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Samuel Alexander [16]Samuel Allen Alexander [6]Samuel A. Alexander [5]
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Samuel Allen Alexander
Ohio State University (PhD)
  1. AGI and the Knight-Darwin Law: Why Idealized AGI Reproduction Requires Collaboration.Samuel Alexander - forthcoming - In International Conference on Artificial General Intelligence. Springer.
    Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and (...)
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  2. Measuring the Intelligence of an Idealized Mechanical Knowing Agent.Samuel Alexander - 2020 - Lecture Notes in Computer Science 12226.
    We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define real-number intelligence levels of compli- cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows to be codes (...)
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  3. Intelligence Via Ultrafilters: Structural Properties of Some Intelligence Comparators of Deterministic Legg-Hutter Agents.Samuel Alexander - 2019 - Journal of Artificial General Intelligence 10 (1):24-45.
    Legg and Hutter, as well as subsequent authors, considered intelligent agents through the lens of interaction with reward-giving environments, attempting to assign numeric intelligence measures to such agents, with the guiding principle that a more intelligent agent should gain higher rewards from environments in some aggregate sense. In this paper, we consider a related question: rather than measure numeric intelligence of one Legg- Hutter agent, how can we compare the relative intelligence of two Legg-Hutter agents? We propose an elegant answer (...)
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  4. The Archimedean Trap: Why Traditional Reinforcement Learning Will Probably Not Yield AGI.Samuel Allen Alexander - 2020 - Journal of Artificial General Intelligence 11 (1):70-85.
    After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to AGI. We indicate two possible ways (...)
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  5. Short-Circuiting the Definition of Mathematical Knowledge for an Artificial General Intelligence.Samuel Alexander - forthcoming - Lecture Notes in Computer Science.
    We propose that, for the purpose of studying theoretical properties of the knowledge of an agent with Artificial General Intelligence (that is, the knowledge of an AGI), a pragmatic way to define such an agent’s knowledge (restricted to the language of Epistemic Arithmetic, or EA) is as follows. We declare an AGI to know an EA-statement φ if and only if that AGI would include φ in the resulting enumeration if that AGI were commanded: “Enumerate all the EA-sentences which you (...)
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  6. A Machine That Knows Its Own Code.Samuel A. Alexander - 2014 - Studia Logica 102 (3):567-576.
    We construct a machine that knows its own code, at the price of not knowing its own factivity.
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  7.  91
    Reward-Punishment Symmetric Universal Intelligence.Samuel Allen Alexander & Marcus Hutter - forthcoming - In AGI-21.
    Can an agent's intelligence level be negative? We extend the Legg-Hutter agent-environment framework to include punishments and argue for an affirmative answer to that question. We show that if the background encodings and Universal Turing Machine (UTM) admit certain Kolmogorov complexity symmetries, then the resulting Legg-Hutter intelligence measure is symmetric about the origin. In particular, this implies reward-ignoring agents have Legg-Hutter intelligence 0 according to such UTMs.
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  8. An Axiomatic Version of Fitch’s Paradox.Samuel Alexander - 2013 - Synthese 190 (12):2015-2020.
    A variation of Fitch’s paradox is given, where no special rules of inference are assumed, only axioms. These axioms follow from the familiar assumptions which involve rules of inference. We show (by constructing a model) that by allowing that possibly the knower doesn’t know his own soundness (while still requiring he be sound), Fitch’s paradox is avoided. Provided one is willing to admit that sound knowers may be ignorant of their own soundness, this might offer a way out of the (...)
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  9. Measuring Intelligence and Growth Rate: Variations on Hibbard's Intelligence Measure.Samuel Alexander & Bill Hibbard - 2021 - Journal of Artificial General Intelligence 12 (1):1-25.
    In 2011, Hibbard suggested an intelligence measure for agents who compete in an adversarial sequence prediction game. We argue that Hibbard’s idea should actually be considered as two separate ideas: first, that the intelligence of such agents can be measured based on the growth rates of the runtimes of the competitors that they defeat; and second, one specific (somewhat arbitrary) method for measuring said growth rates. Whereas Hibbard’s intelligence measure is based on the latter growth-rate-measuring method, we survey other methods (...)
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  10. A Type of Simulation Which Some Experimental Evidence Suggests We Don't Live In.Samuel Alexander - 2018 - The Reasoner 12 (7):56-56.
    Do we live in a computer simulation? I will present an argument that the results of a certain experiment constitute empirical evidence that we do not live in, at least, one type of simulation. The type of simulation ruled out is very specific. Perhaps that is the price one must pay to make any kind of Popperian progress.
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  11. Can Reinforcement Learning Learn Itself? A Reply to 'Reward is Enough'.Samuel Allen Alexander - forthcoming - CIFMA 2021.
    In their paper 'Reward is enough', Silver et al conjecture that the creation of sufficiently good reinforcement learning (RL) agents is a path to artificial general intelligence (AGI). We consider one aspect of intelligence Silver et al did not consider in their paper, namely, that aspect of intelligence involved in designing RL agents. If that is within human reach, then it should also be within AGI's reach. This raises the question: is there an RL environment which incentivises RL agents to (...)
