Relational theories with null values and non-herbrand stable models

AbstractIn this note, we consider the problem of introducing variables in temporal logic programs under the formalism of Temporal Equilibrium Logic, an extension of Answer Set Programming for dealing with linear-time modal operators. To this aim, we provide a definition of a first-order version of Temporal Equilibrium Logic that shares the syntax of first-order Linear-time Temporal Logic but has different semantics, selecting some Linear-time Temporal Logic models we call temporal stable models. Then, we consider a subclass of theories (called splittable temporal logic programs) that are close to usual logic programs but allowing a restricted use of temporal operators. In this setting, we provide a syntactic definition of safe variables that suffices to show the property of domain independence – that is, addition of arbitrary elements in the universe does not vary the set of temporal stable models. Finally, we present a method for computing the derivable facts by constructing a non-temporal logic program with variables that is fed to a standard Answer Set Programming grounder. The information provided by the grounder is then used to generate a subset of ground temporal rules which is equivalent to (and generally smaller than) the full program instantiation.

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A Roadmap on Updates

Database Technologies ◽

10.4018/978-1-60566-058-5.ch137 ◽

2009 ◽

pp. 2261-2267

Author(s):

Fernando Zacarías Flores ◽

Dionicio Zacarías Flores ◽

Rosalba Cuapa Canto ◽

Luis Miguel Guzmán Muñoz

Keyword(s):

Standard Model ◽

Logic Programming ◽

Structural Properties ◽

Relational Databases ◽

Logic Program ◽

Answer Set Programming ◽

Central Issue ◽

The Standard Model ◽

Answer Set

Updates, is a central issue in relational databases and knowledge databases. In the last years, it has been well studied in the non-monotonic reasoning paradigm. Several semantics for logic program updates have been proposed (Brewka, Dix, & Knonolige 1997), (De Schreye, Hermenegildo, & Pereira, 1999) (Katsumo & Mendelzon, 1991). However, recently a set of proposals has been characterized to propose mechanisms of updates based on logic and logic programming. All these mechanisms are built on semantics based on structural properties (Eiter, Fink, Sabattini & Thompits, 2000) (Leite, 2002) (Banti, Alferes & Brogi, 2003) (Zacarias, 2005). Furthermore, all these semantic ones coincide in considering the AGM proposal as the standard model in the update theory, for their wealth in properties. The AGM approach, introduced in (Alchourron, Gardenfors & Makinson, 1985) is the dominating paradigm in the area, but in the context of monotonic logic. All these proposals analyze and reinterpret the AGM postulates under the Answer Set Programming (ASP) such as (Eiter, Fink, Sabattini & Thompits, 2000). However, the majority of the adapted AGM and update postulates are violated by update programs, as shown in(De Schreye, Hermenegildo, & Pereira, 1999).

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A Roadmap on Updates

Encyclopedia of Artificial Intelligence ◽

10.4018/978-1-59904-849-9.ch201 ◽

2011 ◽

pp. 1370-1375

Author(s):

Fernando Zacarías Flores ◽

Dionicio Zacarías Flores ◽

Rosalba Cuapa Canto ◽

Luis Miguel Guzmán Muñoz

Keyword(s):

Standard Model ◽

Logic Programming ◽

Structural Properties ◽

Relational Databases ◽

Logic Program ◽

Answer Set Programming ◽

Central Issue ◽

The Standard Model ◽

Answer Set

Updates, is a central issue in relational databases and knowledge databases. In the last years, it has been well studied in the non-monotonic reasoning paradigm. Several semantics for logic program updates have been proposed (Brewka, Dix, & Knonolige 1997), (De Schreye, Hermenegildo, & Pereira, 1999) (Katsumo & Mendelzon, 1991). However, recently a set of proposals has been characterized to propose mechanisms of updates based on logic and logic programming. All these mechanisms are built on semantics based on structural properties (Eiter, Fink, Sabattini & Thompits, 2000) (Leite, 2002) (Banti, Alferes & Brogi, 2003) (Zacarias, 2005). Furthermore, all these semantic ones coincide in considering the AGM proposal as the standard model in the update theory, for their wealth in properties. The AGM approach, introduced in (Alchourron, Gardenfors & Makinson, 1985) is the dominating paradigm in the area, but in the context of monotonic logic. All these proposals analyze and reinterpret the AGM postulates under the Answer Set Programming (ASP) such as (Eiter, Fink, Sabattini & Thompits, 2000). However, the majority of the adapted AGM and update postulates are violated by update programs, as shown in (De Schreye, Hermenegildo, & Pereira, 1999).

