scholarly journals Strong Equivalence of Qualitative Optimization Problems

2013 ◽  
Vol 47 ◽  
pp. 351-391
Author(s):  
W. Faber ◽  
M. Truszczyński ◽  
S. Woltran
Keyword(s):  
Equilibrium Model ◽  
Propositional Logic ◽  
Optimization Problems ◽  
Modular Representation ◽  
Set Optimization ◽  
Propositional Theory ◽  
Reasoning Tasks ◽  
Answer Set Semantics ◽  
The Way ◽  
Answer Set

We introduce the framework of qualitative optimization problems (or, simply, optimization problems) to represent preference theories. The formalism uses separate modules to describe the space of outcomes to be compared (the generator) and the preferences on outcomes (the selector). We consider two types of optimization problems. They differ in the way the generator, which we model by a propositional theory, is interpreted: by the standard propositional logic semantics, and by the equilibrium-model (answer-set) semantics. Under the latter interpretation of generators, optimization problems directly generalize answer-set optimization programs proposed previously. We study strong equivalence of optimization problems, which guarantees their interchangeability within any larger context. We characterize several versions of strong equivalence obtained by restricting the class of optimization problems that can be used as extensions and establish the complexity of associated reasoning tasks. Understanding strong equivalence is essential for modular representation of optimization problems and rewriting techniques to simplify them without changing their inherent properties.

scholarly journals Characterising equilibrium logic and nested logic programs: Reductions and complexity,

2009 ◽  
Vol 9 (05) ◽  
pp. 565-616 ◽  
Author(s):  
DAVID PEARCE ◽  
HANS TOMPITS ◽  
STEFAN WOLTRAN
Keyword(s):  
Propositional Logic ◽  
Stable Model ◽  
Logic Programs ◽  
Complexity Bounds ◽  
Inference Engines ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Reasoning Tasks ◽  
Answer Set Semantics ◽  
Quantified Propositional Logic

AbstractEquilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case ofnested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs intoquantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to asquantified Boolean formulas(QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are theconsistency problem,brave reasoningandskeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz.ordinary,stronganduniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds. Hereby, our encodings yield upper bounds in a direct manner. Besides this useful feature, our approach has the following benefits: First, our encodings yield auniform axiomatisationfor a variety of problems in a common language. Second, extant solvers for QBFs can be used as back-end inference engines to realise implementations of the encoded task in a rapid prototyping manner. Third, our axiomatisations also allow us to straightforwardly relate equilibrium logic with circumscription.

scholarly journals Better Paracoherent Answer Sets with Less Resources

2019 ◽  
Vol 19 (5-6) ◽  
pp. 757-772 ◽  
Author(s):  
GIOVANNI AMENDOLA ◽  
CARMINE DODARO ◽  
FRANCESCO RICCA
Keyword(s):  
Problem Solving ◽  
Equilibrium Model ◽  
State Of The Art ◽  
Answer Set Programming ◽  
Evaluation Technique ◽  
Answer Set Semantics ◽  
Computational Resources ◽  
Answer Sets ◽  
Modeling Feature ◽  
Answer Set

AbstractAnswer Set Programming (ASP) is a well-established formalism for logic programming. Problem solving in ASP requires to write an ASP program whose answers sets correspond to solutions. Albeit the non-existence of answer sets for some ASP programs can be considered as a modeling feature, it turns out to be a weakness in many other cases, and especially for query answering. Paracoherent answer set semantics extend the classical semantics of ASP to draw meaningful conclusions also from incoherent programs, with the result of increasing the range of applications of ASP. State of the art implementations of paracoherent ASP adopt the semi-equilibrium semantics, but cannot be lifted straightforwardly to compute efficiently the (better) split semi-equilibrium semantics that discards undesirable semi-equilibrium models. In this paper an efficient evaluation technique for computing a split semi-equilibrium model is presented. An experiment on hard benchmarks shows that better paracoherent answer sets can be computed consuming less computational resources than existing methods.

scholarly journals Transition systems for model generators—A unifying approach

2011 ◽  
Vol 11 (4-5) ◽  
pp. 629-646 ◽  
Author(s):  
YULIYA LIERLER ◽  
MIROSLAW TRUSZCZYNSKI
Keyword(s):  
Logic Programming ◽  
Propositional Logic ◽  
Answer Set Programming ◽  
Transition Systems ◽  
Satisfiability Solvers ◽  
Answer Set Semantics ◽  
Computing Models ◽  
Answer Set

AbstractA fundamental task for propositional logic is to compute models of propositional formulas. Programs developed for this task are called satisfiability solvers. We show that transition systems introduced by Nieuwenhuis, Oliveras, and Tinelli to model and analyze satisfiability solvers can be adapted for solvers developed for two other propositional formalisms: logic programming under the answer-set semantics, and the logic PC(ID). We show that in each case the task of computing models can be seen as “satisfiability modulo answer-set programming,” where the goal is to find a model of a theory that also is an answer set of a certain program. The unifying perspective we develop shows, in particular, that solvers clasp and minisat(id) are closely related despite being developed for different formalisms, one for answer-set programming and the latter for the logic PC(ID).

