Probabilistic Argumentation. An Equational Approach

There is a generic way to add any new feature to a system. It involves 1) identifying the basic units which build up the system and 2) introducing the new feature to each of these basic units. In the case where the system is argumentation and the feature is probabilistic we have the following. The basic units are: a. the nature of the arguments involved; b. the membership relation in the set S of arguments; c. the attack relation; and d. the choice of extensions. Generically to add a new aspect (probabilistic, or fuzzy, or temporal, etc) to an argumentation network can be done by adding this feature to each component a-d. This is a brute-force method and may yield a non-intuitive or meaningful result. A better way is to meaningfully translate the object system into another target system which does have the aspect required and then let the target system endow the aspect on the initial system. In our case we translate argumentation into classical propositional logic and get probabilistic argumentation from the translation. Of course what we get depends on how we translate. In fact, in this paper we introduce probabilistic semantics to abstract argumentation theory based on the equational approach to argumentation networks. We then compare our semantics with existing proposals in the literature including the approaches by M. Thimm and by A. Hunter. Our methodology in general is discussed in the conclusion

On Argumentation Logic and Propositional Logic

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies

Algorithms for computational argumentation in artificial intelligence

Argumentation is a vital aspect of intelligent behaviour by humans. It provides the means for comparing information by analysing pros and cons when trying to make a decision. Formalising argumentation in computational environment has become a topic of increasing interest in artificial intelligence research over the last decade. Computational argumentation involves reasoning with uncertainty by making use of logic in order to formalize the presentation of arguments and counterarguments and deal with conflicting information. A common assumption for logic-based argumentation is that an argument is a pair where Φ is a consistent set which is minimal for entailing a claim α. Different logics provide different definitions for consistency and entailment and hence give different options for formalising arguments and counterarguments. The expressivity of classical propositional logic allows for complicated knowledge to be represented but its computational cost is an issue. This thesis is based on monological argumentation using classical propositional logic [12] and aims in developing algorithms that are viable despite the computational cost. The proposed solution adapts well established techniques for automated theorem proving, based on resolution and connection graphs. A connection graph is a graph where each node is a clause and each arc denotes there exist complementary disjuncts between nodes. A connection graph allows for a substantially reduced search space to be used when seeking all the arguments for a claim from a given knowledgebase. In addition, its structure provides information on how its nodes can be linked with each other by resolution, providing this way the basis for applying algorithms which search for arguments by traversing the graph. The correctness of this approach is supported by theoretical results, while experimental evaluation demonstrates the viability of the algorithms developed. In addition, an extension of the theoretical work for propositional logic to first-order logic is introduced