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Declarative Programming of Search Problems with Builtin Arithmetic
"... We address the problem of providing a logical formalization of arithmetic in declarative modelling languages for NP search problems. The challenge is to simultaneously allow quantification over an infinite domain such as the natural numbers, provide natural modelling facilities, and control expressi ..."
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We address the problem of providing a logical formalization of arithmetic in declarative modelling languages for NP search problems. The challenge is to simultaneously allow quantification over an infinite domain such as the natural numbers, provide natural modelling facilities, and control expressive power of the language. To address the problem, we introduce an extension of the model expansion (MX) based framework to finite structures embedded in an infinite secondary structure, together with “doubleguarded ” logics for representing MX specifications for these structures. The logics also contain multiset functions (aggregate operations). Our main result is that these logics capture the complexity class NP on “smallcost ” arithmetical structures. 1
GIDL: A Grounder for FO
 in ‘Proceedings of the Twelfth International Workshop on NonMonotonic Reasoning
"... In this paper, we present GIDL, a grounder for FO+. FO+ is a very expressive extension of firstorder logic with several constructs such as inductive definitions, aggregates and arithmetic. We describe the input and output language of GIDL, and provide details about its architecture. In particular ..."
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In this paper, we present GIDL, a grounder for FO+. FO+ is a very expressive extension of firstorder logic with several constructs such as inductive definitions, aggregates and arithmetic. We describe the input and output language of GIDL, and provide details about its architecture. In particular, the core grounding algorithm implemented in GIDL is presented. We compare GIDL with other FO+ grounders and with grounders for Answer Set Programming.
Building a knowledge base system for an integration of logic programming and classical logic
 In ICLP. 71–76
"... Abstract. This paper presents a Knowledge Base project for FO(ID), an extension of classical logic with inductive definitions. This logic is a natural integration of classical logic and logic programming based on the view of a logic program as a definition. We discuss the relationship between induc ..."
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Abstract. This paper presents a Knowledge Base project for FO(ID), an extension of classical logic with inductive definitions. This logic is a natural integration of classical logic and logic programming based on the view of a logic program as a definition. We discuss the relationship between inductive definitions and common sense reasoning and the strong similarities and striking differences with ASP and Abductive LP. We report on inference systems that combine stateoftheart techniques of SAT and ASP. Experiments show that FO(ID) model expansion systems are competitive with the best ASPsolvers. 1
Approximate reasoning in firstorder logic theories
"... Many computational settings are concerned with finding (all) models of a firstorder logic theory for a fixed, finite domain. In this paper, we present a method to compute from a given theory and finite domain an approximate structure: a structure that approximates all models. We show confluence of ..."
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Many computational settings are concerned with finding (all) models of a firstorder logic theory for a fixed, finite domain. In this paper, we present a method to compute from a given theory and finite domain an approximate structure: a structure that approximates all models. We show confluence of this method and investigate its complexity. We discuss some applications, including 3valued query answering in integrated and partially incomplete databases, and improved grounding in the context of model expansion for firstorder logic.
Model Expansion and the Expressiveness of FO(ID) and Other Logics
"... Model expansion problem is a question of determining, given a formula and a structure for a part of the vocabulary of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problemsolving paradigm based on model expansion by (Mitchell & ..."
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Model expansion problem is a question of determining, given a formula and a structure for a part of the vocabulary of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problemsolving paradigm based on model expansion by (Mitchell & Ternovska, 2005; Mitchell, Ternovska, Hach, & Mohebali, 2006) posed the question of complexity of this problem for logics used in the paradigm. We discuss the complexity of the model expansion problem for a number of logics, alongside that of satisfiability and model checking. As the task is equivalent to witnessing leading existential secondorder quantifiers in a model checking setting, the paper is in large part a survey of this area together with some new results. In particular, we describe the combined and data complexity of model expansion for FO(ID) (Denecker & Ternovska, 2008), as well as guarded and kguarded logics of (Andréka, van Benthem, & Németi, 1998) and (Gottlob, Leone, & Scarcello, 2001).
Lazy Model Expansion: Interleaving Grounding with Search
"... Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The stateoftheart approach for bounded model generation for rich knowledge representation languages like A ..."
