby anold herringisnot correlatedwith thecapacityto think thata child with anold coatisdistractedby a redherring.Thethoughtsoughtto becorrelated,though,if the fact that one is a syntacticre-combinationof the otherwassuf-ficient for systematiccorrelation.Notice that both thoughtsaresyntac-tically combinedfrom exactly the sameprimitivesby exactly the sameoperations.One may, however, well have the capacityto think of redcoatsandold herringseventhoughonelacksthecapacityto think of redherrings.The two thoughtsfail to be correlatedbecausered herring isidiomaticand–asaconsequence–semanticcompositionalityis violated.

Formally speaking,a languageis semanticallycompositionalif andonly if its semanticsis a homomorphic image of its syntax (cf.[PartMeWalä 90]): Let å ã æ ì � ç��Æ��ç ì ��ç @ � ç��Æ��ç @ èJé be the syntax alge-bra of the languagewith the syntacticcategories

ì � ç��Æ�Æ��ç ì � (e.g., setsof adjectivesor nouns)andwith thesyntacticoperations@ � ç ��Æ��ç @�è (e.g.,adjective-nouncombination);andlet ê ã æ ë � ç ��Æ��ç ë �)ç v � ç��Æ��ç�v è é bethe semanticalgebraof the languagewith the semanticcategoriesandoperations.In the caseof a homomorphismwe have a family of func-tions æ Õ>Î 5 ì Î3î ë Î � t}ã C ç ��Æ��ç'� é thatallows usto definethefunction

Õ

1 I take Smolensky’s approachonly asa representative for a variety of modelsthat pursueasimilar strategy. For a survey of relatedmodelssee[Wer01].

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of semanticevaluationin thefollowing way:

Õ 5 ì � ôI��Æ��ô ì � ô´ä @ � ç��Æ��ç @ è é î ë � ôI�Æ�Æ�*ô ë � ô´äµv � ç��Æ��ç�v è é (1.1)

suchthatÕhóMì õ{ã ÕµÎ�óMì õ if

ìlñ ì Îfor every tEã C ç��Æ�Æ��ç��

andÕhó @ × õ~ã�v ×

for every íàã C ç��Æ��ç�Úµ�Usingthedefinitionof homomorphism,we cannow saythata languageiscompositionalif andonly if thesemanticevaluationfunctiondistributesover thesyntacticstructureof any of thelanguage’s formulas:

Õhó @ ×*ókì�� ç ��Æ��ç ìÈqMî õ�õ~ã Õhó @ × õ ó2Õwókì�� õ ç��Æ��ç ÕhóMìïqMî õ�õJ� (1.2)

This equationnicely interpretstheinformal definitionof compositional-ity accordingto which the semanticvalueof a complex formula is de-terminedby its syntacticstructureandthesemanticvaluesof its syntac-tic components.Syntacticoperationsthat violate (1.2) generateidioms.Thatsomeidioms,at least,underminesemanticcompositionalitycanbedemonstratedwith regardto the examplered herring. Provided,firstly,that thesemanticvaluesof red herring andnot-not-redherring differ –thevalueof theformerrelatesto a maneuver of drawing attentionawayfrom themain issue,whereasthatof thelatterrelatesto someredly col-oredfish–,provided,secondly, that the semanticvaluesof red andnot-not-redarethesame,andprovided,thirdly, thatredherringandnot-not-redherring areoutcomesof thesamesyntacticoperation–with redandherringasargumentsin theoneandnot-not-red andherring in theothercase–,thissyntacticoperation,then,hasnosemanticfunctionascounter-part.Recallthatno functionoutputsdifferentvaluesif it takesthesameitemsasarguments.With thesethreeratherplausibleprovisions,the id-iomaticcharacterof redherringconstitutesaviolationof semanticcom-positionalityasdefinedabove.2 As we have seen,idiomsunderminethe

2 I do not intendto make any substantialstatementsaboutidioms,here.In anobjectionto thereceived view, which is reflectedin [NunSagWas94] andaccordingto which someidiomsviolatesemanticcompositionality, Westerstahl [Wesá ] arguesthatidiomscanalwaysbeem-beddedin compositionallanguages.He proposesthreewaysof doing so: (i) extendthe setof atomicexpressionsby a holophrasticreadingof the idiom, (ii) extendthe list of syntacticoperationssothattheliteral andtheidiomaticreadingof theidiom turnout to beoutcomesofdifferentsyntacticoperations,or (iii) take thecomponentsof theidiom ashomonymsof theiroccurrencesin its literal readingandaddthemto thesetof atomicexpressions.Noneof thethreeoptionsafflict our argumentation,though,becausein eachcasea child with an old coatis distractedby a redherring wouldno longerbea syntacticre-combinationof a child with a


systematiccorrelationof two thoughtseven whenonethoughtis noth-ing but asyntacticre-combinationof theother. Wemay, hence,infer thatsemanticcompositionalityis necessaryfor systematicity–its violationwouldallow for idioms–andthatsyntacticcombinationis not sufficient.Smolensky’sstrategy to implementthesyntaxof a languageontoa con-nectionistnetwork doesnot suffice to establishthat the network itselfsubservessystematicrepresentationalcapacities.

