Marjon Blondeel

Complexity of fuzzy answer set programming under Łukasiewicz semantics

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules - a syntactic generalization which does not affect the expressivity of ASP in the classical case - the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution. The complexity of the main reasoning tasks for disjunctive FASP is NP-complete.There is a reduction from reasoning in FASP to bilevel linear programming.We connect the complexity to an existing open problem about integer equations.

Fuzzy answer set programming : an introduction

In this chapter, we present a tutorial about fuzzy answer set programming (FASP); we give a gentle introduction to its basic ideas and definitions. FASP is a combination of answer set programming and fuzzy logics which has recently been proposed. From the answer set semantics, FASP inherits the declarative nonmonotonic reasoning capabilities, while fuzzy logic adds the power to model continuous problems. FASP can be tailored towards different applications since fuzzy logics gives a great flexibility, e.g. by the possibility to use different generalizations of the classical connectives. In this chapter, we consider a rather general form of FASP programs; the connectives can in principal be interpreted by arbitrary [0,1] n → [0,1]-mappings. Despite that very general connectives are allowed, the presented framework turns out to be an intuitive extension of answer set programming.

Fuzzy autoepistemic logic and its relation to fuzzy answer set programming

Autoepistemic logic is an important formalism for nonmonotonic reasoning. It extends propositional logic by offering the ability to reason about an agents (lack of) beliefs. Moreover, it is well known to generalize the stable model semantics of answer set programming. Fuzzy logics on the other hand are multi-valued logics, which allow to model the intensity to which properties are satisfied. We combine these ideas to a fuzzy autoepistemic logic which can be used to reason about ones beliefs in the degrees to which properties are satisfied. We show that many properties from classical autoepistemic logic, e.g. the equivalence between autoepistemic models and stable expansions, remain valid under this generalization. In this paper, we consider a version of fuzzy answer set programming and show that its answer sets can be equivalently described as models in fuzzy autoepistemic logic. We also define a fuzzy logic of minimal belief and negation-as-failure and use this as a tool to show that fuzzy autoepistemic logic generalizes fuzzy answer set programming.

Fuzzy autoepistemic logic: reflecting about knowledge of truth degrees

Autoepistemic logic is one of the principal formalisms for nonmonotonic reasoning. It extends propositional logic by offering the ability to reason about an agents (lack of) knowledge or beliefs. Moreover, it is well known to generalize the stable model semantics of answer set programming. Fuzzy logics on the other hand are multi-valued logics, which allow to model the intensity with which a property is satisfied. We combine these ideas to a fuzzy autoepistemic logic which can be used to reason about ones knowledge about the degrees to which proporties are satisfied. In this paper we show that many properties from classical autoepistemic logic remain valid under this generalization and that the important relation between autoepistemic logic and answer set programming is preserved in the sense that fuzzy autoepistemic logic generalizes fuzzy answer set programming.

On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief

Autoepistemic logic is an important formalism for nonmonotonic reasoning originally intended to model an ideal rational agent reflecting upon his own beliefs. Fuzzy autoepistemic logic is a generalization of autoepistemic logic that allows to represent an agents rational beliefs on gradable propositions. It has recently been shown that, in the same way as autoepistemic logic generalizes answer set programming, fuzzy autoepistemic logic generalizes fuzzy answer set programming as well. Besides being related to answer set programming, autoepistemic logic is also closely related to several modal logics. To investigate whether a similar relationship holds in a fuzzy logical setting, we firstly generalize the main modal logics for belief to the setting of finitely-valued Łukasiewicz logic with truth constants Ł k c , and secondly we relate them with fuzzy autoepistemic logics. Moreover, we show that the problem of satisfiability checking in these logics is NP-complete. Finally, we generalize Levesques results on stable expansions, belief sets, and only knowing operators to our setting, and provide a complete axiomatization for a logic of only knowing in the Ł k c framework.

Repairing Inconsistent Taxonomies Using MAP Inference and Rules of Thumb

Several authors have developed relation extraction methods for automatically learning or refining taxonomies from large text corpora such as the Web. However, without appropriate post-processing, such taxonomies are often inconsistent (e.g. they contain cycles). A standard approach to repairing such inconsistencies is to identify a minimally consistent subset of the extracted facts. For example, we could aim to minimize the sum of the confidence weights of the facts that are removed for restoring consistency. In this paper, we present MAP inference as a base method for this approach, and analyze how it can be improved by taking into account dependencies between the extracted facts. These dependencies correspond to rules of thumb such as if a given fact is wrong then all facts that have been extracted from the same sentence are also likely to be wrong, which we encode in Markov logic. We present experimental results to demonstrate the potential of this idea.

Statistical relational learning with soft quantifiers

Quantification in statistical relational learning (SRL) is either existential or universal, however humans might be more inclined to express knowledge using soft quantifiers, such as “most” and “a few”. In this paper, we define the syntax and semantics of PSL(^Q), a new SRL framework that supports reasoning with soft quantifiers, and present its most probable explanation (MPE) inference algorithm. To the best of our knowledge, PSL(^Q) is the first SRL framework that combines soft quantifiers with first-order logic rules for modeling uncertain relational data. Our experimental results for link prediction in social trust networks demonstrate that the use of soft quantifiers not only allows for a natural and intuitive formulation of domain knowledge, but also improves the accuracy of inferred results.