Argumentation Papers¶

Rationality Postulates: Applying Argumentation Theory for Non-monotonic Reasoning (2017)
On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games (1995)
On the Acceptability of Arguments in Bipolar Argumentation Frameworks
An argumentation-based approach for reasoning about trust in information sources
Fuzzy Argumentation Frameworks
A probabilistic semantics for abstract argumentation
Modular Semantics and Characteristics for Bipolar Weighted Argumentation Graphs
Interpretability of Gradual Semantics in Abstract Argumentation
Axiomatic Foundations of Acceptability Semantics
Argumentation-based reasoning in agents with varying degrees of trust
Bipolarity in argumentation graphs: Towards a better understanding
Gradual Valuation for Bipolar Argumentation Frameworks
Evaluation of Arguments from Support Relations: Axioms and Semantics
Strategical Argumentative Agent for Human Persuasion
ArgTrust: Decision Making with Information from Sources of Varying Trustworthiness
Support in abstract argumentation
From fine-grained properties to broad principles for gradual argumentation: A principled spectrum
Social Abstract Argumentation
Ranking-based Semantics for Argumentation Frameworks

Rationality Postulates: Applying Argumentation Theory for Non-monotonic Reasoning (2017)¶

(M. Caminada)

Summary:
- Looks at how Dung’s abstract argumentation framework can be used for non-monotonic reasoning with strict and defeasible logical rules
- Rules correspond to arguments, which have logical conclusions
- Do the aggregated conclusions of an extension make sense? This is the question that the rationality postulates address
Formal preliminaries:
- \(\mathcal{L}\) is a propositional language
- \(\mathcal{R}_s\) and \(\mathcal{R}_d\) are strict and defeasible rules of the form \(\phi_1,\ldots,\phi_n \to \phi\) and \(\psi_1,\ldots,\psi_m \implies \psi\) respectively (\(n,m=0\) is allowed, in which case \(\phi\) and \(\psi\) are strict or defeasible constraints)
- \(n: \mathcal{R}_d \to \mathcal{L}\) (what is the intuition behind \(n\)?!)
- \(\le\) is a partial pre-order on \(\mathcal{R}_d\)
- Arguments are defined recursively:
  - If \(\to \phi\) is a strict rule, then \(A=\to \phi\) is an argument. We write
    - \(\text{conc}(A) = \phi\) for the conclusion of \(A\)
    - \(\text{sub}(A) = \{A\}\) for its sub-arguments
    - \(\text{defrules}(A) = \emptyset\)
    - \(\text{toprule}(A) = \to \phi\)
  - If \(A_1,\ldots,A_n\) are arguments and \(r=\text{conc}(A_1),\ldots,\text{conc}(A_n) \to \phi\) is a strict rule, then \(A=A_1,\ldots,A_n \to \phi\) is an argument, and
    - \(\text{conc}(A) = \phi\)
    - \(\text{sub}(A) = \text{sub}(A_1) \cup \cdots \cup \text{sub}(A_n) \cup \{A\}\)
    - \(\text{defrules}(A) = \text{defrules}(A_1) \cup \cdots \cup \text{defrules}(A_n)\)
    - \(\text{toprule}(A) = r\)
  - Arguments for defeasible rules are defined similarly, except that \(\text{defrules}(A)\) also includes the top rule
- An argument \(A\) is called strict if \(\text{defrules}(A) = \emptyset\), and is called defeasible otherwise
- Attacks:
  - \(A\) undercuts \(B\) is there is some \(B' \in \text{sub}(B)\) with \(\text{toprule}(B) = r \in \mathcal{R}_d\) and \(\text{conc}(A) = -n(r)\)
  - \(A\) restrivively rebuts \(B\) if \(\text{conc}(A) = -\text{conc}(B')\) for some \(B' \in \text{sub}(B)\) with \(\text{toprule}(B') \in \mathcal{R}_d\). That is, if the conclusion of \(A\) contradicts the conclusion of some sub-argument of \(B\) which was formed through application of a defeasible rule
  - \(A\) unrestrictedly rebuts \(B\) if \(\text{conc}(A) = -\text{conc}(B')\) for some \(B' \in \text{sub}(B)\) where \(B'\) is a defeasible argument.
- (Note: for restrictive rebut attacks, the conclusion of \(A\) contradicts the conclusion of some sub-argument \(B'\) of \(B\), where \(B'\) was obtained by applying a defeasible rule. For the unrestricted rebut, \(B'\) is allowed to have come from a strict rule, so long as a defeasible rule is used somewhere in the construction of \(B'\))
- We extend the preference order \(\le\) to sets of defeasible rules (see the paper for two example definitions); let \(\trianglelefteq\) denote this (partial?) order. We can now form a preference order \(\preceq\) on arguments by comparing their defeasible rules (exactly which set of rules used may vary: see the paper for weakest link and last link variants).
- Can finally form an abstract argumentation framework. \(def_{ur}\) is the attack relation where \((A, B) \in def_{ur}\) if \(A\) undercuts or unrestrictedly rebuts \(B\) and \(A \not\prec B\). \(def_{rr}\) is defined similarly but for restricted rebut.
Rationality postulates:
- After forming the AF, we can apply semantics to find extensions. For each extension \(E\), we have the est of justified conclusions \(S = \{\text{conc}(A) : A \in E\}\)
- We want \(S\) to have the following properties:
  - Direct consistency: \(\not\exists x \in S : -x \in S\)
  - Closure under strict rules: write \(Cl(S)\) for the smallest superset of \(S\) such that \(\phi \in Cl(S)\) whenever there is a strict rule \(\phi_1,\ldots,\phi_n \to \phi \in \mathcal{R}_s\) and each \(\phi_i\) is in \(Cl(S)\).
    
