A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory
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The Monad.Reader Issue 6 by Bernie Pope [email protected] and Dan Piponi [email protected] and Russell O’Connor [email protected]
The Monad.Reader Issue 6 by Bernie Pope [email protected] and Dan Piponi [email protected] and Russell O’Connor [email protected] Wouter Swierstra, editor. 1 Contents Wouter Swierstra Editorial 3 Bernie Pope Getting a Fix from the Right Fold 5 Dan Piponi Adventures in Classical-Land 17 Russell O’Connor Assembly: Circular Programming with Recursive do 35 2 Editorial by Wouter Swierstra [email protected] It has been many months since the last issue of The Monad.Reader. Quite a few things have changed since Issue Five. For better or for worse, we have moved from wikipublishing to LATEX. I, for one, am pleased with the result. This issue consists of three top-notch articles on a variety of subjects: Bernie Pope explores just how expressive foldr is; Dan Piponi shows how to compile proofs in classical logic to Haskell programs; Russell O’Connor has written an embedded assembly language in Haskell. Besides the authors, I would like to acknowledge several other people for their contributions to this issue. Andres L¨oh provided a tremendous amount of TEXnical support and wrote the class files. Peter Morris helped design the logo. Finally, I’d like to thank Shae Erisson for starting up The Monad.Reader – without his limitless enthusiasm for Haskell this magazine would never even have gotten off the ground. 3 Getting a Fix from the Right Fold by Bernie Pope [email protected] What can you do with foldr? This is a seemingly innocent question that will confront most functional programmers at some point in their life. -
The Busy Beaver Frontier
Electronic Colloquium on Computational Complexity, Report No. 115 (2020) The Busy Beaver Frontier Scott Aaronson∗ Abstract The Busy Beaver function, with its incomprehensibly rapid growth, has captivated genera- tions of computer scientists, mathematicians, and hobbyists. In this survey, I offer a personal view of the BB function 58 years after its introduction, emphasizing lesser-known insights, re- cent progress, and especially favorite open problems. Examples of such problems include: when does the BB function first exceed the Ackermann function? Is the value of BB(20) independent of set theory? Can we prove that BB(n + 1) > 2BB(n) for large enough n? Given BB(n), how many advice bits are needed to compute BB(n + 1)? Do all Busy Beavers halt on all inputs, not just the 0 input? Is it decidable, given n, whether BB(n) is even or odd? 1 Introduction The Busy Beaver1 function, defined by Tibor Rad´o[13] in 1962, is an extremely rapidly-growing function, defined by maximizing over the running times of all n-state Turing machines that eventu- ally halt. In my opinion, the BB function makes the concepts of computability and uncomputability more vivid than anything else ever invented. When I recently taught my 7-year-old daughter Lily about computability, I did so almost entirely through the lens of the ancient quest to name the biggest numbers one can. Crucially, that childlike goal, if pursued doggedly enough, inevitably leads to something like the BB function, and hence to abstract reasoning about the space of possi- ble computer programs, the halting problem, and so on.2 Let's give a more careful definition. -
The Busy Beaver Competition: a Historical Survey Pascal Michel
The Busy Beaver Competition: a historical survey Pascal Michel To cite this version: Pascal Michel. The Busy Beaver Competition: a historical survey. 2017. hal-00396880v6 HAL Id: hal-00396880 https://hal.archives-ouvertes.fr/hal-00396880v6 Preprint submitted on 4 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Busy Beaver Competition: a historical survey Pascal MICHEL∗ Equipe´ de Logique Math´ematique, Institut de Math´ematiques de Jussieu–Paris Rive Gauche, UMR 7586, Bˆatiment Sophie Germain, case 7012, 75205 Paris Cedex 13, France and Universit´ede Cergy-Pontoise, ESPE, F-95000 Cergy-Pontoise, France [email protected] Version 6 September 4, 2019 Abstract Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, . .. More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. -
A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory
