What is Computation (Jack Copeland).pdf

8/10/2019 What is Computation (Jack Copeland).pdf

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B. JAC K COPELAND

W H A T IS C O M P U TAT I O N ?

ABSTRACT. To compute is to execute an algorithm. More precisely, o say that a device ororgan computes is to say that there exists a modelling relationship of a certain kind betweenit and a formal specification of an algorithm and supporting architecture. The key issue isto delimit the phrase ' of a certain kind'. I call this the problem of distinguishing betweenstandard and nonstandard models of computation. The successfuldrawing of his distinctionguards Turing's 1936 analysis of computation against a difficulty hat has persistently been

raised against it, and undercuts various objections hat have been made to the computationaltheory o f mind.

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In 1936 Turing pub l i shed h i s now fam ous ana lysi s o f the concep t o fcomp utat ion. 1 I t i s t rue to say that th is analys is has b eco me s tandardin mathemat ica l logic and the sc iences . How ever, there i s in the phi losoph-

ica l li tera ture a cer ta in c lass of pro blem cases which forms the bas is of anobjec t ion to Tur ing 's analys is . Th e thrus t o f the object ion is tha t a l thoughTuring 's accoun t ma y be n ecess ary i t is not suff icient. I f it i s taken to besuff ic ient then too many ent i t ies turn out to be computers . The object ioncarr ies an embarrass ing impl ica t ion for computat ional theor ies o f mind:such theor ies are devoid of empir ica l content . I f v i r tual ly anything meetsthe requirements for being a computat ional sys tem then wherein l ies theexplan atory fo rce of the c la im that the bra in i s such a sys tem?

I a im to m eet the object ion. According to the point of v iew to be put

forward here , to compute i s to execute an a lgor i thm. More precise ly, tosay that a device or organ com putes i s to s ay that there exis ts a model l ingre la t ionship of a cer ta in k ind betw een i t and a formal speci f ica tion ofan algori thm and su pporting architecture. T he ke y issue is to delimit thephrase 'o f a cer ta in sor t ' . Fo r want of a bet ter terminolog y I descr ibe thepro blem to be addressed as that of d is t inguishing b etw een s tandard andnonstan dard mo dels o f computat ion. Pro minent a t tempts to deal wi th thepro blem involve modif ica tion of Tur ing 's analys is ( for examp le Goel 1991and 1992, Searle 1990 and 1992, Smith forthcoming, Sterelny 1990). In

m y view this i s both undesi rable and u nnecessary. M y in tent ion here i s touph old the suff ic iency of Tur ing 's analys is . 2

Synthese 108: 335-359, 1996.(~) 1996Kluw er Academ ic Publishers Printed in the Netherlands


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WHAT IS COMPUTATION 337

Let f be the fu nctio n and let o~ be a n architecture-specific algorith m thattakes argum ents of f as inputs and delivers values of f as outputs.

An algor i thm is a mec hanic al or mor onic procedure for achieving aspecified result ( in the case of o~, of cou rse, the specified result is arrivingat the valu es of f) . That is to say, an algorithm is a f inite list of mach ine-executable ins t ructions such that anyone or anything that correct ly fol lowsthe instructions in the specified order is certain to achieve the result inque stio n. To say t hat oz isspe ifi to an architecture is to say not only thata ma chi ne w ith this arch itecture c an run oL but also that ea ch instructio n ino~ cal ls expl ic i tly for the p erforma nce of so me sequence of the pr imit ive(or a tom ic ) operat ions mad e avai lable in the archi tecture. (The sequence

ma y consis t of a s ingle operat ion.) Thus an a lgor i thm at least one o f whoseinstructions calls explicit ly for a multiplication cannot be specific to anarchitecture tha t has addi t ion but not multiplicatio n available as a primitive.A program cal l ing for mul t ipl icat ions can run on such an archi tectureonly because the compiler (or equivalent) replaces each mult ipl icat ioninstructio n in the prog ram by a series of addit ion instructions.

The primitive operations that are available vary from architecture toarchitecture. To give som e example s: - If the architecture in questio n isa Turing m ach ine th en oe will be a Turing mach ine table. The primit ive

operat ions include: mo ve the tape lef t one square; read the sym bol beneaththe head; replace the sym bol beneath the he ad by 0 . I f the archi tecture inquest ion is that def ined by a par t icular assembly language (an assemblylanguage is an archi tecture-specif ic programming language) then a wi l lbe a progra m in that language. The pr imit ive operat ions that are avai lablema y include: adding the binary n umb ers in a specif ied pair of regis tersand storing the result in a specified register; shift ing the bits in a specifiedregister one place to the right ( the bit at the far r ight falls out and 0 isfed in at the left); taking the logical c onjun ction of wha tever binary str ings

occur in a specified pair of registers and storing the result in a specifiedregister ( the logical conjun ctio n of, for exam ple, 1100 and 1010 is 1000).Wh ere th e architecture in questio n is a neural net with a particular pattern ofconn ectiv ity and certain w eights on the conne ctions, c~ consists of step-by-step applica tions o f a certain propag ation rule and a certain activation rule.(A propagation rule calculates the total weighted input into an art if icialneu ron and an act ivat ion rule calcula tes what the act ivi ty level of thene ur on is to be, given its total we igh ted input.) In such an architecture

steps are, of course, carried out in parallel : the execution of an algorithmis not necessar i ly a sequent ia l procedure . 3

Suppo se one has a forma l specif icat ion of the archi tecture in qu est ionand o f the algor ithm a , call i t SPEC. For definiteness, let SPE C take the


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338 13. JACKCOPELAND

form o f a se t of axioms, a l though nothing in what fol lows turns on the useof the axiomat ic m etho d as opposed to som e other s tyle of formal isa tion.

( In the ne xt sect ion I g ive a s impl i f ied example o f such an axiom set. ) Soon the one hand w e have SPEC, a de scr ipt ion of a mach ine, and on theo ther we have the en t i ty e . How do we br idge the gap and say tha t e / ssuch a ma chin e (a t the t ime in quest ion)? The br idge is effected by m eanso f a sy s t em o f l belling for e . (The c once pt of a label l ing o f a device isa lso used by G and y 1980 and Deutsc h 1985.) For example , labels m ay beeight binary digi ts long and be associa ted with groupings o f subdevices inthe fol lowing way: the possible s tates - me asure d in volts , say - of thesesubdevice s are divided into two mutu al ly exclusive classes and a s ta te is

labelled 0 if i t fal ls into the first , 1 if i t fal ls into the second. (No privilegei s accorded to the b inary sys tem: l abel s m ay equa l ly wel l be n - tup les o freal numb ers . ) Th us a label l ing sch em e for an ent i ty consists o f two parts :(1) the d esignat ion o f cer ta in par ts o f the ent i ty as label-bearers , and (2)the meth od for specifyin g the label borne by ea ch label-bear ing par t a t anygiven t ime.

The idea, of course , is that the labels const i tu te a cod e such thatspatia l or tem poral sequen ces of labels have sem ant ical in terpreta t ions . Inexplaining the beh aviour and funct ion of the label led ent i ty one ascr ibes the

semant ical in terpreta t ion associa ted with the labels d i rect ly to the label ledsta tes themselves . To take a s imple exam ple , i f the vol tage across one m em -ber of a pair off l ipf lops is 600 mV and across the second is 100 m V the pairm ay be label led (High, Low) or (1, 0) and be descr ibed as represen t ingor s to r ing the num ber two .

W he n the formal axio ms in SPEC are true of an ent i ty e under a label l ingsche me L the orde red pair (e , L) wi l l be cal led amodel of SPEC. Here ,then, i s the prom ised intuit ive account :

Entitye is computingfunction if and only f there exist a labelling schemeL and a formalspecification SPEC (of an architecture and an algorithm specific to the architecture hattakes arguments of f as inputs and delivers values of f as outputs) such tha t (e, L) is amodel o f SPEC.

(A sen tence o f the fo rm There exi st s a func t ion f such tha t . . . ma y betrue i r respect ive of wh ethe r it is kno wn to be t rue. I use the phrase thereexis t a label ling sch em e and a formal specif icat ion such that in the sam eway.)