    The rationality postulate says that \(Cl(S) = S\).
  - Indirect consistency: \(\not\exists x \in Cl(S) : -x \in Cl(S)\)
- ASPIC+ has these three properties: it achieves them by using restricted rebut, complete semantics and ensuring strict rules are closed under transposition: if \(\phi_1,\ldots,\phi_n \to \phi\) is a strict rule then so too is \(\phi_1,\ldots,\phi_{i-1},-\phi,\phi_{i+1},\ldots,\phi_n \to -\phi_i\) for any \(i\)
- Allowing unrestricted rebut makes things difficult. Beyond complete semantics one needs strongly admissisble extensions. The grounded extension does the job here, and has all three properties.
- (Table 1 in the chapter is a good reference)
Non-interference:
- A defeasible theory gives rise to an argumentation system
- Non-interference: merging a syntactically disjoint theory does not affect the conclusions drawn regarding the propositional atoms in the original theory
- Crash resistance (hard to get one’s head around, but it’s a weaker property than non-interference… ‘Crashing’ is a severe situation where a syntactically disjoint theory not only changes the entailments to the other one, but causes the other theory to be completely ignored).

On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games (1995)¶

(Dung, Phan Minh)

Seminal paper in argumentation in AI. According to Google Scholar as of 2nd April 2020, it has 4,285 citations (!)
Summary:
- Studies argumentation for reasoning in a way that can be performed computationally
- Studies acceptability of arguments. Idea: an argument is acceptable if it successfully defends against its attackers
- Makes the link between argumentation and nonmonotonic reasoning and logic programming
Acceptability of arguments:
- Defines an argumentation framework: \(AF = \langle AR, attacks \rangle\)
- Defines notions of conflict-freeness, acceptability, admissibility
- Defines extensions and semantics: preferred, stable, grounded, complete
- Some results on relations between different extensions
- Notions of coherent and controversial frameworks, and some results about extensions of such frameworks
Argumentation in \(n\)-person games:
- Makes a link between cooperative game theory and argumentation: existing notions of ‘stable’ solutions in cooperatives games coincide with stable extensions in a corresponding argumentation framework
Stable marriage problem:
- One-to-one correspondence between stable marriage solutions and stable extensions of the corresponding argumentation framework
Nonmonotonic reasoning:
- Abstract argumentation framework can be seen to generalise two different approaches to nmr
- Reiter’s default logic: preferred extensions of the AF generalise extensions already considered in the default logic
- Relation to Pollock’s defeasible reasoning (skimmed)
Logic programming: TODO: read

On the Acceptability of Arguments in Bipolar Argumentation Frameworks¶

(C. Cayrol, M.C. Lagasquie-Schiex)

Dung’s framework is extended to add an independent support relation \(\mathcal{R}_{sup}\) in addition to the defeat relation \(\mathcal{R}_{def}\).
They only work with acyclic graphs. This rules out the idea above with bi-directional support/defeat between arguments for the TD case.
New notions of extensions defined which take support into account.