A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory Adam Yedidia Scott Aaronson MIT MIT [email protected] [email protected] May 17, 2016 Abstract Since the definition of the Busy Beaver function by Rad´oin 1962, an interesting open question has been the smallest value of n for which BB(n) is independent of ZFC set theory. Is this n approximately 10, or closer to 1,000,000, or is it even larger? In this paper, we show that it is at most 7,910 by presenting an explicit description of a 7,910-state Turing machine Z with 1 tape and a 2-symbol alphabet that cannot be proved to run forever in ZFC (even though it presumably does), assuming ZFC is consistent. The machine is based on work of Harvey Friedman on independent statements involving order-invariant graphs. In doing so, we give the first known upper bound on the highest provable Busy Beaver number in ZFC. To create Z, we develop and use a higher-level language, Laconic, which is much more convenient than direct state manipulation. We also use Laconic to design two Turing machines, G and R, that halt if and only if there are counterexamples to Goldbach’s Conjecture and the Riemann Hypothesis, respectively. 1 Introduction 1.1 Background and Motivation Zermelo-Fraenkel set theory with the axiom of choice, more commonly known as ZFC, is an ax- iomatic system invented in the twentieth which has since been used as the foundation of most of modern mathematics. It encodes arithmetic by describing natural numbers as increasing sets of sets. -
Programming in XML”
Growing a Language from the Inside Out Theo D’Hondt Growing a Language from the Inside Out Francqui Chair - 2010 Inaugural Lecture - 19 March 2010 Theo D’Hondt Software Languages Lab Faculty of Sciences - Vrije Universiteit Brussel http://soft.vub.ac.be Francqui Chair - 2010 1 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Francqui Chair - 2010 2 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Francqui Chair - 2010 2 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Converting to MP3 Francqui Chair - 2010 2 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Francqui Chair - 2010 3 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Compiling a big project Francqui Chair - 2010 3 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Francqui Chair - 2010 4 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Compiling a small project Francqui Chair - 2010 4 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Francqui Chair - 2010 5 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Application with green threads Francqui Chair - 2010 5 Inaugural Lecture Growing a Language from the Inside Out Theo D’Hondt A Simple Experiment Francqui Chair - 2010 6 Inaugural Lecture Growing a Language from the -
Handout 2: October 25 Three Proofs of Undecidability Of
COS 487: Theory of Computation Fall 2000 Handout 2: October 25 Sanjeev Arora Three proofs of Undecidability of ATM (From G. Chaitin, “G¨o del’s Theorem and Information” International J. Theor. Physics 22 (1982), pp. 941-954.) 1. Define a function F where F (N) is defined to be either 1 more than the value of the Nth computable function applied to the natural number N, or zero if this value is undefined because the Nth computer program does not halt on input N. F cannot be a computable function, for if program M calculated it, then one would have F (M)= F (M) + 1, which is impossible. But the only way that F can fail to be computable is because one cannot decide in general if the Nth program ever halts when given input N. 2. This proof runs along the lines of Bertrand Russell’s paradox of the set of all things that are not members of themselves. Consider programs for enumerating sets of natural numbers, and number these computer programs. Define a set of natural numbers consisting of the numbers of all programs which do not include their own number in their output set. This set of natural numbers cannot be recursively enumerable, for if it were listed by computer program N, one arrives at Russell’s paradox of the barber in a small town who shaves all those and only those who do not shave themselves, and can neither shave himself nor avoid doing so. But the only way that this set can fail to be recursively enumerable is if it is impossible to decide whether or not a program ever outputs a specific natural number, and this is a variant of the halting problem. -
Problems in Number Theory from Busy Beaver Competition