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Lloyd-Topor completion and general stable models

Theory and Practice of Logic Programming ◽

10.1017/s1471068413000318 ◽

2013 ◽

Vol 13 (4-5) ◽

pp. 503-515 ◽

Cited By ~ 2

Author(s):

VLADIMIR LIFSCHITZ ◽

FANGKAI YANG

Keyword(s):

Logic Program ◽

Program Completion ◽

Stable Model ◽

First Order ◽

The Relationship ◽

Stable Models

AbstractWe investigate the relationship between the generalization of program completion defined in 1984 by Lloyd and Topor and the generalization of the stable model semantics introduced recently by Ferraris et al. The main theorem can be used to characterize, in some cases, the general stable models of a logic program by a first-order formula. The proof uses Truszczynski's stable model semantics of infinitary propositional formulas.

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Modeling and Language Extensions

AI Magazine ◽

10.1609/aimag.v37i3.2673 ◽

2016 ◽

Vol 37 (3) ◽

pp. 33-44 ◽

Cited By ~ 4

Author(s):

Martin Gebser ◽

Torsten Schaub

Keyword(s):

Expressive Language ◽

Logic Program ◽

Answer Set Programming ◽

Stable Model ◽

Logic Programs ◽

First Order ◽

High Level Modeling ◽

High Level ◽

Answer Sets ◽

First Order Variables

Answer set programming (ASP) has emerged as an approach to declarative problem solving based on the stable model semantics for logic programs. The basic idea is to represent a computational problem by a logic program, formulating constraints in terms of rules, such that its answer sets correspond to problem solutions. To this end, ASP combines an expressive language for high-level modeling with powerful low-level reasoning capacities, provided by off-the-shelf tools. Compact problem representations take advantage of genuine modeling features of ASP, including (first-order) variables, negation by default, and recursion. In this article, we demonstrate the ASP methodology on two example scenarios, illustrating basic as well as advanced modeling and solving concepts. We also discuss mechanisms to represent and implement extended kinds of preferences and optimization. An overview of further available extensions concludes the article.

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Stable models for infinitary formulas with extensional atoms

Theory and Practice of Logic Programming ◽

10.1017/s1471068416000314 ◽

2016 ◽

Vol 16 (5-6) ◽

pp. 771-786

Author(s):

AMELIA HARRISON ◽

VLADIMIR LIFSCHITZ

Keyword(s):

Programming Languages ◽

Answer Set Programming ◽

Splitting Theorem ◽

First Order ◽

Definition Of ◽

Stable Models ◽

Answer Set

AbstractThe definition of stable models for propositional formulas with infinite conjunctions and disjunctions can be used to describe the semantics of answer set programming languages. In this note, we enhance that definition by introducing a distinction between intensional and extensional atoms. The symmetric splitting theorem for first-order formulas is then extended to infinitary formulas and used to reason about infinitary definitions.

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Do We Need Many-valued Logics for Incomplete Information?