To gain a feel for wha t is being said here consider the ho ary problem

of wheth er the solar system comp utes solut ions to it s ow n equat ions o fmo tion (Fodor 1975, 74) . (A ga n d er vers ion asks wheth er the universe isa com pute r dedic ated to com put ing i ts own behaviour. ) Cer ta inly in someloose sense the planets fol low the re levant law of mot ion (Kepler s law,


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WHAT IS COMPUTATION? 339

say). However, this hardly suffices to show that the planets execute analgorithm. Fo r one thing , Kep ler s law simp ly is not a l ist of instructions

each o f which cal ls for the performance o f one or another of the pr imi-tive operations of so me given architecture. Kepler s law states a functionalrelat ionship f betwe en cer ta in magnitudes but i t is obviously not i tself analgo rithm that takes argum ents of f as inputs and delivers values of f asoutputs. S uch algorithm s do exist, of course. At least one o f the m requiresa sup portin g architecture that ma kes available the op eration of shift ing thebits in a register one place to the right. Let SPEC be a specification o f thisalgorithm and its supporting architecture. Is the solar system an exampleof the type of compu t ing machine descr ibed by SPEC? To answer yes is

to sup pose that the solar system is a register machin e - is to sup pose thatthe solar system consis ts o f an interconnected s tructure of binary regis tersthat are respo nsive to the shift operation an d the various other operationsdem and ed by the algori thm, for exam ple binary addi t ion and logical con-junct ion. Such a sup posi t ion no doubt s t r ikes you as ludicrous. Perhapsthose w ho ser iously enter ta in the though t that the solar system com puteswil l respond that i t was never an algori thm l ike this one that they hadin mind. W ell and good. T he foregoing account of com putat ion presentsthe m with a chal lenge: i f they want to pers ist in the claim that the solar

system is com puting the funct ion f then they mu st describe for us the solarsystem s co mpu tat ional archi tecture and detai l the aIgorithm by wh ich thesolar system arr ives a t values of f .

Despi te i ts naturalness the above account o f comp utat ion wil l not do.This follows imm edia tely from a result that I will callSearle s Theorem:

For any entity x with a sufficiently large number of discriminabl e parts) and for any

architect ure-algo rithm specificatio n y there exists a labelling scheme L such that x, L} isa model o f y.

In Se ct ion 41 wil l prove Searle s Theorem . Sear le s chal lenge to those wh oth ink the co ncept o f computa t ion s ign i f ican t - and i t is a good c ha l len ge- i sto m od ify the foreg oing acco unt in such a w ay as to avoid this trivialisation.I show in Sect ion 5 that this can be done w ithout departing from Turing scharacterisation o f com putat ion.

Interestingly eno ugh Turing s f riend and col league Max Ne wm an used

wh at is essentially a notational variant of Searle s T heo rem as the basisof a devastat ing object ion to Russel l s causal theory o f percept ion (New-m an 1928). 4 In The Analysis o f MatterRussel l argued that a l though ourpercept ions te l l us nothing about the intrinsic character of the external


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3 4 0 B. JACKCOPELAND

s t i m u l i t h a t o c c a s i o n t h e m , w e m a y n e v e r t h e l e s s i n f e r a g r e a t d e a l a b o u tt h e s t r u c t u r e o f s t i m u l i ( R u s s e l l 1 9 2 7 , 2 2 7 ) .

[I]t would seem that, wherever we infer from perceptions, it is only structure that we canvalidly infer; and structure is what can be expressed by mathematical logic . .. (Russell1927, 254)

N e w m a n s o b j e c t i o n i s a s f o l lo w s .

A point to be emphasised is th at .. , no ... information about the aggregate A, except itscardinal number, is contained in the statement that there exists a system of relations, withA as field, whose structure is an assigned one. For given any aggregate A, a system ofrelations between its members can be found having any assigned structure compatible withthe cardinal numbe r o f A. (N ewman 1928, 140)These statements [of Russell s] can only mean, I think, that our knowledge o f the externalworld takes this form: The w orld consists of objects, forming an aggregate who se structurewith regard to a certain relatien R is known, say W; but of the relation R nothin g is known

but its existence; that is, all we can sa y is, here is a relation R such that the structureof the external world with reference to R is W . N o w .. . such a statement expresses onlya trivial property of the world. Any collection of things can be organised so as to have thestructure W , provided there are the right number o f them. (Newman 1928, 144)

We k n o w o f R u s s e ll s r e s p o n s e t o N e w m a n s o b j e c ti o n f r o m a le tt eri n c l u d e d i n t h e s e c o n d v o l u m e o f R u s s e l l s a u t o b i o g r a p h y :

Dear NewmanMany thanks for sending me the off -pr int ... I read it with great interest and some dismay.You make it entirely obvious that my statements to the effect that nothing is known aboutthe phy sical world exc ept its structure are either false or trivial, and I am somew hat asha medat not having noticed the point for myself. It is of course obvious, as you point out, that theonly e ffective assertion abou t the physical w orld involved in saying that it is susceptible tosuch and such a structure is an assertion ab out its cardinal number (Russell 1968, 176)

.

T h i s s e c t io n e x p l a i n s b y m e a n s o f a n e x a m p l e w h a t is m e a n t b y a n a x i o m a t -

i c s p e c i f i c a t i o n o f a m a c h i n e a r c h i t e c t u r e . I w i l l c o n s i d e r a s i m p l e m a c h i n e

M w h o s e c e n t r a l p r o c e s s o r c o n s i s t s o f t h r e e e i g h t -b i t r e g i s t er s : a n i n s tr u c -t i o n r eg i s t e r I , a d a t a b u f fe r D a n d a n a c c u m u l a t o r A . To g iv e a n e x a m p l eo f t h e s o r t o f t h i n g t h a t M d o e s , i f t h e i n s t r u c t io n t o p e r f o r m a n a d d i t i o ne n t e r s I , t h e m a c h i n e a d d s t h e c o n t e n t s o f D t o t h e c o n t e n t s o f A a n ds t o r e s t h e r e s u l t i n A . F o r s i m p l i c i t y I w i l l i g n o r e a l l m a t t e r s c o n c e r n i n g

i n p u t , o u t p u t , d a t a t r a n s f e r b e t w e e n t h e c p u a n d m e m o r y, a n d p r o g r a ms t o r a g e a n d c o n t r o l , a n d I w i l l o m i t t h e s p e c i f i c a t i o n o f a n y p a r t i c u l a r a l g o -r i t h m o r p r o g r a m . M i s a v o n N e u m a n n m a c h i n e ( i. e. a p a r t i cu l a r t y p eo f s e r ia l p r o c e s s o r ) . H o w e v e r , I e m p h a s i s e t h a t M a p p e a r s h e r e m e r e l y a s


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an i llustration. As I have already said, the accou nt of com putat ion unde rinvestigation applies both to serial and parallel architectures, including

con nec tionist netw orks. (It is interesting, incidentally, that Turing h im sel fseems to have b een the f irst to co nsider bui lding compu ting mach ines out ofs imple, neuron-l ike elements co nnected together into netw orks in a largelyrand om m anner; see Turing 1948, Copeland and Proudfoot 1996.)

The beh aviour of M is descr ibed by me ans of a pr imit ive term ACTIO N-IS . ( AC TION -IS is a me m ber of the sam e family as Belnap s s t it con-struction and Sege rberg s ~ operator. 5) The axiom s for M (or rather for Msimplified in the wa ys just me ntion ed) are as follows. T he functio n Y~ is tobe read the contents of ( register) x and ~ is to be read bec om es .

A x l

A x 2

A x 3

A x 4

I f f = 00000001 ACT ION-IS ( .4 =~/ ))

If i = 0000 0010 A CTIO N-IS (A ==> .~ + / ) )

If i = 00000011 ACTION -IS (A ~ A x/7))

I f = 00000100 ACTION-IS (.~ ~ A + / ) )

A nd so on for the remain ing instruct ions in the mach ine s order code. Thus

A x 1 says that the displayed binary digits are the instruction to wipe theaccum ulator and t ransfer the contents of D to the accumulator ; andA x 4says that the disp layed digits are the instruction to a dd the co ntents of theregis ter whose address is s tored in D to the contents of the accum ulatorand store the result in the accu mulator.

