Generation of Conditional Structure

In this chapter we will investigate whether it is possible to constructively define a function $\gred$ as described in section 3.1, i.e. a function which allows to generate the CS $\gamma(\kb)$ of a knowledge base $\kb$ from a reduced set of possible worlds of $\kb$ . The definition of this function will be based on the findings in chapters 5, 7 and 8.

We will restrict ourselves to the generation of conditional contributions of atomic conditionals, i.e. to atomic knowledge bases which only include a single conditional.

In section 9.1 we will first show how the CA-table of two conditionals can be generated from their RCS-generated conditional contributions. In section 9.2 we then will use the CA-tables of all conditionals to create the related CS. In section 9.3 we will summarize and discuss the findings of this chapter.

9.1 Generation of Common Appearances

In this section we will combine the reduced vf truth tables of two conditionals into a combined vf truth table. We will see that this is especially easy for cases when the two atomic conditionals don’t share any common predicates. We will investigate those cases more carefully when the two atomic conditionals share a common predicate and will see that only the sharing of a antecedence predicate will require special handling.

In this section we will extend the definitions of the common antecedence set and the (reduced) vf truth table without formally re-defining them. This will help us to create the common appearance tables for two atomic conditionals. Unfortunately the proposed method does not work in all cases and we will discuss those types of scenarios where the RCS mechanisms are not sufficient.

Definitions

In order to clearly distinguish the vf-pairs of different conditionals we introduce the indexed vf-pair.

Definition 45 (Indexed vf-Pair)

Let $\R_i$ be a conditional.

Then the indexed vf-pair $(x,y)_i$ with $x,y\in\IN_0$ is a vf-pair which is part of the conditional contribution of conditional $\R_i$ .

The following definition extends definition 39 of the countedby set.

Definition 46 (Combined Counted By Set)

Let $\Rcac_1$ and $\Rcac_2$ be cac-conditional. Let $c\in\cat(\Rcac_1)\cup\cat(\Rcac_2)$ .

Then the combined set of a-atoms counting $c$ is

\ccountedby(\Rcac_1,\Rcac_2,c) = \{~a~|~\grounding{c}{a}\in\gnd(\Rca_1)\cup\gnd(\Rca_2)~\}

Note that we did not change the name of the set compared to definition 39, we just added another cac-conditional as an additional parameter to it.

With definition 46 we can now also re-use definition 40 of the common antecedence set in order to combine the common antecedence sets of two conditionals in a meaningful way.

Different Antecedence, Same Conclusion Predicates

For the case when the two atomic conditionals share the same conclusion predicate it is necessary to combine the two vf truth tables, as we will see in the following example 54 . This example informally defines the combined common antecedence table as well as the combined vf truth table.

Example 54 (Creating CA-Tables – Different Antecedence Predicates)

Let $s_1 = \{~a,~b~\}$ and $s_2 = \{~x,~y~\}$ be sorts, $V_1$ and $W$ variables over $s_1$ , $V_2$ a variable over $s_2$ , and let $C, A_1$ and $A_2$ be predicates.

Let $\Rcac_1=\acond{C(V_1,V_2)}{A_1(W)}{V_1\neq W}$
and $\Rcac_1=\acond{C(V_1,V_2)}{A_2(W)}{V_1\neq W,~V_1\neq a,~V_2\neq x}$ .

Then we get the combined common antecedence table as show in table 9.1.

Table 9.1
	$\Rcac_1$		$\Rcac_2$
	$A_1(a)$	$A_1(b)$	$A_2(a)$	$A_2(b)$
$C(a,x)$		$r_1$			$\Big\} \coan_1$
$C(a,y)$		$r_2$			$\Big\} \coan_1$
$C(b,x)$	$r_3$				$\}\coan_2$
$C(b,y)$	$r_4$		$r_5$	$r_6$	$\}\coan_3$

Table 9.1: Combined Common Antecedence Table of Example 54

If there would have been only $\Rcac_1$ in a normal common antecedence table, the last two rows would have formed a single common antecedence set, as $C(b,x)$ and $C(b,y)$ are both counted by $A_1(a)$ and no other a-atom of $\Rcac_1$ . Due to the instantiation restrictions of $\Rcac_2$ it is necessary to have individual common antecedence sets for each of the last two rows, as $C(b,x)$ is not counted at all by $\Rcac_2$ , but $C(b,y)$ is.

From the above combined common antecedence table we can derive the combined vf truth table¹ as shown in table 9.3.