2 Constituency

A furtherargumentprovidesuswith a deeperinsight into what’s wrongwith connectionistapproachestowardrepresentationalism.Most seman-tic theoriesexplain the semanticpropertiesof internal representationseitherin termsof co-variance,in termsof inferentialrelations,in termsof associations,or by a combinationof thethree.Some,e.g., hold thatacertaininternalstateis a representationof rednessbecausethestateco-varieswith nearbyinstancesof redness.This co-variancerelationis, ofcourse,backedby theintrinsicandextrinsiccausalpropertiesof thered-nessrepresentation.Othersholdthatsomerepresentations–e.g., bachelor–characteristicallyaresuchthat thesubjectis disposedto infer otherrep-resentations–e.g., unmarried– from it. Thosedispositions,again,aregroundedin thecausalpropertiesof therepresentationsin question.Onemay, thirdly, hold that the semanticvalueof a representationlike cowis determinedby thefact that it is associatedwith otherrepresentations,e.g., milk, leather, mammal, grass, etc. The mechanismof association,too, superveneson the causalpropertiesof the representationin ques-tion. All of thesetheorieshave one principle in common:An internalrepresentationhasits semanticvaluebecauseit hasa certaincausalrolewithin thesystem(and–perhaps–therestof theworld).

The questionof how the semanticvalue of an internal representa-tion is determined,and perforce,how it is determinedby the seman-tic valuesof its syntacticcomponents,hence,leadsto the questionofhow thecausalpropertiesof aninternalrepresentationaredetermined—and perforcehow they are determinedby the causalpropertiesof thesyntacticcomponents.Fromchemistryandotherscienceswe know that

redcoat is distractedby an old herring. This,however, would simply negatetheassumptionthat it is. The assumptionhasbeenmadefor the sake of the argumentwith the intentiontoshow thatsyntacticre-combinationis not sufficient for systematicity.

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atomsdeterminethe causalpropertiesof moleculesbecauseatomsareconstituentsof molecules.A state

1is commonlyregardedto bea con-

stituentof astate3 if andonly if it is necessarilyandgenerallytruethat,if 3 occursat a certainregion of spaceat a certaintime, then

1occurs

at the sameregion at the sametime. Independentlyfrom sciences,onecanevenmake it a hardmetaphysicalpoint: If thecausalpropertiesof astate

øaredeterminedby the causalpropertiesof the states

í�� ç��Æ��ç í�andtheir relationsto eachother, then

í�� ç ��Æ��ç í�areconstituentsof

ø.3

We mayconcludethat thesemanticvaluesof thesyntacticcomponentsof an internalrepresentationdeterminethe semanticvalueof the inter-nal representationjust in casethesyntacticcomponentsareconstituentsof the internalrepresentation.Two remarksshouldbeadded:First, syn-tactic componentsaren’t constituentsper se. The article “le” is a syn-tactic component,but not a constituentof the French“l’homme”. Sec-ond, the requirementthat syntacticcomponentsof internal representa-tions be constituentsof the latter doesnot follow from the constraintof compositionalityalone.Theremay well be compositionallanguages(in thesensedefinedabove) for which syntacticcomponentsaren’t con-stituents.However, therequirementis justifiedby theconstraintof com-positionalitytogetherwith thepremisethat internalrepresentationsowetheir semanticvaluesto thecausalrole they play for therepresentationalsystem.Thispremisehighlightsaparticularityof internal representationanddoesnot generalizeto otherrepresentationalmedialike naturallan-guages.Thewordsandphrasesof English,e.g., owe their semanticval-

3 Thereis anindependentargumentfor this principle,which however requiresKim’s [Kim89]principleof explanatoryexclusion.Theprinciple roughlysaysthatno two independentphe-nomenaeach(completely)determineoneandthe samephenomenon.Given the truism thatthecausalpropertiesof a whole ¼ aredeterminedby thecausalpropertiesof anexhaustivesampleð yàñOò(ò`ò`ñ ð ~ of constituentsof ¼ (plusstructure),it followsthatthecausalpropertiesofthestatesó y ñdò`ò`ò(ñ ópô (plusstructure)determinethecausalpropertiesof ¼ only if ó y ñOò(ò`ò`ñ ó§ôarenot independentfrom ð y ñdò`ò`ò(ñ ð ~ . Sincethereis a limited repertoireof relevantmetaphys-ical dependency relations,viz. identity, reduction,supervenienceandconstituency, onemayconcludethateachó ¹ is either(i) identicalwith, (ii) reducibleto, (iii) supervenienton, (iv) aconstituentsof, (v) or composedof oneor moreof the ð î . In all five casesevery ó ¹ wouldbea constituentsof ¼ . In thefirst case,this is trivial. In thesecondandthe third case,if ó ¹reducesto, or is supervenienton,oneor moreof the ð î , ó ¹ necessarilyco-occurswith the ð îin question.Sincethe latter, asconstituentsof ¼ , necessarilyoccurwhenever andwherever¼ does,also ó ¹ necessarilyoccurswhenever andwherever ¼ doesandis, thus,a constituentof ¼ . In thefourth case,it follows becausetherelationof constituency is transitive.Thefifthcaseholdsbecauseevery compositionof constituentsof a wholeis itself a constituentof thewhole.