The intende d interpretation of a sta tement of the form if X ACT ION-IS Y is that the occurrence of X produc es or br ings about the act ion Y.Such a s ta tement is of the sam e logical s t rength as the s ta tement thatX causes Y, in that each supports the counterfactual i f X had occurred

then Y wo uld have o ccurred . However, the t ruth-condi tion of s ta tementso f t he fo rm i f X ACTION-ISY cannot be couched in terms o fphysicalcausation, for the con struct ion must be appl icable not on ly to real hardw arebut a lso to me rely conceptua l machines . For example, we wish to say thateach action o f a Turing ma chin e is the result of i ts configuration (i .e. thecombinat ion of i ts s ta te and the scanned symbol) . Unless we intend tospeak metaphorically, the phrase is the result of cannot be replaced hereby is cau sed by , for the mac hine is a purely abstract entity.

I t i s worth emphasis ing that i f X ACT ION-IS Y expresses a not ion

of con seque nce or dep end ency that is s t ronger than that expressed by themater ia l implicat ion X D Y. Suppo se that in the course of M s operat ionsthe instruct ion 00000100 happ ens never to enter I ; the axioms neverthelesstell us wh ich act ion wo uld have en sued i f i t had. Yet s ince any mater ia l


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342 B. JACKCOPELAND

implication with a false antecedent is true, the following implications areall true in the envisaged circumstances:

T = 00000 1 O0 D .4 ~ .4 + / )

f = 000 001 00 D .4 =. D

_T = 000 001 00 D -4 =~ A +/ )

/~ = 000 00 100 D .4 =~ -~ x / )

Clearly the set of material implications of this form that are true of M

gives no guid ance as to how M is designed to behave. Fur thermore, s inceany material implication with a true consequent is true i t is equally_thecase that i f a t some point in the comp utat ion the act ion _~ ~ A + /) i sperfo rmed then the fol lowing are a l l t rue:

i = 00 00 00 01 D 42. ~ _~ + / }

i = 00 00 00 10 D A ~ -4 + D

etc.

Yet the correspond ing ' i f . . . ACTION -IS - - - ' s ta tements are a ll fa lse .As is wel l know n, the problem o f analysing counterfactual-support ing

statem ents is a hard one - so hard, indeed, that ma ny logicians l ike topretend that a f irst-order extensional language suffices for al l scientif icpurpose s (see for example Quine 1960) . Yet an adequate theory of compu-ta t ion cannot be co uched in a purely extensional language .

Quine p roposes to paraphrase c ounterfactual-supporting s ta tements bymea ns of a relative term ' M ' ( 'al ike in structure ') . For exam ple, 'z is

soluble ' i s to be paraphrased as ( ' in rough outl ine ' ): 3y (Mx y y dissolves)(Quin e 1960, 224). This proposa l cannot work in the present context, sincethe com put ing dev iceh s n o relevant structure over and above that detailedby the very axiom s in the archi tecture-algor i thm specif icat ion that are tobe paraphrased. In the jargon, there are 'mul t ip le physical realisa t ions ' ofone and the same com put ing device. For example , M ma y be real ised bymea ns o f valves or t ransis tors or cogs and levers h la Babbage. Th e on lyfeature that such disparate physical ent i t ies need have in co mm on is thatall are realisations of M .

Turing him self g ives express ion to th is s t rong de pend ency re la t ionshipby means o f the phrase ' comple te ly de te rmined by ' : each ac tion ofa Tur ingmach ine is co mplete ly determined by i ts conf igurat ion. Here is h is e legantpresentat ion of his analysis o f computat ion.


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W H AT I S C O M P U T AT I O N ? 343

We may compare a man in the process of computing a real number to a machine whichis only capable o f a finite number o f conditions ql, q2,. . . qk which will be called m -configurations . The machine is supplied with a tape (the analogue of paper) runningthrough it, and divided into sections (called squares ) each capable of bearing a symbol .At any mom ent there is just one square, say the r- th, bearing the symbolS r) which is inthe machine . We may call this square the scanned square . The symbol on the scannedsquare may be called the scanned symbol .. . The possible behaviour o f the machineat any moment is determined by the m-configuration q,~ and the scanned symbolS r).This pair q,~, S (r ) will be called the configuration : thus the configuration determinesthe possible behaviour of the ma ch in e. .. If at each stage the motion of a machine . ..is completelydetermined by the configuration, we shall call the machine an automaticmachine .. . In some of the configurations in which the scanned square is blank (i.e.bears no symbol) the machine writes down a new symbol on the scanned square: in other

configurations it erases the scanned sy mb ol. The machine may also change the squarewhich is being scanned, but only by shifting it one place to the fight or left. In addition toany of these operations the m -configuration may be ch an ge d. .. It is my contention thatthese operations include all those which are used in the computation of a number. (Turing1936, 231-2 )

.

I n t hi s s e c t i o n I o u t l in e t h e p r o o f o f t h e t h e o r e m t o w a r d w h i c h S e a r leges tu re s i n t he qu o ta t i on g iven in Sec t io n 1 , v i z .:

For any entity x (with a sufficiently larg e number o f discriminable parts) and for anyarchitecture-algorithm specification y there exists a labelling scheme L such that (x , L) isa model o f y.

Th e s t r a t egy i s t o p i ck so me a rb i t r a ry phys i c a l en t i t y e w i th a comfo r t -a b l y l a rge n u m b e r o f d i s c r im i n a b l e p a r ts - S e a r l e 's w a l l o r H i n c k ' s p a i lw i l l do n i ce l y - and sh ow tha t t he re ex is t s a labe l l i ng o f e wh ich enab le sone to i n t e rp re t t he ax io ms fo r M in such a wa y tha t each o f t hem i s t rue

o f e . O n e t h e n p e r f o r m s a u n i v e rs a l g e n e ra l is a ti o n : t h e p r o o f m a k e s n oes sen t i a l u se o f f ea tu re s pecu l i a r t o e o r M and so t he r e su l t ho lds fo r anya rch i t ec tu re -a lgo r i t hm spec i f i ca t i on and fo r any en t i t y (w i th a su ff i c i en t lyl a rg e n u m b e r o f d i s c r i m i n a b l e p a r ts ) .

Th e f i rs t t h ing to be d one i s t o s e t tl e on a way o f co r r e l a t i ng b ina ryn u m b e r s w i t h p h y s i c a l s t r u c t u r e . L e t ' s s i m p l y g r a n t S e a r l e , H i n c k f u s s ,Ly c a n e t al . a m e t h o d t h a t e n a b l e s o n e to c o r r e l a te u n i q u e b i n a r y n u m b e r swi th r eg ions o f wh a tev e r phys i ca l o b j ec t is in ques t ion . Fo r i n s t ance , i f t hew a l l h a s a h i g h p o l y m e r c o n t e n t t h e n th e f o l l o w i n g s im p l e m e t h o d c a n b e

u s e d : w h e n t h e n u m b e r o f p o l y m e r c h a i n s t h a t e n d i n a g i v e n s p a c e S iso d d t h e n S t o k e n s 0 , a n d w h e n t h e n u m b e r i s e v e n S t o k e n s 1.6

Th e re i s no h ope o f t ak ing the th r ee r eg i s te r s 1 , D and A to be th r eep a r t i c u l a r r e g i o n s o f t h e w a ll . T h e r e a r e t w o f a i rl y o b v i o u s r e a s o n s f o r


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44 B. JACK COPELAND

TABL E I

Cycle i D 4 /~

i l d l al x l

2 i2 d2 a2 x2

n i~ d n an x ~

this. (1) We require the co ntents of the registers to remain co nstant unless

al tered by an ins t ruct ion. M may, for example , be running a progra m wh osefunct ion is to raise the num ber in D to a given power. The program doesthis by repeatedly add ing the contents of D to A. Clear ly the contents ofD must rema in constant throughout the computat ion. There is no reason toexpect the se lected physical proper ty (or proper t ies) to rema in constant ina given region for precisely the required period. (2) We require the con tentsof the registers to chan ge appropria te ly dur ing the execut ion of a program.We can not expect a m ere wal l to have physical proper ties that change ina way responsive to the sequencing of ins t ruct ions in whatever program

M happe ns to be running. Thus there is no region that can serve as thereferent of 1 . I t woul d be equal ly absurd to expect the physics of thewal l to be respon sive to the d eman ds of binary ar i thmet ic ; so there is noregion that can serve as the referent of A . The so lution is to use enti t iesof greater abst raction as referents of the terms I , A and D . ( I callth is procedure Sear l i f icat ion . ) Ea ch of these terms is to be interpretedas a funct ion who se value a t any point in the com putat ion is the contentsof the register at that point . (A similar constr uction serves to interpret theindirect address D . ) I f an expl ic i t t reatment were being given of those of

M s axioms that contain terms designat ing an input regis ter and an outputregis ter then these terms w ould be deal t wi th in jus t the same way.

