COMPARACIÓN DE LAS TRES EDICIONES DE LOS

it is told or knows otherwise.

6.3

Fact-supported Arguments

We generalise for our AABA framework the notion of an argument that has a support consisting of assumptions only as in conventional ABA (see Chapter 2, Section 2.4.2). We will define arguments as deductions for sentences with a support consisting of assumptions as well as personal facts. We define first finite (deduction) trees adapted from Dung et al. [DKT09] and then a specific kind of such trees that is of interest in this chapter. The symbol τ represents an extra-logical symbol not in L.

Definition 6.2 (finite deduction tree) Given an AABA framework AFAg, a finite (deduc-

tion) tree T for p ∈ L (the conclusion or claim) is as follows:

• the root of T is labelled by p and denoted root(T ) • for every node N of T

– if N is a leaf then N is labelled by a sentence in L or by τ ;

– if N is not a leaf and lN is the label of N , then there is an inference rule lN ←

b1, . . . , bm (m ≥ 0) in Rc and

either m = 0 and the child of N is labelled by τ

or m > 0 and N has m children, labelled by b1, . . . , bm (respectively)

• the set of sentences, not including τ , labelling the leaves of T is denoted leaves(T ).

Nodes in a tree as above are connected by the inference rules, with sentences matching the conclusion of an inference rule connected as parent nodes to sentences matching the premises of the inference rule as children nodes. Finite deduction trees as above differ from a deduction tree in Definition 2 of [DKT09] in that, firstly, only the inference rules in the common knowledge are

l(x, app2)

@ @@

r(x, seeDrAli) f r(app2, seeDrAli)

Figure 6.1: A f-argument for l(x, app2) with support {r(x, seeDrAli), f r(app2, seeDrAli)} wrt AFx

as defined in Section 6.2.1.

used in the definition above to produce new branches and, secondly, the leaves in the definition above may be non-assumptions as well as assumptions. This is different from [DKT09] where, firstly, all rules are used to produce new branches and, secondly, only assumptions are leaves. We define next fact-supported arguments (abbreviated ‘f-arguments’) for a claim p with support S consisting only of assumptions and personal facts as finite trees with p at the root and S at the leaves.

Definition 6.3 (f-argument) Given an AABA framework AFAg, a f-argument for p ∈ L

(the conclusion or claim) with support S ⊆ A ∪ Rp_Ag is a finite (deduction) tree T where root(T ) = p and leaves(T ) = S.

We illustrate in Figure 6.1 a finite tree that is a f-argument. F-arguments for a claim can be computed by means of backward deductions defined as follows:

Definition 6.4 (backward deduction) Given an AABA framework AFAg and a selection

function f : 2L

→ L,1 _{a backward deduction of S ⊆ A ∪ R}p

Ag for a claim p ∈ L according to

AFAg (wrt f ) is a finite sequence of tuples

hC0, S0i, . . . , hCi, Sii, . . . , hCn, Sni

where n ≥ 1, C0 = {p}, S0 = Cn = ∅, Sn= S, and, for every 0 ≤ i < n, if f (Ci) = σ, then

1. If σ ∈ Rp_Ag or σ ∈ A, then • Ci+1 = Ci− {σ}

1

6.3. Fact-supported Arguments 137 • Si+1 = Si∪ {σ}

2. Otherwise, choose σ ← b1, . . . , bm ∈ Rc and

• Ci+1 = (Ci − {σ}) ∪ {b1, . . . , bm}

• Si+1 = Si

We write S ⊢fAF_Ag p if there exists a backward deduction of S for p according to AFAg (wrt

f ).2

Backward deductions, as defined above, are a generalisation of SLD resolution (and also deduc- tions of conclusions based on sets of premises as in Defition 2.2 of [DKT06]), which is the basis of proof procedures for logic programming. As in SLD, if there is a backward deduction using one selection function for picking sentences in Ci, then there is a backward deduction using any

other selection function. Thus, different selection functions are simply different but equivalent ways of generating the same argument for a claim. A backward deduction as above differs from an argument (“a deduction whose premises are all assumptions”) as in [DKT06] in that arguments in [DKT06] are only supported by assumptions whereas our backward deductions are supported by non-assumptions (personal facts of the agent) as well as assumptions.

The following example illustrates the notion of backward deduction.

Example 6.1 (backward deduction) Given the AABA framework AFx from Section 6.2.1,

a backward deduction

{r(x, seeDrAli), f r(app2, seeDrAli)} ⊢AF_x l(x, app2)

built by x affirming that it likes app2 is obtained as shown in Table 6.1 and described here:

• At step i = 0, R2 ∈ Rc is chosen in applying Case 2 of Definition 6.4.

2

We write ⊢AF_Ag instead of ⊢fAF_Ag where the particular selection function is unimportant in the given

i Ci Si Apply. . .

0 {l(x, app2)} ∅ Case 2

1 {r(x, seeDrAli), f r(app2, seeDrAli)} ∅ Case 1 2 {f r(app2, seeDrAli)} {r(x, seeDrAli)} Case 1 3 ∅ {r(x, seeDrAli), f r(app2, seeDrAli)}

Table 6.1: Backward deduction for l(x, app2) according to AFx where AFx is as defined in Sec-

tion 6.2.1. The underlined atoms are picked by the selection function.

• At step i = 1, Case 1 is applied since r(x, seeDrAli) ∈ Rp x.

• At step i = 2, Case 1 is applied since f r(app2, seeDrAli) ∈ Rp x.

In Definition 6.4, we use the word “choose” to identify backtrackable points in the search for a backward deduction. The need for this is illustrated by the following example:

Example 6.2 (backtracking in the search for a backward deduction) Consider an AABA framework AFx for some agent x with

L = {p, q, r}, A = ∅, Rc = {p ← q} ∪ {p ← r}, Rpx = {r}

If in the search for a backward deduction for p, p ← q ∈ Rc is chosen when applying case 2 of

Definition 6.4, then, since q /∈ Rp

x, q /∈ A and there is no inference rule q ← b1, . . . , bm ∈ Rc, the

search for a backward deduction needs to backtrack and choose p ← r ∈ Rc, leading eventually

to {r} ⊢AFx p.

We end this section by proving the relationship between f-arguments and backward deductions.

Lemma 6.1 If, for some selection function f , there is a backward deduction S ⊢fAF_Ag p, then

there is a f-argument for p with support S wrt AFAg.

Lemma 6.2 If there is a f-argument for p with support S wrt AFAg, then there is a backward

deduction S ⊢fAF_Ag p for any selection function f .