uesmainly to the interpretationof Englishspeakers.Theremaywell bea languagewhosetokenshave thesamecausalproperties(sound,loud-ness,etc.)asthoseof English,but differ with respectto their semanticvalues.For internalrepresentation,in contrast,causalpropertiesaredeci-sive with regardto semanticsbecauseinternalrepresentationsrepresentautonomously, i.e., withoutbeinginterpretedby any othersystem.4

Connectionistattemptsto rendersystematicity, we may now diag-nose,fail becausethe mappingbetweenthe language’s syntaxand thenetwork doesnot preserve theconstituentrelationswithin thelanguage.Thus,even if the languageto be syntacticallyimplementedis itself se-manticallycompositionalandeven if every syntacticcomponentin thelanguageis a constituent(asis thecasefor many formal languages),themappingdoesnot transfersemanticcompositionality.5 In Smolensky’sarchitecture,thenetwork counterpartsof, say, brownandcowaren’t con-stituentsof the network counterpartof brown cow. Although the syn-tactic operationthat maps(brown, cow) onto brown cow may satisfythe principle of semanticcompositionality, the network operationthatmaps

óMõöó*ö ÷�ø>ù�ú õ.ç õAó�û�ø>ù õ�õ onto h(browncow)–withõ

beingthehomo-morphismbetweenthe languageandthenetwork– maywell violatese-manticcompositionality. If

õöó*ö ÷�ø>ù�ú õ andõAó�û�ø>ù õ aren’t constituentsofõAó*ö ÷�ø>ù�úXû�ø>ù õ you,e.g., cannotsay:

õAó*ö ÷�ø>ù�úeû�ø>ù õ co-varieswith browncows because

õöó*ö ÷�ø>ù�ú õ co-varieswith brown things andõAó�û�ø>ù õ co-

varieswith cows. If the semanticvaluesof internal representationsareto be determinedby the semanticvaluesof their syntacticcomponents(plus structure)andif semanticevaluationis doneby co-variation,yououghtto beableto saythis.If theconstituentrelations,ontheotherhand,had indeedbeenpreserved, you could have said this.6 For similar rea-son,you will bedeprivedof thepossibilityto explain theinferentialandthe associative propertiesof the complex representationon the basisoftheinferentialandtheassociativepropertiesof theprimitive representa-tions if constituency structuresarenot preserved andcausalpropertiesare,therefore,notdeterminedbottom-upfrom theprimitivesto thecom-plex. Thus,if semanticevaluationcorrespondsto inferentialroleor asso-

4 Thispoint is madein a moreelaboratewayby Dretske [Dre88a].5 Thisholdsevenif themappingis anisomorphismratherthana homomorphism.6 FodorandMcLaughlin [FodMcL90,Fod97]alsoseea connectionbetweentheideasof com-

positionality, co-variationandconstituency.

262 MarkusWerning

ciativenets,theprincipleof compositionalitywill againnotbewarrantedby Smolensky’sarchitecture.

3 Synchrony

Constituency is asynchronicrelation,while causalconnectednessis adi-achronicrelation.Wholeandpartco-exist in time,whereascausesandef-fectssucceedin time.Thereferenceto causalconnectionsandtheflow ofactivationwithin thenetwork will, therefore,notsuffice to establishcon-stituentrelations.Whatwe, in addition,needis anadequatesynchronicrelation.Oscillatorynetworksprovidea framework to definesucharela-tion: therelationof synchrony betweenoscillations.

An elementaryoscillator is realizedby coupling an excitatory unitwith an inhibitory unit usingdelayconnections.An additionalunit al-lows for external input (seeFigure1a).Within the network, oscillatoryelementsarecoupledby eithershort-rangesynchronizingconnectionsorlong-rangedesynchronizingconnections(seeFigure1b).A multitudeofoscillatorscanbe arrangedin featuremodules(e.g., the color module),employing appropriatepatternsof connectivity. Given a certainselec-tivity of the input unit, eachoscillator is designedto indicatea certainproperty(e.g., redness)within the featuredomain.Oscillatorsfor likepropertiesareconnectedsynchronizingly, thosefor unlikepropertiesareconnecteddesynchronizingly. The behavior of oscillatorynetworks hasbeenstudiedin detail elsewhere(cf. [Sch/ Kon

�94]).7 Stimulatedoscil-

latory networks,characteristically, show object-specificpatternsof syn-chronizedanddesynchronizedoscillatorswithin andacrossfeaturemod-ules.Oscillatorsthatrepresentpropertiesof thesameobjectsynchronize,while oscillatorsthatrepresentpropertiesof differentobjectsdesynchro-nize. We observe that for eachrepresentedobject a certainoscillationspreadsthroughthenetworks.Theoscillationpertainsonly to oscillatorsthatrepresentthepropertiesof theobjectin question(seeFigure2).