To ma ke matters specific let s supp ose that M (the real M ) is runn inga particular progra m, say Wordstar, and let s cons ider a particular run ofthe program last ing, say, for n of M s c lock-cycles . (M performs oneinstruction per cycle or beat of its internal clock.) Table I shows theconten t s o f I , A , D a n d / ) a t the end of each cyc le o f the run ( thus thecol um n hea ded _A shows the contents o f the accumu latorafter that cycle sinstruction has be en executed ). The result is a tableof labels Columns two

through f ive display not physical s tates of com pone nts of M but binarynumbers that label such states.

Each cycle c ends (we m ay suppose) a t a precise mo me nt to . This wi l lbe a mo me nt in the his tory of the wal l, Cal l such mom ents the designated


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W H A T I S C O M P U T A T I O N ?

T A B L E I I

345

T Iil IZ l lal 151

tl li [ Idll lall Ix;I

t,~ [in[ Idol la,~l Ix,~l

mome nts . Us ing the hypothes i sed method for cor rela ting b inary numberswith regions of the wal l one can ass ign dis t inct regions of the wal l as i ti s a t a des igna ted momenttc to each of ic, dc ac and xc. Making theseassignments yields a second table. Unlike Table I this is a table not oflabels but of regions of the wall . I write 1i1[ to nam e the reg ion a ssignedto i l , and so on; [ il1 may be prono unce d the mole cular encoding of thebinary number i lk The terms in the f i rs t column refer to the designatedmoments for th is par t icular run of the program, in the second column tothe mo lecu la r encodings o f the con ten t so f I and so on.

Tables 1 and 2 taken together const i tu te the desi red label l ing of thewal l. Th e next three paragraphs develop some termin ology required for thedemo nstra t ion that there exis ts an interpreta t ion of M s axioms such thateach is true o f the wall so labelled. (Readers who are prepa red to take thedetai ls for granted m ay w ish to skip them.)

Let I be the funct ion who se d oma in is the se t T ( i.e. the se t of designatedmo me nts for th is run of the program) and whose range is the se t [ I[ , andsimilarly for D, A, and X. I w ill be used to interpret occurrences o f thesymbo l I in the axioms, D to in terpret occurrences of D and A of A .

X interprets the indirec t address used inA x 4 .t t wi ll be use d to denote the des igna ted mom ent immedia te ly p reced ing

a given designate d m om en t t . Not ice that A(t ~) represents the s tate o f theaccum ulator whic h imme diate ly precedes the s ta te A(t) .

Le t t j be any des igna ted mo men t such tha ti j = 00000010 (i .e. suchthat i j i s the ins t ruct ion to add the contents of D to the contents of A) .Consid er the se t of a ll ordered pairs ( (A(t) ) , D( t j ) ) , A( t j ) ) , one for eachs u c h m o m e n t t j . By construction, this set is a function. Call i t PLUS.Similar ly, le t tk be a designated m om en t such thatie = 0000 0011 (i.e. the

inst ruct ion to mul t iply the contents of D by the contents of A). Co nsiderthe set of al l ordered pairs ((A(t~), D(tk)) , A(tk)) . Again this set is afunct ion; cal l i t TIMES. Last ly, le t t l be a designated moment such thatit = 0000 010 0 (i.e. the instruction to add the contents of the register wh ose


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346 B. JACKCOPELAND

address is stored in D to the contents of A). The func tion PL U St is the set( (A( tl ) , X( t t )) , A (h) ) .

The disp layed axiom s for M are interpreted as follows (the quantifierranges over designated moments):

Axl : Vt I t )= 1000000011 ~ A t) = D t))

Az2: Vt(I(t) = 1000000101D A(t) = PLUS((A(t ) ,D(t))) )

Ax3: Vt(I(t) = 1000000111 D A(t) = TIM ES ((A (t') , D( t)) ))

Ax4: Vt(I(t) = 100000100[ D A(t) = P LU St( (A (t ') , D(t) ))) .

It is easily verified that e ach o f these statements is true.Under the interpretation provided, then, M's axioms are true of the

wall. This is the technical analogue of Searle's claim that 'there is somepattern of molecu le move ments that is isomorphic with the formal struc-ture of Word star' (Searle 1992, 209). Since the construction I am callin gSearlification is quite gen erally applicable the argum ent will go through nomatter w hat entity and what architecture-algorithm specification are unde r

consideration (subject only to the usual fine print about the cardinality ofthe parts). This completes the pro of of Searle 's Theorem.

Reflection on the m odel presented in this section will yield the promisedcriteria for guarding the analysis given above of the predicate ' is comp utingthe function f ' from mod els of an architecture-algorithm specification thatare of the wro ng ki nd to sustain the analysis.

.

A nonstandard interpretation of a theory is an interpretation that doesnot respect the intended mean ings of the terms of the theory. 7 (Whenmathematical logicians speak of nonstandard interpretations of numbertheory, analysis and set theory they have in mind interpretations who sedomain s are not isomorphic to the dom ain o f the intended interpretation.In the more g eneral sense of the term 'nonstandard interpretation' in usehere the doma in of such an interpretation m ay or ma y not be isomorphicto the do main of the intended interpretation.) For example, a nonstandard

interpretation of some statements concerning European geography mightassign the numbe r 1 as referent of the symbol 'Lon don ' and the num ber 16as the referent of 'Mo scow '. Sentences of the form 'a is north of b' mightbe assigned truth conditions of the form ' the referent of b < the referent


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of a ' . In th is mode l l ing the sen tence 'Mo scow i s nor th o f Lond on ' i st rue , bu t is no longer about Mosco w and Lond on .

What is perhaps the most spectacular nonstandard interpreta t ion ofthem a ll i s due to Skolem. Bui ld ing upon ear l ie r work by L6 wen he im heprove d that i f there is some interpreta t ion unde r which the axiom s o f atheory are t rue , then they are t rue under an interpreta t ion whose domainis the se t of natural nu mbe rs ( for any f i rs t -order theory wh ose la nguage iscountable and contains the ident i ty predicate). 8 The result of applying thistheo rem to the theory of real numb ers is known as Skolem 's paradox. I t i san ax iom (or theorem ) o f real numb er theory that the cardinal i ty of N (theset of al l natural num bers) is less than the cardinali ty of lI~ (the set of al l

real numb ers) . (That is , the se t of a l l real numbe rs is larger than any set ofna tura l num bers . ) Ye t the L6 we nhe im- Sko lem theorem te l ls us tha t th isaxiom is t rue unde r an interpreta tion that countenanc es nothing but naturalnum ber s and sets thereof. How, one m ight ask, can a sentence enta i l ing theexis tence o f a se t larger than any set of natural num bers be t rue in a universewhere there are no sets o ther than sets of natural numbers? Or to take amore mundane example , how can the sen tence 'Sko lem i s Norwegian ' ,which enta i ls '3z (z is Norwegian) ' , be t rue under an interpreta t ion thatcountenances no Norwegians , on ly Swedes?