Qualitative properties of solutions to set optimization problems

2021 ◽  
Vol 40 (2) ◽  
Author(s):  
Lam Quoc Anh ◽  
Nguyen Huu Danh ◽  
Pham Thanh Duoc ◽  
Tran Ngoc Tam
Keyword(s):  
Optimization Problems ◽  
Set Optimization ◽  
Qualitative Properties ◽  
Qualitative Properties Of Solutions

scholarly journals ACO with Intuitionistic Fuzzy Pheromone Updating Applied on Multiple-Constraint Knapsack Problem

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1456
Author(s):  
Stefka Fidanova ◽  
Krassimir Todorov Atanassov
Keyword(s):  
Decision Making ◽  
Knapsack Problem ◽  
Propositional Logic ◽  
Optimization Problems ◽  
Real Life ◽  
Combinatorial Optimization Problems ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Multiple Constraint

Some of industrial and real life problems are difficult to be solved by traditional methods, because they need exponential number of calculations. As an example, we can mention decision-making problems. They can be defined as optimization problems. Ant Colony Optimization (ACO) is between the best methods, that solves combinatorial optimization problems. The method mimics behavior of the ants in the nature, when they look for a food. One of the algorithm parameters is called pheromone, and it is updated every iteration according quality of the achieved solutions. The intuitionistic fuzzy (propositional) logic was introduced as an extension of Zadeh’s fuzzy logic. In it, each proposition is estimated by two values: degree of validity and degree of non-validity. In this paper, we propose two variants of intuitionistic fuzzy pheromone updating. We apply our ideas on Multiple-Constraint Knapsack Problem (MKP) and compare achieved results with traditional ACO.

scholarly journals Grounding and Solving in Answer Set Programming

AI Magazine ◽  
2016 ◽  
Vol 37 (3) ◽  
pp. 25-32 ◽  
Author(s):  
Benjamin Kaufmann ◽  
Nicola Leone ◽  
Simona Perri ◽  
Torsten Schaub
Keyword(s):  
Problem Solving ◽  
Propositional Logic ◽  
Logic Program ◽  
Answer Set Programming ◽  
Key Issues ◽  
Answer Sets ◽  
Answer Set

Answer set programming is a declarative problem solving paradigm that rests upon a workflow involving modeling, grounding, and solving. While the former is described by Gebser and Schaub (2016), we focus here on key issues in grounding, or how to systematically replace object variables by ground terms in a effective way, and solving, or how to compute the answer sets of a propositional logic program obtained by grounding.

scholarly journals Negation as a Resource: a Novel View on Answer Set Semantics*

2015 ◽  
Vol 140 (3-4) ◽  
pp. 279-305 ◽  
Author(s):  
Stefania Costantini ◽  
Andrea Formisano
Keyword(s):  
Answer Set Semantics ◽  
Answer Set

scholarly journals An ASP Approach to Generate Minimal Countermodels in Intuitionistic Propositional Logic

2019 ◽  
Author(s):  
Camillo Fiorentini
Keyword(s):  
Propositional Logic ◽  
Answer Set Programming ◽  
Kripke Model ◽  
Kripke Semantics ◽  
Minimal Models ◽  
Intuitionistic Propositional Logic ◽  
Answer Set

Intuitionistic Propositional Logic is complete w.r.t. Kripke semantics: if a formula is not intuitionistically valid, then there exists a finite Kripke model falsifying it. The problem of obtaining concise models has been scarcely investigated in the literature. We present a procedure to generate minimal models in the number of worlds relying on Answer Set Programming (ASP).

scholarly journals A Formal Approach for Cautious Reasoning in Answer Set Programming (Extended Abstract)

2020 ◽  
Author(s):  
Giovanni Amendola ◽  
Carmine Dodaro ◽  
Marco Maratea
Keyword(s):  
Answer Set Programming ◽  
Satisfiability Modulo Theories ◽  
Propositional Satisfiability ◽  
Formal Approach ◽  
Boolean Formulas ◽  
Reasoning Tasks ◽  
Answer Set

The issue of describing in a formal way solving algorithms in various fields such as Propositional Satisfiability (SAT), Quantified SAT, Satisfiability Modulo Theories, Answer Set Programming (ASP), and Constraint ASP, has been relatively recently solved employing abstract solvers. In this paper we deal with cautious reasoning tasks in ASP, and design, implement and test novel abstract solutions, borrowed from backbone computation in SAT. By employing abstract solvers, we also formally show that the algorithms for solving cautious reasoning tasks in ASP are strongly related to those for computing backbones of Boolean formulas. Some of the new solutions have been implemented in the ASP solver WASP, and tested.

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