An argumentation-based approach for reasoning about trust in information sources¶

(L. Amgoud, R. Demolombe)

Talks about what trust means between agents, and different types of trust, from a sort of philosophical point of view. Has good references for other work on this question.
Defines a modal logic with modal operators for belief and the act of informing: \(\text{Bel}_i \phi\) means that agent \(i\) believes \(\phi\), and \(\text{Inf}_{j,i} \phi\) means that agent \(j\) has informed \(i\) that \(\phi\) holds.
Defines different types of trust between agents in the logical language (trust in sincerity, validity, completeness, cooperativity, competence and vigilance).
Describes how to build different kinds of argument from a belief base in the logical language (from the point of view of a particular agent). Usual semantics are used to find the acceptable arguments and the inferred sentences which (are defined to) follow from the belief base.
Moves on to discuss graded trust, as opposed to the binary trust or no-trust situation above. Operators for graded trust are introduced to the logical language, and axioms for the language given.
Describe again how to build arguments, but the arguments themselves are now ordered according to the strength of the trust involved
Argumentation framework built using the attack relation and argument ordering. If \(a \mathcal{R} b\) and \(a\) is weaker than \(b\) we swap the order and say that \(b\) defeats \(a\). Argumentation then proceeds using the defeat relation instead of \(\mathcal{R}\).

Fuzzy Argumentation Frameworks¶

(J. Janssen, M. De Cock)

Dung’s framework assumes attacks are “all or nothing”: there is no possibility of an attack being stronger or weaker than another.
Similarly, arguments are either accepted or rejected; there is no notion of accepting one argument to a higher degree than another
Dung’s framework is extended to a fuzzy version. The attack relation is a fuzzy subset of \(\mathcal{A} \times \mathcal{A}\)
Extensions become fuzzy sets rather than crisp
Fuzzy analogues of conflict-freeness, admissibility, stability are defined
Can identify fuzzy sets of arguments that have certain degrees of conflict-freeness and stability, in place of finding stable extensions
There is a trade-off between conflict-freeness and stability: e.g. in the paper there is an example of 0.99-conflict-free set which is only 0.17-stable, and a 0.66-stable set which is only 0.8-conflict-free
Paper goes on to talk about fuzzy answer set programming, which I did not read fully

A probabilistic semantics for abstract argumentation¶

(M. Thimm)

Like the fuzzy approach, the aim is to have more fine-grained argumentation than just accept/reject.
Considers probability functions that assign a probability to sets of extensions (i.e. sets of sets of arguments). This means probabilities can also be assigned to individual extensions and arguments.
Probabilities here represent degress of belief
An in/out/undecided labelling is a special instance of a probability function. They give results that links properties of the labelling (grounded, complete, stable) with properties of the probability function (e.g. to do with minimum/maximum entropy)
Extra results about probability functions related to complete labellings

Modular Semantics and Characteristics for Bipolar Weighted Argumentation Graphs¶

(T. Mossakowski, F. Neuhaus)

Considers weighted bipolar argumentation frameworks. “Weighted” means that each argument is assigned an initial weight (typically a real number)
Semantics for such frameworks assign each argument an acceptability degree (also typically a real number). Note that semantics do not assign degrees to sets of arguments (extensions), but to individual arguments.
Considers “modular” semantics that consist of two parts:
- An aggregation function takes the attackers/supports for an argument and their degrees, and produces some real number
- An influence function takes this number along with the initial weight, and produces an acceptability degree
I think application of the two above functions is iterated until convergence (TODO: check the details)
Loads of axioms, some seemingly similar to ours (TODO: understand them properly).
Results giving criteria for convergence/divergence
TODO: read the technical bits in more detail

