Algorithms, nymphs, and shepherds

Computability and complexity are the two major lines along which the science of algorithms has evolved; but the same concepts guide many human activities. We try to catch some glimpses of the connections between these two worlds, to meet the expectations of the audience. We will encounter nymphs and universal Turing Machines, ancient traditions and randomized procedures, all tending to the same end: having fun with algorithms. c © 2002 Elsevier Science B.V. All rights reserved. Although the formalization of algorithms was essentially unknown before the present century, a precise organization of actions has played an important role in the human behavior much before landing in mathematics. Clearly a similar phenomenon has occurred in all sciences. With the great profundity found in his entire work, Ernst Mach insisted that scienti2c thought derives from common thought of people; and that the development of scienti2c thought consists of a continuous and punctual correction of common thought [5]. The integration between scienti2c and common thought is indeed strong in the 2eld of algorithms, although scientists seem to have been too busy to notice this point (Mach himself made only a few remarks about calculus, but at his times algorithmica was still to be born). We will move in this direction without pursuing philosophic claims, with the sole aim of enlightening the unplanned relevance of some mathematical concepts in human (and, as we shall see, divine) behavior. Due to space constraints, we will restrict our discussion to two speci2c topics: the role of self-reference in computing a function and the growth of con2dence in algorithmic results for an increasing number of independent testimonies. 1. Choe and the diagonal language “Cretans lie”. This ancient statement stroke the bases of thought more than the population of Crete (that, incidentally, was known for its sense of humor). Epimenides E-mail address: [email protected] (F. Luccio). 0304-3975/02/$ see front matter c © 2002 Elsevier Science B.V. All rights reserved. PII: S0304 -3975(01)00053 -6 224 F. Luccio / Theoretical Computer Science 282 (2002) 223–229 of Cnossus is credited as being the author of the provocation, as testi2ed even by Apostole Paul (Titus I-12). A Cretan himself, Epimenedes thus introduced self-reference into the statement: this would have not borne any consequence if not mixed with negation (lie), as it instead was. In fact this disruptive combination gave rise to a famous paradox: if the sentence were true (Cretans indeed lie) the author would have laid, thereby implying that Cretans do not lie; and vice versa. The same combination of self-reference and negation has been used along centuries for many other paradoxical constructions, gradually moving from philosophic speculations to mathematical logic. The most striking example of application in the latter area is GD odel’s proof that any formal system containing the arithmetic of natural numbers is incomplete [3]. With a clever construction of a mathematical environment, GD odel gave a formula on natural numbers essentially equivalent to the statement: “This statement cannot be proved”. The statement is provable in a chosen formal system if and only if it is false in the same system. Therefore either we prove something untrue (that is clearly contradictory) or the system is incomplete in the sense that contains statements that are true but non-provable. However, before proceeding into the theory of computing we shall again search through the roots of our knowledge. Self-reference recalls the sorrowful existence of Echo and Narcissus, in one of the most delicate stories of Greek mythology. Narcissus was a beautiful youth, but inaccessible to love. Nemesis caused him to walk to a clear fountain where he could see his own image reHected in the water; and Narcissus became so enamoured of himself that he drowned trying to reach his image. Echo (a nymph of questionable honor, as she used to divert the attention of Hera while Zeus was having fun with the other nymphs) had a pure feeling of love for Narcissus. When he died she pined away in grief, leaving to us, as sole remains, her voice changed into an echo. Narcissus loved himself in such a lyrical manner to induce us to speculate on the nature of love: a subtle question thoroughly examined by Plato [7]. Following his analysis we must preliminarily investigate the development of mankind, with crucial attention to the geometric shape of their bodies. Originally each person was round all over, with four arms and legs, two faces perfectly alike, two privy members, and all the other parts in proportion. So there were three sexes: man–man (the male), woman–woman (the female), and man–woman (the androgynous, sharing equally in male and