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Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The stateoftheart approach for bounded model generation for rich knowledge representation languages like Answer Set Programming (ASP) and FO(·) and a CSP modeling language such as Zinc, is groundandsolve: reduce the theory to a ground or propositional one and apply a search algorithm to the resulting theory. An important bottleneck is the blowup of the size of the theory caused by the grounding phase. Lazily grounding the theory during search is a way to overcome this bottleneck. We present a theoretical framework and an implementation in the context of the FO(·) knowledge representation language. Instead of grounding all parts of a theory, justifications are derived for some parts of it. Given a partial assignment for the grounded part of the theory and valid justifications for the formulas of the nongrounded part, the justifications provide a recipe to construct a complete assignment that satisfies the nongrounded part. When a justification for a particular formula becomes invalid during search, a new one is derived; if that fails, the formula is split in a part to be grounded and a part that can be justified. Experimental results illustrate the power and generality of this approach. 1.
The SAT Solver MXC, version 0.75 (2008 SAT Race Version)
, 2008
"... MXC is a complete, clauselearning SAT solver, written in C++. Development of MXC began in 2006, primarily to have an inhouse solver to support the research project described in [6, 7]. Since then, we have been keeping MXC up to date with recent developments in “industrial ” SAT solver algorithms, ..."
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MXC is a complete, clauselearning SAT solver, written in C++. Development of MXC began in 2006, primarily to have an inhouse solver to support the research project described in [6, 7]. Since then, we have been keeping MXC up to date with recent developments in “industrial ” SAT solver algorithms, and submitting to the competitions. The first released version, MXC v. 0.1 [2], was entered in the 2006 Sat Race
Search Problems as Model Finding
"... ABSTRACT. Arguments for logicbased knowledge representation often emphasize the primacy of entailment in reasoning, and traditional logicbased formulations of AI tasks were frequently in terms of entailment. More recently, practical progress in satisfiabilitybased methods has encouraged formulati ..."
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ABSTRACT. Arguments for logicbased knowledge representation often emphasize the primacy of entailment in reasoning, and traditional logicbased formulations of AI tasks were frequently in terms of entailment. More recently, practical progress in satisfiabilitybased methods has encouraged formulation of problems as model finding. Here, we argue for the formalization of search problems, which abound in AI as well as other areas, as a particular form of model finding called model expansion. An important conceptual part of this proposal is the formalization of the problem instance as a structure, rather than as a formula. Adopting this view leads naturally to taking descriptive complexity theory as the starting point for developing a theory of languages for representing search problems. We explain the formalization of search as model expansion, and the reasons we consider it an appropriate basis for such a theory. We emphasize the role of model expansion for first order logic, with extensions, in specifying NP search problems, and describe the formalization of arithmetic in this context. This paper is dedicated to Hector J. Levesque.
Lazy Model Expansion: Interleaving Grounding with Search
, 2014
"... Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The stateoftheart approach for bounded model generation for rich knowledge representation languages, like A ..."
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Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The stateoftheart approach for bounded model generation for rich knowledge representation languages, like Answer Set Programming (ASP), FO(·) and Zinc, is groundandsolve: reduce the theory to a ground or propositional one and apply a search algorithm to the resulting theory. An important bottleneck is the blowup of the size of the theory caused by the reduction phase. Lazily grounding the theory during search is a way to overcome this bottleneck. We present a theoretical framework and an implementation in the context of the FO(·) knowledge representation language. Instead of grounding all parts of a theory, justifications are derived for some parts of it. Given a partial assignment for the grounded part of the theory and valid justifications for the formulas of the nongrounded part, the justifications provide a recipe to construct a complete assignment that satisfies the nongrounded part. When a justification for a particular formula becomes invalid during search, a new one is derived; if that fails, the formula is split in a part to be grounded and a part that can be justified. The theoretical framework captures existing approaches for tackling the grounding bottleneck such as lazy clause generation, groundingonthefly and presents a generalization of the 2watched literal scheme. We present an algorithm for lazy model expansion and integrate it in a model generator for FO(ID), a language extending firstorder logic with inductive definitions. The algorithm is implemented as part of the