Argumentation frameworks as constraint satisfaction problems

Argumentation is a promising approach for defeasible reasoning. It consists of justifying each plausible conclusion by arguments. Since the available information may be inconsistent, a conclusion and its negation may both be justified. The arguments are thus said to be conflicting. The main issue is how to evaluate the arguments. Several semantics were proposed for that purpose. The most important ones are: stable, preferred, complete, grounded and admissible. A semantics is a set of criteria that should be satisfied by a set of arguments, called extension, in order to be acceptable. Different decision problems related to these semantics were defined (like whether an argumentation framework has a stable extension). It was also shown that most of these problems are intractable. Consequently, developing algorithms for these problems is not trivial and thus the implementation of argumentation systems not obvious. Recently, some solutions to this problem were found. The idea is to use a reduction method where a given problem is translated in another one like SAT or ASP. This paper follows this line of research. It studies how to encode the problem of computing the extensions of an argumentation framework (under each of the previous semantics) as a constraint satisfaction problem (CSP). Such encoding is of great importance since it makes it possible to use the very efficient solvers (developed by the CSP community) for computing the extensions. Our encodings take advantage of existing reductions to SAT problems in the case of Dung’s abstract framework. Among the various ways of translating a SAT problem into a CSP one, we propose the most appropriate one in the argumentation context. We also provide encodings in case two other families of argumentation frameworks: the constrained version of Dung’s abstract framework and preference-based argumentation framework.     

Minimal hypotheses: extension-based semantics to argumentation

The emptiness problem of the preferred semantics and the non-existence problem of the stable semantics are well recognized for argumentation frameworks. In this paper, we introduce two strong semantics, named s-preferred semantics and s-stable semantics, to guarantee the non-emptiness of the preferred extensions and the existence of the stable extensions respectively. Our semantics are defined by two concepts of extensions of argumentation frameworks, namely s-preferred extension and s-stable extension. Each is constructed in a similar way to the original semantics. The novelty of our semantics is that an extension of an argumentation framework is considered as a pair of sets of arguments, in which the second element of an extension is viewed as a kind of hypotheses that should be minimized. The s-preferred semantics not only solves the emptiness problem of the preferred semantics, but also coincides with the preferred semantics when nonempty preferred extensions exist. Meanwhile, the s-stable semantics ensures the existence of extensions, and coincides with the stable semantics when the stable extensions exist as well. The relations among various semantics for argumentation frameworks are discussed.     

Coalitions of arguments: A tool for handling bipolar argumentation frameworks

Bipolar argumentation frameworks enable to represent two kinds of interaction between arguments: support and conflict. In this paper, we turn a bipolar argumentation framework into a meta‐argumentation framework where conflicts occur between sets of arguments, characterized as coalitions of supporting arguments. So, Dung's well‐known semantics can be used on this meta‐argumentation framework to select the acceptable arguments. © 2009 Wiley Periodicals, Inc.     

分级论辩系统的逻辑研究

近年来,形式论证已逐渐成为人工智能领域的研究热点之一。自Dung于1995年提出抽象辩论框架起,学术界普遍认为论辩的核心任务是在各种基于外延的语义下对论点集进行评估,以确定其辩护状态。分级论辩系统(Graded Argumentation System,GAS)是对经典Dung型论辩系统(Dung-style Argumentation System,DAS)的推广,通过一般化DAS语义的两个核心性质,即无冲突性和可接受性,来提供更细化的论点状态概念。当前的论辩系统语义等效性研究主要集中在框架和论点层次上,可为其结构约简提供有力的保证。针对两个不同分级论辩系统中论点的语义等效问题,首先运用分级模态逻辑(Graded Modal Logic,GML)形式化分级论辩系统的片段,然后建立并证明了分级论辩系统基于外延的语义和GML公式之间的一一对应关系,最后定义分级互模拟关系并证明其蕴含分级论辩系统的4个重要的语义等价性。     

