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Book Review

Exploring and The Unknowable Reviewed by Panu Raatikainen

Exploring Randomness What Gödel proved in 1931 is that in any finitely Gregory Chaitin presented system of mathematical axioms there are Springer-Verlag, 2000 sentences that are true but that cannot be proved ISBN 1-85233-417-7 to be true in the system. Church showed in 1936 176 pages, $34.95 that there is no general mechanical method for deciding whether a given sentence is logically valid The Unknowable or not and, similarly, that there is no method for Gregory Chaitin deciding whether a given sentence is a theorem of Springer-Verlag, 1999 a given axiomatized mathematical theory. Such an ISBN 9-814-02172-5 impossibility proof required an exact mathemati- 122 pages, $29.00 cal substitute for the informal, intuitive notion of a mechanical procedure; Church used his own λ- In the early twentieth century two extremely in- definable functions. Turing arrived independently fluential research programs aimed to establish at the same results at the same time. Moreover, he solid foundations for with the help gave a superior philosophical explication of the con- of new formal logic. The logicism of Gottlob Frege cept of mechanical procedure in terms of abstract and Bertrand Russell claimed that all mathemat- imaginary machines, known today as Turing ma- ics can be shown to be reducible to logic. David chines; this advance made it possible to prove ab- Hilbert and his school in turn intended to demon- solute unsolvability results and to develop Gödel’s strate, using logical formalization, that the use of incompleteness theorem in its full generality. This infinistic, set-theoretical methods in mathemat- identification of the intuitive notion of mechanical ics—viewed with suspicion by many—can never method and an exact mathematical notion is usu- lead to finitistically meaningful but false state- ally called Church’s thesis or, more properly, the ments and is thus safe. This came to be known as Church-Turing thesis. It is the fundamental basis Hilbert’s program. of all proofs of absolute unsolvability. These grand aims were shown to be impossible One of the greatest achievements of modern by applying the exact methods of logic to itself: was certainly the proof by Yuri the limitative results of Kurt Gödel, Alonzo Matiyasevich in 1970, based on earlier work by Church, and Alan Turing in the 1930s revolution- Julia Robinson, Martin Davis, and Hilary Putnam, ized the whole understanding of logic and that the tenth problem of Hilbert’s famous list of mathematics (the key papers are reprinted in [5]). open mathematical problems from 1900 is in fact Panu Raatikainen is a fellow in the Helsinki Collegium for unsolvable; i.e., there is no general method for de- Advanced Study and a docent of theoretical ciding whether a given Diophantine equation has at the University of Helsinki. His e-mail address is a solution or not [11]. This result implies that in [email protected]. any axiomatized theory there exist Diophantine

