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 (QELF). 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 QELF 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
QELF 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. (Under consideration for
acceptance in TPLP)

本文提出了一种基于可计算函数的 Answer Set Programming（ASP）变体，通过 intensional sets 的应用，扩展了集合对象的应用，提供了全逻辑处理聚合的能力，并且给出了含义明确、可组合的聚合语义。

使用有意义集合的函数式 ASP：应用于 Gelfond-Zhang 聚合

Functional ASP with Intensional Sets: Application to Gelfond-Zhang  Aggregates

In this paper we present an extension of Peirce's existential graphs to
provide a diagrammatic representation of expressions in Quantified Equilibrium
Logic (QEL). Using this formalisation, logical connectives are replaced by
encircled regions (circles and squares) and quantified variables are
represented as "identity" lines. Although the expressive power is equivalent to
that of QEL, the new representation can be useful for illustrative or
educational purposes.

本文提出一种 Peirce 存在图的扩展，以提供 Quantified Equilibrium Logic（QEL）表达式的图表表示。使用这种形式化方法，逻辑连接词被圆圈和正方形等区域所代替，量化变量则用 “身份” 线表示。虽然表达能力和 QEL 一样，但新的表示方法可用于说明或教育目的。

平衡图

Equilibrium Graphs

In this paper we propose an extension of Answer Set Programming (ASP), and in
particular, of its most general logical counterpart, Quantified Equilibrium
Logic (QEL), to deal with partial functions. Although the treatment of equality
in QEL can be established in different ways, we first analyse the choice of
decidable equality with complete functions and Herbrand models, recently
proposed in the literature. We argue that this choice yields some
counterintuitive effects from a logic programming and knowledge representation
point of view. We then propose a variant called QELF where the set of functions
is partitioned into partial and Herbrand functions (we also call constructors).
In the rest of the paper, we show a direct connection to Scott's Logic of
Existence and present a practical application, proposing an extension of normal
logic programs to deal with partial functions and equality, so that they can be
translated into function-free normal programs, being possible in this way to
compute their answer sets with any standard ASP solver.

本文提出了 QELF 来处理部分函数，并且将其与 Scott 存在逻辑做了直接联系，还介绍了一个实际应用，即将扩展后的常规逻辑程序转换成无函数的常规程序，可以使用任何标准 ASP 求解器计算其答案集。