Non-Computability of Consciousness

Transcription

1 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 382 Original Article Non-Computabilit of Concioune Daegene Song Abtract Wit te great ucce in imulating man intelligent beavior uing computing device tere a been an ongoing debate weter all conciou activitie are computational procee. In ti paper te anwer to ti quetion i own to be no. A certain penomenon of concioune i demontrated to be full repreented a a computational proce uing a quantum computer. Baed on te computabilit criterion dicued wit Turing macine te model contructed i own to necearil involve a noncomputable element. Te concept tat ti i olel a quantum effect and doe not work for a claical cae i alo dicued. Ke Word: computabilit concioune Turing macine NeuroQuantolog 2007; 4: Introduction Reearc in te field of artificial intelligence wic attempt to imitate and imulate intelligent activitie uing a macine a bloomed along wit te development of information tecnolog (Ruell 2002). Becaue te tud of artificial intelligence a provided man inigt into intelligent beavior uc a pattern recognition deciion teor etc. tere i a quetion weter concioune or elf-awarene could emerge out of a computational tem a view termed a trong artificial intelligence. Ti quetion can be repraed and tated a follow: Are all Correponding autor: Daegene Song Addre: Scool of Computational Science Korea Intitute for Advanced Stud Seoul Korea dong@kia.re.kr conciou activitie computational procee? In ti paper te anwer to ti quetion i own to be no. In order to examine te computabilit of a pical penomenon te penomenon ould firt be repreented a a computational model; ubequentl te computabilit of ti particular model can be examined. Te pical penomenon can ten be claimed to be computable or not baed on ti examination. A imilar approac will be taken in order to examine te computabilit of concioune. Becaue concioune i a penomenon experienced b an oberver repreentation of concioune a a computational proce will be attempted and it computabilit will be examined. Altoug traditional approace for

2 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 383 tuding concioune ave included neurocience (Koc 2004) or neural network modeling (Tononi and Edelman 998; arve 994) it i demontrated tat a quantum tem to be preented below it necearil involve a conciou a oppoed to a pical activit of an oberver oberving te unitar dnamic of a quantum tate. Baed on ti obervation a particular quantum computer can be built uc tat it ield a computational model involving concioune. Uing logic imilar to tat in Turing' aling problem it can be own tat ti computational model necearil run into a contradiction. A a reult ti effectivel provide a counter-example to te aumption tat all conciou activitie are computational procee. In ti paper it i not claimed tat all conciou activitie can be contructed uing a quantum computer nor tat te are quantum mecanical. Intead it will be argued tat te quantum tem to be preented necearil involve a certain conciou activit and tat quantum teor provide a full decription of ti particular conciou activit. Ti argument will be ued to build a quantum computing macine uc tat it uffice to provide a counter-example. A ingle counter-example i ufficient to prove te aumption i incorrect. 2. Computabilit and Turing macine. In order to dicu computabilit of concioune let u firt conider wat it mean to be computable. Ti can be done uing te notion of Turing macine. A Turing macine denoted a i a teoretical model of dnamic computing tem configured wit an internal tate a tape containing a mbol in eac cell and identification of te poition of te ead on te tape (Fig. ). Te time evolution of te i decribed b (Ia) (I a d) were I i te internal tate a a mbol in te cell of te tape and d= ±. Terefore at te it cell te ead read te mbol a. Ten given te current internal tate I wic provide an intruction te new mbol a =I(a) i written and te internal tate i updated to I and te ead move eiter one cell to te rigt (d=) or one cell to te left (d=-) i.e. to te (id)t cell. Fig. Turing macine. A Turing macine i an abtract model of a computing tem coniting of internal tate a tape containing mbol in eac cell and a ead tat read and write te mbol. Evolution in time of te Turing macine i decribed b (Ia) (I a d) were I i te internal tate and a i a mbol written on te tape. At te it cell i.e. te ead' poition te ead read te mbol a and wit te intruction I it write a new mbol a and move eiter one cell to te left (d=-) or to te rigt (d=) wit an updated internal tate I. Te initiation and termination of computation are indicated b internal tate 0 and repectivel. Among te function of te internal tate i te indication of initiation and termination of computation. Initiall ti particular tate i et to 0 indicating te initiation of te computation. After te computation i completed te tate i et to and te macine ceae it activit. Te output of te computation correpond to te mbol in te cell were te ead i located wen te macine alt. For a given input i te run

