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Full-Text Articles in Mathematics
The Nature Of Numbers: Real Computing, Bradley J. Lucier
The Nature Of Numbers: Real Computing, Bradley J. Lucier
Journal of Humanistic Mathematics
While studying the computable real numbers as a professional mathematician, I came to see the computable reals, and not the real numbers as usually presented in undergraduate real analysis classes, as the natural culmination of my evolving understanding of numbers as a schoolchild. This paper attempts to trace and explain that evolution. The first part recounts the nature of numbers as they were presented to us grade-school children. In particular, the introduction of square roots induced a step change in my understanding of numbers. Another incident gave me insight into the brilliance of Alan Turing in his paper introducing both β¦
Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi
Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi
HMC Senior Theses
In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in π are thought of as 0-turn games, ππ as 1-turn βpuzzleβ games, and in general πΊβπ as π-turn games, in which decision problems answer the binary question, βcan the starting player guarantee a win?β We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of π-turn CIRCUIT SATISFIABILITY games, whose πΊβπ-completeness is assumed from prior work (we briefly justify this assumption β¦