A greatnumberof neurobiologicalstudieshaveby now corroboratedthe view that cortical neuronsareratherplausiblymodelledby oscilla-tory networks (for a survey cf. [Sin

�Gra

�95,Wer01]).Togetherwith the

computersimulationsof [Sch/ Kon�94], thesestudiessupporttwo hy-

potheses:7 Oscillatorynetworksaredynamicalsystemsin thesensethatthey aredescribedby systemsof

differentialequationsthatinvolve time-dependentfunctions(cf. [vGe98]).


-

+F(xe(t - τei))F(xi(t - τie))

wei

wie xe(t)

xi(t)

ie(t)

-

+

-

+

-

+

-

+

-

+

(a)

(b)

Fig.1. (a) Elementaryoscillator. ü , time; ³W²�ü;´ ,unit activity; ¾�²�³µ´ sigmoidaloutput function;ý , coupling weight; þ , delay time; ÿ��'²�ü;´ , ex-ternal input. Subscripts:� , excitatory unit; ÿ ,inhibitory unit. (b) Oscillatory elementscou-pledby short-rangesynchronizingconnections(dashed)andlong-rangedesynchronizingcon-nections(dotted).

H

orientation-module

V

VH

HV

VH

H

HV

VH

HV

VH

H

HV

VH

HV

VH

H

R

color-module

B

BR

YG

GY

R

BR

RB

GY

YG

B

RB

BR

YG

GY

R

Fig.2. Schemeof a typical responsearousedintheappropriatereceptive field by a greenverti-cal stimulusobject and a red horizontalstim-ulus object. Circles with letters signify oscil-lators/neuronswith the property they indicate(H, V: horizontal, vertical; R, G, B, Y: red,green,blue,yellow). Likeshadingssignify syn-chronousactivity.

Hypothesis1 (Indicati vity). Thereareclustersof neuronswhosefunc-tion it is to show activity only whenan object in the receptive field in-stantiatesa certainproperty. Theseclustersarecalled � -clusterswith �beingthepropertyindicated.

Hypothesis2 (Synchrony).Neuronsof different� -clustershavethefunc-tion to show the sameoscillation (i.e., to be activatedsynchronously)only if thepropertiesindicatedby each� -clusterareinstantiatedby thesameobjectin thereceptivefield.

In otherwords, the samenessof oscillationsindicatesthe samenessofobjects,andan oscillation’s pertainingto a � -clusterindicatesthat theobjectindicatedby theoscillationhastheproperty� .

4 Algebra

Oscillatorynetworksthatimplementthetwo hypothesescanbegivenanabstractalgebraicdescription.To definesuchanalgebra,we have to in-troduceanumberof notions.First,we takebrainstatesto besetsof timeslicesof brains.Eachtimeslicecoversatemporalinterval.An individualbrainis in acertainbrainstateduringtheinterval

ã � 7 � R H ç?« � R H>� just

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in caseits timeslicebelongsto theappropriatesetof timeslicesof brains.Ifø Ú is thesetof all possibletime-slicesof brainscoveringtheinterval

, thenthepower set � ó�ø Ú�õ is thesetof all possiblebrainstatesduring . Second,let �� be thesetof all oscillationsduring

. An oscillation0 ó ��õ is the (quasi-periodic)spikingactivity of a neuronasa functionof

time duringa temporalinterval. Mathematicallyspeaking,theseoscilla-tionsarevectorsin theHilbert space��/ � 7 � R H ç?« � R H>� of in theintervalsquare-integrablefunctions.Thisspacehasthecountablebasis� C� �� ó �§t H �}�� õ � � ñ��

(4.1)

andtheinnerproduct�0 ó ��õ � 9 ó ��õ��ã�� n�� /=�� / 0 ó ��õ*9 ó ��õ"!½�?� (4.2)

Thedegreeof synchrony betweentwo oscillationslies between0 and1andis definedas

� ó 0bç�9 õ¹ã�0 � 9��R$#

�0 � 0 �

�9 � 9�� .8 Third, a setof oscilla-

tionscanbeassignedto each� -clusterof neurons.Sucha set–let’s callit � -set–containsall oscillationsthat theneuronsof the � -clustershowduring

. Let

!&%bethesetof all � -sets.We cannow definetheneuronal

algebra' , which comprisesthreecarriersetsandfour operations:' ã æ ��ç !&% ç(� ó�ø Ú�õ ç ã*)�ç Ùã*)pç ñ )&ç?�+) é � (4.3)

By convention,we use ,*0�- and ,'9.- as symbolsfor elementsof �*� ,capitallettersfor elementsof

!&%, and ,0/1- and ,"23- for elementsof � ó�ø Ú�õ .

