Strong extension axioms and Shelah's zero-one law for choiceless polynomial time

Journal of Symbolic Logic 68 (1):65-131 (2003)
Abstract
This paper developed from Shelah's proof of a zero-one law for the complexity class "choice-less polynomial time." defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general
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