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Model Expansion as a Framework for Modelling and Solving Search Problems
"... We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite mod ..."
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We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework. 1.
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
On the complexity of model expansion
 In Proceedings of the 17th international conference on Logic for programming, artificial intelligence, and reasoning, LPAR’10
, 2010
"... Abstract. We study the complexity of model expansion (MX), which is the problem of expanding a given finite structure with additional relations to produce a finite model of a given formula. This is the logical task underlying many practical constraint languages and systems for representing and sol ..."
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Abstract. We study the complexity of model expansion (MX), which is the problem of expanding a given finite structure with additional relations to produce a finite model of a given formula. This is the logical task underlying many practical constraint languages and systems for representing and solving search problems, and our work is motivated by the need to provide theoretical foundations for these. We present results on both data and combined complexity of MX for several fragments and extensions of FO that are relevant for this purpose, in particular the guarded fragment GFk of FO and extensions of FO and GFk with inductive definitions. We present these in the context of the two closely related, but more studied, problems of model checking and finite satisfiability. To obtain results on FO(ID), the extension of FO with inductive definitions, we provide translations between FO(ID) with FO(LFP), which are of independent interest. 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.
Complexity of expanding a finite structure and related tasks
 The 8th Int. Workshop on Logic and Comput. Complexity (LCC
, 2006
"... The authors of [MT05] proposed a declarative constraint programming framework based on classical logic extended with nonmonotone inductive definitions. In the framework, a problem instance is a finite structure, and a problem specification is a formula defining the relationship between an instance ..."
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The authors of [MT05] proposed a declarative constraint programming framework based on classical logic extended with nonmonotone inductive definitions. In the framework, a problem instance is a finite structure, and a problem specification is a formula defining the relationship between an instance and its solutions. Thus, problem solving amounts to expanding a finite structure with new relations, to satisfy the formula. We present here the complexities of model expansion for a number of logics, alongside those of satisfiability and model checking. As the task is equivalent to witnessing the existential quantifiers in ∃SO model checking, the paper is in large part of a survey of this area, together with some new results. In particular, we describe the combined and data complexity of FO(ID), firstorder logic extended with inductive definitions [DT04] and the guarded and kguarded logics of [AvBN98] and [GLS01]. 1
A Method for Solving NP Search Based on Model Expansion and Grounding
, 2007
"... The logical task of model expansion (MX) has been proposed as a declarative constraint programming framework for solving search and decision problems. We present a method for solving NP search problems based on MX for first order logic extended with inductive definitions and cardinality constraints ..."
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The logical task of model expansion (MX) has been proposed as a declarative constraint programming framework for solving search and decision problems. We present a method for solving NP search problems based on MX for first order logic extended with inductive definitions and cardinality constraints. The method involves grounding, and execution of a propositional solver, such as a SAT solver. Our grounding algorithm applies a generalization of the relational algebra to construct a ground formula representing the solutions to an instance. We demonstrate the practical feasibility of our method with an implementation, called MXG. We present axiomatizations of several NPcomplete benchmark problems, and experimental results comparing the performance of MXG with stateoftheart Answer Set programming (ASP) solvers. The performance of MXG is competitive with, and often better than, the ASP solvers on the problems studied.
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).
Computing Science
"... ii We explore the application of MXG, a declarative programming solver for NP search problems based on Model Expansion (MX) for first order logic with inductive definitions. We present specifications for several common NPcomplete benchmark problems in the language of MXG, and describe some modeli ..."
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ii We explore the application of MXG, a declarative programming solver for NP search problems based on Model Expansion (MX) for first order logic with inductive definitions. We present specifications for several common NPcomplete benchmark problems in the language of MXG, and describe some modeling techniques we found useful in obtaining good solver performance. We present an experimental comparison of the performance of MXG with Answer Set Programming (ASP) solvers on these problems, showing that MXG is competitive and often better. As an extended example, we consider an NPcomplete phylogenetic inference problem. We present several specifications for this problem, employing a variety of techniques for obtaining good performance. Our best solution, which combines instance preprocessing, redundant axioms, and symmetry breaking axioms, performs orders of magnitude faster than previously reported declarative programming solutions using ASP solvers. iii iv To my mom and dad. Acknowledgments I would like to thank Dr. David Mitchell, my senior supervisor, for his guidance and support throughout my Masters studies. I thank Dr. Uwe Glasser and Dr. Arvind Gupta for their comments on the final document. I am grateful to Eugenia Ternovska, Jan Manuch, and Sharon (Xiaohong) Zhong for helpful discussions. My thanks to Jonathan Kavanagh for providing his ASP programs. Last but not least I would like to thank Raheleh Mohebali, my beloved partner, and my parents for their endless support. v
APPLICATIONS OF DECLARATIVE PROGRAMMING AND KNOWLEDGE MANAGEMENT (INAP 2011) AND
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