It will beclearfrom context whetherwe usethesymbolsasvariablesorconstantsand whetherthey are interpretedin ' . Quotationmarksareomittedwhereappropriate.Eachof the four operationshasbrain statesasvalues.Let usfirst definetheoperationof synchrony:ã*)Í5 �*�546�� î � ó5ø Ú�õ suchthat (4.4)ó 0bç�9 õ87î äµl ñ�ø Ú � l ¤ 0�ã�9 é¤ã:9�; � 0àã�9 � )�

8 Thedegreeof synchrony, sodefined,correspondsto thecosineof theanglebetweenthevec-tors < and = . Alternative measuresfor synchrony (respectively temporalcoherence)areavail-able,in particularfor discretefunctionsof spikingactivity.


Thisoperationmapstwo oscillations0 and 9 ontoabrainstate� 0àãÑ9 � ) ,

which is thesetof thosetemporalbrainsliceswhich make it truethat 0equals9 . Sincetheequalityof oscillationsis a fuzzynotionanddependson the degreeof synchrony betweenthem, it is useful to furthermoredefinea closenessfunction >�5?� ó5ø Ú�õ î �`A ç CØ� . It tells us how closeaconcretebrain,whichfor reasonsof simplicity is heldconstant,comestoan idealbrainstate.We identify theclosenessof a concretebrain to theidealstate,in which theoscillations0 and 9 areabsolutelysynchronous,with thedegreeof synchrony betweentheoscillations0 and 9 asoccur-ring in theconcretebrain: > ó*� 0àã�9 � ) õ�ã �Ñó 0bç�9 õJ�

Theoperationof asynchronyis definedanalogously:

Ùã*)Í5 �*�546�� î � ó5ø Ú�õ suchthat (4.5)ó 0bç�9 õ87î äµl ñ�ø Ú � l ¤ 0 Ùã�9 é¤ã:9�; � 0 Ùã�9 � ) �Thecorrespondingclosenessvalueis setto: > ó'� 0 ÙãÑ9 � õ ) ã C 7 � ó 0bç�9 õ .

If neuronsof a certain � -clustershow a certainoscillation,we cansay that the oscillationpertainsto the � -cluster. Alternatively, we maysaythat theoscillationis elementof the � -setof oscillationsthatrelatesto the � -clusterof neurons.To referto this state,we definetheoperationof pertaining: ñ ) 5 �*�@4 !&%hî � ó5ø Ú�õ suchthat (4.6)ó 0¡ç��õ+7 î äµl ñ´ø Ú � l ¤ 0 ñ �àé®ã�9; � 0 ñ � � ) �How closea concretebrain comesto the state

� 0 ñ � � ) dependsonthe highestdegreeof synchrony betweenthe oscillation 0 andany os-cillation amongthe clusterof neuronsthat contribute to the � -set � :> ó'� 0 ñ � � ) õÑã v�0Wsdä � ó 0bç'shõ � s ñ �àé�� A further, trivially definedoperationis theco-occurrenceof two states:

�+)+51� ó5ø Ú�õA4B� ó5ø Ú�õ î � ó5ø Ú�õ suchthat (4.7)ó /dçC2�õ+7î /EDF2%ã�9; � /e�F2 � ) �In fuzzylogic it is quitecommonto identify thevalueof aconjunctionastheminimumof thevaluesof eitherconjunct: > ó'� /4�G2 � ) õ&ã-v<tO�EäH> ó /¡õ ,c(q)é . Thefour operationsallow usto giveanalgebraicinterpretationofthe schemeshown in Figure2. Assumingthat the dark-shadedneurons

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show theoscillation 0 andthelight-shadedneurons9 , Figure2 expressesthefollowingbrainstate(Theassociativity of co-occurrencederivesfromtheassociativity of setintersection):

� 0 ñ6I ��0 ñ ÿ+��9 ñKJ ��9 ñL� ��0 Ùã�9 � )� (4.8)

Theclosenessvalueof this stateequals1 only if 0 and 9 areorthogonal.

5 Language

We will now definean algebraL of indexical concepts,propertycon-cepts,andpropositions.It will turnour to beisomorphicto ' . Sinceit iscontroversialwhetherconceptsandpropositionsaresemanticor (in thesenseof Fodor’s [Fod75] languageof thought)syntacticentities,I willremainneutralon this issue,for now, andleave thephilosophicalinter-pretationof L for discussionat theendof this paper. I take propositionsto besetsof possibleworlds.Providedthat MÓÚ bethesetof all possibleworlds, the power set � ó M,Ú�õ is the setof all propositions.9 Let

YONQPbe

a setof indexical conceptslike this and that, which potentiallyrefer toobjects.Let RTS bea setof propertyconceptslike rednessandverticalitywherepropertiesareconceived of merelyassetsof objects.Like ' , Lcomprisesthreecarriersetsandfour operations:LVUXWZY NQP\[ RTS [(]_^ Ma`cb ç U�d [�eU�d [(f d [Zg d é3h (5.1)

In the context of L , we usethe linguistic items“ i ” and“ j ” to expressindexical concepts,capitallettersto expresspropertyconcepts,and kml1nand koHn to expresspropositions.Alternatively, onemaywell useEnglishwordsandphrasesto expressentitiesof p . Noticethatthecomplex con-cept that is expressedby the sentencek�iqUrj.n doesnot meanthat theconceptsi and j areidentical.It ratherexpressesapropositionaboutthe

