“精确实数计算”中的算法代价
Algorithmic Cost in "Exact Real Computation"
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中文总结 AI 辅助
该研究定量强化精确实数计算（ERC）的图灵完备性，为其操作原语分配位代价，证实多项式代价实函数与多项式时间图灵可计算性等价，且iRRAM C++库的运行时测量验证了理论预测。
中文摘要 AI 辅助
编程语言或系统的图灵完备性表征其表达能力；强丘奇-图灵假说将其从定性层面细化为多项式时间等价。精确实数计算（Exact Real Computation, ERC）是一种新型数值编程范型，用于对连续数据进行命令式处理，这类数据以精确实体形式呈现，即无舍入误差[doi: https://doi.org/10.1007/978-3-662-44199-2_107]。ERC被设计为可计算分析[doi: https://doi.org/10.1007/978-3-642-56999-9,doi: https://doi.org/10.1007/978-1-4684-6802-1]底层图灵机的便捷实用替代方案，已被证明在定性层面等价于图灵机。本研究对这种定性图灵完备性进行了定量强化：我们为ERC的操作原语（包括部分/多值测试）分配位代价，使得任何产生多项式代价的实函数都能在多项式时间内图灵可计算，反之亦然。对iRRAM C++库中实现的运行时测量结果证实了我们的理论性能预测。
英文摘要
Turing completeness of a programming language or system characterizes its expressive power; and the strong Church-Turing hypo-/thesis refines such from qualitative to polynomial-time equivalence. Exact Real Computation (ERC) is a novel numerical programming language paradigm: for the imperative processing of continuous data as entities appearing as exact, i.e. devoid of rounding errors [doi:10.1007/978-3-662-44199-2_107]. ERC has been designed [doi:10.46298/lmcs-20(2:17)2024] as convenient and practical alternative, namely proven qualitatively equivalent, to the Turing machines originally underlying Computable Analysis [doi:10.1007/978-3-642-56999-9,doi:10.1007/978-1-4684-6802-1]. The present work quantitatively strengthens this qualitative Turing-completeness: We assign bit-costs to ERC's operational primitives (including partial/multivalued tests) in such a way that any real function incurring polynomial cost becomes Turing-computable in polynomial time, and vice versa. Runtime measurements on implementations in the iRRAM C++ library confirm our theoretical performance predictions.
