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首页> 外文期刊>Theory and Practice of Logic Programming >Functional ASP with Intensional Sets: Application to Gelfond-Zhang Aggregates
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Functional ASP with Intensional Sets: Application to Gelfond-Zhang Aggregates

机译：带有内涵集的功能ASP：在Gelfond-Zhang集料中的应用

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In this paper, we propose a variant of Answer Set Programming (ASP) with evaluable functions that extends their application to sets of objects, something that allows a fully logical treatment of aggregates. Formally, we start from the syntax of First Order Logic with equality and the semantics of Quantified Equilibrium Logic with evaluable functions (QEL(F)(=)). Then, we proceed to incorporate a new kind of logical term, intensional set (a construct commonly used to denote the set of objects characterised by a given formula), and to extend QEL(F)(=) semantics for this new type of expression. In our extended approach, intensional sets can be arbitrarily used as predicate or function arguments or even nested inside other intensional sets, just as regular first-order logical terms. As a result, aggregates can be naturally formed by the application of some evaluable function (count, sum, maximum, etc) to a set of objects expressed as an intensional set. This approach has several advantages. First, while other semantics for aggregates depend on some syntactic transformation (either via a reduct or a formula translation), the QEL(F)(=) interpretation treats them as regular evaluable functions, providing a compositional semantics and avoiding any kind of syntactic restriction. Second, aggregates can be explicitly defined now within the logical language by the simple addition of formulas that fix their meaning in terms of multiple applications of some (commutative and associative) binary operation. For instance, we can use recursive rules to define sum in terms of integer addition. Last, but not least, we prove that the semantics we obtain for aggregates coincides with the one defined by Gelfond and Zhang for the Alog language, when we restrict to that syntactic fragment.
机译：在本文中，我们提出了一种具有可评估功能的答案集编程（ASP）的变体，该变体将其应用程序扩展到对象集，从而可以对集合进行完全逻辑的处理。形式上，我们从具有相等性的一阶逻辑的语法和具有可评估函数（QEL（F）（=））的量化平衡逻辑的语义开始。然后，我们继续合并一种新的逻辑术语，即内涵集（一种通常用于表示由给定公式表征的对象集的结构），并为这种新型表达扩展QEL（F）（=）语义。在我们的扩展方法中，可以像常规的一阶逻辑项一样将内涵集任意用作谓词或函数自变量，甚至嵌套在其他内涵集内。结果，可以通过将某些可评估函数（计数，总和，最大值等）应用于表示为内涵集的一组对象来自然形成聚集体。这种方法有几个优点。首先，虽然聚合的其他语义依赖于某种语法转换（通过归约或公式转换），但QEL（F）（=）解释将它们视为常规的可评估函数，提供了构成语义，并避免了任何形式的语法限制。其次，现在可以通过简单地添加公式来在逻辑语言中显式定义聚合，这些公式根据某些（可交换和关联）二进制运算的多种应用来固定其含义。例如，我们可以使用递归规则根据整数加法定义和。最后但并非最不重要的一点是，当我们限制于该句法片段时，我们证明为聚合所获得的语义与Gelfond和Zhang为Alog语言所定义的语义一致。

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