9 Theassumptionthattheclassof possibleworldsis a setmayimposesomerestrictionson theuniverse.It is debatable,furthermore,whetherit makessenseto saythatevery setof possibleworlds is a proposition.An analogousobjectionmay apply to the view that identifieseverysetof time slicesof brainswith a brainstate.Notice,however, thatonly thesetsof possibleworlds(andtheir intersections)–andthesetsof time slicesof brains(andtheir intersections),respectively– which are in the rangesof the first threealgebraicoperationsmatterfor ourconsiderations,anyway. For anappropriaterestrictionof thealgebrasseep. 267.


identity of the objectsreferredto by the conceptsi and j .10 The firstoperationof p is definedasfollows:

Sameness:U dts YOuQvxw�YOuQvzy ]{^}| `~b suchthat (5.2)^ i [ j�b8�y �(� f6| `��i�U�j(�5U:��T��i�U�j.� d hThesamenessoperationmapstwo conceptsi and j ontoa proposition.Thelatteris thesetof thosepossibleworldsthatmake thecomplex con-ceptwhich is expressedby the sentencek�i�U�j.n true. The remainingoperationsaredefinedanalogously:

Dif ference:eU�d s YOuQvxwGYOuQv�y ]{^}| `~b suchthat (5.3)^ i [ j�b8�y �(� f6| `��i eU�j(�5U:��T��i eU�j.� d h

Copula:f dts YOuQvxwK�T��y ]_^�| `cb suchthat (5.4)^ i [.� b+�y �(� f6| `��xi fG� ��U:��i fK� � d h

Conjunction:g d�s ]_^�| `~b&w ]{^}| `~b+y ]{^}| `~b suchthat (5.5)^ l [ ocb+�y�lE�xo5U:�� l g o�� d h

The operationsenableus to denotethe propositionwhich the Englishsentence“This is a greenvertical and that is a red horizontalobject”expresses(We assumethat “this” and“that” expresstheconceptsi andj , and “green”, “red”, “vertical”, “horizontal” expressthe concepts� ,�

, � , and � ; the associativity of the conjunctiong d derivesfrom the

associativity of setintersection.):11��i f � g i f � g j f � g j f � g i eU�j.� d h (5.6)

6 Isomorphism

To establishthe isomorphism,we, first, reducethe third carrier set ofeachalgebrato that oneof its subsetsthat is the closureof the united

10 A crucialdifferencebetweenthenotionsof expressingandreferringshouldnotbeoverlookedhere.In the English sentence“this is the sameas that”, “this” and “that” do not refer toconcepts,but to objects.They, nevertheless,expresstheconceptsthis andthat.

11 For reasonsof simplicity, we have assumed,furthermore,that naturallanguagesobey a set-theoreticrather than a predicative logic. Thus, “this is red” is analyzedas “this” ¡@¢£ “red” ratherthan, in a predicative way, as “red(this)”. Alternatively, one may changedefinition5.4by substituting“ ¤_¥§¦©¨ ” for “ ¦T¢&¤ ”.

268 MarkusWerning

rangesof thefirst threeoperationsunderthe forth operation(Thesore-ducedalgebrasaremarkedbysuperscriptk � n ). Let thesoattainedreduc-tion of

]_^0ª `cb betheset «{¬"i�¬ , which,hence,comprisesonly brainstatesconstructiblein the neuronalalgebra;and let the reductionof

]{^}| `cbbe the set 5`~®.l , which, thus,is restrictedto propositionsconstructiblein theconceptualalgebra.Secondly, we will treatbothalgebrasmoduloequivalence: ¯�°±O² U W´³�µ�¶ [Z·&¸0[ ��«{¬"i�¬O� ²{¹ U*º [�eU*º [©f º [Zg º¼» and p¼°±O² UW YOuQv [ �+� [ ��5`3®ZlQ� ²½¹ U d [�eU d [(f d [.g d » . ¯ °±O² is isomorphicto p °±O² , providedthat(i) thereareasmany oscillationsin ¯ asthereareindexical conceptsin p (i.e., � ³�µ¶ �{U � YOu¾v{� ) and(ii) each ¿ -cluster, respectively, eachre-latedsetof oscillationsin ¯ is assignedto exactly onepropertyconceptof p (i.e., � ·&¸ �3UÀ�Á�+�H� ).

