Credulous Reasoning Research Articles

Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics, including credulous reasoning. When asking for projected counts, we are interested in counting the number of extensions of a given argumentation framework, while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by the treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming. Our algorithms run in double or triple exponential time in the treewidth, depending on the semantics under consideration. Finally, we establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension under the exponential time hypothesis (ETH).

A weighted causal theory for acquiring and utilizing open knowledge

Motivated by enabling intelligent robots/agents to take advantage of open-source knowledge resources to solve open-ended tasks, a weighted causal theory is introduced as the formal basis for the development of these robots/agents. The action model of a robot/agent is specified as a causal theory following McCain and Turner's nonmonotonic causal theories. New knowledge is needed when the robot/agent is given a user task that cannot be accomplished only with the action model. This problem is cast as a variant of abduction, that is, to find the most suitable set of causal rules from open-source knowledge resources, so that a plan for accomplishing the task can be computed using the action model together with the acquired knowledge. The core part of our theory is constructed based on credulous reasoning and the complexity of corresponding abductive reasoning is analyzed. The entire theory is established by adding weights to hypothetical causal rules and using them to compare competing explanations which induce causal models satisfying the task. Moreover, we sketch a model theoretic semantics for the weighted causal theory and present an algorithm for computing a weighted-abductive explanation. An application of the techniques proposed in this paper is illustrated in an example on our service robot, KeJia, in which the robot tries to acquire proper knowledge from OMICS, a large-scale open-source knowledge resource, and solve new tasks with the knowledge.

Adaptively applying modus ponens in conditional logics of normality

This paper presents an adaptive logic enhancement of conditional logics of normality that allows for defeasible applications of Modus Ponens to conditionals. In addition to the possibilities these logics already offer in terms of reasoning about conditionals, this way they are enriched by the ability to perform default inferencing. The idea is to apply Modus Ponens defeasibly to a conditional and a fact on the condition that it is ‘safe' to do so concerning the factual and conditional knowledge at hand. It is for instance not safe if the given information describes exceptional circumstances: although birds usually fly, penguins are exceptional to this rule. The two adaptive standard strategies are shown to correspond to different intuitions, a skeptical and a credulous reasoning type, which manifest themselves in the handling of so-called floating conclusions.

Proof complexity of propositional default logic

Default logic is one of the most popular and successful formalisms for non-monotonic reasoning. In 2002, Bonatti and Olivetti introduced several sequent calculi for credulous and skeptical reasoning in propositional default logic. In this paper we examine these calculi from a proof-complexity perspective. In particular, we show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen’s system LK. Hence proving lower bounds for credulous reasoning will be as hard as proving lower bounds for LK. On the other hand, we show an exponential lower bound to the proof size in Bonatti and Olivetti’s enhanced calculus for skeptical default reasoning.

Regular disjunction-free default theories

In this paper, the class of regular disjunction-free default theories is introduced and investigated. A transformation from regular default theories to normal default theories is established. The initial theory and the transformed theory have the same extensions when restricted to old variables. Hence, regular default theories enjoy some similar properties (e.g., existence of extensions, semi-monotonicity) as normal default theories. Then, a new algorithm for credulous reasoning of regular theories is developed. This algorithm runs in a time not more than O(1.45n), where n is the number of defaults. In case of regular prerequisite-free or semi-2CNF default theories, the credulous reasoning can be solved in polynomial time. However, credulous reasoning for semi-Horn default theories is shown to be NP-complete although it is tractable for Horn default theories. Moreover, skeptical reasoning for regular unary default theories is co-NP-complete.

Logic knowledge bases with two default rules

Logic knowledge based systems (LKBS) containing at most one form of default negation and explicit (or “classical”) negation have been studied in the literature. In this paper we describe a class of LKBS containing multiple forms of default negation in addition to explicit negation. We define a semantics for these systems in terms of the welldfounded semantics defined by Van Gelder et al. (1988) and the stable semantics introduced by Gelfond and Lifschitz (1988) and later extended to the 3dvalued case by Przymusinski (1991). We investigate properties of the new combined semantics and calculate the computational complexity of three main reasoning tasks for this semantics, namely existence of models, skeptical and credulous reasoning. An effective procedure to construct the collection of models characterizing the semantics of such a system is given. Applications to knowledge representation and knowledge base merging are presented.

Default reasoning by deductive planning

This paper deals with the automation of reasoning from incomplete information by means of default logics. We provide proof procedures for default logics' major reasoning modes, namely, credulous and skeptical reasoning. We start by reformulating the task of credulous reasoning in default logics as deductive planning problems. This interpretation supplies us with several interesting and valuable insights into the proof theory of default logics. Foremost, it allows us to take advantage of the large number of available methods, algorithms, and implementations for solving deductive planning problems. As an example, we demonstrate how credulous reasoning in certain variants of default logic is implementable by means of a planning method based on equational logic programming. In addition, our interpretation allows us to transfer theoretical results, such as complexity results, from the field of planning to that of default logics. In this way, we have isolated two yet unknown classes of default theories for which deciding credulous entailment is polynomial.

The complexity of path-based defeasible inheritance

Touretzky [22] proposed a formalism for nonmonotonic multiple inheritance reasoning which is sound in the presence of ambiguities and redundant links. We show that Touretzky's inheritance notion is NP-hard, and thus, provided P ≠ NP, computationally intractable. This result holds even when one only considers unambiguous, totally acyclic inheritance networks. A direct consequence of this result is that the conditioning strategy proposed by Touretzky to allow for fast parallel inference is also intractable. We also analyze the influence of various design choices made by Touretzky. We show that all versions of so-called downward inheritance, i.e., on-path or off-path preemption and skeptical or credulous reasoning, are intractable. However, on the positive side, tractability can be achieved when using upward inheritance. Finally, we consider the problem of determining what holds in all extensions of an ambiguous inheritance network. Without any preemption—corresponding to a very cautious form of inheritance—this problem is tractable, but when allowing even the most basic forms of preemption, it becomes intractable.