Interpretability of Gradual Semantics in Abstract Argumentation¶

(Jérôme Delobelle, Serena Villata)

Looks at how to interpret the results of gradual semantics: understand what factors led to the result.
In particular, which arguments were impactful for \(x\) receiving the acceptability degree it did
- “… so that the reasons leading to the acceptability of one or a set of arguments in a framework may be explicitly assessed.”
Looks at two concrete gradual semantics: h-categorizer semantics and counting semantics (see paper for references)
Defines impact of a set of arguments \(X\) on a particular argument \(y\) as (roughly) the difference between the acceptability of \(y\) and the acceptability after removing \(X\) (and removing attacks from/to \(X\))
- If this difference is positive, \(y\)’s acceptability goes down after removing arguments in \(X\). Therefore \(X\) has a positive impact on the acceptability of \(y\)
- Similar interpretation when the impact is negative or zero
Gives a method for transforming an AF with cycles into an (infinite) acyclic one
- This could be useful in the TD context for applying ideas in other papers which require an acyclic graph
Defines, for each argument \(y\), the impact ranking, which ranks other arguments by how much they influence the acceptability of \(y\)

Axiomatic Foundations of Acceptability Semantics¶

(Leila Amgoud and Jonathan Ben-Naim)

TODO: read

Argumentation-based reasoning in agents with varying degrees of trust¶

(S. Parsons et. al.)

First argumentation paper I’ve seen which links explicitly to truth discovery (cites TruthFinder paper)
Looks at “a group of agents with varying degrees of trust of each other”
The model for trust:
- Set of agents \(Ags\) and a (non-symmetric) trust relation \(\tau\)
- Agents and \(\tau\) are seen as a directed graph
- Numeric trust values are given by a function \(tr: Ags \times Ags \to \R\) which is consistent with \(\tau\)
- Trust is extended in a sort of transitive way: if there is a directed path of trust from \(a\) to \(b\) then some operation \(\otimes\) is used to combine trust values along the path
- For each agent, trust values in other agents induce a trust ranking
Model for argumentation
- Each agent has a “private” knowledge base, and a “public” set of “past utterances” (both formulae of some propositional language). Other agents have knowledge of the public KB.
- Arguments are formed a standard way: an argument is a pair \((S, p)\) where \(S\) is consistent, \(S \vdash p\), and \(S\) is the minimal set with this property.
- An argument \(A_1\) undercuts (attacks) \(A_2\) if the conclusion of \(A_1\) is (logically equivalent to) some element of the support of \(A_2\)
- Agents have a belief value \(bel(\phi)\) for each formulae in its knowledge base
- Belief in an argument \((S, p)\) is defined by combining belief in the formulae in \(S\) using some operation \(\otimes\)
- An argumentation system is constructed from the perspective of each agent: it consists of arguments formed from the KB, the undercutting relation, and a preference relation on the arguments.
Combining trust and argumentation:
- Consider a trust network as above and fix an agent \(a\). \(a\)’s KB is its private knowledge together with the public knowledge of all other agents
- Belief in formulae from another agent \(b\) is obtained from \(a\)’s trust in \(b\) by means of a “trust to belief” function \(ttb\). This extends to trust in arguments as above.
- End result (I think) is that each agent forms an argumentation framework consisting of arguments constructed from the beliefs of all agents. Belief in arguments from other agents depends on the trust \(a\) has in that agent
TODO: finish reading

Bipolarity in argumentation graphs: Towards a better understanding¶

(Claudette Cayrol, Marie-Christine Lagasquie-Schiex)