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2019/851 ◽

2019 ◽

Author(s):

Marco Console ◽

Paolo Guagliardo ◽

Leonid Libkin

Keyword(s):

Incomplete Information ◽

Relational Databases ◽

Predicate Logic ◽

Truth Values ◽

Design Choice ◽

Null Values ◽

Epistemic Approach ◽

The Many ◽

The Right ◽

Sql Query

One of the most common scenarios of handling incomplete information occurs in relational databases. They describe incomplete knowledge with three truth values, using Kleene's logic for propositional formulae and a rather peculiar extension to predicate calculus. This design by a committee from several decades ago is now part of the standard adopted by vendors of database management systems. But is it really the right way to handle incompleteness in propositional and predicate logics? Our goal is to answer this question. Using an epistemic approach, we first characterize possible levels of partial knowledge about propositions, which leads to six truth values. We impose rationality conditions on the semantics of the connectives of the propositional logic, and prove that Kleene's logic is the maximal sublogic to which the standard optimization rules apply, thereby justifying this design choice. For extensions to predicate logic, however, we show that the additional truth values are not necessary: every many-valued extension of first-order logic over databases with incomplete information represented by null values is no more powerful than the usual two-valued logic with the standard Boolean interpretation of the connectives. We use this observation to analyze the logic underlying SQL query evaluation, and conclude that the many-valued extension for handling incompleteness does not add any expressiveness to it.

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Towards the Temporal Approach to Abstract Data Types

Fundamenta Informaticae ◽

10.3233/fi-1988-11105 ◽

1988 ◽

Vol 11 (1) ◽

pp. 49-63

Author(s):

Andrzej Szalas

Keyword(s):

Temporal Logic ◽

Abstract Data Types ◽

Data Types ◽

First Order ◽

Abstract Data ◽

Order Completeness ◽

General Method

In this paper we deal with a well known problem of specifying abstract data types. Up to now there were many approaches to this problem. We follow the axiomatic style of specifying abstract data types (cf. e.g. [1, 2, 6, 8, 9, 10]). We apply, however, the first-order temporal logic. We introduce a notion of first-order completeness of axiomatic specifications and show a general method for obtaining first-order complete axiomatizations. Some examples illustrate the method.

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Closed Sets of Boolean Terms in Relational Databases

Fundamenta Informaticae ◽

10.3233/fi-1991-14310 ◽

1991 ◽

Vol 14 (3) ◽

pp. 367-385

Author(s):

Andrzej Jankowski ◽

Zbigniew Michalewicz

Keyword(s):

Computational Complexity ◽

Incomplete Information ◽

Relational Databases ◽

The Other ◽

New Approach ◽

Closed Sets ◽

Null Value ◽

Algebraic Operations

A number of approaches have been taken to represent compound, structured values in relational databases. We review a few such approaches and discuss a new approach, in which every set is represented as a Boolean term. We show that this approach generalizes the other approaches, leading to more flexible representation. Boolean term representation seems to be appropriate in handling incomplete information: this approach generalizes some other approaches (e.g. null value mark, null variables, etc). We consider definitions of algebraic operations on such sets, like join, union, selection, etc. Moreover, we introduce a measure of computational complexity of these operations.

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Unsatisfiable Core Analysis and Aggregates for Optimum Stable Model Search

Fundamenta Informaticae ◽

10.3233/fi-2020-1974 ◽

2020 ◽

Vol 176 (3-4) ◽

pp. 271-297

Author(s):

Mario Alviano ◽

Carmine Dodaro

Keyword(s):

State Of The Art ◽

Answer Set Programming ◽

Efficient Algorithms ◽

Satisfiability Problem ◽

Core Analysis ◽

Stable Model ◽

Maximum Satisfiability ◽

Model Search ◽

Unsatisfiable Core ◽

Stable Models

Many efficient algorithms for the computation of optimum stable models in the context of Answer Set Programming (ASP) are based on unsatisfiable core analysis. Among them, algorithm OLL was the first introduced in the context of ASP, whereas algorithms ONE and PMRES were first introduced for solving the Maximum Satisfiability problem (MaxSAT) and later on adapted to ASP. In this paper, we present the porting to ASP of another state-of-the-art algorithm introduced for MaxSAT, namely K, which generalizes ONE and PMRES. Moreover, we present a new algorithm called OLL-IN-ONE that compactly encodes all aggregates of OLL by taking advantage of shared aggregate sets propagators. The performance of the algorithms have been empirically compared on instances taken from the latest ASP Competition.

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