Logical Methods in Computer Science Vol. 11(4:10)2015, pp. 1–35 Submitted Mar. 11, 2015 www.lmcs-online.org Published Dec. 14, 2015 PROBLEMS IN NUMBER THEORY FROM BUSY BEAVER COMPETITION PASCAL MICHEL Equipe´ de Logique Math´ematique, Institut de Math´ematiques de Jussieu–Paris Rive Gauche, UMR 7586, Bˆatiment Sophie Germain, case 7012, 75205 Paris Cedex 13, France and Universit´ede Cergy- Pontoise, IUFM, F-95000 Cergy-Pontoise, France e-mail address: [email protected] Abstract. By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory. 1. Introduction 1.1. A well defined noncomputable function. It is easy to define a noncomputable function on nonnegative integers. Indeed, given a programming language, you produce a systematic list of the programs for functions, and, by diagonalization, you define a function whose output, on input n, is different from the output of the nth program. This simple definition raises many problems: Which programming language? How to list the programs? How to choose the output? In 1962, Rado [Ra62] gave a practical solution by defining the busy beaver game, also called now the busy beaver competition. -
On the Hardness of Knowing Busy Beaver Values BB (15) and BB (5, 4)
On the hardness of knowing busy beaver values BB(15) and BB(5; 4) Tristan St´erin∗ Damien Woods∗ Abstract The busy beaver value BB(n) is the maximum number of steps made by any n-state, 2-symbol deterministic halting Turing machine starting on blank tape, and BB(n; k) denotes its k-symbol generalisation to k ≥ 2. The busy beaver function n 7! BB(n) is uncomputable and its values have been linked to hard open problems in mathematics and notions of unprovability. In this paper, we show that there are two explicit Turing machines, one with 15 states and 2 symbols, the other with 5 states and 4 symbols, that halt if and only if the following Collatz-related conjecture by Erd}os[7] does not hold: for all n > 8 there is at least one digit 2 in the base 3 representation of 2n. This result implies that knowing the values of BB(15) or BB(5,4) is at least as hard as solving Erd}os'conjecture and makes, to date, BB(15) the smallest busy beaver value that is related to a natural open problem in mathematics. For comparison, Yedidia and Aaronson [20] show that knowing BB(4,888) and BB(5,372) are as hard as solving Goldbach's conjecture and the Riemann hypothesis, respectively (later informally improved to BB(27) and BB(744)). Finally, our result puts a finite, albeit large, bound on Erd}os'conjecture, by making it equivalent to the following finite statement: for all 8 < n ≤ min(BB(15); BB(5; 4)) there is at least one digit 2 in the base 3 representation of 2n. -
Recovery, Convergence and Documentation of Languages
Recovery, Convergence and Documentation of Languages by Vadim Zaytsev September 14, 2010 VRIJE UNIVERSITEIT Recovery, Convergence and Documentation of Languages ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Exacte Wetenschappen op woensdag 27 oktober 2010 om 15.45 uur in de aula van de universiteit, De Boelelaan 1105 door Vadim Valerievich Zaytsev geboren te Rostov aan de Don, Rusland promotoren: prof.dr. R. Lammel¨ prof.dr. C. Verhoef Dit onderzoek werd ondersteund door de Nederlandse Organisatie voor Wetenschappelijk Onderzoek via: This research has been sponsored by the Dutch Organisation of Scientific Research via: NWO 612.063.304 LPPR: Language-Parametric Program Restructuring Acknowledgements Working on a PhD is supposed to be an endeavour completed in seclusion, but in practice one cannot survive without the help and support from others, fruitful scientific discus- sions, collaborative development of tools and papers and valuable pieces of advice. My work was supervised by Prof. Dr. Ralf Lammel¨ and Prof. Dr. Chris Verhoef, who often believed in me more than I did and were always open to questions and ready to give expert advice. They have made my development possible. LPPR colleagues — Jan Heering, Prof. Dr. Paul Klint, Prof. Dr. Mark van den Brand — have been a rare yet useful source of new research ideas. All thesis reading committee members have dedicated a lot of attention to my work and delivered exceptionally useful feedback on the late stage of the research: Prof. -
Computability Theory and Big Numbers