3 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 384 following te time evolution decribed above and eiter (A) produce an outcome f=(i) and alt wit te internal tate et to or (B) loop forever and te internal tate never reace. A tem i called computable if it correpond to a uc tat it follow eiter (A) or (B) for a given input i and i called noncomputable oterwie. Te iue of computabilit i conidered in te following etup: uppoe T i defined to ave te following two propertie:. Computable: T(i) wen i T 2. Non-computable: T(i) wen i = T Tat i T correpond to a tat follow eiter (A) or (B) except wen te input i T i.e. te decription of T itelf (Fig. 2). In te following approac te manner in wic te computational model involving concioune ma be defined troug T will be own uing a quantum computing macine uc tat te two condition regarding computabilit are atified i.e. necearil containing noncomputabilit. 3. Te alting Problem Before proceeding wit te dicuion of concioune it i intructive to review Turing' alting problem (Turing 936) and to examine it ue of a propert imilar to tat een in T uc tat te problem wa own to be non-computable. Te ituation for te alting problem i a follow: for ome an outcome indicated b te alt internal tate 0 i computed wile for ome oter wit a given input te computation loop forever indicated b a contant internal tate 0. Turing' alting problem ak if tere i a tat can ditingui between te two tpe given an arbitrar and an input. Suppoe tere i uc a tat perform a calculation given te decription of and i uc tat it i able to determine if alt or not. Ti aumption ten make it poible to contruct a particular uc tat te macine doe not alt for an input if and onl if () alt. owever a contradiction follow for wen te input i itelf becaue ( ) doe not alt if and onl if Fig 2. Computabilit and Non-computabilit. T i defined to ave te propert correponding to a Turing macine tat eiter alt or not unle it i given an input of T itelf. Te alting problem can be defined in aociation wit T uc tat it necearil i non-computable. Similarl quantum teor allow concioune to be repreented a a computational proce in term of T uc tat it would necearil conit of a non-computable element wen te input i T itelf. ( ) alt. Terefore b identifing te aociated wit te potetical tat could decide if an arbitrar would alt on a given input it i poible to ow tat te contain an element tat neiter alt after completing te computation nor loop forever a ould a valid. Te contructed a te ame propert a T in Fig. 2 i.e. it i

4 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 385 computable except wen te input i itelf. 4. Conciou Activit in Quantum Stem In order to repreent a penomenon of concioune a a computational model te manner in wic a conciou activit i involved in a quantum tem i firt dicued. Ti will be conducted uing te notation of a qubit a two-level quantum tem. A qubit in a denit matrix form i written a r ψ ψ = ( µ ˆ σ ) / 2 were µ ˆ = ( µ x µ µ z ) = (inθ coφinθ inφcoθ ) r and σ = σ σ σ ) wit σ x σ ( x z = 0 0 = i 0 i 0 σ = 0 0. z and Terefore a qubit ψ ψ can be repreented a a unit vector µ ˆ = ( µ µ ) pointing in x µ z ( θ φ) of a pere wit 0 θ π0 φ 2π. In quantum teor tere i anoter important variable called an obervable. For a ingle qubit an obervable can alo be written a a unit vector ν ˆ ( ν ν ν ) = (inϑ coϕinϑ inϕcoϕ) = x z pointing ( ϑ ϕ) direction in a pere. Terefore if one i to make a meaurement in ( ϑ ϕ) direction te obervable would be σ r. Repreenting a qubit and an obervable a unit vector in te Bloc pere will make teir viualization eaier wic will be elpful in te following dicuion. Let u conider one particular penomenon denoted a P and decribed a follow: an oberver oberve te unitar evolution of a qubit wit repect to te obervable. Te oberver i oberving te evolution of indirectl and a meaurement can be followed in order to confirm te evolution. Wen a meaurement on wit te obervable i made it ield a real eigenvalue tat can be directl oberved b te oberver. Before dicuing te decription of te penomenon P uing te dnamic of quantum teor it i necear to illutrate w te penomenon P necearil involve a conciou activit of te oberver. An obervable erve a a coordinate or a reference frame wen te meaurement i made on a given tate vector (Pere 99). Ti concept i eaier to viualize wit two unit vector and. Te unit vector repreenting an obervable i.e. i erving te role of a coordinate for te unit vector repreenting a qubit. Becaue te meaurement i performed b an oberver te obervable i conidered to be a coordinate or a reference frame of te oberver for a given qubit. owever in quantum teor obervable being a reference frame of te oberver are fundamentall different from reference frame in claical pic. In quantum teor te tate vector ave repreentation in and evolve in te ilbert pace a complex vector pace. Ti decription wa invented in order to correctl predict te outcome of meaurement performed on a tate

