Rebuttal and Undercutting Attacks

For the descriptions below assume we have a system of only two agents, x1 and x2, with knowledge bases
KB(x1) = a <- b, a <- c, c
KB(x2) = ~a, ~b
where the set of candidate assumptions is {b}. The two agents are engaged in a dialectical argumentation process wherein x1 is attempting to defend an argument for the proposition 'a' and x2 is attempting to attack it.

Undercutting Attack
Assume agent x1 presents the argument
({a <- b, b}, a)
where 'b' is an assumption made by x1. Agent x2 can and will successfully undercut/attack this argument by putting forward the assumption-free argument
({~b}, ~b).

Rebuttal Attack
Assume agent x1 presents the assumption-free argument
({a <- c, c}, a).
Agent x2 can rebut/attack this argument by putting forward its own assumption-free argument
({~a}, ~a).
Note that x1 could potentially counter-attack this attack by putting forward the original argument again, unless there is a restriction specified by the argumentation protocol. Likewise, x2 could attack the counter-attack with the same argument again, and so on indefinitely.

Reducing a rebuttal to an undercutting attack
If we redefine the knowledge bases of agents x1 and x2 as follows:
KB(x1) = a <- b, a <- c ^ alpha, c
KB(x2) = ~a <- beta, ~b <- gamma
where the set of candidate assumptions is {b, alpha, beta, gamma} and the contrary of b is ~b, the contrary of alpha is ~a, the contrary of beta is a and the contrary of gamma is b.

Now if agent x1 presents the argument
({a <- c ^ alpha, c, alpha}, a)
Agent x2 can undercut/attack this argument by putting forward the argument
({~a <- beta, beta}, ~a).
Note that the culprit here is not the literal 'a' but the assumption 'alpha'.