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equations that have no solution but cannot be a comprehensive survey of the theory and its proved in the theory to have no solution. applications). However, it is now a widespread view, espe- Chaitin was active in developing this approach cially in circles, that certain into a systematic theory (although one should variants of incompleteness and unsolvability not ignore the important contributions by many results by the American computer scientist Gregory others). From the 1970s onwards Chaitin’s inter- Chaitin are the last word in this field. These vari- est has focused more and more on incompleteness ants are claimed to both explain the true reason and unsolvability phenomena related to the notion for Gödel’s incompleteness theorem and to be the of program-size complexity. Indeed, he now ultimate, or the strongest possible, incomplete- says that “the most fundamental application” of ness results. Chaitin’s results emerge from the the theory is in “the new light that it sheds on theory of algorithmic complexity or program-size the incompleteness phenomenon” (The Unknow- complexity (also known as “Kolmogorov com- able, pp. 86–7). plexity”); Chaitin himself was, in fact, one of the It was known from the beginning that program- founders of the theory. size complexity is unsolvable. Chaitin, however, The classical work on unsolvability dealt solely made in the early 1970s an interesting observation: with solvability in principle: one abstracted from Although there are strings with arbitrarily large the practical limits of space and time and required program-size complexity, for any mathematical only finiteness. From the late 1950s onward, how- axiom system there is a finite limit c such that in ever, more and more attention has been paid to dif- that system one cannot prove that any particular ferent kinds of complexity questions—at least in string has a program-size complexity larger than part because of the emergence of computing ma- c [1]. Later Chaitin attempted to extend his “in- chines and the practical resource problems that ac- formation-theoretic” approach to incompleteness companied them. In logic and computer science theorems in order to obtain “the strongest possi- various different notions of complexity have been ble version of Gödel’s incompleteness theorem” ([3], studied intensively. First, computational complex- p. v). For this purpose he has defined a specific ity measures the complexity of a problem in terms infinite “random” sequence Ω. of resources, such as space and time, required to As was noted, one of the major sources that solve the problem relative to a given machine originally motivated the development of the model of computation. Second, descriptive com- theory of program-size complexity, especially in plexity analyzes the complexity of a problem in Kolmogorov’s case, was a problem in the theory terms of logical resources, such as the number of of probability, viz. that of giving a precise and variables, the kinds of quantifiers, or the length of plausible definition for the notion of a random a formula required to define the problem. And fi- string. The problem is related to the paradox of nally, by the algorithmic complexity, or the pro- randomness, which may be explained as follows: gram-size complexity (or ), Assume we are given two binary strings of 20 of a number or a string, one means the size of the digits each, and we are informed that they were shortest program that computes as output that both obtained by flipping a coin. Let these two number or string. strings be: x = 00000000000000000000 Theory of Algorithmic Complexity The basic idea of the theory of algorithmic com- and plexity was suggested in the 1960s independently y = 01001110100111101000. by Ray J. Solomonoff, Andrei N. Kolmogorov, and Gregory Chaitin. Solomonoff used it in his com- Now according to the standard theory of putational approach to scientific inference, probability, these strings are equally probable. Kolmogorov aimed initially to give a satisfactory And yet intuitively one tends to think that x definition for the problematic notion of a random cannot possibly be a randomly generated string— sequence in probability theory, and Chaitin there is too much regularity in it—whereas y first studied the program-size complexity of appears to be genuinely irregular and random Turing machines for its own sake. Kolmogorov and may well be the result of a toss of a coin. The went on to suggest that this notion also provides algorithmic theory of randomness explicates this a good explication of the concept of the informa- idea of regularity with the help of Turing machine tion content of a string of symbols. Later Chaitin programs. One considers a finite string as regular, followed him in this interpretation. Consequently, or nonrandom, if it can be generated by a simple the name “algorithmic information content” has program, i.e., if its program-size complexity is frequently been used for program-size complex- considerably smaller than its length. Accordingly, ity, and the whole field of study is very often a finite string is defined to be random if its called “algorithmic ” ([10] is program-size complexity is roughly equal to its