Computing ideal sceptical argumentation

We present two dialectic procedures for the sceptical ideal semantics for argumentation. The first procedure is defined in terms of dispute trees, for abstract argumentation frameworks. The second procedure is defined in dialectical terms, for assumption-based argumentation frameworks. The procedures are adapted from (variants of) corresponding procedures for computing the credulous admissible semantics for assumption-based argumentation, proposed in [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114–159]. We prove that the first procedure is sound and complete, and the second procedure is sound in general and complete for a special but natural class of assumption-based argumentation frameworks, that we refer to as p-acyclic. We also prove that in the case of p-acyclic assumption-based argumentation frameworks (a variant of) the procedure of [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114–159] for the admissible semantics is complete. Finally, we present a variant of the procedure of [P.M. Dung, R.A. Kowalski, F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence 170 (2006) 114–159] that is sound for the sceptical grounded semantics.     

Reasoning about preferences in argumentation frameworks

The abstract nature of Dung's seminal theory of argumentation accounts for its widespread application as a general framework for various species of non-monotonic reasoning, and, more generally, reasoning in the presence of conflict. A Dung argumentation framework is instantiated by arguments and a binary conflict based attack relation, defined by some underlying logical theory. The justified arguments under different extensional semantics are then evaluated, and the claims of these arguments define the inferences of the underlying theory. To determine a unique set of justified arguments often requires a preference relation on arguments to determine the success of attacks between arguments. However, preference information is often itself defeasible, conflicting and so subject to argumentation. Hence, in this paper we extend Dung's theory to accommodate arguments that claim preferences between other arguments, thus incorporating meta-level argumentation based reasoning about preferences in the object level. We then define and study application of the full range of Dung's extensional semantics to the extended framework, and study special classes of the extended framework. The extended theory preserves the abstract nature of Dung's approach, thus aiming at a general framework for non-monotonic formalisms that accommodate defeasible reasoning about as well as with preference information. We illustrate by formalising argument based logic programming with defeasible priorities in the extended theory.     

Strong order equivalence

Recently, strong equivalence for Answer Set Programming has been studied intensively, and was shown to be beneficial for modular programming and automated optimization. In this paper we define the novel notion of strong order equivalence for logic programs with preferences (ordered logic programs). Based on this definition we give, for several semantics for preference handling, necessary and sufficient conditions for programs to be strongly order equivalent. These results allow us also to associate a so-called SOE structure to each ordered logic program, such that two ordered logic programs are strongly order equivalent if and only if their SOE structures coincide. We also present the relationships among the studied semantics with respect to strong order equivalence, which differs considerably from their relationships with respect to preferred answer sets. Furthermore, we study the computational complexity of several reasoning tasks associated to strong order equivalence. Finally, based on the obtained results, we present – for the first time – simplification methods for ordered logic programs.     

Encoding deductive argumentation in quantified Boolean formulae

There are a number of frameworks for modelling argumentation in logic. They incorporate a formal representation of individual arguments and techniques for comparing conflicting arguments. A common assumption for logic-based argumentation is that an argument is a pair 〈Φ,α〉 where Φ is minimal subset of the knowledge-base such that Φ is consistent and Φ entails the claim α. Different logics provide different definitions for consistency and entailment and hence give us different options for argumentation. Classical propositional logic is an appealing option for argumentation but the computational viability of generating an argument is an issue. To better explore this issue, we use quantified Boolean formulae to characterise an approach to argumentation based on classical logic.     

The computational complexity of ideal semantics

We analyse the computational complexity of the recently proposed ideal semantics within both abstract argumentation frameworks (afs) and assumption-based argumentation frameworks (abfs). It is shown that while typically less tractable than credulous admissibi-lity semantics, the natural decision problems arising with this extension-based model can, perhaps surprisingly, be decided more efficiently than sceptical preferred semantics. In particular the task of finding the unique ideal extension is easier than that of deciding if a given argument is accepted under the sceptical semantics. We provide efficient algorithmic approaches for the class of bipartite argumentation frameworks and, finally, present a number of technical results which offer strong indications that typical problems in ideal argumentation are complete for the class of languages decidable by polynomial time algorithms allowed to make non-adaptive queries to a C oracle, where C is an upper bound on the computational complexity of deciding credulous acceptance: C=np for afs and logic programming (lp) instantiations of abfs; for abfs modelling default theories.     

