We elaborate upon the formal foundations of hybrid Answer Set Programming (ASP) and extend its underlying logical framework with aggregate functions over constraint values and variables. This is achieved by introducing the construct of conditional expressions, which allow for considering two alternatives while evaluating constraints. Which alternative is considered is interpretation-dependent and chosen according to an associated condition. We put some emphasis on logic programs with linear constraints and show how common ASP aggregates can be regarded as particular cases of so-called conditional linear constraints. Finally, we introduce a polynomial-size, modular and faithful translation from our framework into regular (condition-free) Constraint ASP, outlining an implementation of conditional aggregates on top of existing hybrid ASP solvers.
Weighted knowledge bases for description logics with typicality have been recently considered under a "concept-wise" multipreference semantics (in both the two-valued and fuzzy case), as the basis of a logical semantics of Multilayer Perceptrons. In this paper we consider weighted conditional EL^bot knowledge bases in the two-valued case, and exploit ASP and asprin for encoding concept-wise multipreference entailment for weighted KBs with integer weights.
Aggregates provide a concise way to express complex knowledge. The problem of selecting an appropriate formalisation of aggregates for answer set programming (ASP) remains unsettled. This paper revisits it from the viewpoint of Approximation Fixpoint Theory (AFT). We introduce an AFT formalisation equivalent with the Gelfond-Lifschitz reduct for basic ASP programs and we extend it to handle aggregates. We analyse how existing approaches relate to our framework. We hope this work sheds some new light on the issue of a proper formalisation of aggregates. This paper is under consideration for acc…
This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator $Φ^{aggr}_P$ for aggregate programs, independently proposed Pelov et al. This operator allows us to closely tie the computational complexity of the answer set checking and answer sets existence problems to the cost of checking a solution of the aggregates in the program. Finally, we relate th…
University of Calabria（卡拉布里亚大学）；University of Klagenfurt（克拉根福大学）；Graz University of Technology（格拉茨工业大学）
Answer Set Programming (ASP) emerged in the late 1990ies as a paradigm for Knowledge Representation and Reasoning. The attractiveness of ASP builds on an expressive high-level modeling language along with the availability of powerful off-the-shelf solving systems. While the utility of incorporating aggregate expressions in the modeling language has been realized almost simultaneously with the inception of the first ASP solving systems, a general semantics of aggregates and its efficient implementation have been long-standing challenges. Aggregates have been proposed and widely used in database…
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 exte…
We introduce a parallel offline algorithm for computing hybrid conditional plans, called HCP-ASP, oriented towards robotics applications. HCP-ASP relies on modeling actuation actions and sensing actions in an expressive nonmonotonic language of answer set programming (ASP), and computation of the branches of a conditional plan in parallel using an ASP solver. In particular, thanks to external atoms, continuous feasibility checks (like collision checks) are embedded into formal representations of actuation actions and sensing actions in ASP; and thus each branch of a hybrid conditional plan des…
Answer set programming (ASP) is a logic programming paradigm that can be used to solve complex combinatorial search problems. Aggregates are an ASP construct that plays an important role in many applications. Defining a satisfactory semantics of aggregates turned out to be a difficult problem, and in this paper we propose a new approach, based on an analogy between aggregates and propositional connectives. First, we extend the definition of an answer set/stable model to cover arbitrary propositional theories; then we define aggregates on top of them both as primitive constructs and as abbrevia…
Università del Piemonte Orientale（东皮埃蒙特大学）；Università di Torino（都灵大学）
Weighted knowledge bases for description logics with typicality have been recently considered under a "concept-wise" multipreference semantics (in both the two-valued and fuzzy case), as the basis of a logical semantics of MultiLayer Perceptrons (MLPs). In this paper we consider weighted conditional ALC knowledge bases with typicality in the finitely many-valued case, through three different semantic constructions. For the boolean fragment LC of ALC we exploit ASP and "asprin" for reasoning with the concept-wise multipreference entailment under a phi-coherent semantics, suitable to characteriz…
University of Corunna（拉科鲁尼亚大学）；Institut de Recherche en Informatique de Toulouse（图卢兹计算机科学研究所）；CNRS（法国国家科学研究中心）
Dynamic Epistemic Logic (DEL) is a family of multimodal logics that has proved to be very successful for epistemic reasoning in planning tasks. In this logic, the agent's knowledge is captured by modal epistemic operators whereas the system evolution is described in terms of (some subset of) dynamic logic modalities in which actions are usually represented as semantic objects called event models. In this paper, we study a variant of DEL, that wecall DEL[ASP], where actions are syntactically described by using an Answer Set Programming (ASP) representation instead of event models. This represen…
The paper presents a knowledge representation language $\mathcal{A}log$ which extends ASP with aggregates. The goal is to have a language based on simple syntax and clear intuitive and mathematical semantics. We give some properties of $\mathcal{A}log$, an algorithm for computing its answer sets, and comparison with other approaches.
