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

Annotated revision programs

Revision programming is a formalism to describe and enforce updates of belief sets and databases. That formalism was extended by Fitting who assigned annotations to revision atoms. Annotations provide a way to quantify the confidence (probability) that a revision atom holds. The main goal of our paper is to reexamine the work of Fitting, argue that his semantics does not always provide results consistent with intuition, and to propose an alternative treatment of annotated revision programs. Our approach differs from that proposed by Fitting in two key aspects: we change the notion of a model of a program and we change the notion of a justified revision. We show that under this new approach fundamental properties of justified revisions of standard revision programs extend to the annotated case.Comment: 30 pages, to appear in Artificial Intelligence Journa

Generalized Paxos Made Byzantine (and Less Complex)

One of the most recent members of the Paxos family of protocols is Generalized Paxos. This variant of Paxos has the characteristic that it departs from the original specification of consensus, allowing for a weaker safety condition where different processes can have a different views on a sequence being agreed upon. However, much like the original Paxos counterpart, Generalized Paxos does not have a simple implementation. Furthermore, with the recent practical adoption of Byzantine fault tolerant protocols, it is timely and important to understand how Generalized Paxos can be implemented in the Byzantine model. In this paper, we make two main contributions. First, we provide a description of Generalized Paxos that is easier to understand, based on a simpler specification and the pseudocode for a solution that can be readily implemented. Second, we extend the protocol to the Byzantine fault model

A Denotational Semantics for First-Order Logic

In Apt and Bezem [AB99] (see cs.LO/9811017) we provided a computational interpretation of first-order formulas over arbitrary interpretations. Here we complement this work by introducing a denotational semantics for first-order logic. Additionally, by allowing an assignment of a non-ground term to a variable we introduce in this framework logical variables. The semantics combines a number of well-known ideas from the areas of semantics of imperative programming languages and logic programming. In the resulting computational view conjunction corresponds to sequential composition, disjunction to ``don't know'' nondeterminism, existential quantification to declaration of a local variable, and negation to the ``negation as finite failure'' rule. The soundness result shows correctness of the semantics with respect to the notion of truth. The proof resembles in some aspects the proof of the soundness of the SLDNF-resolution.Comment: 17 pages. Invited talk at the Computational Logic Conference (CL 2000). To appear in Springer-Verlag Lecture Notes in Computer Scienc

On the Semantics of Gringo

Input languages of answer set solvers are based on the mathematically simple concept of a stable model. But many useful constructs available in these languages, including local variables, conditional literals, and aggregates, cannot be easily explained in terms of stable models in the sense of the original definition of this concept and its straightforward generalizations. Manuals written by designers of answer set solvers usually explain such constructs using examples and informal comments that appeal to the user's intuition, without references to any precise semantics. We propose to approach the problem of defining the semantics of gringo programs by translating them into the language of infinitary propositional formulas. This semantics allows us to study equivalent transformations of gringo programs using natural deduction in infinitary propositional logic.Comment: Proceedings of Answer Set Programming and Other Computing Paradigms (ASPOCP 2013), 6th International Workshop, August 25, 2013, Istanbul, Turke

Fifty years of Hoare's Logic

We present a history of Hoare's logic.Comment: 79 pages. To appear in Formal Aspects of Computin