# Gregory J. Chaitin's research while affiliated with Federal University of Rio de Janeiro and other places

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## Publications (96)

This note gives some information about the magical number Ω and why it is of interest. Our purpose is to explain the significance of recent work by Calude and Dinneen attempting to compute Ω. Furthermore, we propose measuring human intellectual progress (not scientific progress) via the number of bits of Ω that can be determined at any given moment...

This note gives some information about the magical number and why it is of interest. Our purpose is to explain the signicance of recent work by Calude and Dinneen attempting to compute . Furthermore, we propose measuring human intellectual progress (not scientic progress) via the number of bits of that can be determined at any given moment in time...

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel (1871-1956).

In this essay we present an information-theoretic perspective on epistemology using software models. We shall use the notion of algorithmic information to discuss what is a physical law, to determine the limits of the axiomatic method, and to analyze Darwin’s theory of evolution. Weyl, Leibniz, complexity and the principle of sufficient reason The...

This essay is intended as a contribution to The Once and Future Turing edited by S. Barry Cooper and Andrew Hodges, which will be published by Cambridge University Press as part of the Turing Centenary celebration.

Leibniz in 1686 in his Discours de mtaphysique points out that if an arbitrarily complex theory is permitted then the notion
of theory” becomes vacuous because there is always a theory. This idea is developed in the modern theory of algorithmic information,
which deals with the size of computer programs and provides a new view of Gdel’s work on inc...

Turing's famous 1936 paper \On computable numbers, with an application to the Entschei- dungsproblem" denes a computable real num- ber and uses Cantor's diagonal argument to ex- hibit an uncomputable real. Roughly speaking, a computable real is one that one can calculate digit by digit, that there is an algorithm for approximating as closely as one...

We propose using random walks in software space as abstract formal models of biological evolution. The goal is to shed light on biological creativity using toy models of evolution that are simple enough to prove theorems about them. We consider two models: a single mutating piece of software, and a population of mutating software. The fitness funct...

Using work of Hao Wang, we exhibit a tiling characterization of the bits of the halting probability.

Some Gödel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Introduction: What is mathematics? It is a pleasure for me to be here today giving this talk in a lecture series in honor of Frederigo Enriq...

Using 1947 work of Post showing that the word problem for semi- groups is unsolvable, we explicitly exhibit an algebraic characteriza- tion of the bits of the halting probability . Our proof closely follows a 1978 formulation of Post's work by M. Davis. The proof is self- contained and not very complicated.

We discuss the halting probability Ω, whose bits are irreducible mathematical facts, that is, facts which cannot be derived from any principles simpler than they are. In other words, you need a mathematical theory with N bits of axioms in order to be able to determine N bits of Ω. This pathological property of Ω is difficult to reconcile with tradi...

We show that unlike the general case of the relationship between algorithmic probability and program-size for enumerating sets, in the case of the graphs of total functions these two quantities are closely related.

We discuss mathematical and physical arguments against continu- ity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871-1956).

In 1686 in his Discours de Metaphysique, Leibniz points out that if an arbitrarily complex theory is permitted then the notion of "theory" becomes vacuous because there is always a theory. This idea is developed in the modern theory of algorithmic information, which deals with the size of computer programs and provides a new view of Godel's work on...

We discuss how to compute the halting probability Omega in the limit in

By using ideas on complexity and randomness originally suggested by the
mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of
algorithmic information is able to show that there can never be a "theory of
everything" for all of mathematics.

This paper, which is dedicated to Alan Turing on the 50th anniversary of his death, gives an overview and discusses the philosophical implications of incompleteness, uncomputability and randomness.

This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.

The information-theoretic point of view proposed by Leibniz in 1686 and developed by algorithmic information theory (AIT) suggests that mathematics and physics are not that different. This will be a first-person account of some doubts and speculations about the nature of mathematics that I have entertained for the past three decades, and which have...

In his celebrated 1936 paper Turing defined a machine to be circular iff it performs an infinite computation outputting only finitely many symbols. We define ( as the probability that an arbitrary machine be circular and we prove that is a random number that goes beyond $2, the probability that a universal self alelimiting machine halts. The algori...

We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as the halting probability of a universal machine equip...