5 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 386 vector wic ield a real eigenvalue outcome. Not onl do tate vector reide and evolve in te ilbert pace o do te obervable. Becaue te obervable correpond to te reference frame of te oberver and te exit in te complex ilbert pace it mut be concluded tat unlike reference frame in claical pic quantum obervable correpond to an oberver' reference frame in tougt. Tat i an obervable ould be conidered to repreent te conciou tatu of an oberver wile oberving a given tate vector. Ti argument explain w te penomenon P necearil involve a conciou activit. Te qubit in P can be an 2-level quantum tem for example a pin /2- particle or an quantum tem in 2-level etc. owever it i not necear to pecif all propertie of te pical tem oter tan becaue i a pure tate and i dientangled from te tate tat repreent oter propertie of tat quantum tem. Terefore a far a te penomenon P i concerned provide a full decription of te pical tem. Te ame logic applie to a well. Te vector i not entangled wit vector repreenting oter obervable. Terefore mut provide a full decription of te conciou tatu of te oberver in penomenon P. Tat i imilarl to te cae wit it i not necear to be concerned wit oter conciou activitie of te oberver becaue i dientangled from tem. Terefore P not onl necearil involve a conciou activit of te oberver give a full decription of te conciou activit a far a te penomenon P i concerned. Quantum teor provide two approace in decribing te natural penomenon P. Given and te firt i b appling a unitar operation to te qubit wit µ ˆ = Uµ ˆU were a meaurement would ield te expectation value of ν ˆ µ ˆ. Te econd i b appling a unitar operation to te obervable a = U U and a meaurement would ield µ ˆ. Tat i quantum teor init tat in order to ave an oberver oberve te unitar tranformation of wit repect to eiter a unitar tranformation i applied to te qubit i.e. te firt approac or te oberver' reference frame i canged i.e. te econd approac. Te firt approac i called te Scrödinger picture and te econd correpond to te eienberg picture. In te econd approac it wa te obervable tat went troug a unitar tranformation wic ould decribe te ame penomenon P a te firt approac. Becaue te evolution of obervable troug unitar tranformation i performed in te ilbert pace and te obervable i te oberver' conciou tatu in P an obervable tat i being canged mut correpond to a conciou activit of an oberver. owever wile te oberver' conciou tatu i being canged te oberver i not oberving te obervable but te tate vector. Terefore ti approac alo ield te decription of te natural penomenon P jut a in te firt approac.

6 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 387 (ee Fig. ). Te ead in te claical ma 5. Conciou Activit in Quantum Computing Proce So far it a been argued tat te penomenon P necearil involve an oberver' conciou activit and quantum teor provide a full decription of te conciou tatu of te oberver regarding P. Baed on tee obervation a quantum computational model i to be contructed uc tat it repreent a penomenon involving a conciou activit and it computabilit will be examined. In particular T will be defined in term of ti computational model and te computabilit for a given input i will be examined. In te next ection it will be argued tat wen te input i T itelf it repreent concioune and will be proven to be non-computable imilarl to te alting problem. Let u review baic element of quantum computation b following te dicuion in (Benioff 980; Deutc 985). Te particular cla of quantum computer to be conidered i aumed to perform a computation on an input of a ingle qubit i.e. a unit vector in te Bloc pere in wic te ubcript i placed in order to ditingui it from te alt qubit to be defined ortl. In ti particular cla te computation i aumed to be conducted troug a unitar proce on a given ingle input qubit a rotation about te -axi b δ i.e. U co( δ /2) 0 0 in( δ /2) 0 in( δ /2) 0 co( δ /2). Among te component of te claical te ead exit wic read eac cell on te tape correpond to te obervable in te quantum computer. A demontrated earlier for a ingle qubit te obervable can alo be written a a unit vector in te Bloc pere wic will be denoted a. A uggeted in (Deutc 985) in addition to te tem input qubit an additional qubit i placed wic indicate if te computation on te tem qubit a uccefull ended b 0 after a valid computation on te ingle tem qubit wic remain 0 oterwie. Ti i equivalent to te claical in wic it internal tate indicate if te macine completed it computation b 0. Te alt qubit i et to point at te z- direction i.e. = (00). Te correponding obervable = (00) alo et to point at te z-direction initiall. Terefore te quantum computer contructed for a given input cloed tem coniting of i a and. Becaue tere i freedom to et te obervable it can be ued to identif one particular quantum computer wic work on a given input coice of. Among te infinitel man aume tat one particular quantum computer exit wit te obervable = (00). Becaue te unitar evolution will be U onl te initial obervable full caracterize ti particular quantum computer and it will be defined a T. Te quantum model contructed operate on a ingle qubit and onl a ingle

