Unfounded Sets and Well-Founded Semantics of Answer Set Programs with Aggregates

by   Mario Alviano, et al.

Logic programs with aggregates (LPA) are one of the major linguistic extensions to Logic Programming (LP). In this work, we propose a generalization of the notions of unfounded set and well-founded semantics for programs with monotone and antimonotone aggregates (LPAma programs). In particular, we present a new notion of unfounded set for LPAma programs, which is a sound generalization of the original definition for standard (aggregate-free) LP. On this basis, we define a well-founded operator for LPAma programs, the fixpoint of which is called well-founded model (or well-founded semantics) for LPAma programs. The most important properties of unfounded sets and the well-founded semantics for standard LP are retained by this generalization, notably existence and uniqueness of the well-founded model, together with a strong relationship to the answer set semantics for LPAma programs. We show that one of the D-well-founded semantics, defined by Pelov, Denecker, and Bruynooghe for a broader class of aggregates using approximating operators, coincides with the well-founded model as defined in this work on LPAma programs. We also discuss some complexity issues, most importantly we give a formal proof of tractable computation of the well-founded model for LPA programs. Moreover, we prove that for general LPA programs, which may contain aggregates that are neither monotone nor antimonotone, deciding satisfaction of aggregate expressions with respect to partial interpretations is coNP-complete. As a consequence, a well-founded semantics for general LPA programs that allows for tractable computation is unlikely to exist, which justifies the restriction on LPAma programs. Finally, we present a prototype system extending DLV, which supports the well-founded semantics for LPAma programs, at the time of writing the only implemented system that does so. Experiments with this prototype show significant computational advantages of aggregate constructs over equivalent aggregate-free encodings.


page 1

page 2

page 3

page 4


Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer ...

Properties of Answer Set Programming with Convex Generalized Atoms

In recent years, Answer Set Programming (ASP), logic programming under t...

Semantics and Compilation of Answer Set Programming with Generalized Atoms

Answer Set Programming (ASP) is logic programming under the stable model...

Probability Aggregates in Probability Answer Set Programming

Probability answer set programming is a declarative programming that has...

A Logical Charaterisation of Ordered Disjunction

In this paper we consider a logical treatment for the ordered disjunctio...

Linear Programs with Conjunctive Database Queries

In this paper, we study the problem of optimizing a linear program whose...

Automated Expected Value Analysis of Recursive Programs

In this work, we study the fully automated inference of expected result ...

Please sign up or login with your details

Forgot password? Click here to reset