Assumption-Based Argumentation

The abstract view of argumentation in particular [Dun95] does not deal with the problem of actually finding arguments and attacks amongst them. This section introduces Assumption- Based Argumentation (ABA), an approach for representing argumentation in logic, giving the arguments themselves structure, giving a precise meaning to the notion of “attack” between arguments and presenting a notion of semantics for arguments in terms of ABA. Further details to that presented here can be found in [DKT06, DMT07, DKT09].

ABA Framework

Typically, as in Defeasible Logic Programming [GS04] and Deductive Argumentation [BH08a], arguments are built by connecting rules in the belief set, and attacks arise from conflicts among such arguments. In ABA, arguments are (implicitly meant to be) obtained by reasoning back- wards with a given set of inference rules (the belief set), from conclusions to premises that are assumptions, and attacks are defined in terms of a notion of contrary of assumptions. The ABA approach for representing argumentation, as presented in this section, as well as approaches such as Defeasible Logic Programming [GS04] and Deductive Argumentation [BH08a], deal with cases where there is a single agreed upon knowledge-base. Thus, these approaches can be seen as intra-agent argumentation, whereby an agent engages in a dialectical argumentation process with itself to determine its “acceptable” beliefs (desires, intentions).

An agent’s belief set and backward reasoning in ABA are defined in terms of a deductive system, which is defined as follows:

Definition 2.6 (Deductive System) A deductive system is a pair (L, R) where

• L is a formal language consisting of countably many sentences, and • R is a countable set of inference rules of the form

x1, . . . , xn

2.4. Computational Argumentation 43 where x ∈ L is called the conclusion and x1, . . . , xn ∈ L are called the premises of the

inference rule, and n ≥ 0.

A deduction of a conclusion by means of a deductive system (L, R) then is obtained by applying backwards the rules in R, as follows:

Definition 2.7 (Deduction of a conclusion) Given a deductive system (L, R) and a se- lection function f (any function that maps from sets of elements to elements), a (backward) deduction of a conclusion x based on (or supported by) a set of premises P , denoted P ⊢ x, is a sequence of multi-sets S1, . . . , Sm, where S1 = x, Sm = P , and for every 1 ≤ i < m, where σ

is the sentence occurrence in Si selected by f :

1. If σ is not in P then Si+1 = Si− {σ} ∪ S for some inference rule of the form σ ← S ∈ R.

2. If σ is in P then Si+1 = Si.

Each Si is referred to as a step in the deduction.

Deductions are the basis for the construction of arguments in ABA. To obtain an argument from a backward deduction the premises are restricted to ones that are assumptions. Indeed a set of assumptions and a contrary mapping, together with a deductive system as above, make up an ABA framework, as follows:

Definition 2.8 (ABA framework) An ABA framework is a tuple hL, R, A, ci where

• (L, R) is a deductive system, consisting of a language L and a set R of inference rules; • A ⊆ L, A 6= {}. A is referred to as the set of (candidate) assumptions;

• If x ∈ A, then there is no inference rule of the form x ← x1, . . . , xn ∈ R;

Intuitively, assumptions are sentences that can be assumed to hold but can be questioned and disputed (as opposed to axioms that are instead beyond dispute), and the contrary of an assumption stands for the reason why that assumption may be undermined and thus may need to be dropped. Indeed an inference rule containing one or more assumptions in the premises can be seen as a “defeasible rule” in the sense of [GS04], used for representing defeasible knowledge, i.e. weak or tentative information that may be used if nothing could be posted against it. On the other hand, an inference rule containing no assumptions in the premises can be seen as a “strict rule” in the sense of [GS04], used for representing non-defeasible information, i.e. strict (sound) knowledge.

We assume in this thesis that the inference rules in R have the syntax l0 ← l1, . . . ln (for n ≥ 0)

where li ∈ L. We will represent l0 ← simply as l0, and refer to this as a fact. As in [DMT07] we

restrict attention to flat ABA frameworks, such that if l0 ∈ A, then there exists no inference

rule of the form l0 ← l1, . . . , ln ∈ R, for any n ≥ 0. Lastly, an argument in favour of a sentence

p ∈ L is a backward deduction A ⊢ p such that A ⊆ A

Attack-relationship between arguments

Unlike Dung’s abstract argumentation framework [Dun95] wherein the attack-relationship be- tween arguments is explicit, in ABA this relationship is determined by the internal structure of the arguments. In ABA, as defined above, arguments are deductions from conclusions based solely upon assumptions. A set of assumptions attacks another set of assumptions as follows:

Definition 2.9 (Attack in ABA) A set of assumptions A attacks a set of assumptions B iff there exists an assumption x in B and, for some y ∈ c(x), a deduction A′

⊢ y such that A′

is a subset of A: if this is the case, it is said that A attacks B on x.

This notion of attack between sets of assumptions implicitly gives a notion of attack between ar- guments supported by sets of assumptions: the attacking argument needs to have as conclusion the contrary of an assumption in the support of the attacked argument.

2.4. Computational Argumentation 45 Argument Semantics

Standard argument semantics as per abstract argumentation (see Section 2.4.1) can be ascribed to ABA frameworks, as follows: A set of assumptions A is

• conflict-free iff it does not attack itself;

• admissible iff A is conflict-free and A attacks every set of assumptions B that attacks A; • preferred iff it is maximally admissible;

• sceptically preferred iff A is the intersection of all preferred sets of assumptions;

• complete iff A is admissible and contains all assumptions x such that A attacks all attacks against {x};

• grounded iff it is minimally complete;

• ideal iff A is admissible and it is contained in every preferred set of assumptions.

