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Short lists with short programs for functions
[article]

2014
*
arXiv
*
pre-print

Let {ϕ_p} be an

arXiv:1409.5906v3
fatcat:njjmld5erjfq7hren6jdt26x4q
*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. ...##
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On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT
[chapter]

2010
*
Lecture Notes in Computer Science
*

For a reasonable sound

doi:10.1007/978-3-642-15205-4_18
fatcat:4ge5ewg3graapmfn7fdqtvb5za
*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 ...##
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Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations

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). ...

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

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). ...

##
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On Approximating Real-World Halting Problems
[chapter]

2005
*
Lecture Notes in Computer Science
*

induces a both efficient

doi:10.1007/11537311_40
fatcat:7nlozpqxkrgw5cr3vjjyn3avga
*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] . ...##
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Optimal Acceptors and Optimal Proof Systems
[chapter]

2010
*
Lecture Notes in Computer Science
*

" proof (called

doi:10.1007/978-3-642-13562-0_4
fatcat:33vuki4rsvbhvhlikdoowabqwy
*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 ...##
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Gödel Machines: Towards a Technical Justification of Consciousness
[chapter]

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 ...

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

2003
*
arXiv
*
pre-print

The new millennium, however, has brought substantial progress in the field of theoretically

arXiv:cs/0302012v2
fatcat:styszpjydbfr5d27ch7rcbbwxu
*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." ...##
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Goedel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements
[article]

2006
*
arXiv
*
pre-print

Unlike previous non-self-referential methods based on hardwired proof searchers, ours not only boasts an

arXiv:cs/0309048v5
fatcat:umofevbwhnbrdpeahzqneig5cu
*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. ...##
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Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories

2016
*
Journal of Logic and Computation
*

It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively

doi:10.1093/logcom/exw025
fatcat:s6hq6scyzvapfbtwuj4xvdzl5u
*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] ). ...##
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Ultimate Cognition à la Gödel

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. ...

##
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On the Turing degrees of minimal index sets

2007
*
Annals of Pure and Applied Logic
*

We identify sets of m-minimal

doi:10.1016/j.apal.2007.07.003
fatcat:b2jz6jin6vgwdidqmwy47y3f54
*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). ...##
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Enumerations including laconic enumerators

2017
*
Theoretical Computer Science
*

In contrast to this positive result for strings, we find that, in any Kolmogorov

doi:10.1016/j.tcs.2017.08.001
fatcat:ouinchuyzjhpjfax5bsjjf4p2a
*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. ...##
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Logical Approaches to Computational Barriers: CiE 2006

2007
*
Journal of Logic and Computation
*

XII Table of Contents of

doi:10.1093/logcom/exm031
fatcat:lv36duerfvd25inr2f2be56qvq
*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 ...##
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Generating structured test data with specific properties using nested Monte-Carlo search

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,

doi:10.1145/2576768.2598339
dblp:conf/gecco/PouldingF14
fatcat:d2lqigauprewjkw7qxctlgxgxm
*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. ...
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