We present a logical system CFP (Concurrent Fixed Point Logic) from whose proofs one can extract nondeterministic and concurrent programs that are provably total and correct with respect to the proven formula. CFP is an intuitionistic first-order logic with inductive and coinductive definitions extended by two propositional operators, A || B (restriction, a strengthening of the implication B -> A) and $\ddownarrow(A)$ (total concurrency). The target of the extraction is a lambda calculus with constructors and recursion extended by a constructor Amb (for McCarthy's amb) which is interpreted operationally as globally angelic choice. The correctness of extracted programs is proven via an intermediate domain-theoretic denotational semantics. We demonstrate the usefulness of our system by extracting a concurrent program that translates infinite Gray code into the signed digit representation. A noteworthy feature of our system is that the proof rules for restriction and concurrency involve variants of the classical law of excluded middle that would not be interpretable computationally without Amb.

中文翻译：

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从证明中提取总的Amb程序

我们提出了一种逻辑系统CFP（并发不动点逻辑），其证明可以从中提取不确定性和并发性程序，这些程序可以证明是合计的，并且对于经过验证的公式而言是正确的。CFP是直觉的一阶逻辑，具有两个命题运算符A ||扩展的归纳和共归定义。B（限制，含义B的加强-> A）和$ \ ddownarrow（A）$（并发总数）。提取的目标是具有构造函数的Lambda演算，并通过构造函数Amb（对于McCarthy的amb）扩展递归，该构造函数在操作上被解释为全局天使选择。提取的程序的正确性是通过中间域理论的指称语义来证明的。通过提取将无限格雷码转换为带符号的数字表示形式的并发程序，我们证明了系统的实用性。我们系统的一个值得注意的特征是，限制和并发的证明规则涉及被排除中间语言的经典定律的变体，如果没有Amb，这些定律就无法在计算上进行解释。