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length, i.e., if it cannot be compressed to a shorter language provides exactly what is needed, so he program. (Note that this notion is relative to a invented a new version of LISP. Chaitin begins The chosen programming language or coding system; Unknowable by saying that what is new in the book a finite string may be random in one but nonran- is the following: “I compare and contrast Gödel’s, dom in another.) Turing’s and my work in a very simple and straight- Extending this approach to infinite strings forward manner using LISP” (p. v). According to turned out to be, however, more difficult than was Chaitin this book is a “prequel” to his previous thought. Kolmogorov’s first book, The Limits of Mathematics (Springer-Verlag, idea was that an infinite string 1998), and is an easier introduction to his work be considered random if all of on incompleteness. In Exploring Randomness its finite initial segments are Chaitin in turn writes that “[t]he purpose of this random. But Per Martin-Löf book is to show how to program the proofs of the showed that this definition does main theorems about program-size complexity, not work and then gave a more so that we can see the in these proofs satisfactory definition in mea- running on the computer” (p. 29). sure-theoretic terms. In 1975 Such an approach may be attractive for pro- Chaitin presented a definition gramming enthusiasts and engineers, but for the (which is equivalent to Martin- rest of us its value is less clear. It is quite doubt- Löf’s definition) in terms of ful whether it manages to increase the under- program-size complexity: An standing of the basic notions and results and infinite string is defined to be whether it really makes the fundamental issues, random if the program-size which are rather theoretical and conceptual, more complexity of an initial segment accessible. All these can be, and have been, ex- of length n does not drop arbi- plained quite easily and elegantly in terms of sim- trarily far below n [2]. One ply describable Turing machines, and it is ques- should add that it is not indis- tionable that it is really easier to understand them putable that this really provides by first learning Chaitin’s specially modified LISP in all respects an unproblematic and then programming them. And after all, the key explication of the notion of randomness. point here is that there is no program for decid- Also in 1975 Chaitin presented for the first ing the basic properties—that they are not pro- time his (since then much-advertised) number Ω: grammable. In The Unknowable (p. 27) Chaitin it is the halting probability of the universal says that “[r]eaders who hate computer program- Turing machine U, i.e., the probability that U halts ming should skip directly to Chapter VI”—that is, when its binary input is chosen randomly bit by should skip half of the book (and similarly with Ex- bit, such as by flipping a coin. The infinite string ploring Randomness). What is left is two popular Ω is, according to the above definition, random [2]. surveys of the field. Somewhat analogously to his earlier incomplete- What is totally missing from Chaitin’s accounts ness result, Chaitin has demonstrated that no is the link between his particular programming axiomatic mathematical theory enables one to language and the intuitive notion of mechanical determine infinitely many digits of Ω ([3], [4]; procedure, that is, an analogue of the Church- cf. [6], [8]). Turing thesis. Turing’s conceptual analysis of what a mechanical procedure is and the resulting Chaitin’s New Approach via LISP Programs Church-Turing thesis are indispensable for the This is the general theoretical background of the proper understanding of the fundamental books under review. How do these two new books unsolvability results: only the thesis gives them by Chaitin relate to these older works? In terms of their absolute character (in contradistinction to results there is hardly anything new compared to unsolvability by some fixed, restricted methods). the older work by Chaitin and others reviewed Consequently, without some extra knowledge, above. Rather, these books aim to popularize that the theoretical relevance of the limitations of the work. In The Unknowable the emphasis is on the LISP programs that Chaitin demonstrates may incompleteness phenomena related to program- remain unclear to the reader: one may wonder size complexity. Exploring Randomness aims to whether perhaps some other programming explain the program-size complexity approach language would do better. This is a serious to randomness. weakness of these presentations as first intro- What is new is Chaitin’s approach via LISP- ductions to the unsolvability phenomena. programming; he has “translated” his own earlier work, which was in terms of abstract, Problematic Philosophical Conclusions idealized Turing machines, into actual programs The most controversial parts of Chaitin’s work in the LISP computer language. Well, not exactly: are certainly the highly ambitious philosophical according to Chaitin, no existing programming conclusions he has drawn from his mathematical

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work. Recall that according to Chaitin the most but their relevance for the foundations of mathe- fundamental application of the theory is in the matics has been greatly exaggerated. new light that it sheds on the incompleteness Further, Chaitin has often stated that he has phenomenon. He writes: “Gödel and Turing were shown that mathematical truth is random: “But the only the tip of the iceberg. AIT [algorithmic infor- bits of this number Ω, whether they’re 0 or 1, are mation theory] provides a much deeper analysis mathematical truths that are true by accident! of the limits of the formal axiomatic method. It …they’re true by no reason…there is no reason that provides a deeper source of incompleteness, a individual bits are 0 or 1!” (Ex- more natural explanation for the reason that no ploring Randomness, pp. 23–4) finite set of axioms is complete” (Exploring This is false too. The individual Randomness, p. 163). bits of Ω are 0 or 1 depending But why does Chaitin think so? It is because on whether certain Turing ma- he interprets his own variants of incompleteness chines halt or not—that is the theorems as follows: “The general flavor of my reason. It is an objective matter work is like this. You compare the complexity of of fact; the truth here is com- the axioms with the complexity of the result you’re pletely determined, and trying to derive, and if the result is more complex Chaitin’s interpretation of the than the axioms, then you can’t get it from those situation is quite misleading. axioms” (The Unknowable, p. 24). Or, in other Chaitin also claims that Ω is “maximally unknowable” and words: “my approach makes incompleteness more that in his setting one gets in- natural, because you see how what you can do completeness and unsolvability depends on the axioms. The more complex the “in the worst possible way” (Ex- axioms, the better you can do” (The Unknowable, ploring Randomness, p. 19). But p. 26). contrary to what Chaitin’s own But appearances notwithstanding, this is sim- interpretations suggest, his re- ply wrong. In fact, there is no direct dependence sults can in fact be derived as between the complexity of an axiom system and quite easy corollaries of Tur- its power to prove theorems. On the one hand, ing’s classical unsolvability result and are not es- there are extremely complex systems of axioms sentially stronger than it. Moreover, there are many that are very weak and enable one to prove only incompleteness and unsolvability results in the trivial theorems. Consider, for example, an enor- literature of mathematical logic that are in various mously complex finite collection of axioms with the ways stronger than Chaitin’s results; many of them form n

