Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to behaviors of GKAT expressions. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms.

中文翻译：

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带有测试的守卫克莱恩代数：方程，共导和完备性

带测试的守卫克莱恩代数（GKAT）是KAT的有效片段，因为它允许几乎线性地确定等价性。在本文中，我们研究了GKAT的（共）代数性质。我们最初的重点是通过省略所谓的提前终止公理，可以区分执行不同操作的失败程序的片段。我们开发了一种可操作的（代数的）和指称的（代数的）语义，并证明它们是一致的。然后，我们以这种语义来表征GKAT表达式的行为，从而导致捕获与GKAT表达式的行为相对应的自动机协变量的共等式。最后，我们证明了缩减片段的公理是合理的，并且具有完整的语义，然后基于此结果恢​​复了合理且完整的语义。