female). Their round shape gave them incredible vigor and speed, so they got to conspire against the gods. In punishment Zeus ordered to split them in two halves. Since then each human being seeks desperately the missing half. In Plato’s own words: “thus anciently is mutual love ingrained in mankind, reassembling our early estate and endeavouring to combine two in one and heal the human sore”. Now all this explains the diKerent natures of love. The halves of the original male, or of the female, search and love beings of their own sex. The halves of the androgynous, instead, search the missing half in the opposite sex, until they 2nd it in happiness, or proceed for ever in an adulterous life. Although the sympathy of Plato does not seem to be directed to the third gender, much can be learned from heterosexual behavior. While arrogance F. Luccio / Theoretical Computer Science 282 (2002) 223–229 225 Fig. 1. Part of the in2nite matrix of mythological loves. has condemned mankind to the perpetual search of a lost nature, the androgynous was happy in his completeness. Much earlier than Narcissus, the androgynous loved himself. Let us now return to Echo. She loved Narcissus with sincerity, but her questionable behavior in the Zeus business may induce a legitimate suspicion of additional, more relaxed sentiments. So we extend the de2nition of the nymph to one who loves all the ones who love themselves (Narcissus and the androgynous, among others). But does Echo love herself? The question is logically harmless: since self-reference is not mixed with negation, both answers, yes or no, are consistent with the de2nition of Echo. The paradox arises with a nymph of opposite feelings: Choe (pronounce ′k M o −′ e), who loves the ones who do not love themselves. Now the question of whether Choe loves herself or not is autocontradictory, as in Epimenides’ paradox. To depict the whole situation we make use of the matrix of mythological loves (Fig. 1) whose interest is enhanced by the numerous loving habits of those creatures [1] and to which Cantor’s diagonal argument applies naturally. Deities were numerable but virtually in2nite in number, as they were personages of past anecdotes and tales yet to be told, materializations of natural events, protectors of homes and 2elds, inhabitants of fountains, trees and mountains. Each one with in2nite variations according to place 226 F. Luccio / Theoretical Computer Science 282 (2002) 223–229 or time. Fig. 1 shows only a part of an in2nite table, where Y , or N , appears in cell (i; j) to indicate that the creature in row i loves, or does not love, the creature in column j. Here we 2nd reciprocal eternal loves, as the one of nymph Arethusa merging with Alpheus, god of rivers, in the conHuence of two streams in Siracusa. Alpheus has Y only for Arethusa, and Arethusa has Y only for Alpheus. Non-reciprocal violent loves, as the one of Zeuss in the form of a bull who raped Europa. Y and N are non-symmetrical in this case. The reHexive loves of Narcissus and Androgynous, with a Y on the diagonal of the matrix. The love of Echo for them and only for them. Blank cells of the matrix may denote lack of information on our side, as in the case of the cell (Echo, Echo) where both Y and N would be legitimate. But blanks also denote a subtler situation. Apollo, for example, may regularly go round with Europa without yet having a precise feeling for her. Nobody knows if he will eventually decide for Y or N , or go on for ever. A blank here denotes that a procedure is still running. And now, what is the position of Choe into the matrix? A diagonal argument shows that Choe cannot be a member of the celestial company, because her row would not have a consistent entry at the intersection with the diagonal. That is, although the behavior of Choe is well de2ned, she could exist only as a living paradox (this is probably the reason why Choe is not found in any book of mythology). If we insist to be living in a world free of paradoxes Choe does not exist, although her behavior does. Indeed behaviors are functions D → {Y; N; }, where D = { | is a deity} is the in2nite set of deities, and denotes a blank; and deities are Turing Machines. In the matrix, behaviors are represented as rows of in2nite length, each assumed by the deity speci2ed in the row label. Unlike labels, all possible rows are not numerable: it is then not surprising that there exist legitimate behaviors that cannot be assumed by anybody. Mathematicians have certainly realized that Choe is the (non-existing) Turing Machine that should recognize the diagonal language Ld (see [4]) Ld = { | ∈ D; does not love }: From this we can rephrase a classical result in terms of mythology. Let Lu = { | and ∈ D; loves } be the universal language. We have Theorem 1. Lu is recursively