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  12. Infinite Graphs in Systematic Biology, with an Application to the Species Problem.Samuel A. Alexander - 2013 - Acta Biotheoretica 61 (2):181--201.
    We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a decomposition result (...)
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  13. A Purely Epistemological Version of Fitch's Paradox.Samuel Alexander - 2012 - The Reasoner 6 (4):59-60.
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  14. Fast-Collapsing Theories.Samuel A. Alexander - 2013 - Studia Logica (1):1-21.
    Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.
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  15. Mathematical Shortcomings in a Simulated Universe.Samuel Alexander - 2018 - The Reasoner 12 (9):71-72.
    I present an argument that for any computer-simulated civilization we design, the mathematical knowledge recorded by that civilization has one of two limitations. It is untrustworthy, or it is weaker than our own mathematical knowledge. This is paradoxical because it seems that nothing prevents us from building in all sorts of advantages for the inhabitants of said simulation.
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  16.  61
    Arithmetical Algorithms for Elementary Patterns.Samuel A. Alexander - 2015 - Archive for Mathematical Logic 54 (1-2):113-132.
    Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.
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  17. Self-Referential Theories.Samuel A. Alexander - 2020 - Journal of Symbolic Logic 85 (4):1687-1716.
    We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.
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  18. Legg-Hutter Universal Intelligence Implies Classical Music is Better Than Pop Music for Intellectual Training.Samuel Alexander - 2019 - The Reasoner 13 (11):71-72.
    In their thought-provoking paper, Legg and Hutter consider a certain abstrac- tion of an intelligent agent, and define a universal intelligence measure, which assigns every such agent a numerical intelligence rating. We will briefly summarize Legg and Hutter’s paper, and then give a tongue-in-cheek argument that if one’s goal is to become more intelligent by cultivating music appreciation, then it is bet- ter to use classical music (such as Bach, Mozart, and Beethoven) than to use more recent pop music. The (...)
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  19. Self-Graphing Equations.Samuel Alexander - manuscript
    Can you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper’s self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on fonts) and it is trivial. We fix these flaws by formalizing the problem.
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  20. Guessing, Mind-Changing, and the Second Ambiguous Class.Samuel Alexander - 2016 - Notre Dame Journal of Formal Logic 57 (2):209-220.
    In his dissertation, Wadge defined a notion of guessability on subsets of the Baire space and gave two characterizations of guessable sets. A set is guessable if and only if it is in the second ambiguous class, if and only if it is eventually annihilated by a certain remainder. We simplify this remainder and give a new proof of the latter equivalence. We then introduce a notion of guessing with an ordinal limit on how often one can change one’s mind. (...)
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  21. This Sentence Does Not Contain the Symbol X.Samuel Alexander - 2013 - The Reasoner 7 (9):108.
    A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".
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  22. Biologically Unavoidable Sequences.Samuel Alexander - 2013 - Electronic Journal of Combinatorics 20 (1):1-13.
    A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes König's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.
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  23. A Paradox Related to the Turing Test.Samuel Alexander - 2011 - The Reasoner 5 (6):90-90.
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  24. Formulas for Computable and Non-Computable Functions.Samuel Alexander - 2006 - Rose-Hulman Undergraduate Mathematics Journal 7 (2).
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  25. Did Socrates Know How to See Your Middle Eye?Samuel Allen Alexander & Christopher Yang - 2021 - The Reasoner 15 (4):30-31.
    We describe in our own words a visual phenomenon first described by Gallagher and Tsuchiya in 2020. The key to the phenomenon (as we describe it) is to direct one’s left eye at the image of one's left eye, while simultaneously directing one's right eye at the image of one's right eye. We suggest that one would naturally arrive at this phenomenon if one took a sufficiently literal reading of certain words of Socrates preserved in Plato's Alcibiades. We speculate that (...)
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  26.  59
    Extending Environments To Measure Self-Reflection In Reinforcement Learning.Samuel Allen Alexander, Michael Castaneda, Kevin Compher & Oscar Martinez - manuscript
    We consider an extended notion of reinforcement learning in which the environment can simulate the agent and base its outputs on the agent's hypothetical behavior. Since good performance usually requires paying attention to whatever things the environment's outputs are based on, we argue that for an agent to achieve on-average good performance across many such extended environments, it is necessary for the agent to self-reflect. Thus, an agent's self-reflection ability can be numerically estimated by running the agent through a battery (...)
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  27.  53
    An Alternative Construction of Internodons: The Emergence of a Multi-Level Tree of Life.Samuel Allen Alexander, Arie de Bruin & D. J. Kornet - 2015 - Bulletin of Mathematical Biology 77 (1):23-45.
    Internodons are a formalization of Hennig's concept of species. We present an alternative construction of internodons imposing a tree structure on the genealogical network. We prove that the segments (trivial unary trees) from this tree structure are precisely the internodons. We obtain the following spin-offs. First, the generated tree turns out to be an organismal tree of life. Second, this organismal tree is homeomorphic to the phylogenetic Hennigian species tree of life, implying the discovery of a multi-level tree of life: (...)
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