In discussions of the nature of life, the terms "complexity," "organism, " and "information content," are sometimes used in ways remarkably analogous to the approach of algorithmic information theory, a mathematical discipline which studies the amount of information necessary for computations. We submit that this is not a coincidence and that it is...

Efforts to calculate values of the noncomputable Busy Beaver function are discussed in the light of algorithmic information theory.

An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the di#culty o...

Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. I'll discuss how one can use this complexity measure to study what can and cannot be achieved by formal axiomatic mathematical theorie...

Two philosophical applications of the concept of program-size complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second, we propose that the notion of algorithmic independence sheds light on the question of being and how the world...

We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that comprehension is compression, i.e., explaining many facts using few theoretical assumptions, and that a theory may be vie...

We consider the notion of randomness relative to an oracle: a real number is random in A if and only if its initial segments are algorithmically incompressible in a self-delimiting universal machine equipped with an oracle A. We prove that the probability that a program for infinite computations outputs a cofinite set is random in the second jump o...

We recently celebrated the hundredth birthdays of the discovery of the electron, and of the publication of Planck’s paper that started the quantum revolution.1 Who knows what this new century will bring?!

[Copenhagen, 1996, and my second TV interview. Tor Nørretranders is a Danish science journalist and intellectual who is extremely well-known in Denmark. Only one of his many books, the Danish best-seller Mark Verden, on information theory and consciousness, is available in English, as The User Illusion. This interview was part of Tor’s fabulous Min...

[A “Milênio” (Millennium) program interview (interviews with thinkers who are shaping the third millennium) broadcast by Globo News TV in Brazil in June 2001. This is my third and latest TV interview, and it was filmed at my home, mostly in the back garden, and begins and ends with a close-up of the cover of my book The Unknowable. The Globo News T...

I have shown that God not only plays dice in physics, but even in pure mathematics, in elementary number theory, in arithmetic! My work is a fundamental extension of the work of Gödel and Turing on undecidability in pure mathematics. I show that not only does undecidability occur, but in fact sometimes there is complete randomness, and mathematical...

[This lecture was given at a three-day meeting on math, physics and theology at the Technical University of Vienna where I was welcomed in 1991 as Gödel’s successor by Hans-Christian Reichel. The Vienna newspaper Der Standard printed my photo with a full-page article by John Casti entitled (in German) “Out Gödeling Gödel”, and Casti’s article was b...

[Interview on philosophy, Annandale-on-Hudson, March 2001. Kate Mullen is an undergraduate at Bard College, and she published the bulk of this interview in the Bard Free Press, the student newspaper.]

We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that is exactly as random as the halting probability of a universal machine equipped...

This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Gödel’s proof; the first modern incompleteness theorem, Turing’s halting problem; and a piece of postmodern metamathematics, the halting probability Ω. The...

[Roissy/CDG Airport, October 2000. This interview took place in a café at the airport, and began at 5:30 in the morning. It’s part of a “Bridge the gap” series of discussions and exchanges on science, art and humanity organized by the Center for Contemporary Art in Kitakyushu, Japan. Hans-Ulrich Obrist is a curator at the Museum of Modern Art of th...

[My first TV interview, broadcast by the BBC in. 1990. It had the amazing effect of convincing some Brits that I believe that 2 + 2 is sometimes 5 and sometimes 3, which is precisely what I deny in the interview. This was in a weekly program on the arts, Arena: Numbers, and it was the first program of its season and therefore quite visible. My incl...

[Held in the historic Café Tortoni, Buenos Aires, 1998. An extremely lively interview by a talented young Argentine mathematician and writer, Guillermo Martínez. Among other things, he is the author of a remarkable novel about genius, Regarding Roderer, and has translated Tasić’s book on postmodernism into Spanish. This interview was the main artic...

The purpose of this book is to show you how to program the proofs of the main theorems about program-size complexity, so that we can see the algorithms in these proofs running on the computer. And in order to write these programs, we need to use a programming language. Unfortunately, no existing programming language is exactly what is needed. So I...

In this chapter I’ll show you that Solovay randomness is equivalent to strong Chaitin randomness. Recall that an infinite binary sequence x is strong Chaitin random iff (H(xn
), the complexity of its n-bit prefix xn
) − n goes to infinity as n increases. I’ll break the proof into two parts.

