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Samuel Alexander

Publications and source records attributed to Samuel Alexander.

13 recordsLinked to original sources
arXiv2021

Representation and Invariance in Reinforcement Learning

Researchers have formalized reinforcement learning (RL) in different ways. If an agent in one RL framework is to run within another RL framework's environments, the agent must first be converted, or mapped, into that other framework. In this paper, we lay foundations for studying relative-intelligence-preserving mappability between RL frameworks. We introduce a criterion which is sufficient for relative intelligence to be preserved according to one particular method of measuring intelligence. We show that this criterion cannot be met when mapping between certain deterministic and stochastic RL frameworks, suggesting inherent fundamental differences between these different versions of RL.

cs.AI↗
arXiv2021

Measuring Intelligence and Growth Rate: Variations on Hibbard's Intelligence Measure

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 for measuring function growth rates, and exhibit the resulting Hibbard-like intelligence measures and taxonomies. Of particular interest, we obtain intelligence taxonomies based on Big-O and Big-Theta notation systems, which taxonomies are novel in that they challenge conventional notions of what an intelligence measure should look like. We discuss how intelligence measurement of sequence predictors can indirectly serve as intelligence measurement for agents with Artificial General Intelligence (AGIs).

cs.AI↗
arXiv2018

A type of simulation which some experimental evidence suggests we don't live in

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.

physics.pop-ph↗
arXiv2014

Guessing, Mind-changing, and the Second Ambiguous Class

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 iff it is in the second ambiguous class (boldface Delta^0_2), iff 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. We show that for every ordinal alpha, a guessable set is annihilated by alpha applications of the simplified remainder if and only if it is guessable with fewer than alpha mind changes. We use guessability with fewer than alpha mind changes to give a semi-characterization of the Hausdorff difference hierarchy, and indicate how Wadge's notion of guessability can be generalized to higher-order guessability, providing characterizations for boldface Delta^0_alpha for all successor ordinals alpha>1.

math.LO↗
arXiv2013

Fast-collapsing theories

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.

math.LO↗
arXiv2012

Biologically Unavoidable Sequences

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 Koenig's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata.

math.CO↗
arXiv2012

Infinite graphs in systematic biology, with an application to the species problem

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 involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels' question, "Can biology lead to new theorems?"

q-bio.PE↗
arXiv2011

A Dichotomy in Machine Knowledge

We show that a machine, which knows basic logic and arithmetic and basic axioms of knowledge, and which is factive (knows nothing false), can either know that it is factive, or know its own Goedel number, but not both.

math.LO↗
arXiv2011

A Cantor-Bendixson-like process which detects Delta_2^0

For each subset of Baire space, we define, in away similar to a common proof of the Cantor-Bendixson Theorem, a sequence of decreasing subsets S_alpha of N^N, indexed by ordinals. We use this to obtain two new characterizations of the boldface Delta_2^0 Borel pointclass. ADDENDUM: In January 2012 we learned that the notion of guessability appeared in an equivalent form, and even with the same name, in the doctoral dissertation of William Wadge [4]. As for the main result of this paper, Wadge proved one direction and gave a proof for the other direction which he attributed to Hausdorff. The proofs in this paper present an alternate means to those results.

math.LO↗
arXiv2011

Highly lopsided information and the Borel hierarchy

In a game where both contestants have perfect information, there is a strict limit on how perfect that information can be. By contrast, when one player is deprived of all information, the limit on the other player's information disappears, admitting a hierarchy of levels of lopsided perfection of information. We turn toward the question of when the player with super-perfect information has a winning strategy, and we exactly answer this question for a specific family of lopsided-information games which we call guessing games.

math.LO↗
arXiv2011

The First-Order Syntax of Variadic Functions

We extend first-order logic to include variadic function symbols, and prove a substitution lemma. Two applications are given: one to bounded quantifier elimination and one to the definability of certain Borel sets.

math.LO↗
arXiv2010

On Guessing Whether A Sequence Has A Certain Property

A concept of "guessability" is defined for sets of sequences of naturals. Eventually, these sets are thoroughly characterized. To do this, a nonstandard logic is developed, a logic containing symbols for the ellipsis as well as for functions without fixed arity. New proofs are given for some seemingly-unrelated known results.

math.LO↗