The paper presents a knowledge representation language $\mathcal{A}log$ which extends ASP with aggregates. The goal is to have a language based on simple syntax and clear intuitive and mathematical semantics. We give some properties of $\mathcal{A}log$, an algorithm for computing its answer sets, and comparison with other approaches.
We provide a comprehensive elaboration of the theoretical foundations of variable instantiation, or grounding, in Answer Set Programming (ASP). Building on the semantics of ASP's modeling language, we introduce a formal characterization of grounding algorithms in terms of (fixed point) operators. A major role is played by dedicated well-founded operators whose associated models provide semantic guidance for delineating the result of grounding along with on-the-fly simplifications. We address an expressive class of logic programs that incorporates recursive aggregates and thus amounts to the sc…
Gelfond and Zhang recently proposed a new stable model semantics based on Vicious Circle Principle in order to improve the interpretation of logic programs with aggregates. The paper focuses on this proposal, and analyzes the complexity of both coherence testing and cautious reasoning under the new semantics. Some surprising results highlight similarities and differences versus mainstream stable model semantics for aggregates. Moreover, the paper reports on the design of compilation techniques for implementing the new semantics on top of existing ASP solvers, which eventually lead to realize a…
Weighted Logic is a powerful tool for the specification of calculations over semirings that depend on qualitative information. Using a novel combination of Weighted Logic and Here-and-There (HT) Logic, in which this dependence is based on intuitionistic grounds, we introduce Answer Set Programming with Algebraic Constraints (ASP(AC)), where rules may contain constraints that compare semiring values to weighted formula evaluations. Such constraints provide streamlined access to a manifold of constructs available in ASP, like aggregates, choice constraints, and arithmetic operators. They extend …
The paper continues the investigation of Poincare and Russel's Vicious Circle Principle (VCP) in the context of the design of logic programming languages with sets. We expand previously introduced language Alog with aggregates by allowing infinite sets and several additional set related constructs useful for knowledge representation and teaching. In addition, we propose an alternative formalization of the original VCP and incorporate it into the semantics of new language, Slog+, which allows more liberal construction of sets and their use in programming rules. We show that, for programs withou…
University of Corunna（拉科鲁尼亚大学）；University of Potsdam（波茨坦大学）
Characterizing hybrid ASP solving in a generic way is difficult since one needs to abstract from specific theories. Inspired by lazy SMT solving, this is usually addressed by treating theory atoms as opaque. Unlike this, we propose a slightly more transparent approach that includes an abstract notion of a term. Rather than imposing a syntax on terms, we keep them abstract by stipulating only some basic properties. With this, we further develop a semantic framework for hybrid ASP solving and provide aggregate functions for theory variables that adhere to different semantic principles, show that…
The paper presents two equivalent definitions of answer sets for logic programs with aggregates. These definitions build on the notion of unfolding of aggregates, and they are aimed at creating methodologies to translate logic programs with aggregates to normal logic programs or positive programs, whose answer set semantics can be used to defined the semantics of the original programs. The first definition provides an alternative view of the semantics for logic programming with aggregates described by Pelov et al. The second definition is similar to the traditional answer set definition for no…
New Mexico State University（新墨西哥州立大学）；Omaha State University（奥马哈州立大学）；University of Potsdam（波茨坦大学）
We present a general approach to planning with incomplete information in Answer Set Programming (ASP). More precisely, we consider the problems of conformant and conditional planning with sensing actions and assumptions. We represent planning problems using a simple formalism where logic programs describe the transition function between states, the initial states and the goal states. For solving planning problems, we use Quantified Answer Set Programming (QASP), an extension of ASP with existential and universal quantifiers over atoms that is analogous to Quantified Boolean Formulas (QBFs). We…
