Learning Concepts: A Learning-Theoretic Solution to the Complex-First Paradox

Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

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Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method

Applied Mathematics and Mechanics ◽

10.1007/s10483-006-0614-y ◽

2006 ◽

Vol 27 (6) ◽

pp. 833-839 ◽

Cited By ~ 8

Author(s):

Yang Zhong ◽

Yong-shan Zhang

Keyword(s):

Thin Plate ◽

Symplectic Geometry ◽

Rectangular Thin Plate ◽

Theoretic Solution ◽

Geometry Method

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ERRATUM: "RATIONAL DYNAMICS AND EPISTEMIC LOGIC IN GAMES"

International Game Theory Review ◽

10.1142/s0219198907001485 ◽

2007 ◽

Vol 09 (02) ◽

pp. 377-409 ◽

Cited By ~ 5

Author(s):

JOHAN VAN BENTHEM

Keyword(s):

Backward Induction ◽

Recursive Algorithms ◽

Strategic Games ◽

Solution Sets ◽

Solution Algorithms ◽

Mutual Knowledge ◽

Dominated Strategies ◽

Game Theoretic ◽

Theoretic Solution ◽

Extensive Games

Game-theoretic solution concepts describe sets of strategy profiles that are optimal for all players in some plausible sense. Such sets are often found by recursive algorithms like iterated removal of strictly dominated strategies in strategic games, or backward induction in extensive games. Standard logical analyses of solution sets use assumptions about players in fixed epistemic models for a given game, such as mutual knowledge of rationality. In this paper, we propose a different perspective, analyzing solution algorithms as processes of learning which change game models. Thus, strategic equilibrium gets linked to fixed-points of operations of repeated announcement of suitable epistemic statements. This dynamic stance provides a new look at the current interface of games, logic, and computation.

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