TOWARD A COMPUTATIONAL ANALYSIS OF PROBABILISTIC ARGUMENTATION FRAMEWORKS

In this article we analyze probabilistic argumentation frameworks (PAFs), defined as an extension of Dung abstract argumentation frameworks in which each argument n is asserted with a probability p_n. The debate around PAFs has so far centered on their theoretical definition and basic properties. This work contributes to their computational analysis by proposing a first recursive algorithm to compute the probability of acceptance of each argument under grounded and preferred semantics and by studying the behavior of PAFs with respect to reinstatement, cycles, and changes in argument structure. The computational tools proposed may provide strategic information for agents selecting the next step in an open argumentation process and they represent a contribution in the debate about gradualism in abstract argumentation.     

Modelling defeasible and prioritized support in bipolar argumentation

Cayrol and Lagasquie-Schiex introduce bipolar argumentation frameworks by introducing a second relation on the arguments for representing the support among them. The main drawback of their approach is that they cannot encode defeasible support, for instance they cannot model an attack towards a support relation. In this paper, we introduce a way to model defeasible support in bipolar argumentation frameworks. We use the methodology of meta-argumentation in which Dung??s theory is used to reason about itself. Dung??s well-known admissibility semantics can be used on this meta-argumentation framework to compute the acceptable arguments, and all properties of Dung??s classical theory are preserved. Moreover, we show how different contexts can lead to the alternative strengthening of the support relation over the attack relation, and converse. Finally, we present two applications of our methodology for modeling support, the case of arguments provided with an internal structure and the case of abstract dialectical frameworks.     

Verification of programs that destructively manipulate data

We investigate various equivalence relations between expressions in a first-order functional programming language augmented with the ability to destructively alter the underlying data. To define the semantics we introduce the notion of a memory structure. A computation theory for lexically scoped functional language is then defined over these structures. The equivalence relations are then defined within this model theoretic framework. A distinction is made between intensional relations and extensional relations. The former class turn out to have a much more manageable theory than the latter. The principal intensional relation studied is strong isomorphism, its properties allow for elegant verification proofs in a style similar to that of purely applicative languages. In particular the relation is preserved under many standard syntactic manipulations and transformations.     

The logic of message-passing

Message-passing is a key ingredient of concurrent programming. The purpose of this paper is to describe the equivalence between the proof theory, the categorical semantics, and term calculus of message-passing. In order to achieve this we introduce the categorical notion of a linear actegory and the related polycategorical notion of a poly-actegory. Not surprisingly the notation used for the term calculus borrows heavily from the (synchronous) π-calculus. The cut-elimination procedure for the system provides an operational semantics.     

Operational domain theory and topology of sequential programming languages

A number of authors have exported domain-theoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of non-trivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features.     

Computing Arguments and Attacks in Assumption-Based Argumentation

Most computational frameworks for argumentation are based on abstract argumentation, which determines an argument's acceptability on the basis of its ability to counterattack all arguments attacking it. However, this view of argumentation doesn't address how to find arguments, identify attacks, and exploit premises. Assumption-based argumentation addresses these three issues. It's a refinement of abstract argumentation but remains general purpose, nonetheless. Rather than considering arguments to be a primitive concept, assumption-based argumentation defines them as backward deductions (using sets of rules in an underlying logic) supported by sets of assumptions. This approach reduces the notion of an attack against an argument to that of deduction of a contrary of an assumption.     

