This article deals with advanced notions of equivalence between nonmonotonic logic programs under the answer-set semantics, a topic of considerable interest, because such notions form the basis for program verification and are useful for program optimisation, debugging, and modular programming. In fact, there is extensive research in answer-set programming (ASP) dealing with different notions of equivalence between programs. Prominent among these notions is uniform equivalence , which checks whether two programs have the same semantics when joined with an arbitrary set of facts. In this article, we study a family of more fine-grained versions of uniform equivalence, viz. relativised uniform equivalence with projection , which extends standard uniform equivalence in terms of two additional parameters: one for specifying the input alphabet and one for specifying the output alphabet for programs. In particular, the second parameter is used for projecting answer sets to a set of designated output atoms. Answer-set projection, in particular, allows to compare programs that make use of different auxiliary atoms, which is important for practical programming aspects. We introduce novel semantic characterisations for the program correspondence problems under consideration and analyse their computational complexity. In the general case, deciding these problems lies on the third level of the polynomial hierarchy. Therefore, this task cannot be efficiently reduced to propositional answer-set programs itself (under the usual complexity-theoretic assumptions). However, reductions to quantified Boolean formulas (QBFs) are feasible. Indeed, we provide efficient (in fact, linear-time constructible) reductions to QBFs and discuss simplifications for certain special cases. These QBF reductions yield the basis for a prototype implementation, the system cc ⊤, for deciding correspondence problems by using off-the-shelf QBF solvers. We discuss an application of cc ⊤ for verifying the correctness of solutions by students drawn from a laboratory course on logic programming and knowledge representation at the Technische Universität Wien, employing relativised uniform equivalence with projection as the underlying program correspondence notion.

中文翻译：

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超越答案集程序之间的统一等价性

本文讨论了答案集语义下的非单调逻辑程序之间等价的高级概念，这是一个相当有趣的话题，因为这些概念构成了程序验证的基础，并且对程序优化、调试和模块化编程很有用。事实上，在处理程序之间等价的不同概念的答案集编程 (ASP) 方面进行了广泛的研究。这些概念中最突出的是统一等价，它检查两个程序在与任意一组事实连接时是否具有相同的语义。在本文中，我们研究了一系列更细粒度的统一等价，即。投影的相对统一等价，它在两个附加参数方面扩展了标准统一等价：一个用于指定输入字母表，另一个用于指定程序的输出字母表。特别是，第二个参数用于投影答案集到一组指定的输出原子。特别是答案集投影允许比较使用不同辅助原子的程序，这对于实际编程方面很重要。我们为正在考虑的程序对应问题引入了新的语义特征，并分析了它们的计算复杂性。在一般情况下，决定这些问题取决于多项式层次结构的第三级。因此，该任务不能有效地简化为命题答案集程序本身（在通常的复杂性理论假设下）。然而，减少到量化的布尔公式（QBF）是可行的。实际上，我们为 QBF 提供了有效的（实际上是线性时间可构造的）约简，并讨论了某些特殊情况的简化。这些 QBF 减少为原型实现系统 cc ⊤ 提供了基础，用于通过使用现成的 QBF 求解器来确定对应问题。我们讨论了 cc ⊤ 的应用，用于验证来自维也纳工业大学逻辑编程和知识表示实验室课程的学生的解决方案的正确性，采用具有投影的相对化统一等价作为基础程序对应概念。