What Is a Random Sequence? Sergio B. Volchan 1. INTRODUCTION. Whatis randomness?Are thererandom events in nature?Are therelaws of randomness? Theseold anddeep philosophical questions still stircontroversy today. Some schol- ars have suggestedthat our difficultyin dealing with notions of randomnesscould be gaugedby the comparativelylate developmentof probabilitytheory, which had a somewhathampered development [20], [21]. Historiansgenerally agree upon the year 1654 as a convenientlandmark for the birthof mathematicalprobability. At thattime, some reasonablywell-articulated ideas on the subjectwere advancedin the famous correspondenceof Pascaland Fermat regarding the divisionof stakesin certaingames of chance.However, it was only in 1933 that a universallyaccepted axiomatization of the theorywas proposedby A. N. Kolmogorov[28], with many contributionsin between.That is, almostthree hundred years after its beginnings,and a hundredyears afterCauchy's work on the rigorizationof analysis,probability theory finally reached maturity.It achievedthe statusof an autonomousdiscipline of puremathematics, in- steadof being viewed as a mixedbranch of appliedmathematics and physics. In contrast,the uses of notions of randomnessare as old as civilizationitself. It appearedin a varietyof gamesof chance(coin-tossing, dice, etc.), as well as in divina- tion, decision-making,insurance, and law. Manyreasons for this discrepancybetween theoryand applicationhave been put forward.One suggestionis that a full develop- ment of the theory,going beyond combinatorics,had to wait for the creationof the very sophisticatedmathematical tools andconcepts of set theoryand measuretheory. A moreplausible reason could be thatour cognitive(and even psychological)consti- tution,which mighthave evolvedto look for patternsand trends even wherethere are none, is not well suitedto grasprandomness.1 In supportof that last idea, many psychologicalstudies have shown that people (even experts)perform poorly when using intuitionto deal with randomness[2]. One classical example2is the 'gambler'sfallacy': the common (false) belief that, after a sequenceof losses in a gameof chance,there will follow a sequenceof gains,and vice versa,in a kindof self-compensation. Whatare the characteristicsusually associatedwith randomness?A commonidea is to identifyrandomness with unpredictability.This intuitionoriginates in people's experiencewith games of chance.For example,a sequenceof coin tosses looks very irregular,and no matterhow manytimes we've tossed the coin, say a thousandtimes, no one seems to be ableto predictthe outcomeof the next toss. Thatarguably explains the widespreaduse of randomizingdevices, like coins, dice, and bones, to guarantee fairness in gambling3and decision-making. However,one could questionwhether these are examplesof "really"random phe- nomena.After all, actualcoin-tossing (for example)is a purelymechanical process, governedtherefore by Newton'slaws of motion.Hence, its outcomeis as predictable, 'The mathematicianEmile Borel claimed the human mind is not able to simulate randomness [34]. 2A more subtle one is the Monty Hall problem (see Snell and Vanderbei[32]). 3The development of the mathematicalanalysis of games of chance seems to have been motivated not only by the desire to devise winning strategiesfor the games but also by the desire to detect fraud in them [4]. 46 ? THE MATHEMATICALASSOCIATION OF AMERICA [Monthly109 in principle, as the motion of the planets,once the initial conditionsare given.4The observedunpredictability results from a peculiarcombination of circumstances[24], [49]. First, there is a kind of "instability"built into the system, of the kind usually associatedwith meteorologicalsystems; i.e., it is a dynamicalsystem displaying sen- sitive dependenceon (some set of) initialconditions. That, coupled with our inability to knowthese conditionswith infiniteprecision, results in unpredictabilityin practice, even thoughthe processis totallylawful in the sense of classicalmechanics. In other words,we have an instanceof the phenomenonof "chaos." It is reasonableto ask whetherthere are "intrinsic"(or ontological)notions of ran- domness.The usual suggestionis the notion of "lawlessness,"also conceived of as "disorder,""irregularity," "structurelessness," or "patternlessness,"that we'll discuss later.It certainlyincludes unpredictability in some sense. But one needs to be care- ful here. To begin with, one needs to distinguishbetween "local"irregularity versus "global"(or statistical)regularities observed in many chancephenomena [43], [42]. Forexample, although we cannotpredict the outcomeof individualcoin tosses, it is an empiricalfact thatthe proportionof heads (or tails) obtainedafter a great numberof tosses seems to convergeto or