Ther e is no real difficulty here. Perhap s the lat ter interpretation assignsthe fol lowing t ruth-condi t ion:

'Sko le m i s Norwegian ' i s t rue i ff the re fe ren t o f 'Sko lem ' C K

wher e K is some subset of the domai n (perhaps the se t of unma rr iedmembers of the domain) . Provided the object ass igned as the referent ofthe symbol 'Sko lem ' i s indeed in K, the sen tence 'Sko lem i s Norwegian 'is t rue und er the in terpretat ion; but i t i s c lear ly no longer about S kolem a nd

Norw egians . In the case of an interpreta t ion within the natural numb ers o freal num ber theory, som e set of naturals is ass igned as the referent of thesym bol '1~'. Th e sente nce 'card N < card II~' is ma de true by the expe dien tof ensur ing that the model contains no 1- 1 funct ion that maps this se t ontowhatev er se t is ass igned as referent of 'N ' . (Similar ly one could mak e thesentence 'Moscow is the most nor ther ly European c i ty ' t rue in the modelme nt io ned ear l ier by choosing not to include any num bers greater than 16in the domain of the model . ) In summary, whi le axioms containing theterm 'N:' are true un der the nonsta nda rd interpretation I have ske tched, the

axiom s thus in terpreted are no longer abo ut the se t of a ll real numbers .The interpreta t ion I have descr ibed for the axioms o f M is a nonstan dard

one . We may say tha t the wal l computes and mean by th i s no th ing morethan that the axiom s for M, a computer, are t rue unde r the in terpreta tion I


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34 8 B. JACKCOPELAND

have descr ibed. However, i t wou ld be an e lem enta ry error to infer f rom thistha t the wal l computes in any genuine se n se - j u s t as i t would be an e r ro r

to infer that some countable se ts are uncountable f rom the fact that theaxio m 'ca rd N < card I~' is true in Skole m's countable mo del of num bertheory. The intend ed me anin g o f th is axiom is that lt~ is uncou ntable ( i .e. i slarger than an y set of natural numb ers) , but th is does not me an that i f theaxio m is t rue u nde r a nonstan dard interpreta t ion then wha tever II~ refersto und er that in terpreta t ion is uncountable . Similar ly, under their in tendedinterpreta t ion the axiom s for M add up to the proposi t ion that M is a vonNe um ann compute r, bu t thi s does no t mea n tha t i f the ax ioms a re t rueunde r a no ns tandard in te rpre ta tion then w hatever they a re t rue o f under

tha t inte rpre ta tion is av on Ne uma nn compute r.Wh ere the pair (e , L) is a mod el for some SPE Cbut only under a

nonstandard interpretation of SPEC I wi ll say that (e , L) is a nonstan dardmod el o f SPEC. A mod el that i s no t nons tandard wil l be ca l l edhonestSear le ' s T heo rem shows that according to the analysis given in Sec t ion 2the p red ica te ' i s com put ing the func tion f ' i s t r iv ia lly t rue o f very man yentit ies. I sugg est the follow ing modifica tion to the analysis:

Entitye is computing unctionf if and only if there exist a labelling schemeL and a formalspecification SPEC (of an architectureand an algorithm specific to the architecture hattakes arguments of f as inputs and delivers values of f as outputs) such that (e, L) is anhonestmodel o f SPEC.

The c la im, then, i s that the mod el constructed in Sect ion 4 is a nonstan-dard one. W hy exact ly? What are the grounds for insis ting that the mode lfa il s to respec t the in tended m eanings o f the t e rms o f the ax iomat ic theory?These are threefold . Firs t , the axiomat ic theory under considerat ion andthose l ike i t - the var ious dyn am ic logics , for exam ple - are in tended ( in aphrase o f Segerberg 's (1989, 248)) as logics of com puteraction However,

a l l the computat ional act ivi ty occurredoutside the wal l , in the course ofobtaining Table I , whic h is s imply a rec ord o f the act ivi ty wi thin the cpuof the mac hine that actual ly perfo rme d the com putat ion. Once Table I i ssecure d the labell ing sch em e is constructed f rom i t ex post facto . The wal lunder th is novel descr ipt ion is a t most a pass ive ' scoreboard ' and is nomore an act ive par t ic ipant in the computat ion than the scoreboard is anact ive play er in a game o f bi ll iards. T he axiom s for M cer ta inly containthe term 'ACT ION -IS ' but the fact that the axioms (as in terpreted) are t rueof the wal l goes no wa y toward showing tha t the wal lac ted in accordance

with the ins t ruct ions in the a lgor ithm. Th e wal l so acted on ly i f the referento f 'I~ ' in Skole m's countable mo del is uncountab le

Second , the non standard interpreta tion int roduces unintended temporalspecif ic i ty in to the theory. The label ling sc hem e used in the m odel is ,


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of course , incomplete in that the scheme ident i f ies the regions of thewal l that bear labels a t the designated momentst l , . . . , t n but provides

no informat i on conce rning whi ch regions of the wal l are the label-bearersat t imes pr ior to t l or subsequent totn. This is in sharp contrast to theunm ark ed case , where the label ling scheme remains appl icable throughou tthe l i fe t ime o f the ent i ty (assuming no hardware modif icat ions) , certa inf ixed regions of the ent i ty, or d is junct ions of such, being designated abinit io as label-bearers. Th is incom plet ene ss in the labell ing schem e showsup in the in terpreta t ion of the construct ion i f . . . ACTION -IS - - - , v ia thepresence of the universal quant if ier ranging over the designated mom ent st l ,. ., tn . The axioms as in terpreted have no enta i lments concernin g t imes

lying outs ide this range, whereas under their in tended interpreta t ion theaxioms are (and entail ) condi t ional s ta tements that are t rue a t any mom en tduring the n ormal funct ioning o f the device . ( Is the wal l perhaps in a s ta teof malfunct ion a t a ll t imes pr ior to t l and sub sequent totn? Hardly. Atthe wh im of the mode l ler the wal l can be made to run the progra m again,say th rough the mome ntst n + l , . . . , t 2 n . Moreover, under the nonstand ardinterpretation the whole axiomatisation is , so to speak, in the past tense,wherea s to descr ibe a physical ent i ty as a comput in g machine of a certa inkind is to envisage being able topredic t aspects of i t s physical behaviour

on the basis of i ts architecture-algorithm specification and i ts labell ingscheme. That is to say, the in tended mea ning of the term ACT ION-IScertainly involves no restrict ion to past actions. Yet i t is necessari ly the casethat under the nonsta ndard interpreta t ion each axio m is a s ta tement aboutthe past . I t is, of course, the ex post facto nature of the labell ing sch em e thatint roduces these unw ante d tem poral specif ic i t ies in to the in terpretat ion.

Third , the construct ion i f . . . ACTION -IS - - - i s in terpreted by meansof mater ia l impl ica t ion ( the t ruth-condi t ion ass igned to each axi om is aunive rsally quantified material implication). Nor can a better interpretation

be fo und for th is construct ion within the context of an a t tempt to show thatthe wal l serves to mod el M s axioms. There are possible worlds di ffer ingmini mal ly f ro m the actual world in which the axioms as in terpreted havetrue antece dents and fa lse consequents . Consider :

Vt(I(t ) = I000000111 D A(t) = T IM ES ((A (t ) , D( t))) ) .

Sup pose that in the actual wo rld I(t) = [00000010 I. That is to say, theregion whic h is the value of I a t the mom ent t i s a molecu lar encodin g of

the binary num ber 000000 I0 ( th is numb er being M s ins t ruct ion to add/) to .4). Take a possib le world w in wh ich the properties of this sameregion are jus t suff ic ient ly di fferent to make i t a molecular encoding ofthe binary num ber di ffer ing f rom the forego ing only in i ts least s ignif icant


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350 13. JACKCOPELAND

digi t . The regions den oted by A(t) , A(t I ) and D(t) code the same nu mbe rsin w as they do in the actual world . ( I f quan tum me chanics is even roughly

true then such a w orld is not only nomolo gica l ly possible but may alsobe indis t inguishable f rom the actual world throug h al l h is tory up to a t imet min us delta.) Say for d efiniteness that A (t ~) =J 1J and D(t) =1000000101 So fro m Table II A(t ) = 1000000111. Sup pos e further that thecomp utat ion represented in Table I I is such that

3u (I (u ) = 1000000111 A (u ') = 1000000011 D (u) = 10000 0010 l).

Since TIM ES is a functio n it follows that

TI ME S( (A (t ' ) , D(t ))) = 1000000101.

So it is true in w that I(t) = 100000011[ and false in w that A(t ) =TIM ES( (A( t ' ) , D(t ) ) ) . There is then, no condi t ional connect ive R-+ witha possible worlds semant ics such that

Vt(I(t)= 1 0 0 0 0 0 0 11[ 1 ~ A ( t ) : TIMES((A(t ),D(t))))

i s t rue . The same goe s for the remaining axioms.