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he is dealing with. It is regrettable that Chaitin does of these matters, it is questionable whether these not respond to criticism of his work but simply books are really a good place to start. evades difficult questions and keeps on writing as (There is an errata for Exploring Randomness if they did not exist. Chaitin’s own attitude begins on Chaitin’s home page: http://www.cs. to resemble the dogmatism he accuses his oppo- auckland.ac.nz/CDMTCS/chaitin/). nents of. At worst, Chaitin’s claims are nearly megalo- References maniacal. What else can one think of statements [1] GREGORY J. CHAITIN, Randomness and mathematical such as the following?: “AIT will lead to the major proof, Sci. Amer. 232:5 (1975), 47–52. [2] , A theory of program size complexity for- breakthrough of 21st century mathematics, which ——— mally identical to information theory, J. Assoc. Com- will be information-theoretic and complexity based put. Mach. 22 (1975), 329–340. characterizations and analyses of what is life, what [3] ——— , Algorithmic Information Theory, Cambridge is mind, what is intelligence, what is , University Press, Cambridge, 1987. of why life has to appear spontaneously and then [4] ——— , Randomness in arithmetic, Sci. Amer. 259:1 to evolve” (Exploring Randomness, p. 163). (1988), 80–85. [5] MARTIN DAVIS (ed.), The Undecidable, Raven Press, Problems of Popularization New York, 1965. [6] JEAN-PAUL DELAHAYE, Chaitin’s equation: An exten- The historical surveys of the theory of program- sion of Gödel’s theorem, Notices Amer. Math. Soc. size complexity that Chaitin gives are sometimes 36:8 (1989), 984–987. rather distorted. Some have even complained that [7] PETER GACS, Review of Gregory J. Chaitin, Algorith- Chaitin is “rewriting the history of the field” and mic Information Theory, J. Symbolic Logic 54 (1989), “presenting himself as the sole inventor of its main 624–627. concepts and results” [7]. This complaint also fits [8] , Mathematical games: The random to a considerable degree the present books. They number Omega bids fair to hold the mysteries of the are quite idiosyncratic. universe, Sci. Amer. 241:5 (1979), 20–34. [9] MICHIEL VAN LAMBALGEN, Algorithmic information the- The style of these books is very loose and pop- ory, J. Symbolic Logic 54 (1989), 1389–1400. ular. Large parts of the text are directly transcribed [10] MING LI and PAUL VITANYI, An Introduction to Kol- from oral lectures and include asides like “Thanks mogorov Complexity and Its Applications, Springer- very much, Manuel! It’s a great pleasure to be Verlag, New York, 1993. here!” It is perhaps a matter of taste whether one [11] YURI V. MATIYASEVICH, Hilbert’s Tenth Problem, MIT finds this entertaining or annoying. Personally, I Press, Cambridge, MA, 1993. don’t think that this is a proper style for books in [12] PANU RAATIKAINEN, On interpreting Chaitin’s incom- a scientific series. It certainly does not decrease the pleteness theorem, J. Philos. Logic 27 (1998), 269–586. sloppiness of the text. [13] ——— , Algorithmic information theory and unde- The books bring together popular and intro- cidability, Synthese 123 (2000), 217–225. ductory talks given on different occasions, and some of this material has already appeared else- where. There is also a lot of redundancy and over- lap between the books. Both books (as well as the earlier The Limits of Mathematics) start with a loose and not very reliable historical survey— Chaitin himself calls it “a cartoon summary”— beginning with Cantor’s set theory, going through Gödel’s and Turing’s path-breaking results, and culminating, unsurprisingly, in Chaitin’s own work. All three books then present an introduction to LISP and Chaitin’s modification of it. Both The Un- knowable and Exploring Randomness contain a more theoretical section on algorithmic informa- tion theory and randomness. And finally, both books end with rather speculative remarks on the future of mathematics. Would it have been better to do some editing and publish just one book instead of three? To a considerable degree, Exploring Randomness and The Unknowable just recycle the same old ideas. Consequently, for those with some knowl- edge of this field, these books do not offer anything really new. For those with no previous knowledge

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