In this chapter I’ll present a very general technique, which I’ll call Kraft, for constructing self-delimiting computers by generating their input/output graphs. This is our most basic tool. We’ll use it many times: over and over again in what’s left of Part II and in Part III. In fact, using this technique is at the heart of the next two chapters,...

Okay, so now we’ve got a fairly simple version of LISP. Its interpreter is only three hundred lines of Mathematica code, and it’s less than a thousand lines C and Java. So let’s use it!

This entire chapter will be devoted to the proof of one major theorem:
$$
H(y|x) = H(x,y) - H(x) + O(1)

Now that you, dear reader! have worked your way through this difficult book. and some of you have probably worked your way through all three of my Springer volumes, I would like to state my conclusions, my views, much more emphatically. I would like to summarize as forcefully as possible my new viewpoint. These three books are my justification for...

Thanks very much Manuel! It's a great pleasure to be here!

A lot remains to be done! Hopefully this is just the beginning of AIT! The higher you go, the more mountains you can see to climb!

In his brilliant book-length manuscript on AIT dated May 1975, which unfortunately was never published, Solovay reformulated Martin-Löf’s statistical definition of a random real. The resulting definition is, in my opinion, a substantial improvement: it’s much easier to apply. And in the next chapter we’ll be able to use it to show that Ω is strong...

Okay, in the previous chapter I showed you my definitions of random-ness based on program-size complexity—four of them actually! In this and the next chapter I’ll show you two statistical definitions of a random real that look very different from my program-size irreducibility definition, but which actually turn out to be equivalent. These are P. M...

The first half of the main theorem of this chapter is trivial:
$$
P(x) \geqslant 2^{ - H(x)} ,
$$
therefore
$$
- \log _2 P(x) \leqslant H(x).

There’s only one definition of randomness (divided into the finite and the infinite case for technical reasons): something is random if it is algorithmically incompressible or irreducible. More precisely, a member of a set of objects is random if it has the highest complexity that is possible within this set. In other words, the random objects in a...

U is just one of many possible self-delimiting binary computers C Each such C can be simulated by U by adding a LISP prefix σC
$$
U(\sigma _C p) = C(p).

This is the transcript of a lecture given at UMass-Lowell in which I compare and contrast the work of Godel and of Turing and my own work on incompleteness. I also discuss randomness in physics vs randomness in pure mathematics.

In a famous lecture in 1900, David Hilbert listed 23 difficult problems he felt deserved the attention of mathematicians in the coming century. His conviction of the solvability of every mathematical problem was a powerful incentive to future generations: "Wir mussen wissen. Wir werden wissen". (We must know. We will know.) Some of these problems w...

We present the information-theoretic incompleteness theorems that arise in a theory of program-size complexity based on something close to real lisp. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a lisp definition of the proof-checking function associated with the formal system. Using this concrete a...

A new definition of program-size complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest self-delimiting program for calculating strings A and B if one is given a minimal-size selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be self-delimiting, i.e....

There are a number of questions regarding the size of programs for calculating natural numbers, sequences, sets, and functions, which are best answered by considering computations in which one is allowed to consult an oracle for the halting problem. Questions of this kind suggested by work of T. Kamae and D. W. Loveland are treated. 2 G. J. Chaitin...

We obtain some dramatic results using statistical mechanics--thermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property that if one varies a parameter and asks whether the n...

We outline our construction of a single equation involving only addition, multiplication, and exponentiation of non-negative integer constants and variables with the following remarkable property. One of the variables is considered to be a parameter. Take the parameter to be 0; 1; 2; : : : obtaining an infinite series of equations from the original...

We present a new "character-string" oriented dialect of LISP in which the natural LISP halting probability asymptotically has maximum possible LISP complexity. 1. Introduction This paper continues the study of LISP program-size complexity in my monograph [1, Chapter 5] and the series of papers [2--4]. In this paper we consider a dialect of LISP hal...

An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty...

I'll outline the latest version of my limits of math course. The purpose of this course is to illustrate the proofs of the key information-theoretic incompleteness theorems of algorithmic information theory by means of algorithms written in a specially designed version of LISP. The course is now written in HTML with Java applets, and is available a...

