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Short lists with short programs for functions [article]

Nikolay Vereshchagin
2014 arXiv   pre-print
Let {ϕ_p} be an optimal Gödel numbering of the family of computable functions (in Schnorr's sense), where p ranges over binary strings.  ...  Assume that a list of strings L(p) is computable from p and for all p contains a ϕ-program for ϕ_p whose length is at most ε bits larger that the length of the shortest ϕ-program for ϕ_p.  ...  The author is sincerely grateful to Alexander Shen for asking the question and hearing the preliminary version of the proof of the result.  ... 
arXiv:1409.5906v3 fatcat:njjmld5erjfq7hren6jdt26x4q

On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT [chapter]

Yijia Chen, Jörg Flum
2010 Lecture Notes in Computer Science  
For a reasonable sound and complete proof calculus for first-order logic consider the problem to decide, given a sentence ϕ of first-order logic and a natural number n, whether ϕ has no proof of length  ...  We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT  ...  This research has been partially supported by the National Nature Science Foundation of China (60970011), the Sino-German Center for Research Promotion (GZ400), and the John Templeton Foundation through  ... 
doi:10.1007/978-3-642-15205-4_18 fatcat:4ge5ewg3graapmfn7fdqtvb5za

Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations

Sheri M. Markose
2017 Journal of Dynamics & Games  
and digital agents become entrained within fixed repertoires.  ...  This overturns the view of game theorists that surprise and innovation cannot be a Nash equilibrium of a game.  ...  All objects constructed from a countable alphabet can be put into 1 − 1 mapping with the set of natural numbers referred to as their Gödel numbers (g.ns, for short).  ... 
doi:10.3934/jdg.2017015 fatcat:joapkhn7kfgdzgnxm7jsil3use

Complex Type 4 Structure Changing Dynamics of Digital Agents: Nash Equilibria of a Game with Arms Race in Innovations

Sheri M. Markose
2016 Social Science Research Network  
and digital agents become entrained within fixed repertoires.  ...  This overturns the view of game theorists that surprise and innovation cannot be a Nash equilibrium of a game.  ...  All objects constructed from a countable alphabet can be put into 1 − 1 mapping with the set of natural numbers referred to as their Gödel numbers (g.ns, for short).  ... 
doi:10.2139/ssrn.2900533 fatcat:hv4x3by2wffitnhok5lz5prwve

On Approximating Real-World Halting Problems [chapter]

Sven Köhler, Christian Schindelhauer, Martin Ziegler
2005 Lecture Notes in Computer Science  
induces a both efficient and dense Gödelization in the sense of [Jakoby&Schindelhauer'99].  ...  We consider BrainF*ck (BF), a simple yet Turing-complete real-world programming language over an eight letter alphabet, and prove that the natural enumeration of its syntactically correct sources codes  ...  Also, this dense Gödelization is obviously non-optimal in the sense of [Schn75] .  ... 
doi:10.1007/11537311_40 fatcat:7nlozpqxkrgw5cr3vjjyn3avga

Optimal Acceptors and Optimal Proof Systems [chapter]

Edward A. Hirsch
2010 Lecture Notes in Computer Science  
" proof (called optimal proof system) for every tautology.  ...  In such a situation, it is typical for complexity theorists to search for "universal" objects; here, it could be the "fastest" acceptor (called optimal acceptor ) and a proof system that has the "shortest  ...  Acknowledgements The author is very grateful to Olaf Beyersdorff, Dmitry Itsykson, Hunter Monroe, Sergey Nikolenko, Natalia Tsilevich, Zenon Sadowski, and Alexander Smal for valuable comments, and to all  ... 
doi:10.1007/978-3-642-13562-0_4 fatcat:33vuki4rsvbhvhlikdoowabqwy

Gödel Machines: Towards a Technical Justification of Consciousness [chapter]

Jürgen Schmidhuber
2005 Lecture Notes in Computer Science  
Optimality Theorem 1, and Section 5 an O()-optimal (Theorem 2) initial proof searcher.  ...  Section 6 provides examples and additional relations to previous work, and lists answers to several frequently asked questions about Gödel machines. Section 7 wraps up.  ...  But the Gödel machine is more flexible as we may plug in any type of formalizable utility function (e.g., worst case reward), and unlike AIXI(t,l) it does not require an enumerable environmental distribution  ... 
doi:10.1007/978-3-540-32274-0_1 fatcat:twyyq4cqmzeednza23ilvqu2tm

The New AI: General & Sound & Relevant for Physics [article]

Juergen Schmidhuber
2003 arXiv   pre-print
The new millennium, however, has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction, search, inductive inference based on Occam's razor  ...  , problem solving, decision making, and reinforcement learning in environments of a very general type.  ...  Acknowledgments Hutter's frequently mentioned work was funded through the author's SNF grant 2000-061847 "Unification of universal inductive inference and sequential decision theory."  ... 
arXiv:cs/0302012v2 fatcat:styszpjydbfr5d27ch7rcbbwxu