enumerable and non-recursive. Proof. Let a deity of deities be able to understand if another deity will eventually fall in love with , for arbitrary and . (We like to think that is indeed Hera, mother of all deities and always interested in love stories.) can induce and to meet, and watch the behavior of . If gives clear signs of enamourement, or disgust, decides for Y or N . This is enough to prove that Lu is recursively enumerable. However, if lingers, may not decide by simple observation if is going to eventually develop F. Luccio / Theoretical Computer Science 282 (2002) 223–229 227 a stable sentiment, or keeps in suspense for ever. Nevertheless, we now prove by contradiction that has no other means for deciding. Assume that can decide in 2nite time if falls or not in love with , or will linger inde2nitely. and being arbitrary, can also decide if falls in love with . Complementing the output (where N and perpetual lingering are both seen as negative answers), would recognize Ld thereby taking the role of Choe, hence proving her existence. Such subtle arguments, we believe, should rightfully induce the sensation of being fooled in any person of good sense. The responsibility of such an unpleasant condition is totally on the side of the mathematicians, whose arguments are nevertheless to be accepted because they are always validated by distinguished persons. 2. Massud and primality testing “Nihil enim est tam contrarium rationi et constantiae quam fortuna, ut mihi ne in deum quidem cadere videatur ut sciat quid casu et fortuito futurum sit”. (Nothing indeed is as contrary to reason and regularity as chance. In fact not even the gods, I believe, know what will happen by chance and fortune) [2]. Cicero underlined the power of chance with these words. Twenty centuries later, computer scientists started using that power to solve diPcult problems in a new original manner. The story of chance is not completely clear. Democritus is often credited as being the initiator, although no ancient thinker was possibly more mechanistic than he was. Only in the sixteenth century Cardano picked up the concept again, until a correspondence between Pascal and Fermat on the game of dice, one century later, gave oPcial birth to the study of probability, and raised the discussion on chance again. Philosophers, biologists, physicists have been deeply interested since then in the habits of this crazy personage who mixes up a well ordered universe. They even discussed ardently if chance existed at all (Carl Gustav Jung had no doubt of its existence as a threat to rational behavior). Mathematicians went on more slowly: for long time they were unable to satisfactorily formalize the rules of the game, until, a few decades ago, Kolmogorov laid down the bases of a sound probability theory. Discussing this matter is not our purpose. We will illustrate, instead, how the power of chance can be exploited to give satisfactory answers to important and diPcult questions. The mathematical aspects of randomization can be studied in [6]. We examine a more concrete fact taken from an ancient chronicle of Central Asia (oral tradition). Massud, a young shepherd of the steppe, is consumed by the desire of knowing if Pardis, his betrothed, is beautiful as all their relatives swear. By modesty and tradition Pardis always appears in public with a veiled face, and only women can see her in privacy. Massud wishes to inquire delicately with some of them, but speculates on who are the right women to ask. He fears the opinion of relatives to be exceedingly 228 F. Luccio / Theoretical Computer Science 282 (2002) 223–229 positive; the judgment of Fatima (the belly dancer who used to be his secret girl friend) to be exaggeratedly negative; the opinions of persons of similar taste and habits to be meaninglessly similar. In essence, Massud wishes to collect reliable opinions from independent witnesses chosen at random. To proceed with scienti2c method, Massud establishes some rules based on his knowledge of human psychology. A woman’s opinion that Pardis is unpleasant to the sight must unfortunately be taken for sure. An opinion that Pardis is beautiful, instead, has to be trusted with a certain probability p. In the 2rst case Massud will terminate his enquiry, trying to escape his commitment for marriage without major damages. In front of a positive opinion, instead, he will continue the enquiry with other women, until he is convinced that his betrothed is really beautiful. The algorithm is as follows: Algorithm 1. TEST FOR BEAUTY choose a positive integer k to establish the level of con8dence on the answer while less than k women have expressed an opinion do choose a new woman W independently at random if OPINION(W;Pardis)= ugly then return NO (and run away) else continue return YES (and stay).