Wh y do es i t mat ter that ' i f . . ACTION -IS - - - ' has been interpreted asmater ia l impl icat ion? Because, as previously remarked, the axiom s unde rthat interpretation fail to supp ort assert ions about the co unterfac tual behav-iour o f M . Whereas g iven the in tended meaning of the ' i f . . . ACTION-IS

- - ' construct ion the axioms do l icence assert ions about the counterfactualbehaviour of M . (For example , under i t s in tended interpreta t ion axiom 3licence s the assert ion that if 00000011 had bee n in the instruction registera t t then the machine would have performed the replacement shown inthe axiom.) ' I f . . - ACTIO N-IS - - - ' expresses a s tronger re la t ionship than

material implic ation and that is the third reason to say that the interpretationfai ls to respect the in tended meanings of terms occurr ing in the axioms.The s t rong depe nde ncy re la t ionship that Turing refers to in the course ofexplaining the concept of a comput in g mach ine is s imply absent f rom the'machine ' consis t ing of label led regions of the wal l . Only i f Tur ing hadmad e no m ent i on of th is s t rong depe nde ncy re la t ionship would Sear le ' scr i t ic ism o f him in the above quotat ion be jus t .

In su mmary, I suggest two n ecessary con di t ions for honesty. First , thelabel l ing sc heme must not be ex post facto . This requirement gu ards the

class of honest models f rom intruders that fa i l to respect the in tendedmean ings o f the terms of the axiomat ic theory in ways of the sor t out l inedin the first and seco nd of the above three cri t icisms. Second , the inter-preta t ion associa ted with the model must secure the t ruth of appropria te


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WHAT IS COMPUTATION? 3 51

counterfactuals con cerning the mach ine s behaviour. Ei ther of these tworequirem ents suffices to de bunk the a l leged proble m cases .

.

This section cri t icises some alternative answ ers to the questio n Wh at iscom p u t a t i on? .

1. Deut sch gives the fol lowing characterisa t ion of computat ion:

[A] comp uting mach ine is any physical system w hose dynamical evolution takes i t from

one of a set of inp ut states to one of a set of outp ut states. The states are labelled insome canon ical way, the mach ine is prepared in a state with a given input label and then,fol lowing some motion, the output s tate is mea su re d. . . [T]he measured output label is adefinite function f of the prepared input label; . . . the mac hine is said to com put e thefunction f. (Deutsch 1985, 97)

This accoun t fa i ls easy prey to Sear le s Theorem . Hin ck s pai l i s aphysical syste m who se dynam ical evolut ion takes i t f rom an qnp ut s ta teto an outp ut s tate . The s ta tes are label led in some canonical way by meansof a sui table enco ding/ decod ing funct ion. Som eon e who is , so to speak,

told exact ly where to look in the bucket can measure the outpu t s tate anddetermin e i ts label . So the bucket is compu t ing the funct ion that re la testhe input labels to the output labels. Of course, the labell ing scheme inques tion is entire ly ex post facto (as in Tables I and II) .

There is so met hing e lse very wrong about Deutsc h s analysis : i t make sno ment ion of the not ion of an a lgor i thm. In my view this not ion shouldl ie a t the hear t of an accou nt of computat ion. Consider a c lockw ork c lock.The hand s show 12. This is the input s ta te. The label i s s imply the numer-ical specif icat ion of the t ime shown. The c lock runs for n minutes . The

ensuin g posi t ion of the hands is the outpu t s ta te . Is the c lock com put in gthe f imct ion that re la tes the input label to the output label? Accordingto Deutsc h s a ccount Yes; according to my account No. The e lem entarysteps of an a lgor i thm for der iving the output labels f rom the inp ut labelsmig ht consis t , for exam ple , o f mul t ipl icat ions , addi t ions and subtract ions .No honest way of model l ing a specif icat ion of such an a lgor i thm andan arc hitecture for imp lem ent ing i t :(i.e. an architecture that ma kes thes eelementary operations available) will be able to locate such steps in thefunct ioning of the c lock. Similar remarks apply to the issue of whether

the solar system com putes solut ions to i t s own equat ions of mot ion (Sec-t ion 2 above) . Acco rding to Deutsc h s accou nt i t does but according to anaccoun t that takes the not ion of an a lgor i thm and i ts support ing architec-ture ser iously the solar system does not compute . Nonstandard m odels of


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352 B. JACKCOPELAND

any architecture-alg ori thm specification are easi ly obtain ed but there is noreason to th ink that the chal lenge of providing an honest mo del base d on

the solar sys tem of a sui table a lgori thm and suppor t ing archi tec ture can bemet.The exam ples o f the c lock and the solar sys tem generalise . Any ph ysical

sys tem that i s desc r ibab le as p roduc ing ou tpu t f rom inpu t - a n d i s thereany that is not? - has a labell ing in De utsc h s sense, and so on his accou ntthe sys tem computes the funct ion re la t ing the input labels to the outputlabels. In consequence the claim that the brain computes loses i ts s tatus asa ser ious empir ica l hypothes is . Protect Deutsch s account f rom the radicalt r iv ia li sa tion i t suffers a t the hand of Sear le s Theo rem (by mean s of the

apparatu s d etai led in Section 5) and the accoun t wil l s ti ll be too wide.

2 . The analys is of comp utat ion presented by Cumm ins is responsive tointuit ions such as these concerning the clock and solar system. He writes:

[F]unctions need not be computed to be satisfied. Set mousetraps satisfy a function fromtrippings to snappings without computing it, and physical objects of all kinds satisfymechanical functions withou t computing them. The planets stay in their orbits withoutcomputing them. (Cumm ins 1989, 91)

H e glo sses the notion of a function s b eing satisf ied: a device sat isf ies afunc tion 9 whe n the argumen ts and values o f 9 are l i teral ly states of i t(Cummins 1989, 89) .

Functions associate values with arguments. To see a device as satisfying a function, there-fore, is to see it as having inputs and ou tputs.. , and to see these asarguments and values.(Cummins 1989, 164)

Cum min s p ropo ses the fo l lowing analys is of when a function is beingcomputed :

Computing reduces to program execution, so our problem reduces to explaining what it isto execute a program. The obvious strategy to exploit is the idea that program executioninvolvessteps and to treat each elementary step as a function that the executing systemsimply satisfies... Program executionreduces o s tep satisfaction. (Cummins 1989, 91-92)

This analys is too is t r iv ia li sed by Sear le s Theorem, as i s eas i ly shownby consider ing the e lemen tary s teps of a program for the machine M . Atypical example of such a s tep is

Ad d the contents o f D to the contents o f A and s tore the resul tin A.

We m ay regard th is s tep as taking two binary num bers as input and del iv-er ing a s ingle b inary num ber as output . Cum mins invites us to t rea t each


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WHA T IS COMPUTATION 353

elem enta ry s tep as a fu nct ion that an execu t ing system can satisfy. In th iscase the fu nct ion wil l be on e who se values are label-bear ing s ta tes of the

execu t ing sys tem and w hose a rguments a re pa ir s o f such s ta tes. Un der thelabel l ing prop osed in Sec t ion 4 the wal l sa t isf ies the funct ion in quest ion.On eac h c loc k cycle x a t wh ich the regis ter I contains the ins t ruct ion forperfo rmi ng the abov e s tep it i s the case that

P L U S ( ( A ( t ~ ) , D ( t ~ ) ) ) = A ( t z ) .

The sam e goes for a l l the other e lem enta ry s teps of the program. Soaccord ing to Cu mm ins ' account the wal l i s execu ting the p rogram.

Mo reove r Cumm ins regards computa t ion as a causa l process :[A] computation [isl a causal process specifiedabstractly as an algorithm. (Cummins andSchwarz 1991, 63)

As I rema rked ea r li e r, a causa l account o f compu ta t ion lacks the necessa rygeneral i ty. Abstract devices such as Turing machines compute . Considerthe case o f a p rogrammed s imula t ion of a Tur ing machine . The v i r tua lTuring ma chin e is com put ing, but there are no cau sal re la t ionships betwe enthe con tents of the vi rtual mac hine 's tape and the vi r tual ma chin e 's act ions .