This material was presented in a series of lectures at the Santa Fe Institute, the Los Alamos National Laboratory, and the University of New Mexico, during a one-month visit to the Santa Fe Institute, April 1995.

This is a shortened version of "The Limits of Mathematics--Course Outline &
Software" (IBM Research Report RC 19324, December 1993) in which all
Mathematica code has either been deleted or, if absolutely necessary, replaced
by C code. The intention is to make this material available to a wider
audience.

Lecture given Friday 7 April 1995 at the Santa Fe Institute, Santa Fe, New Mexico. The lecture was videotaped; this is an edited transcript.

The latest in a series of reports presenting the information-theoretic incompleteness theorems of algorithmic information theory via algorithms written in specially designed versions of LISP. Previously in this LISP code only one-character identifiers were allowed, and arithmetic had to be programmed out. Now identifiers can be many characters long...

Can the halting problem be solved if one could compute program-size complexity? The answer is yes and here are two different proofs.

We summarize four different versions of our course notes on the limits of mathematics.

This is yet another version of the course notes in chao-dyn/9407003. Here we use m-expressions more aggressively to further reduce the constants in our information-theoretic incompleteness theorems. Our main theorems are: 1) an N-bit formal axiomatic system cannot enable one to exhibit any specific object with program-size complexity greater than N...

This is yet another version of the course notes in chao-dyn/9407003. Here we change the universal Turing machine that is used to measure program-size complexity so that the constants in our information-theoretic incompleteness theorems are further reduced. This is done by inventing a more complicated version of lisp in which the parentheses associa...

This is an alternative version of the course notes in chao-dyn/9407003. The previous version is based on measuring the size of lisp s-expressions. This version is based on measuring the size of what I call lisp m-expressions, which are lisp s-expressions with most parentheses omitted. This formulation of algorithmic information theory is harder to...

This is a revised version of the course notes handed to each participant at the limits of mathematics short course, Orono, Maine, June 1994.

Lecture given Wednesday 27 October 1993 at a Physics -- Computer Science
Colloquium at the University of New Mexico. The lecture was videotaped; this is
an edited transcript. It also incorporates remarks made at the Limits to
Scientific Knowledge meeting held at the Santa Fe Institute 24--26 May 1994.

This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics. Comment: 30 pages. Abstract added in migration.

A remarkable new definition of a self-delimiting universal Turing machine is presented that is easy to program and runs very quickly. This provides a new foundation for algorithmic information theory. This new universal Turing machine is implemented via software written in Mathematica and C. Using this new software, it is now possible give a self-c...

Consider the number of n-bit strings that have exactly the maximum possible program-size complexity that an n-bit string can have. We show that this number is itself an n-bit string with nearly the maximum possible complexity. From this it follows that at least 2n−cn-bit strings have exactly the maximum complexity that it is possible for an n-bit s...

In my book "Algorithmic Information Theory" I explain how I constructed a million-character equation that proves that there is randomness in arithmetic. My book only includes a few pages from the monster equation, and omits the software used to construct it. This software has now been rewritten in Mathematica. The Mathematica software for my book,...

Lecture given Thursday 22 October 1992 at a Mathematics-Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript.

We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a self-delimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a specific object with complexity greater than n+c,...

It is a great pleasure for me to be speaking today here in Vienna. It’s a particularly great pleasure for me to be here because Vienna is where the great work of Gödel and Boltzmann was done, and their work is a necessary prerequisite for my own ideas. Of course the connection with Gödel was explained in Prof. Reichel’s beautiful lecture.1 What may...

A theory of program-size complexity for something close to real LISP is sketched.

In a previous paper we reported the successful use of graph coloring techniques for doing global register allocation in an experimental PL/I optimizing compiler. When the compiler cannot color the register conflict graph with a number of colors equal to the number of available machine registers, it must add code to spill and reload registers to and...

In a previous paper we reported the successful use of graph coloring techniques for doing global register allocation in an experimental PL/I optimizing compiler. When the compiler cannot color the register conflict graph with a number of colors equal to the number of available machine registers, it must add code to spill and reload registers to and...