Has been work on abstract bipolar argumentation, where the support relation is also abstract
There have been different notions put forward for support between arguments, based on different intuitions and with different interpretations
This paper: frame these different kinds of support (identified in other papers) in a common setting for comparison
” Basically, the idea is to keep the original arguments, to add complex attacks defined by the combination of the original attack and the support, and to modify the classical notions of acceptability.”
Reviews definition of bipolar frameworks (BAFs)
Reviews definition of complex attacks: supported attacks and secondary attacks
Reviews definition of notions for coherence of arguments in BAFs: conflict-freeness wrt complex attacks, safety and closure under the support relation
On to the different support relations… Let \(A(a)\) mean (informally) that an argument \(a\) is accepted.
- Deductive support: \(a {\mathcal{R}_{\text{sup}}} b\) means that \(A(a) \implies A(b)\) (we can deduce \(b\) from the acceptance of \(a\))
- Necessary support: \(a {\mathcal{R}_{\text{sup}}} b\) means that \(A(b) \implies A(a)\) (the acceptance of \(a\) is necessary for the acceptance of \(b\))
- Evidential support: distinguishes between prima-facie arguments (do not need support from others to stand) and standard arguments (must be supported by at least on prima-facie argument)
For each kind of support, extra complex attacks may be defined (which use the support relation in various ways). This means a BAF induces a Dung AF by considering the complex attacks
Propositions linking properties of sets of arguments in the BAF with properties of the set in the corresponding Dung AF
There is a duality between deductive and necessary support
(Glossed over section 5, which is to do with evidential support in more detail…)

Gradual Valuation for Bipolar Argumentation Frameworks¶

(C. Cayrol and M.C. Lagasquie-Schiex)

Introduces the abstract bipolar framework in the same way as On the Acceptability of Arguments in Bipolar Argumentation Frameworks
Requires acyclic graphs
Defines how one may instantiate the bipolar framework (including supports) from propositional formulae
Defines gradual valuation in Dung’s framework: a local gradual valuation assigns each argument a value (from some totally ordered set \(\mathcal{V}\)) based on the value of its parents
- The recursive aspect is probably why acyclicity is required
Extends this to the abstract bipolar framework
- We could use this general definition to create TD operators, if we have a mapping between TD networks and (acyclic) bipolar frameworks

Evaluation of Arguments from Support Relations: Axioms and Semantics¶

(Leila Amgoud, Jonathan Ben-Naim)

Considers abstract framework with only the support relation: no attacks
- Our TD networks are already instances of this (and they are acyclic!)
Framework is also weighted: each argument has an “intrinsic strength” (weight on \([0, 1]\))
A semantics is defines as a mapping which assigns each argument a value in \([0, 1]\) (so we are talking about gradual semantics)
Puts forward 17 (!) axioms for semantics
- 14 “mandatory” axioms, and 4 “optional” ones. The last 3 optional ones are (in some sense?) mutually incompatible
- Many are highly comparable to axioms for TD
Proves some properties of semantics satisfying the axioms
Defines three semantics and analyses their satisfaction of the axioms
- Would be great to apply these to TD networks

Strategical Argumentative Agent for Human Persuasion¶

(Ariel Rosenfeld and Sarit Kraus)

Create artificial agent which persuades people to change their beliefs through dialog (interchange of arguments)
Based on bipolar argumentation frameworks with extra bits; called a WBAF:
- Weighted interactions: a weight function \(W: \mathcal{R} \cup \mathcal{S} \to \mathcal{V}\) (for some totally ordered set \(\mathcal{V}\)) which gives the strength of the attacks and supports
- An argument belief function \(B: \mathcal{A} \to \mathcal{V}\)
- A designated argument \(\omega \in \mathcal{A}\): represents the issue being discussed
Defines a gradual belief valuation function in a very similar way to (Gradual Valuation for Bipolar Argumentation Frameworks)
For the persuasion agent:
- Agent is the persuader, the human is the persuadee
- A dialog is a finite sequence of arguments
- Persuader knows all arguments relating to \(\omega\), but persuadee does not (necessarily)
- Persuadee therefore has their own WBAF
- Persuader tries to estimate the persuadees WBAF based on the arguments put forward, and chooses new arguments to put forward accordingly
- Specifics of choosing which argument to posit involve a Markov process model, Monte Carlo search, some kind of ML… TODO: read this stuff
Experiments with the agent and real people

ArgTrust: Decision Making with Information from Sources of Varying Trustworthiness¶