Computability Theory and Big Numbers Alexander Davydov August 30, 2018 Alexander Davydov Computability Theory and Big Numbers August 30, 2018 1 / 42 Classic Large Numbers Factorial 100! ≈ 9:33 × 10157 (100!)! >> 100! Power towers 33 3 "" 4 = 33 = 37625597500000 Alexander Davydov Computability Theory and Big Numbers August 30, 2018 2 / 42 Graham's Number g64 ( 3 """" 3 if n = 1 gn = 3 "gn−1 3 if n ≥ 2 and n 2 N So large that the observable universe is too small to contain the digital representation of Grahams Number, even if each digit were one Planck volume 1 Planck volume = 4:2217 × 10−105m3 Alexander Davydov Computability Theory and Big Numbers August 30, 2018 3 / 42 Turing Machines A Turing machine serves as a mathematical model for computation Informally: One-dimensional tape of cells that extends infinitely in either direction Each cell contains a symbol from the \alphabet" of the machine Typically 0, 1, and possibly a blank symbol Machine contains a \head" that reads the symbol underneath it Machine is in one of finitely many \states" that determine what the machine does At each step, based on the symbol that the head reads, the head will overwrite the symbol that it just read and then move either to the left or right and then enter a new state The machine has a qHALT state. When the machine reaches the qHALT state, the machine halts and whatever is written on the tape is outputted Alexander Davydov Computability Theory and Big Numbers August 30, 2018 4 / 42 Formal Definition of a Turing Machine A Turing Machine M is described by a tuple (Γ; Q; δ) containing: A set Γ of the symbols that M's tape can contain. -
Who Can Name the Bigger Number? by Scott Aaronson [Author's Blog] Source
Who Can Name the Bigger Number? by Scott Aaronson [Author's blog] Source: http://www.scottaaronson.com/writings/bignumbers.html In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces "Eighty-three!" The second, mightily impressed, replies "You win." A biggest number contest is clearly pointless when the contestants take turns. But what if the contestants write down their numbers simultaneously, neither aware of the other’s? To introduce a talk on "Big Numbers," I invite two audience volunteers to try exactly this. I tell them the rules: You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature. So contestants can’t say "the number of sand grains in the Sahara," because sand drifts in and out of the Sahara regularly. Nor can they say "my opponent’s number plus one," or "the biggest number anyone’s ever thought of plus one"—again, these are ill-defined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins. Are you ready? Get set. Go. The contest’s results are never quite what I’d hope. Once, a seventh-grade boy filled his card with a string of successive 9’s. Like many other big-number tyros, he sought to maximize his number by stuffing a 9 into every place value. -
DM820 Advanced Topics in Programming Languages Peter Schneider-Kamp [email protected]! COURSE ORGANIZATION
DM820 Advanced Topics in Programming Languages Peter Schneider-Kamp [email protected]! http://imada.sdu.dk/~petersk/DM820/! COURSE ORGANIZATION • slid e 2 Course Elements § 7 lectures Thursday 08-10 (Weeks 15–17, 19–22) § 3 lectures Monday 16-18 (Weeks 16–18) § 7 discussion sections Wednesday 08-10 (Weeks 16–22) § 4 presentation sessions Monday 16-18 (Weeks 19–22) § exam = oral presentation + practical project § oral presentation is internal pass/fail § practical project is external with grades • 3 • June 2009 Course Goals § Learn more ways to program! § To this end, you will learn § to use advanced concepts in standard programming languages § to use non-standard programming languages § to use programming in widely differing contexts § Focus of the course on broadly covering (parts of) the field § Depth added through individually assigned topics • 4 • June 2009 Course Contract § I am offering you the following: 1. I will help you in picking a topic 2. I am always willing to help you § From you I expect the following: 1. You give a good presentation of your assigned topic 2. You have FUN programming! § You and I have the right and duty to call upon the contract! • slid e 5 Topics for Lectures 1–6 § Mostly up to you! § Lectures can be picked from following topics a) scripting languages b) constraint programming c) multi-paradigm programming languages d) string/term/graph rewriting e) domain specific languages f) program verification g) extensible programming h) aspect-oriented programming i) parser generation § Today (& next time): A little history of programming languages • slid e 6 Topics for Lectures 7–10 § Up to me! § Selection criteria: § Topics must supplement your chosen topics in order to guarantee a broad overview § Topics must be fun (in some weird way) • slid e 7 Individually Assigned Topics 1/3 Specific Languages: A.