7 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 388 operation i.e. U i conidered. Terefore tere i no need to pecif an internal tate tat ield an intruction becaue tere i onl one operation. Te onl internal tate needed i te indication of initiation and termination of te computation tat i repreented wit te alt qubit. Moreover indication of te poition of te ead i unnecear becaue tere i onl one qubit wic correpond to a tape wit a ingle cell. Terefore te quantum computer contructed correpond to a ver imple cae of a quantum mecanical. One particular penomenon denoted a P2 i conidered a follow: an oberver oberve a rotation of te input about te -axi b δ wit repect to. A in P te oberver i oberving te rotation indirectl and a meaurement on wit te obervable can be followed to confirm te evolution. Note tat P2 i almot identical to te penomenon P except te unitar operation i pecified a U. Terefore imilarl to te cae wit P te penomenon P2 necearil involve a conciou activit of te oberver repreented a. Moreover a dicued wit te intance of P provide a full decription of te conciou tatu of te oberver in reference to P2. In te following it will be etablied tat te quantum computer contructed T repreent P2 a a computational model and i computable terefore indicating tat te penomenon P2 i computable. A dicued earlier te quantum teor provide two approace for te evolution in time of a quantum tem. Terefore becaue T te quantum computer contructed i a quantum tem it ould alo evolve in bot approace. Te evolution in time of T wit an initial input tate i = = (00) will be examined. Te firt approac i.e. te Scrödinger picture i conidered a follow: te unitar operation U tranform te input a µ ˆ U µ ˆ U and te alt qubit (00) alt b tranforming into. In te econd approac i.e. te eienberg picture it i te obervable tat evolve. Terefore U tranform te vector repreenting te obervable into U U and te obervable for te alt qubit (00) i tranformed into. Terefore in te econd approac te oberver' conciou tatu i being canged wile te oberver oberve. Ti ould ield te ame obervation a te firt approac. It i noted tat te expectation value of ( U U ) µ ˆ for te econd approac i equal to te expectation value in te firt approac ˆ ( U µ ˆ U ν ). Terefore bot te firt and te econd computational procee ultimatel decribe te penomenon P2 b correctl producing an outcome decribed in P2. Initiall it wa dicued tat a tem i tated to be computable wen it atifie one of two criteria i.e. eiter (A) it alt after

8 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 389 completion of a valid computation or (B) it loop forever witout alting. T wa own to ield te decription of P2 wit a given input b following bot picture in quantum teor i.e. bot approace ielded te outcome b wic b δ wit repect to rotated about te -axi and alted. Terefore te penomenon P2 can be claimed to be computable becaue it computational repreentation T wit te input wa own to be computable b atifing te criterion (A). 6. Counter-Example to te Aumption. In cae of te eienberg picture decription of P2 a well a of P it wa dicued tat te oberver i in te conciou tatu undergoing cange and oberve. Ti wa own to ield te penomenon of P2 i.e. te oberver oberving te rotation of. A ligtl different cae can be conidered. Wile te oberver i in te conciou tatu tat i being canged te oberver oberve rater tan. Ti i a peculiar apect of concioune--oberving one' own conciou tatu--tat i not oberved in oter meaurement experience for example in claical dnamic. Ti penomenon can be tated a follow and denoted a P3: an oberver oberve a rotation of te input about te -axi b δ wit repect to. Terefore in P3 wic decribe concioune of te oberver i erving te role of a tate vector becaue it i being oberved and an obervable becaue it i erving a te reference frame of te oberver. Unlike te cae of P and P2 te meaurement confirmation i not needed for P3. Wile te conciou tatu i evolving te oberver i not oberving but. No meaurement i needed in order to confirm te evolution of becaue te oberver i alread experiencing it a concioune. In te previou ection it wa demontrated tat T wit an input provide a computational model for decribing te penomenon of P2 and wa own to be computable. Becaue P3 i exactl te ame a P2 except te input a canged to vector from it follow tat T wit an input mut correpond to a computational model repreenting te penomenon P3 (ee Table ). Te obervable =(00) full caracterize T. Terefore T wit an input can alo be tated a T wit an input of te decription of T or impl a T wit an input T. In te following te computabilit of T for a given input of T wic repreent te penomenon P3 a a computational model i to be examined. A etablied previoul quantum teor provide two approace to te evolution in time of T for te input becaue it i a quantum tem were correpond to bot a tate and an obervable. In te firt approac it i te input tem tat evolve. Since te input i te evolution i