Goedel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements [article]

Juergen Schmidhuber
2006 arXiv   pre-print
Unlike previous non-self-referential methods based on hardwired proof searchers, ours not only boasts an optimal order of complexity but can optimally reduce any slowdowns hidden by the O()-notation, provided  ...  the hardware and the entire initial code are described by axioms encoded in an initial proof searcher which is also part of the initial code.  ...  enumerable set of axioms.  ... 
arXiv:cs/0309048v5 fatcat:umofevbwhnbrdpeahzqneig5cu

Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories

Saeed Salehi, Payam Seraji
2016 Journal of Logic and Computation  
It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense.  ...  theory with the set of all true Π_n-sentences or equivalently the Σ_n-soundness of the theory, and the other is n-consistency the restriction of ω-consistency to the Σ_n-formulas.  ...  We can thus argue that Gödel-Rosser's theorem is optimal in a sense. Some Notation and Conventions We fix the following notation and conventions (mostly from [2, 3, 6, 13, 15] ).  ... 
doi:10.1093/logcom/exw025 fatcat:s6hq6scyzvapfbtwuj4xvdzl5u

Ultimate Cognition à la Gödel

Jürgen Schmidhuber
2009 Cognitive Computation  
Gödel exhibited the limits of mathematics and computation by creating a formula that speaks about itself, claiming to be unprovable by an algorithmic theorem prover: either the formula is true but unprovable  ...  Our self-referential program will optimally speed up its proof searcher and other program parts, but only if the speed up's utility is indeed provable-even ultimate cognition has limits of the Gödelian  ...  I am also grateful to many others who asked questions during Gödel machine talks or sent comments by email.  ... 
doi:10.1007/s12559-009-9014-y fatcat:yctnndvu7verhoac6zmyh3jdim

On the Turing degrees of minimal index sets

Jason Teutsch
2007 Annals of Pure and Applied Logic  
We identify sets of m-minimal and T-minimal indices and characterize their truth-table and Turing degrees. n+4) , and that there exists a Kolmogorov numbering ψ satisfying both MIN m ψ ≡ tt ∅ and MIN T  ...  This Kolmogorov numbering also achieves maximal truth-table degree for other sets of minimal indices. Finally, we show that the set of shortest descriptions, SD, is 2-c.e. but not co-2-c.e.  ...  We now merge the numberings ψ, ν, and ξ into a single Gödel numbering ρ satisfying (i)-(vi).  ... 
doi:10.1016/j.apal.2007.07.003 fatcat:b2jz6jin6vgwdidqmwy47y3f54

Enumerations including laconic enumerators

Sanjay Jain, Jason Teutsch
2017 Theoretical Computer Science  
In contrast to this positive result for strings, we find that, in any Kolmogorov numbering, no enumerator of nontrivial size can generate a list containing the minimal index of a given partial-computable  ...  One cannot even achieve a laconic enumerator for nearly-minimal indices of partial-computable functions.  ...  This result permitted us to solve an open problem in an earlier draft of this work and simplify the proofs of the corollaries.  ... 
doi:10.1016/j.tcs.2017.08.001 fatcat:ouinchuyzjhpjfax5bsjjf4p2a

Logical Approaches to Computational Barriers: CiE 2006

A. Beckmann, B. Lowe, D. Normann
2007 Journal of Logic and Computation  
XII Table of Contents of Gödel and the Origins of Computer ScienceJohn W.  ...  Makowsky 330 Towards a Trichotomy for Quantified TJ-Coloring Barnaby Martin, Florent Madelaine 342 Two Open Problems on Effective Dimension Elvira Mayordomo 353 Optimization and Approximation  ... 
doi:10.1093/logcom/exm031 fatcat:lv36duerfvd25inr2f2be56qvq

Generating structured test data with specific properties using nested Monte-Carlo search

Simon Poulding, Robert Feldt
2014 Proceedings of the 2014 conference on Genetic and evolutionary computation - GECCO '14  
In our previous work we introduced GödelTest, a framework for generating such data structures using non-deterministic programs, and combined it with Differential Evolution to optimize the generation process  ...  In this paper we apply Nested Monte-Carlo Search, a single-player variant of MCTS, to the sequence of decisions made by the generating programs used by GödelTest, and show that this combination can efficiently  ...  There is a choice of two Gödel numbers at a maybe choice point-0 and 1-and a Bernoulli distribution (a biased coin toss) is used during game simulation.  ... 
doi:10.1145/2576768.2598339 dblp:conf/gecco/PouldingF14 fatcat:d2lqigauprewjkw7qxctlgxgxm
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