stabilizearound 0.5. Besides, thereis a recurrentsense of paradoxlingering in the enterpriseof lookingfor laws thatgovern lawlessness [11]: after all, this last propertyseems to mean exactly the absenceof any subjugationto laws. Concerningrandomness in naturalphenomena, it is not quiteclear what one should look for. Conceivably,some quantummechanical phenomenon, like radioactivede- cay [23], would be a good candidateto investigate.In this paperwe won't discuss this very importanttopic. We will focus insteadon the admittedlyless ambitiousbut moremanageable question of whetherit is possibleat least to obtaina mathematically rigorous(and reasonable)definition of randomness.That is, in the hope of clarify- ing the conceptof chance,one tries to examinea mathematicalmodel or idealization thatmight (or mightnot) capturesome of the intuitiveproperties associated with ran- domness.In the processof refiningour intuition and circumscribing our concepts,we might be able to arriveat some fundamentalnotions. With luck (no pun intended), these mightin turnfurnish some insightinto the deeperproblems mentioned. At the very least it could help one to discardsome previousintuitions or to decide upon the need for yet anothermathematical model. The historyof mathematicsshows that this strategyis frequentlyfruitful. An ex- ample of this processwas the clarificationof the concept of 'curve'.Not only did it lead to the discoveryof "pathologicalcurves" (which are interesting mathematical ob- jects in themselves,linked to fractalsand Brownian motion) but also to the realization thatsmoothness is a reasonablerequirement in the formalizationof the intuitivenotion of curve [19]. Anotherexample, which is centralto our discussion,was the clarifica- tion of the intuitivenotion of computability(see the next section). Of course,this is not an easy task. The proposedmodel or idealizationshould be simple,without also being totallytrivial. One idea is to consideran abstractionof the coin-tossingexperiment, the so-calledBernoulli trials. Representing the occurrenceof headsby 0 and tails by 1, we associatea binarystring to each possible outcomeof a successivecoin-tossing experiment. We then ask:When is a binarystring random? To appreciatethe difficultiesinvolved, let's examine the "paradoxof random- ness" [14]. It goes like this. Supposeyou toss an honest coin repeatedly,say twenty- threetimes. Considerthe followingoutcomes: 4The mathematicianand formermagician Persi Diaconis was able consistently to get ten consecutive heads in coin-tossing by carefully controlling the coin's initial velocity and angularmomentum. January2002] WHAT IS A RANDOM SEQUENCE? 47 * OOOOOOOOOOOOOOOOOOOoo * 01101010000010011110011 * 11011110011101011111011. The firstresult is generallyconsidered suspect, while the secondand third "look" ran- dom. However,according to probabilitytheory all threeoutcomes, and in fact all the 223 possibleoutcomes, have the same probabilityof 1/223. Why, then, do the last two outcomesseem randomwhile the firstdoes not? It is conceivablethat the ultimatereason for thatperception "belongs to the domain of psychology"[29], to be found in the structureof our visual-cognitiveapparatus. Such issues notwithstanding,the questionis whetherit is possible to distinguishran- dom fromnonrandom strings in a mathematicallymeaningful way. Note thatour intu- ition cannotbe trustedmuch in this task. It's enoughto observethat the second string aboveconsists of the firsttwenty-three digits of the binaryexpansion of VX - 1. So, althoughit "looks"random, in the sense of exhibitingno obvious pattern,its digits were obtainedby a process (root extraction)that, by all reasonablestandards, is not random.Note the overallsimilarity with the thirdstring, obtained by coin-tossing. Forstrings it is only possibleto developa notionof degreesof randomness,there be- ing no sharpdemarcation of the set of all stringsinto randomand nonrandom ones [7]. In fact, once a certainbinary string with m zeroes is consideredrandom, there is no reasonnot to considerequally random the stringobtained by adding(or subtracting) one morezero to it (or fromit). The situationbecomes clearerif one considersinstead the set of all infinitebinary strings,or sequencesof bits. Althoughin real life applicationswe are bound to en- counteronly finite, albeit very long, strings,it is neverthelessworth considering this furtheridealization. The idea of takinginfinite objects as approximationsto finite'but very largeones is not new. For example,in equilibriumstatistical mechanics, in order to have a sharpnotion of a phasetransition one has to workin the so-calledthermody- namiclimit, in whichthe numberof particlestends to infinity(as does the volume,but in sucha way thatparticle density remains constant).5 The greatadvantage