Selective Revision by Deductive Argumentation

The deductive argumentation framework presented in Section 2.3 (Page 22) allows to decide for each sentence ↵ 2 whether ↵ is justifi- able with respect to . Here, we use this to define selection functions. The framework of deductive argumentation heavily depends on the actual instances of categorizers and accumulators, which are left unin- stantiated in Section 2.3. Here, we demand minimal requirements of both functions to be used in a selection function.

Definition 4.2.6 (Well-behaving categorizer). Let ⌧ and ⌧0_{be argument}

trees. A categorizer is called well-behaving if (⌧) > (⌧0_)whenever

⌧ consists only of one single node and ⌧0 consists of at least two nodes.

In other words, a categorizer is well-behaving if the argument tree that has no undercuts for its root is considered the best justification for the root.

Definition 4.2.7 (Well-behaving accumulator). Let ⌧ and ⌧0 _{be argu-}

ment trees. An accumulator  is called well-behaving if and only if ((T+, T-)) > 0whenever T+ 6= ; and T-_=;.

This means, that if there are no arguments against a claim ↵ and at least one argument for ↵ in then ↵ should be accepted in . Both,

0 and 0 as defined in Section 2.3 are well-behaving as well as all

categorizers and accumulators considered in [26]. If is consistent then every sentence ↵ 2 is accepted by with respect to every well-behaving categorizer and well-behaving accumulator.

Let B ✓ LpropAt be a consistent set of sentences, and let be some

well-behaving categorizer and let  be some well-behaving accumula- tor. We consider a selective revision ?fB of the form (4.2.2). In order

some ✓ Lprop_At we implement a selection function f that checks for every sentence ↵ 2 it ↵ is accepted in B [ . Although B is consis- tent, the union B [ is not necessarily consistent which gives rise to an argumentative evaluation. We consider the following two different selection functions based on deductive argumentation.

Definition 4.2.8 (Skeptical Selection Function). We define the skeptical selection function S_B, via

S_B,( ) = {↵₂ | B_[ |⇠_, ↵} for every ✓ Lprop_At .

Definition 4.2.9 (Credulous Selection Function). We define the credu- lous selection function CB,via

C_B,( ) = {↵2 | B_[ |_6⇠, ¬↵} for every ✓ LpropAt .

In other words, the value of S_B,( )consists of those sentences of that are accepted in B [ and the value of CB,( )consists of those

sentences of that are not rejected in B [ . There is a subtle, but important, difference in the behavior of those two selection functions as the following example shows.

Example 4.2.10. Let B1 = {a} and 1 = {¬a}. There is exactly one

argument tree ⌧1 for ¬a and one argument tree ⌧2 for a in B1[ .

In ⌧1 the root is the argument A = h{¬a}, ¬ai which has the single

canonical undercut B = h{a}, ai. In ⌧2 the situation is reversed and the

root of ⌧2 is the argument B which has the single canonical undercut

A. Therefore, the argument structure for ¬a is given via B[ (¬a) =

({⌧1}, {⌧2}). We use the categorizer 0and accumulator 0we defined

in Example 2.3.11 (Page 2.3.11). It follows that 0(⌧1) = 0(⌧2) = 0

and

0( 0( B_[ (a))) = 0(h0, 0i) = 0.

It follows that B [ is undecided about both ¬a and a. Consequently, it follows that

S 0,0

B1 ( 1) =; C

0,0

B1 ( 1) = {¬a}.

That is, if the evaluation of a formula is undecided, the skeptical selec- tion function rejects the formula and the credulous selection function

accepts it. }

Let ⇤ be some (prioritized) multiple base revision operator, some categorizer, and  some accumulator. Using the skeptical selection

function we can define the skeptical argumentative revision _S, follow- ing (4.2.2) via

B _S, = B_{⇤ SB},( ) (4.2.3)

for every ✓ Lprop_At and using the credulous selection function we can define the credulous argumentative revision _C,via

B _C, = B⇤ CB,( ) (4.2.4)

for every ✓ Lprop_At .

We exemplify these two types of argumentative revision operators by the following simple example.

Example 4.2.11. We continue Example 4.2.10. Let ⇤ be some prioritized multiple base revision operator. Then it follows that B1 S0,0 1 =

{a} and B1 _C0,0 1= {¬a}. }

Next we give a more complex example of a selective revision oper- ation.

Example 4.2.12. We continue and formalize Examples 4.2.1 and 4.2.2. We consider the atoms At = {s, h, l, m, f, v} with the following infor- mal interpretations.

s: Anna is a surf fanatic h: Anna travels to Hawaii

f: Anna has financial problems l: Anna takes a loan

m: Anna has a lot of money

v: There is volcano activity on Hawaii Now consider Anna’s belief base B1 given via

B₁ = {s, s ) h, l, l ) m, m ) h, m ) ¬f} .

Anna is a surf fanatic (s) and believes that a surf fanatic should travel to Hawaii (s ) h). Anna has taken a loan (l), and taking a loan means having money available (l ) m). Having money implies she should travel to Hawaii (m ) h), and having money also implies she does not have financial problems (m ) ¬f). Note that B1 ` h, i. e., from

B₁ Anna concludes she should go to Hawaii.