Our focus in this thesis is the admissibility semantics. We will say that a claim is admissible if it is the conclusion of an argument that is supported by a set of assumptions that can be extended to an admissible set of assumptions. Indeed computational mechanisms exist for computing sets of assumptions corresponding to the various semantics, which we discuss next.

Dialectic proof procedures

Within ABA, implicitly, a set of assumptions stands for the set of all arguments whose premises are contained in the given set of assumptions. Thus, the computation of “acceptable” sets of arguments amounts to computing “acceptable” sets of assumptions. The work of Dung et al. [DMT07] present a family of dialectical proof procedures for ABA that can compute “acceptable” sets of assumptions according to various semantics. The proof procedures find a set of assumptions, to defend a given belief (claim), by starting from an initial set of assumptions that supports an argument for the claim and adding defending assumptions incrementally to

counter-attack all attacks. They can be seen as generating a winning strategy for a proponent to win a dispute against an ideal opponent who attacks in every possible way the initial and defending arguments of the proponent. The proponent wins if it has a counter-attack against every attacking argument by the opponent.

The proof procedures in [DMT07] generate and find arguments by reasoning backwards from conclusions to assumptions. They use backward reasoning both to find an initial argument for a given belief and to find attacking and defending arguments for the contrary of an assumption. Each step in a backward argument is viewed as a partially completed, potential argument. Any assumption in such a potential argument can be attacked (by finding an argument for its contrary) before the argument is completed. We present next the AB-dispute derivations of Dung et al. [DMT07] which prove that a claim is admissible by building an admissible set of assumptions which support/defend the claim. The proponent’s winning strategy is represented as a sequence of quadruples consisting of proponent and opponent nodes labelled by multi-sets of sentences, representing steps of potential arguments, as well as the set of defence assumptions (assumptions used by the proponent) and culprits (opponent’s assumptions that have been attacked) generated so far, as follows:

Definition 2.10 (AB-dispute derivation) Let hL, R, A, ci be an ABA framework. Given a selection function, an AB-dispute derivation of a defence set A for a sentence α is a finite sequence of quadruples

P0, O0, A0, C0, . . . , Pi, Oi, Ai, Ci, . . . , Pn, On, An, Cn

where P0 = {α}, A0 = A ∩ {α}, O0 = C0 = {}, Pn = On = {}, A = An and for every

0 ≤ i < n, only one σ in Pi or one S in Oi is selected, and:

1. If σ ∈ Pi is selected then

(i) if σ is an assumption, then • Pi+1 = Pi− {σ}

2.4. Computational Argumentation 47 • Ai+1 = Ai

• Ci+1 = Ci

• Oi+1 = Oi∪ {{c(σ)}}

(ii) if σ is not an assumption, then there exists some inference rule σ ← R ∈ R such that Ci∩ R = {} (filtering of potential defence arguments by culprits) and

• Pi+1 = Pi− {σ} ∪ (R − Ai) (filtering of defence assumptions by defences)

• Ai+1 = Ai∪ (A ∩ R)

• Ci+1 = Ci

• Oi+1 = Oi

2. If S is selected in Oi and σ is selected in S then

(i) if σ is an assumption, then (a) either σ is ignored, i.e.

• Oi+1 = Oi− {S} ∪ {S − {σ}}

• Pi+1 = Pi

• Ai+1 = Ai

• Ci+1 = Ci

(b) or σ 6∈ Ai (filtering of culprits by defence assumptions) and σ ∈ Ci (filtering of

culprits by culprits) and • Oi+1 = Oi− {S}

• Pi+1 = Pi

• Ai+1 = Ai

• Ci+1 = Ci

(c) or σ 6∈ Ai (filtering of culprits by defence assumptions) and σ 6∈ Ci (filtering of

culprits by culprits) and

(c.1) if c(σ) is not an assumption, then • Oi+1 = Oi− {S}

• Pi+1 = Pi∪ {c(σ)}

• Ai+1 = Ai

• Ci+1 = Ci∪ {σ}

(c.2) if c(σ) is an assumption, then • Oi+1 = Oi− {S}

• Pi+1 = Pi (filtering of defence assumptions by defences)

• Ai+1 = Ai∪ {c(σ)}

• Ci+1 = Ci∪ {σ}

(ii) if σ is not an assumption, then • Pi+1 = Pi • Ai+1 = Ai • Ci+1 = Ci • Oi+1 = Oi− {S} ∪S − {σ} ∪ R  σ ← R ∈ R, and R ∩ Ci = {} (filtering of culprits by culprits)

Note that the dispute derivations as above incorporate filtering of and filtering by defence and culprit assumptions so that the final assumption-defence set constructed by the derivation does not attack itself, and so that the dispute derivations can be more efficient and can (finitely) compute infinite dispute trees. Further details can be found in [DMT07], including discussions of the soundness and completeness of the dispute derivations.

In closing, it is important to note that ABA, and to a large extent argumentation frameworks in general (e.g. [GS04, BH08a]), up to now have been considered in a single-agent setting. ABA, in particular, is such that an agent engages in a dispute (dialectic proof procedure) with itself (an imaginary opponent) to decide whether a claim is acceptable according to some acceptability criteria. Amongst our aims in this thesis are to present a generalised proof procedure for the admissibility semantics of ABA, which is still a dispute by an agent with itself but such that the outcome can be readily communicated to other agents. This is important for applications in multi-agent systems (an example of which is argumentation-based negotiation, see Section 2.3.2)

2.5. Summary 49