Annotated nonmonotonic rule systems

AbstractAnnotated logics were proposed by Subrahmanian as a unified paradigm for representing a wide variety of reasoning tasks including reasoning with uncertainty within a single theoretical framework. Subsequently, Marek, Nerode and Remmel have shown how to provide nonmonotonic extensions of arbitrary languages through their notion of a nonmonotonic rule systems. The primary aim of this paper is to define annotated nonmonotonic rule systems which merge these two frameworks into a general purpose nonmonotonic reasoning framework over arbitrary multiple-valued logics. We then show how Reiter's normal default theories may be generalized to the framework of annotated nonmonotonic rule systems

Possibilistic Stable Models

We present the main lines of a new framework that we have defined in order to improve the knowledge representation power of Answer Set Programming paradigm. Our proposal is to use notions from possibility theory to extend the stable model semantics by taking into account a certainty level, expressed in terms of necessity measure, on each rule of a normal logic program. First of all, we introduce possibilistic definite logic programs and show how to compute the conclusions of such programs both in syntactic and semantic ways. The syntactic handling is done by help of a fix-point operator, the semantic part relies on a possibility distribution on all sets of atoms and the two approaches are shown to be equivalent. In a second part, we define what is a possibilistic stable model for a normal logic program, with default negation. Again, we define a possibility distribution allowing to determine the stable models. We end our presentation by showing how we can use our framework to adressing inconsistency in Answer Set Programming

Aggregated fuzzy answer set programming

Fuzzy Answer Set programming (FASP) is an extension of answer set programming (ASP), based on fuzzy logic. It allows to encode continuous optimization problems in the same concise manner as ASP allows to model combinatorial problems. As a result of its inherent continuity, rules in FASP may be satisfied or violated to certain degrees. Rather than insisting that all rules are fully satisfied, we may only require that they are satisfied partially, to the best extent possible. However, most approaches that feature partial rule satisfaction limit themselves to attaching predefined weights to rules, which is not sufficiently flexible for most real-life applications. In this paper, we develop an alternative, based on aggregator functions that specify which (combination of) rules are most important to satisfy. We extend upon previous work by allowing aggregator expressions to define partially ordered preferences, and by the use of a fixpoint semantics

A core language for fuzzy answer set programming

A number of different Fuzzy Answer Set Programming (FASP) formalisms have been proposed in the last years, which all differ in the language extensions they support. In this paperwe investigate the expressivity of these frameworks. Specificallywe showhowa variety of constructs in these languages can be implemented using a considerably simpler core language. These simulations are important as a compact and simple language is easier to implement and to reason about, while an expressive language offers more options when modeling problems

Foundations of fuzzy answer set programming

Answer set programming (ASP) is a declarative language that is tailored towards combinatorial search problems. Although ASP has been applied to many problems, such as planning, configuration and verification of software, and database repair, it is less suitable for describing continuous problems. In this thesis we therefore studied fuzzy answer set programming (FASP). FASP is a language that combines ASP with ideas from fuzzy logic -- a class of many-valued logics that are able to describe continuous problems. We study the following topics: 1. An important issue when modeling continuous optimization problems is how to cope with overconstrained problems. In many cases we can opt to allow imperfect solutions, i.e. solutions that do not satisfy all constraints, but are sufficiently acceptable. However, the question which one of these imperfect solutions is most suitable then arises. Current approaches to fuzzy answer set programming solve this problem by attaching weights to the rules of the program. However, it is often not clear how these weights should be chosen and moreover weights do not allow to order different solutions. We improve upon this technique by using aggregators, which eliminate the aforementioned problems. This allows a richer modeling language and bridges the gap between FASP and other techniques such as valued constraint satisfaction problems. 2. The wishes of users and implementers of a programming language are often in direct conflict with each other. Users prefer a rich language that is easy to model in, whereas implementers prefer a small language that is easy to implement. We reconcile these differences by identifying a core language for FASP, called core FASP (CFASP), that only consists of non-constraint rules with monotonically increasing functions and negators in the body. We show that CFASP is capable of simulating constraint rules, monotonically decreasing functions, aggregators, S-implicators and classical negation. Moreover we remark that the simulations of constraints and classical negation bear a great resemblance to their simulations in classical ASP, which provides further insight into the relationship between ASP and FASP. 3. As a first step towards the creation of an implementation method for FASP we research whether it is possible to translate a FASP program to a fuzzy SAT problem. We introduce the concept of the completion of a FASP program and show that for programs without loops the models of the completion coincide with the answer sets. Furthermore we show that if a program has loops, we can translate the program to a fuzzy SAT problem by generalizing the concept of loop formulas. We illustrate this on a continuous version of the k-center problem. Such a translation is important because it allows us to solve FASP programs by means of solvers for fuzzy SAT. Under the appropriate conditions it is for example possible to solve FASP programs by means of off-the-shelf solvers for mixed integer programming (MIP)