Consider the new information 1= {f, f ) ¬h, v, v ) ¬h} stem-

ming from communication with Anna’s mother. With 1 the mother

of Anna wants to convince Anna not to travel to Hawaii. In particular,

1 states that Anna has financial problems (f), that having financial

problems Anna should not travel to Hawaii (f ) ¬h), that there is also volcano activity on Hawaii (v), and that given volcano activity Anna should not travel to Hawaii (v ) ¬h).

As one can see there are several arguments for and against h in B_{1[ 1}, e. g., hs, s ) h, hi, hf, f ) ¬h, ¬hi. If follows that B_{1[ 1}

accepts f ) ¬h, but rejects f, v, and v ) ¬h with respect to 0 and

0. Furthermore, B1[ 1 accepts ¬f and rejects ¬v and ¬(v ) ¬h)

with respect to 0 and 0 which means that both v and v ) ¬h

are credulously accepted. Consequently, the values of S 0,0

B1 ( 1)and

C 0,0

B1 ( 1)are given via

S 0,0

B1 ( 1) = 1\ {f, v, v ) ¬h} and C 0,0

B1 ( 1) = 1\ {f}.

Let ⇤ be some prioritized multiple base revision operator and define

0,0

S and 0

,0

C via (4.2.3) and (4.2.4), respectively. Then some possi-

ble revisions of B1 with 1 are given via

B₁ 0,0

S 1 = {s, s ) h, l, l ) m, m ) h, m ) ¬f, f ) ¬h}

and B₁ 0,0

C 1 = {s, l ) m, m ) h, m ) ¬f, f ) ¬h, v ) ¬h, v} .

Note that it holds that B1 _S0,0 1 ` h and B1 <sub>C</sub>0,0 1` ¬h. }

For the evaluation of our approach the sophisticated algorithms presented in [28] can be used. However, the underlying problems of deciding whether a set of propositional formulae is consistent is NP- complete and deciding whether it entails a given formula is co-NP- complete [107] such that no generally effective implementation can be expected.

For the selection functions S_B, and C_B,and the resulting revision operators _S, and _C,we can show the following results.

Proposition 4.2.13. Let be a well-behaving categorizer and  be a well-behaving accumulator. Then the selection functions S_B,and C_B, satisfy Inclusionf, Weak inclusionf, Weak extensionalityf,

Consistency preservationf and Weak maximalityf.

Proof. See Appendix A.1.2 on Page 227.

In particular, note that both S_B, and C_B, do not satisfy either Consistencyf or Maximalityf in general. On the basis of the Proposi-

tions 4.2.5 and 4.2.13 the following corollary can be shown.

Corollary 4.2.14. Let be a well-behaving categorizer and  be a well- behaving accumulator. Then both _S,and _C,are non-prioritized mul- tiple base revision operators.

Proof. See Appendix A.1.2 on Page 228.

Hence, we have shown that our construction of a selection function on the basis of deductive argumentation leads to non-prioritized mul- tiple base revision operator. In the next section we consider multiple base revision operators for answer set programming.

4.3 multiple base revision for asp

We described the state of the art of change operations for ASP in Section 2.5.4. While several approaches to find change operators that satisfy the AGM Postulates for belief sets or epistemic states have been made, there is no work on the consideration of the belief base approach for ASP apart the first publication our approach, which we present in the following, in [154].

We focus on the revision operation and base it on a consolidation operation, which can be used to specify other types of change opera- tions as well. The well developed classical base revision approach has not been considered in the light of ASP before. In fact, we argue that the belief base approach is the intuitive one for ASP. AGM change operations on belief sets can be seen as operations on the knowledge level, abstractly describing how an ideal reasoner would change its be- liefs. This underlies the assumption of a perfect reasoner while ASP’s main features are effective computation of finite programs with finite answer sets. The deductive closure, a crucial property of belief sets, is defined neither for programs nor for answer sets. Belief bases are also more expressive; since on the knowledge level one cannot distin- guish between inferred beliefs and fundamental, or self-supporting, ones. While this abstraction from the fundamental beliefs and their syntactic representation has advantages for the global picture of be- lief change we argue that ASP is primarily a syntax based approach. A key feature of ASP is that beliefs are formulated in form of easily understandable rules that allow for explicit exceptions and the ex- planation of inferences. From the base revision perspective the result of a change operation for ASP should be founded, understandable and close to the original syntax. As we described in Section 2.5.4 the SE-model based approach to the revision of answer set programs in- spired by the AGM belief set approach [72] leads to unintuitive results from the ASP perspective. The example for this taken from [216] is as follows. Consider the programs P1 = {p., q.} and P2 = {p q., q.},

they have the same SE-models and therefore also the results of the revision by the program Q = {¬q.} are the same. It holds that the answer sets for AS(P1⇤SEQ) = AS(P2⇤SEQ) = {{p}}. While for P1

this is a desired result, for P2it is not since p is not justified if q is not

in the answer set.

Here, we present a general exploration of the application of classic base revision theory to change operations on ASP. We discuss ASP specific postulates from the literature in the light of a base revision approach, proofs of the relationships among both and formulation of adapted postulates. Finally, we develop a new base revision con- struction via a screened consolidation operator which is applicable to ASP. We make use of global selection functions that lead to the

definition of general operator that can be used iteratively. We prove a characterization theorem for our construction.

The remainder of this section is structured as follows. In the next section we develop base revision postulates for ASP, discuss them and relate them to specific postulates for change operations ASP from the literature. After that we present our construction of multiple base revision operators and show its applicability to ASP and the corre- spondence to the postulates.