As a visiting professor in the Department of Computer Science of the University of Auckland, Greg Chaitin is a frequent visitor in New Zealand. During his recent visit in July 2006 we had time for a dialogue about mathematics, physics, and philosophy— C.S.C.

## Citations

... Reality in mathematics is mostly understood as effective computability. Effective computable numbers are intended to be real numbers [16] that can be computed by a Turing machine terminating in a finite amount of time and with arbitrary precision, where the Turing machine is specified as a quadruple T = (Q, Σ, s, δ) where Q is a finite set of states qi; Σ is a finite set of symbols sj, (e.g., an alphabet); s is the initial state s ⸦ Q, being Q the set of all the possible states; and δ is a transition function that determines the next acquired state occurring from computation state qi to computation state qi+1 in finite time and with arbitrary finite precision. Different versions of the Turing machine are all computationally equivalent [17]. ...

... As such their emphasis is not on general computations but rather on showing feasibility of specific computations in the laboratory. Articles [8, 9, 13, 31, 48] directly address Turing completeness, but the algorithmic or programming aspects are not easy to see. How our approach is different: Contrary to several existing models , our atomic notion (the " blob " ) carries a fixed amount of data and has a fixed number of possible interaction points with other blobs. ...

... We have as Boltzmann entropy, the equalization of particles giving an evenness. Information entropy is regarded as the ability to predict (Shannon, 1948;Chaitin, 2001). It is uniformity that allows the experimental scientific method (Whewell, 1847), but is not uniformity randomness? ...

... However, in mathematics, randomness is closely tied to undecidability-the existence of propositions that cannot be decided within a given axiomatic framework [13]. ...

... Still others, -like A.V. Aho [2], or Searle (again) [18] require that there must be some computational model supporting the computation. Fredkin [13] has put it like this: The thing about a computational process is that we normally think of it as a bunch of bits that are evolving over time plus an engine -the computer. Deutsch [10] holds an even tougher view: there must also be a physical realization of a computational model. ...

Reference: What is Computation: An Epistemic Approach

... Chaitin is indeed aware of the fact that X is computable in the limit: in his opinion, with computations in the limit, which is equivalent to having an oracle for the halting problem, X seems quite understandable: it becomes a computable sequence. From a more general point of view, as Calude and Chaitin (1999) directly affirm, X is ''computably enumerable''. We have, moreover, to recognize that, at the level of extreme undecidability, the incompleteness results arising from X are in no way the ''strongest''. ...

Reference: Life, cognition and metabiology

... The cumulative evolution model is defined in [36,37] as a sequence of sole (resourceunbounded) Turing machines that evolve over time due to the transformations effected by randomly generated algorithmic mutations. Thus, in accordance with evolutionary biology, not only are these Turing machines subjected to randomly generated mutations and natural selection, but they may also inherit information from their predecessors. ...

... which consists in the concatenation of all binary strings enumerated in quasi-lexicographical order [76]. 19 A a Chaitin world number is given by a Chaitin Ω U number (or halting probability), that is the probability that the universal self-delimiting Turing machine U halts [77]. Chaitin world numbers "hold proofs" for almost all mathematical known results; such as as Fermat Last Theorem (in the 400 initial bits), Goldbach's conjecture, or important conjectures like Riemann Hypothesis (in the 2745 bits initial bits) and P vs. NP (in the 6,495 initial bits; cf. ...

Reference: Spurious, Emergent Laws in Number Worlds

... It is the number of symbols in a standard language that are needed to describe the simplest possible program for computing a given function. Beyond a certain size, however, it is impossible to prove that a program is minimal (Chaitin, 1998). Nonetheless, mAbducer finds minimal solutions to rearrangements of a moderate size, and the programs it developseven if they are not minimalare simple enough for K-complexity to be a feasible measure of their complexity. ...

Reference: Recursion in programs, thought, and language

... Chaitin is a well known mathematician who proved that in mathematics exists an infinite number of statements which can not be proved nor disproved from the existing set of axioms. He discussed this in an article from the Scientific American magazine in 2006 and before that in standard scientific publications [65]. Similarly before him, Gödel presented his incompleteness theorem. ...