The re levan t causa l ne twork has ac tiv it ies o f the under ly ing real m achineat i ts nod es , no t act ivi t ies of the vi r tual mach ine.

An analysis of com putat ion in terms o f causat ion is in tolerably narrow.Ev en i f that fact i s d isregarded Cu mm ins c annot protect h is s tep-sat isfact ionaccount of computat ion f rom tr ivia l isa t ion by adding a r ider to the effectthat i f s ta tes are to sa t isfy a funct ion then the value-s ta te must be acausaloutcome of the argum ent-s ta te(s) . For there exis ts a label l ing o f the wal lsuch that label-bear ing s ta tes sa t is fy the ap propria te funct ions even in th iss t ronger sense of ' sa t is fy ' . ( ,Or i f that ' s no t t rue then pick a larger wal l )

The t r ick, o f course , i s to se lect s ta tes for label ling thatare causal ly re la tedin the requ i red ways . S tay ing wi th the p rev ious example o f an e lementa rystep, the state ]azl must be cho sen in such a way that (as wel l as having ther ight s t ructural proper t ies to bear the label as und er the enco ding /dec odingfunct ion being used) i t i s a causal outco me o f the s ta tes[a x_ 1)l, d~[ Thefunct ion PL US is then sat isfied in the s t ronger sense .

3 . Lyc an 's ow n way o f deal ing with Hinck 's pai l is not , as i t s tands,successful . He wri tes :

[W]e se e .. , why Block's group organismsare admissibleas sentientbeings but Hinckfuss'spail of water is not: the homunculi-head .. incorporate[slff-ers, ~-ers, and "-ers" ofcountless other types, courtesy of the bureaucrats who are do ing all the work; the pailof water does not contain any "-ers" o f any kind that is mentioned n a homunctionalist


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354 B. JACKCOPELAND

program, precisely because it is not organised in the relevant way, even if the de factomotions of some of the molecules n the pail happen to ape the motions hat would be madeby an organism hatw s functionallyorganised on the human model. (Lycan 1981,41; seealso 1987, 34)

As ever, l e t M 's ax ioms se rve as an ex am p le - s impli st ic bu t adequa tefor the purpose in hand - of a funct ional is t archi tectural specification.Ly can 's thought is that the specif icat ion requires cer ta in kinds o fe rs thatthe pail can not supply. But le t er l . . . e rn be a li st of the var ious ersinvolved in M 's archi tecture ( ins truct ion regis ter, accumulator, e tc .) andconsid er a mod el fo und ed on the pai l of the sor t descr ibed in Sect ion 4.Un der the in terpreta t ion associa ted with the mo del a referent for each term'er l ' , . . . , ' e rn ' i s indee d fumish ed by the pai l; so Lycan 's c la im is fa lse .(To be sure each o f M 's regis ters is real ised in a highly dis tr ibuted wa y inthe pail o r wal l, but then so would they be in a connect ionis t mod el l ing ofM .) Lycan ' s a t t empt to defuse the p rob lem cases s imply underest imatesthe Sko leme sque tact ics that under l ie them.

.

I t is the com putat ional the ory of mind, and not extant defini tions of com-putat ion, that i s the pr incipal target of Sear le and Hinckfuss . Numerousother colo urful problem cases are to be fou nd in the cr i t ical l i tera ture oncom putat ional funct ional ism. Al l the var ious cases can be divided into twogroups. Cases in G roup I are typif ied by H inck 's pai l and the W ordstarwal l. Al l c an be d ea r wi th by m eans o f the d i st inc t ion be twee n nonstan-dard and h onest mod els develop ed here . The cases in Group II are typif iedby Block ' s ingen ious and w el l -known thought exper iments invo lving the

ec on om y of Bol ivia and the populat ion of China. Group II cases pose a pri -ma fac ie cha l lenge to computa t iona l func t iona li sm bu t have no te nd en cy -no t even a p r ima fac ie o n e - to show tha t Tur ing ' s ana lys i s o f computa t ionis tr iv ia l isable and so are not ge rma ne to the present discussion. Never the-less i t is wo rthwh ile br ief ly examining on e o f Bloc k 's exam ple s in orderto es tabl ish that i t does in deed pose no c hal lenge to the acc ount of com -putat ion offe red here (and nor, for that mat ter, to Deu tsch 's or C um mi ns 'account) . I shal l suggest mor eove r that the Group II cases in fact have noforce against co mputat ion al funct ional ist theor ies of mind.

I f com putat ional funct ional ism is true then there exis ts an archi tecture-a lgor i thm spec if i ca t ion SPEC-B LOC K such tha t i f fo r any sys tem z andlabel l ing L (z , L) is an honest mode l of SPEC -BLO CK th en z is cogni t ive-ly equivalent to Block. 'Sup pose ' , says Block, 'w e conver t the govern me nt


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of Chin a to funct ional ism, and we convince i ts off icia ls that it would enor-mou sly en hance their in ternational prest ige to real ize a h uma n mi nd for an

hour. We provide each of the bi ll ion people in China . . . wi th a specia l lydesign ed two-way radio that connects them in the appropria te way to otherpersons . . . (Block 1978, 279). I f each of the bil l ion people is to ld toobey an appropria te ins t ruct ion and does so fa i thful ly then the resul t ingreal isa tion of S PEC -BL OC K wil l be a system that is cogni t ively equivalentto Block. ( In Bloc k s o wn descr ipt ion of the scenario each person is g ivenan instructio n corresp ond ing to a single l ine in a Turing mac hine table; theprecise nature o f the ins t ruct ions wi l l vary depend ing on the com putat ionalarchitecture in question.)

Bloc k wri tes :

Rem embe r that a mac hine table specifies a set of conditionals of the form: if the machineis in Si and receives input j it emits output Ok a nd goes into Sz. An y system that has a setof inputs, outputs and states related in the way d escribed realizes that machine table, evenif it exists only for an instant. For the hour the Chinese system is on , itdoes have a set ofinputs, outputs and states of which such conditio nals are true. (Block 1978, 279)

Wh at B loc k says here is unpersua sive. For there exists a labell ing of thewall such that these indic ative conditiona ls are true of the wall , yet the wall

does not real ise a program or machine table . As Sear le s T heor em shows,i t is simply false that any system that has a set of inputs, outputs and statesre la ted in the way descr ibed realises that machine table . The points Blockough t to empha sise are that the label l ing of the Chinese system is not expost facto and that the physical design o f the system underwri tes a ll therelevant counterfactuals.

There c an be no dispute over the fact that Block s system would c om-pute . I t genuinely runs the a lgor i thm, gen uinely compu tes the funct ioninvolved. The cut t ing edge o f hardware engineer ing consists in the search

for hi ther to unconsidered ways of real is ing computat ional archi tectures(witness the progress ion f rom mercury delay l ines to cathode ray tubestorage to semiconductors) . To match Block s fantas tical com pute r wi thanother: if it turns out that the state transit ions in the digestive syste m o f theChine se s i lkw orm can be manipu lated so that - under a cer ta in label ling -the digestive system forms a practicable and ultra cheap realisation of thearchi tecture-algor i thm specif icat ion commonly used by manufacturers ofhand held calcula tors , then this fact wi l l be se ized upon by engineers eagerto exploi t i t, not regarded as a coun terexample to any account o f com-

putat ion that enta ils i t. Block s hi therto unconside red way of physical lyrealising an a rchitec ture-algo rithm specification is simply that .

Is Block s fantas t ical system real ly a counterexam ple to computat io nalfunct ional ism? No. Th e system is a genuine real isa t ion of SPE C-BL OCK ,


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356 . JACKCOPELAND

but i t is a realisation that cou ld exist onl y in fairyland. I have no firm intu-i t ions abou t fa i ryland, save that one should expe ct the bizarre ; yet Blo ck s

argum ent against funct iona l ism is essent ia l ly an appeal to an intui t ion thathe expec ts us to have concernin g the system, na me ly that it lack[s] men-tali ty (Bl ock 1978, 277). I certa inly hav e no inclina tion to insist that infa i ryland this brain-of-s laves wou ld lack m ental ity.

