We present a general methodology of proving the decidability of equational theory of programming language concepts in the framework of second-order algebraic theories. We propose a Haskell-based analysis tool, i.e. Second-Order Laboratory, which assists the proofs of confluence and strong normalisation of computation rules derived from second-order algebraic theories. To cover various examples in programming language theory, we combine and extend both syntactical and semantical results of the second-order computation in a non-trivial manner. We demonstrate how to prove decidability of various algebraic theories in the literature. It includes the equational theories of monad and λ-calculi, Plotkin and Power’s theory of states and bits, and Stark’s theory of π-calculus. We also demonstrate how this methodology can solve the coherence of monoidal categories.

中文翻译：

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如何用二阶计算分析器 SOL 证明方程理论的可判定性

我们提出了一种在二阶代数理论框架中证明编程语言概念的方程理论可判定性的一般方法。我们提出了一种基于Haskell 的分析工具，即Second-Order Laboratory，它有助于证明从二阶代数理论导出的计算规则的汇合和强归一化。为了涵盖编程语言理论中的各种示例，我们以非平凡的方式组合和扩展了二阶计算的句法和语义结果。我们展示了如何证明文献中各种代数理论的可判定性。它包括 monad 和 λ-演算的方程理论，Plotkin 和 Power 的状态和比特理论，以及 Stark 的 π-演算理论。