A logic of argumentation for specification and verification of abstract argumentation frameworks

In this paper, we propose a logic of argumentation for the specification and verification (LA4SV) of requirements on Dung??s abstract argumentation frameworks. We distinguish three kinds of decision problems for argumentation verification, called extension verification, framework verification, and specification verification respectively. For example, given a political requirement like ??if the argument to increase taxes is accepted, then the argument to increase services must be accepted too,?? we can either verify an extension of acceptable arguments, or all extensions of an argumentation framework, or all extensions of all argumentation frameworks satisfying a framework specification. We introduce the logic of argumentation verification to specify such requirements, and we represent the three verification problems of argumentation as model checking and theorem proving properties of the logic. Moreover, we recast the logic of argumentation verification in a modal framework, in order to express multiple extensions, and properties like transitivity and reflexivity of the attack relation. Finally, we introduce a logic of meta-argumentation where abstract argumentation is used to reason about abstract argumentation itself. We define the logic of meta-argumentation using the fibring methodology in such a way to represent attack relations not only among arguments but also among attacks. We show how to use this logic to verify the requirements of argumentation frameworks where higher-order attacks are allowed [A preliminary version of the logic of argumentation compliance was called the logic of abstract argumentation?(2005).]     

Exploring probabilistic bisimulations, part I

We take a fresh look at strong probabilistic bisimulations for processes which exhibit both non-deterministic and probabilistic behaviour. We suggest that it is natural to interpret such processes as distributions over states in a probabilistic labelled transition system, a pLTS; this enables us to adapt the standard notion of contextual equivalence to this setting. We then prove that a novel form of bisimulation equivalence between distributions are both sound and complete with respect to this contextual equivalence. We also show that a very simple extension to HML, Hennessy–Milner Logic, provides finite explanations for inequivalences between distributions. Finally we show that our bisimulations between distributions in a pLTS are simply an alternative characterisation of a standard notion of probabilistic bisimulation equivalence, defined between states in a pLTS.     

Dynamics of argumentation systems: A division-based method

The changing of arguments and their attack relation is an intrinsic property of a variety of argumentation systems. So, it is very important to efficiently figure out how the status of arguments in a system evolves when the system is updated. However, unlike other areas of argumentation that have been deeply explored, such as argumentation semantics, proof theories, and algorithms, etc., dynamics of argumentation systems has been comparatively neglected. In this paper, we formulate a general theory (called a division-based method) to cope with this problem based on a new concept: the division of an argumentation framework. When an argumentation framework is updated, it is divided into three parts: an unaffected, an affected, and a conditioning part. The status of arguments in the unaffected sub-framework remains unchanged, while the status of the affected arguments is computed in a special argumentation framework (called a conditioned argumentation framework, or briefly CAF) that is composed of an affected part and a conditioning part. We have proved that under a certain semantics that satisfies the directionality criterion (complete, preferred, ideal, or grounded semantics), the extensions of the updated framework are equal to the result of a combination of the extensions of an unaffected sub-framework and sets of the extensions of a set of assigned CAFs. Due to the efficiency of the division-based method, it is expected to be very useful in various kinds of argumentation systems where arguments and attacks are dynamics.     

一种新的辩论推理模式及其应用

文中定义了一个新的辩论推理模式,建立了一个形式化的知识表示框架,并把它应用于研究扩展逻辑程序类的说明语义,结果表明,新语义克服了择优语义的不足。作者还根据上述研究结果实现了逻辑程序设计风格下的知识框架。     

Twelve theses on argumentation systems

In this article we consider the partition of a set of statements P = P ^a ∪ P ^b ∪ P′, where P ^a , P ^b , P′ are, correspondingly, sets of statements only being argumented, both being argumented and argumenting, and only argumenting (basic statements). On P are defined two functions of argument and counter-argument selection forming the semantics for logics of argumentation. A four-valued logic of argumentation is proposed. By means of graph theory are formed trees of arguments (argument trees), wood, consisting of them, and some transformations of the wood, such that their result may be both planar and non-planar graph. Argument systems (systems of arguments) are defined together with their characterizations that use analytic tableaux. With the help of argument trees a specification of the idea of hermeneutic (“vicious”) circle is formalized.