9 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 390 a follow ν ˆ = (in δ 0coδ ) wile te alt qubit i tranformed a µ ˆ ˆ. µ Quantum teor provide a econd approac were te ame vector being an obervable i tranformed a ν = ( in δ 0coδ ) wile te obervable for te alt qubit i tranformed a ν ν ˆ. It i noted tat ν ˆ ˆ unle δ = kπ were k=02... Table. Analog between te computational model T and penomena P2 and P3. If te penomenon P2 can be repreented a a computational model T wit an input ten T wit an input ould correpond to a computational model for te penomenon P3. Computational Model Penomenon (T i= $ µ ) P2: Oberver oberve rotation of $ µ wit repect to v $ (T i= v $ ) P3: Oberver oberve te rotation of v $ wit repect to v $ Let u now dicu te computabilit of T(i) were i = T. In order for T(T) to be computable it a to follow eiter te computabilit criterion (A) or (B). Since T alted on bot approace i.e. µ ˆ ˆ µ wit repect to in bot picture it mut follow (A) rater tan (B) in order to be computable. In order to atif (A) te alt qubit of T mut ave alted accompanied b a valid computation i.e. bot approace ould ield te ame outcome predicted in P3. owever te two approace ielded two generall different outcome for a ingle input vector. Te econd approac did not ield te outcome decribed in te penomenon P3 becaue te vector i rotated b δ. Terefore ti reult in a contradiction becaue T alted on te invalid computation. Te contradiction i noted to reult from a peculiar propert of concioune in wic i erving a a reference frame of te oberver and a a tem to be oberved. Te aumption tate tat all conciou activitie are computational procee. Becaue T(i) wit i = being a computational model of te penomenon P3 i a cloed and independent tem ti mut atif te aumption. owever it wa own tat T wit a given input i not computable. Tat i a particular conciou activit of an oberver oberving te cange of an obervable a decribed in P3 i not computable. Terefore ti lead to a concluion tat te aumption i incorrect becaue it uffice to ave a ingle counter-example to invalidate te aumption. Perap b conidering a larger tem tat include te qubit te contradiction ma be removed and ma ield te reult tat concioune i alwa a computational proce. Ti i commonl een in termodnamic in wic a ubtem violate te econd law but ti violation i alwa removed wen te total tem i conidered. owever ti kind of argument would not work becaue te evolution conidered in P2 and P3 are for pure tate. An attacment of ancilla to T and teir interaction wit te tem qubit

10 NeuroQuantolog December 2007 Vol 5 Iue 4 Page Song D. Non-computabilit of concioune 39 would caue entanglement and ti will not properl repreent te pical penomena P2 and P3. 6. Dicuion Te above argument applie onl a a quantum effect. Te claical cannot define concioune uing te ame tecnique. A dicued a reference frame of quantum meaurement wa repreented in complex ilbert pace wic led to te concluion tat it mut correpond to te oberver' conciou tatu. A claical meaurement ield an outcome in term of te difference between te object and te reference frame of an oberver and unlike concioune te oberver cannot oberve te dnamic of it own reference frame alone. Terefore te ame argument ued wit te quantum computing macine involving conciou activitie cannot be ued in a claical cae. In (Penroe 989) Penroe dicued tat a non-computable apect in concioune ma exit at te fundamental level a decribed in Gödel' incompletene teorem. Including Turing' alting problem tere ave been a number of matematical example owing undecidabilit in Gödel' teorem. In ti paper it wa demontrated tat a in Penroe' uggetion concioune i a pical i.e. rater tan matematical example of Gödel- tpe proof. Reference Benioff PA. Te computer a a pical tem: A microcopic quantum mecanical amiltonian model of computer a repreented b Turing macine. J Stat P 980;22: Deutc D. Quantum-teor te Curc-Turing principle and te univeral quantum computer. Proc R Soc London A 985;400:97-7. arve RL. Neural Network Principle Prentice-all Englewood Cliff NJ 994. Koc C. Te Quet for Concioune: A Neurobiological Approac Robert & Compan Publier Pere A. Quantum Teor Kluwer Academic Publier 99. Penroe R. Te Emperor' New Mind Oxford Univerit Pre New York 989. Ruell S Norvig P. Artificial Intelligence: A Modern Approac Prentice all 2nd edition Tononi G Edelman GM. Concioune and complexit. Science 998;282: Turing AM. On computable number Wit an application to te Entceidungproblem. Proc London Mat Soc 936(2):442: