480 research outputs found
Rewriting recursive aggregates in answer set programming: back to monotonicity
Aggregation functions are widely used in answer set programming for representing and reasoning on knowledge involving sets of objects collectively. Current implementations simplify the structure of programs in order to optimize the overall performance. In particular, aggregates are rewritten into simpler forms known as monotone aggregates. Since the evaluation of normal programs with monotone aggregates is in general on a lower complexity level than the evaluation of normal programs with arbitrary aggregates, any faithful translation function must introduce disjunction in rule heads in some cases. However, no function of this kind is known. The paper closes this gap by introducing a polynomial, faithful, and modular translation for rewriting common aggregation functions into the simpler form accepted by current solvers. A prototype system allows for experimenting with arbitrary recursive aggregates, which are also supported in the recent version 4.5 of the grounder gringo, using the methods presented in this paper
Relating Weight Constraint and Aggregate Programs: Semantics and Representation
Weight constraint and aggregate programs are among the most widely used logic
programs with constraints. In this paper, we relate the semantics of these two
classes of programs, namely the stable model semantics for weight constraint
programs and the answer set semantics based on conditional satisfaction for
aggregate programs. Both classes of programs are instances of logic programs
with constraints, and in particular, the answer set semantics for aggregate
programs can be applied to weight constraint programs. We show that the two
semantics are closely related. First, we show that for a broad class of weight
constraint programs, called strongly satisfiable programs, the two semantics
coincide. When they disagree, a stable model admitted by the stable model
semantics may be circularly justified. We show that the gap between the two
semantics can be closed by transforming a weight constraint program to a
strongly satisfiable one, so that no circular models may be generated under the
current implementation of the stable model semantics. We further demonstrate
the close relationship between the two semantics by formulating a
transformation from weight constraint programs to logic programs with nested
expressions which preserves the answer set semantics. Our study on the
semantics leads to an investigation of a methodological issue, namely the
possibility of compact representation of aggregate programs by weight
constraint programs. We show that almost all standard aggregates can be encoded
by weight constraints compactly. This makes it possible to compute the answer
sets of aggregate programs using the ASP solvers for weight constraint
programs. This approach is compared experimentally with the ones where
aggregates are handled more explicitly, which show that the weight constraint
encoding of aggregates enables a competitive approach to answer set computation
for aggregate programs.Comment: To appear in Theory and Practice of Logic Programming (TPLP), 2011.
30 page
Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms
In this paper, we present two alternative approaches to defining answer sets
for logic programs with arbitrary types of abstract constraint atoms (c-atoms).
These approaches generalize the fixpoint-based and the level mapping based
answer set semantics of normal logic programs to the case of logic programs
with arbitrary types of c-atoms. The results are four different answer set
definitions which are equivalent when applied to normal logic programs. The
standard fixpoint-based semantics of logic programs is generalized in two
directions, called answer set by reduct and answer set by complement. These
definitions, which differ from each other in the treatment of
negation-as-failure (naf) atoms, make use of an immediate consequence operator
to perform answer set checking, whose definition relies on the notion of
conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other
two definitions, called strongly and weakly well-supported models, are
generalizations of the notion of well-supported models of normal logic programs
to the case of programs with c-atoms. As for the case of fixpoint-based
semantics, the difference between these two definitions is rooted in the
treatment of naf atoms. We prove that answer sets by reduct (resp. by
complement) are equivalent to weakly (resp. strongly) well-supported models of
a program, thus generalizing the theorem on the correspondence between stable
models and well-supported models of a normal logic program to the class of
programs with c-atoms. We show that the newly defined semantics coincide with
previously introduced semantics for logic programs with monotone c-atoms, and
they extend the original answer set semantics of normal logic programs. We also
study some properties of answer sets of programs with c-atoms, and relate our
definitions to several semantics for logic programs with aggregates presented
in the literature
Sampler Programs: The Stable Model Semantics of Abstract Constraint Programs Revisited
Abstract constraint atoms provide a general framework for the study of aggregates utilized in answer set programming. Such primitives suitably increase the expressive power of rules and enable more concise representation of various domains as answer set programs. However, it is non-trivial to generalize the stable model semantics for programs involving arbitrary abstract constraint atoms. For instance, a nondeterministic variant of the immediate consequence operator is needed, or the definition of stable models cannot be stated directly using primitives of logic programs. In this paper, we propose sampler programs as a relaxation of abstract constraint programs that better lend themselves to the program transformation involved in the definition of stable models. Consequently, the declarative nature of stable models can be restored for sampler programs and abstract constraint programs are also covered if decomposed into sampler programs. Moreover, we study the relationships of the classes of programs involved and provide a characterization in terms of abstract but essentially deterministic computations. This result indicates that all nondeterminism related with abstract constraint atoms can be resolved at the level of program reduct when sampler programs are used as the intermediate representation
On the semantics of hybrid ASP systems based on Clingo
[Abstract]: Over the last decades, the development of Answer Set Programming (ASP) has brought about an expressive modeling language powered by highly performant systems. At the same time, it gets more and more difficult to provide semantic underpinnings capturing the resulting constructs and inferences. This is even more severe when it comes to hybrid ASP languages and systems that are often needed to handle real-world applications. We address this challenge and introduce the concept of abstract and structured theories that allow us to formally elaborate upon their integration with ASP. We then use this concept to make the semantic characterization of clingo’s theory-reasoning framework precise. This provides us with a formal framework in which we can elaborate upon the formal properties of existing hybridizations of clingo, such as clingcon, clingo[dl], and clingo[lp].This work was supported by DFG grant SCHA 550/11, Germany, by grant PID2020-116201GB-I00 funded by MCIN/AEI/ 10.13039/501100011033, Spain, by Xunta de Galicia and the European Union, GPC ED431B 2022/33, by European COST action CA17124 DigForASP, EU, and by the National Science Foundation (NSF 95-3101-0060-402), USA.Xunta de Galicia; ED431B 2022/33Deutsche Forschungsgemeinschaft; SCHA 550/11United States. National Science Foundation; NSF 95-3101-0060-40
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