In previoussectionswe arguedthatanarchitecturemaynot becom-positionaleven if it is syntacticallyhomomorphic(or even isomorphic)to a compositionallanguage.To preserve semanticcompositionality, theisomorphismbetween p °±O² and ¯ °±O² must, in addition, preserve con-stituentstructure:If a primitive conceptis a constituentof a complexconcept,the isomorphiccounterpartof the primitive conceptmustbe aconstituentof theisomorphiccounterpartof thecomplex concept.Thisiswarranted:Theoscillation i is a constituentof thebrainstates��iÂUajC� º ,��i eUÃj.�Äº and ��i fÅ� �Æº becauseit occurswhenever andwherever thebrainstatesoccur.12 Likewise,theclusterof neuronswhich contributetothe ¿ -set

�areconstituentsof the state ��i fV� � º . The fact that primi-

tive conceptsareconstituentsof complex conceptsis, thus,reflectedintheneuronalalgebra.Figure2 illustratesthattheisomorphismpreservesconstituentrelationsfor all operations:Thecomplex stateshowncanonlyoccurif, indeed,certainburstsof activity andcertainclustersof neuronsoccur. Wemayinfer thatoscillatorynetworksarenotonly isomorphictoacompositionallanguage,but maysubserveawayof representationthatis semanticallycompositionalin its own right (For further interpretationseetheconcludingsection).

Having onceshown the isomorphismand the congruencewith re-spectto constituentstructure,we canextendthe rathersimplealgebras¯ and p in parallel,i.e., in a mannerthatperpetuatesthe isomorphismandthecongruenceof constituentstructure.This way, predictionsabout

12 Noticethat“=” and ¡ÈÇÉ £denoterelationsandthatrelationsobtainjust in casetherelataare

tokened:¦ É�Ê$Ë ¥�Ì�Í�¨}¥§Í É�Ê ¨ and ¦ÎÇÉ�Ê1Ë ¥�ÌZÍ�¨}¥§ÍÏÇÉ�Ê ¨ .


the realizationof structurallymoresophisticatedrepresentationsby os-cillatory networks are generated.I will sketch an examplethat hastodo with the representationof relationslike in. On the conceptuallevel,this, in addition to p , requiresconceptsfor pairs, for relations, and ahigher-order copula. If we take conceptsfor relationsasprimitive andaselementsof the set ÐÎÑ ¸ , we candefinethe remainingoperations.WeadoptKuratowski’s [KurMosÒ 76] convention,accordingto which pairsareasymmetricsetsof secondorder:

Pairing: ÓÔÄÔÖÕ dzs YOuQv�wGYOuQv�y ]{^"]{^ YOu¾v×b�b suchthat (6.1)^ i [ j´bT�y ��Hi [ j(� [ �Hj(��*U:��Ó�i [ j´Õ hIf �{Ø~ÙÚ� is the rangeof the pairing operationandif relationsaresetsofpairswith ÐÎÑ ¸ÈÛÜ]{^ �½Ø~ÙÄ��b , thesecond-ordercopulacomesto:

SecondCopula:f dÒ s �½Ø~ÙÄ�ÎwGÐÎÑ ¸ y ]_^�| `cb suchthat (6.2)^mÝÞ[ � bT�y �(� fK| `ß�� Ý�f � �5U:��8� Ý�f Ò � � d h

The additionaloperationsallow us to denotethe propositionexpressedby thesentence“This greenobjectis in thatredobject”:��i f � g j f � g Ó}i [ j�Õ f Ò8à á×�Äd h (6.3)

To capturerelationalrepresentationsby oscillatorynetworks,we simplyhave to proceedin a parallelway with extending ¯ . We definea pair-ing of oscillations.Let therangeof this operationbetheset ³ �{Ø3ÙÄ� . We,furthermore,postulaterelationalmodulesaselementsin theset Ð¼Ñ ¸ãâ sothat Ð¼Ñ ¸ãâÃÛÜ]_^ ³ �½Ø~ÙÄ��b :ÓÔÚÔ§Õ º s ³�µ�¶ w ³�µ�¶ y ]{^"]{^ ³�µ�¶ b�b suchthat (6.4)^ i [ j´bT�y ��Hi [ j(� [ �Hj(��*U:��Ó�i [ j´Õ ¹f ºÒ s ³ �{Ø~ÙÚ�?wKÐÎÑ ¸ãâ y ]_^0ª `~b suchthat (6.5)^mÝÞ[ � b+�y ��ä fKª `��HäK� Ý�f � �*U:��8� Ý�f Ò � � º h

Thisextensionpredictsthat,in orderto representrelations,someneu-ronsfire with a setof two oscillations,ratherthanwith a singleoscilla-tion. Thiskind of duplex activity canbeachievedeitherby superpositionor by modulationof two oscillations.Figure3 providesanillustration.13

13 Thetopographicalarrangementin thein-moduledoesnothave any representationalfunction.Thesurroundingneuronswith simplex activity may, however, helpdrive theembeddedneu-ronsto show duplex activity (cf. [May01]).

270 MarkusWerning

in-module

R

color-module

B

BR

YG

GY

R

BR

RB

GY

YG

B

RB

BR

YG

GY

R

Fig.3. Predictedneuronalrepresentationof relations.The state å ¦Â¢Fæßç Ê ¢�èßçé ¦�ê ÊOë ¢~ìÎí.î�ïñð is shown. The oscillation ¦ of the G-neurons(dark-shading)occursin the in-moduleonly assuperposedwith, or modulatedby, theoscillation Ê of theR-neurons(light-shading),thus forming the duplex oscillation òó¦�ê Êóô (hybrid shading).Since Ê also occursas simplex on the in-module,the situationon the in-module isrenderedby å ò�òó¦�ê ÊOô ê�ò Êóô�ô ¢~ì1í.î�ïÖð . This is equivalentto å é ¦�ê ÊOë ¢~ì1í.î�ïñð .