Seaf le s conce ssion ( in the passage quoted in Sect ion 1) that i t i s poss ible

to b l o ck . . , un iversa l rea li sab il i ty - by which he means b lock the resul ttha t every th ing would be a d igi ta l compu te r (Searle 1992, 2 0 8 ) - seemsto m e to over tu rn h i s m ain a rgument . 9 Here i s Sear le s summ ary of therelevant por t ion o f his argum ent :

The point is not that the claim The brain is a digital computer s simply alse. Rather, t doesnot get up to the levelof falsehood. It does not have a clear sense. The question Is the brain adigital computer? s ill defined. f it asks, Can we assign a computational nterpretation othe brain? the answer s triviallyyes, because we can assign a computational nterpretationto anything. If it asks, Are brain processes intrinsically computational? the answer is

trivially no, because nothing s intrinsicallycomputational . . (Searle 1992, 225)

Sear le offers the c la im that brains are notintrinsicallydigi ta l com put-ers as a cons eque nce of his c la im that syn tax is not in tr insic to physics(Sear le 1992, 208,225 ) . The la t ter c la im is cer ta inly t rue: the labels are notint rins ic to the physics o f the label led device . There are no discrete binarystates in t rins ic to the ph ysics of my Macintosh. Bina ry labels are a t tachedaccord ing to w heth er cer ta in non-discrete var iables (e .g. vol tage, degree ofmagn et isa t ion) fa ll wi thin one or the other o f two ranges . I f the t ruism that

syntax is not in t rins ic to physics impl ies that brains are not in t r ins ical lydigi ta l com puter s then by par i ty i t impl ies that no ent i ty is in t r ins ical lya digi ta l computer. In Sear le s sense there is nothing int r ins ical ly com -putat ional about neuron-f i r ings; and nor is there anything int r ins icallycom putat ional about micro-chip events. Searle is te l l ing us no mo re thanthat i f the brain is a comp uter, then i t is so o nly in the sense in w hich a l lother com puters are com puters . This is hardly interes ting.

I t is Se ar le s f i rs t wa y of in terpret ing the quest ion Is the brain a digi-ta l com pute r? that i s the important one: Ca n we ass ign a comp utat ional

interpreta t ion to the brain? . (Sear le is careful to dis t inguish the quest ionCa n w e ass ign a computat ional in terpreta tion to the brain? f rom the ques-

t ion Ca n the operat ions of the brain be s imulated on a digi tal com pute r?(Sear le 1992, 200) .) H is c la im that the answ er to the que st ion thus in ter-


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WHAT IS COMPUTATION? 3 5 7

pre t e d i s t r i v i a l l y ye s i s mi s t aken : un ive r sa l r ea l i s ab i l i t y i s f a l s e , a s Ih a v e a rg u e d .

T h e r o c k - b o t t o m i s s u e i n c o g n i t i v e s c i e n c e i s p r e c i s e l y w h e t h e r, a n dto wha t ex t en t , the b ra in can b e a s s ig n e d a c o m p u t a t i o n a l i n t e r p r e ta t i o n .I f t he a rgum en t p re sen ted he re i s co r r ec t , t h i s is an empi r i ca l i s sue . I ti s a lways an empi r i ca l ques t i on whe the r o r no t t he re ex i s t s a l abe l l i ngo f s o m e g i v e n n a t u r a ll y o c c u r r i n g s y s t e m s u c h th a t th e s y s t e m f o r m s a nh o n e s t m o d e l o f s o m e a r c h i t e c t u r e - a l g o r i t h m s p e c if i ca t io n , l0 A n d n o t w i t h -s t and ing the t ru i sm tha t syn tax is no t in t r in s i c t o phy s i c s t he d i s co ve r yo f t h is a r c h i t e c t u r e - a l g o r it h m s p e c if i c a ti o n a n d l a b e l l in g m a y b e t h e k e y t ou n d e r s t a n d i n g t h e s y s t e m s o rg a n i s a t io n a n d f u n c t i o n .

NOTES

i I am grateful to Philip Catton, Mike Resnik, John Searle, Krister Segerberg, Kim Sterelnyand Tim Williamson for comm ents on earlier versions o f this material.2 But not the necessity o f Turing s analysisas usually interpreted that is, as embodyingthe so-called Church-Turing thesis. See my 1996b, 1997 and note 3 below.3 Where the weights, thresholds, activation levels etc. are specifiable by means either ofrational numbers o r else real numbers that are computable in Turing s sense of comp utablenumber then the step-by-step procedure is what I call aclassical algorithm; where otherreal numbers may figure as weights etc., the step-by-step procedure is anonclassicalalgo-rithm. A ny Turing-equivalent device can execute only classical algorithms. The class o farchitectures capable o f executing nonclassical algorithms is diverse and includes purelydigital machines. I develop the classical/nonclassical distinction in 1996b and 1997.4 I am g rateful to Philip Catton for drawing Newman s article to my attention. Newman sobjection to Russell is discussed by D emopoulos and Friedman 1985.5 See, for example, Belnap 1996, Segerberg 1996.6 1 owe this suggestion to Philip Catton.7 See also Copeland 1979, 1986 and 1994.8 Chang and Keisler 1973, 66-68.

91 discuss other aspects of Searle s attack on cognitive science in 1993a and 1993b chapters6 and 10.10 For another argument, quite differen t in approach from the one presented here, in supporto f the claim that the question Is the brain a digital computer? is an empirical one, seeProudfoot and Copeland 1994.

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Logic 8, 399-413.


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Copeland, B. J.: 1986, 'W hat is a Semantics for Classical Neg ation ?',MindXC V, 478--90.Copeland, B. J.: 1993a, 'T he Curious Case of the Chinese Gym ',Synthese95, 173-86.Copeland, B. J.: 1993b,Artificial Intelligence: a Ph ilosophical Introduction,Blackwell,

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Goel, V.: 1991, 'Notationality and the Information Processing Mind ',Minds and Machines1, 129-165.

Goel, V.: 1992, 'Are Computational Explanations Vacuous?',Proceedings o f the FourteenthAnnual Conference of the Cognitive Science Society,Erlbaum, Hillsdale, New Jersey, pp.647-52.

Horgan, T. and Tienson, J.: 1991,Connectionism and the Philosophy of Mind,Kluwer,Dordrecht.

Lycan, W. G.: 1981, 'Fo rm, Function, and Feel',Journal o f Philosophy78, 24-50.Lycan, W. G.: 1987, Consciousness,MIT Press, Cambridge, Mass.Newman , M. H. A.: 1928, 'Mr. Russell's Causal Theory of Perception ',Mind37, 137-48.Proudfoot, D. and Copeland, B. J.: 1994, 'Turing, Wittgenstein and the Science of the

Mind', Australasian Journal o f Philosophy72, 497-519.Quine, W. V. O.: 1960, Wordand Object,MIT Press, Cambridge, Mass.Russell, B.: 1927,The Analysis o f Matter,Kegan Paul, Trench, Trubner, London.Russell, B.: 1968,The Autobiography o f Bertrand Russell,Vol. 2, Allen & U nwin, London.Savag e, C. W. (ed.): 1978 ,Perception and Cognition: Issues in the Foundations o f Psychol-

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American Philosophical Association64, 21-37.Searle, J.: 1992, The Rediscovery of the Mind,MIT Press, Cambridge, Mass.Segerberg, K.: 1989, 'Gettin g Started: Beginnings in the Logic of Action'. A tti del Conve gno

Internazionale di Storia della Logica,Le teorie delle modalitd;San Gimignano 5-8dieembre 1987. CLUEB, Bologna, Italy.

Segerberg, K.: 1996, 'To Do and Not To Do', in Copeland 1996a.Smith B. C.: forthcoming,A View rom Somewhere,MIT Press, Cambridge, Mass.Sterelny, K.: 1990,The Representational Theory o f Mind,Blackwell, Oxford.


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Turing, A. M.: 1936, On Computable Numbers, with an Application to the Entscheidung-sproblem , Proceedings of the London Mathematical SocietySeries 2, 42 (1936-37),230--65.

Turing, A. M .: 194 8, Intelligent Machinery , N ational Physic al Laboratory Report,Reprinted in Meltzer, B. and Michie, D. (eds),Machine Intelligence Vol. 5, EdinburghUniversity Press, Edinburgh, pp. 3-23.

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