7 Conclusion

Any comprehensivephilosophicalinterpretationof our resultswould,byfar, go beyondthescopeof this paper. Let me,still, commentbriefly onthenatureof p and ¯ . Are they syntacticor semanticalgebras?First ofall, p hasall thepropertiesthataretypicalfor thesemanticsof alanguagewhosesyntaxallowsusto build simpleset-theoretic(or predicative)sen-tences.14 It can easily be shown that p is the homomorphicimageofsucha syntax.Semanticcompositionalityis, hence,warranted.Since ¯and p areisomorphic(in their reducedformsandmoduloequivalence),this homomorphismtransfersto ¯ .

We maynow interpret p in anexternalisticway, wherepropositionsare treatedasmind-independententities.Sincepropositionshave beendefinedas setsof possibleworlds, this interpretationcorrespondsto arealisticattitudetowardpossibleworlds in thesenseof Lewis [Lew86].Sincethebrainstatesin ¯ wouldbetheisomorphiccounterpartsof theseexternalisticpropositions,we might say that the brain producessimu-lations of them.The closenessvalue õ ^ � lQ� º b could be interpretedas adegreeof resemblancebetweenthe subject’s brain and the proposition� lQ� d .

14 For theadaptionto predicative languagesseefootnote11. Thesimpleset-theoreticlanguageI have in mind is limited to syntacticoperationsregardingtheconnectives“=”, “ ÇÉ ”, “ ¢ ” and“ ç ”, and,with respectto theextendedalgebra,thesymbolsfor pairing.


Alternatively, we could interpret p internalistically, whereproposi-tions are conceived of as the mentalisticmeaningsof sentences.Thismight somehow correspondsto theview thatpossibleworldsaremind-dependentobjects(cf. [Rosö 90,Kri ÷ 72]). The entitiesof the isomorphicneuronalalgebramay thenserve asthe“cortical meanings”of a simplesemanticallycompositional,set-theoretic(or predicative) language.Theclosenessvaluecouldbeinterpretedasaputativedistancebetweenabe-liever anda proposition.If õ ^ � lQ� º b equals1 for thebrainof thesubject,wecouldsaythatthesubjectfully believesor graspstheproposition� lQ� d .

Thirdly, aninterpretationof p asanalgebraof mentalsymbolsis stillnot precluded.In this case,propositionsareregardedas truth-valuablecombinationsof mentalsymbols.Thismightsomehow correspondto theCarnapian[Car÷ 47] view that possibleworlds arenothingbut statede-scriptions.The only concessionwe have to make is that thosementalsymbolsarenolongercombinedby concatenation.For, thecommutationof mentalsymbols–commutationis the only way to produceequiva-lenciesin p – must leadto identicalpropositionsin order to guaranteethe isomorphism.If ��iqUøjC� d and ��j�Uùic� d werenot identicalproposi-tions, this would conflict with the fact that ��iBUrjC� º and ��jzUùic� º areidenticalbrainstates.Theclosenessvalue õ ^ � lQ� º b maybeinterpretedasthedegreeto which thesubject’sbrainrealizesthetruth-valuablementalsymbol � l¾� d .

Finally, let me compareoscillatorynetworks with Turing-styleandconnectionistarchitectures.Cognitivemodelscanbedistinguishedalongthreefeatures:(i) Trees:Thereareoperationsfrom orderedsetsof argu-ment representationsonto target representations.(ii) Constituency:Forevery tree,its argumentrepresentationsareconstituentsof its targetrep-resentation.(iii) Order: For every target representation,thereis a deter-minateorderamongits constituents.

In standardlanguages,there are trees,words are constituentsofphrases,andthewordsfollow adeterminatewordorder. Wecannow askwhich of the threeprinciplesa given cognitive modelrealizes.Turing-style computersrealizeall threebecausethey build complex represen-tationsfrom primitive onesby concatenation.Integratedconnectionist/symbolicarchitecturesonly realizetrees.Oscillatorynetworks,however,realizebothtreesandtheprincipleof constituency, but not theprincipleof order.

272 MarkusWerning

Oscillatorynetworkslie in somesensein betweenclassicalandcon-nectionistarchitectures.They resembleconnectionistnetworks in manyrespects:They mayserve asassociative,contentaddressablememories.They processinformation in parallel,areable to learn from examples,degradegracefully, etc.Still, oscillatorynetworksarestrongerthantra-ditionalconnectionistnetworksbecausein oscillatorynetworksprimitiverepresentationsareconstituentsof complex representations.The primi-tive representationsdeterminethecausalpropertiesof thecomplex rep-resentationsand,thereby, determinetheir semanticproperties.Oscilla-tory networksunite thevirtuesandavoid thevicesof classicalandcon-nectionistnetworks.They maysubservesemanticallycompositionalandsystematicrepresentationalcapacities.

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