Table 9.2
	$\Theta_1$		$\Theta_2$	$\Theta_3$	$\Rcac_1$				$\Rcac_2$
	$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$		$A_1(a)$	$A_1(b)$	$\Sigma$		$A_2(a)$	$A_2(b)$	$\Sigma$
$\Theta_1\timesM\Theta_2\timesM\Theta_3$					$\omega^{a_1}_{()}$	$\omega^{a_1}_{(1)}$	$\omega^{a_1}_{(2)}$	$\omega^{a_1}_{(1,2)}$	$\omega^{a_2}_{()}$	$\omega^{a_2}_{(1)}$	$\omega^{a_2}_{(2)}$	$\omega^{a_2}_{(1,2)}$
$\theta^{\|0\|}_1 \timesM \theta^{\|0\|}_2 \timesM \theta^{\|0\|}_3$	$0$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,0)$	$(0,1)$	$(0,1)$	$(0,2)$
$\theta^{\|0\|}_1 \timesM \theta^{\|0\|}_2 \timesM \theta^{\|1\|}_3$	$0$	$0$	$0$	$1$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(0,0)$	$(1,0)$	$(1,0)$	$(2,0)$
$\theta^{\|0\|}_1 \timesM \theta^{\|1\|}_2 \timesM \theta^{\|0\|}_3$	$0$	$0$	$1$	$0$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(0,0)$	$(0,1)$	$(0,1)$	$(0,2)$
$\theta^{\|0\|}_1 \timesM \theta^{\|1\|}_2 \timesM \theta^{\|1\|}_3$	$0$	$0$	$1$	$1$	$(0,0)$	$(2,0)$	$(0,2)$	$(2,2)$	$(0,0)$	$(1,0)$	$(1,0)$	$(2,0)$
$\theta^{\|1\|}_1 \timesM \theta^{\|0\|}_2 \timesM \theta^{\|0\|}_3$	$0$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(1,1)$	$(1,3)$	$(0,0)$	$(0,1)$	$(0,1)$	$(0,2)$
$\theta^{\|1\|}_1 \timesM \theta^{\|0\|}_2 \timesM \theta^{\|1\|}_3$	$0$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(0,0)$	$(1,0)$	$(1,0)$	$(2,0)$
$\theta^{\|1\|}_1 \timesM \theta^{\|1\|}_2 \timesM \theta^{\|0\|}_3$	$0$	$1$	$1$	$0$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(0,0)$	$(0,1)$	$(0,1)$	$(0,2)$
$\theta^{\|1\|}_1 \timesM \theta^{\|1\|}_2 \timesM \theta^{\|1\|}_3$	$0$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(1,1)$	$(3,1)$	$(0,0)$	$(1,0)$	$(1,0)$	$(2,0)$
$\theta^{\|2\|}_1 \timesM \theta^{\|0\|}_2 \timesM \theta^{\|0\|}_3$	$1$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(2,0)$	$(2,2)$	$(0,0)$	$(0,1)$	$(0,1)$	$(0,2)$
$\theta^{\|2\|}_1 \timesM \theta^{\|0\|}_2 \timesM \theta^{\|1\|}_3$	$1$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(0,0)$	$(1,0)$	$(1,0)$	$(2,0)$
$\theta^{\|2\|}_1 \timesM \theta^{\|1\|}_2 \timesM \theta^{\|0\|}_3$	$1$	$1$	$1$	$0$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(0,0)$	$(0,1)$	$(0,1)$	$(0,2)$
$\theta^{\|2\|}_1 \timesM \theta^{\|1\|}_2 \timesM \theta^{\|1\|}_3$	$1$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(0,0)$	$(1,0)$	$(1,0)$	$(2,0)$
					$[0]$	$[2]$	$[2]$	$[4]$	$[0]$	$[1]$	$[1]$	$2[1]$

Table 9.2: Combined vf Truth Table of Example 54

From the combined vf truth table we can now derive the CA-table, as shown in table 9.3. It holds for this example that if vf-pair $vf_1$ of $\Rcac_1$ appears in the same line as vf-pair $vf_2$ of $\Rcac_2$ , then they have a common appearance.

Table 9.3
		$[0]$	$[1]$		$2[1]$
$CA(\Rcac_1,\Rcac_2)$		$(0,0)$	$(0,1)$	$(1,0)$	$(0,2)$	$(2,0)$
$[0]$	$(0,0)$	$*$	$*$	$*$	$*$	$*$
$[2]$	$(0,2)$	$*$	$*$	$*$	$*$	$*$
	$(1,1)$	$*$	$*$	$*$	$*$	$*$
	$(2,0)$	$*$	$*$	$*$	$*$	$*$
$[4]$	$(0,4)$	$*$	$*$		$*$
	$(1,3)$	$*$	$*$	$*$	$*$	$*$
	$(2,2)$	$*$	$*$	$*$	$*$	$*$
	$(3,1)$	$*$	$*$	$*$	$*$	$*$
	$(4,0)$	$*$		$*$		$*$

Table 9.3: CA-table of Example 54

Different Antecedence, Different Conclusion Predicates

For atomic conditionals which don’t share any predicates (i.e. for which it holds that the sets of their ground atoms are mutually exclusive) it is easy to create the CA-tables as in such cases all vf-pairs of both conditionals will share a common appearance, therefor resulting always in a table which is "full", i.e. which indicates a "*" in every cell. We will further investigate such cases in section 10.2. In such cases it will therefore not be necessary to create the combined vf truth table at all in such cases, as long as we are only interested in the resulting CS. If, on the other hand, the relation between the CS and the possible worlds or the equivalence classes of possible worlds is required, then the combined vf truth table needs to be created.

Same Antecedence, Different Conclusion Predicates

We now look at the case when two atomic conditionals share the same antecedence predicate.

In example 54 we used rule "if vf-pair $vf_1$ of $\Rcac_1$ appears in the same line as vf-pair $vf_2$ of $\Rcac_2$ , then they have a common appearance". This rule seems logical, but does unfortunately not hold for cases where the two atomic conditionals share the same antecedence predicate, as we will see in the next example.

Example 55 (Creating CA-Tables – Same Antecedence Predicates)

Let $s_1 = \{~a,~b~\}$ be a sort, $V$ and $W$ variables over $s_1$ , and let $C_1, C_2$ and $A$ be predicates.

Let $\Rcac_3=\acond{C_1(V)}{A(W)}{\emptyset}$
and $\Rcac_4=\acond{C_1(V)}{A(W)}{\emptyset}$ .

Then we get the combined common antecedence table as show in table 9.4.

Table 9.4
	$\Rcac_3$		$\Rcac_4$
	$A(a)$	$A(b)$	$A(a)$	$A(b)$
$C_1(a)$	$r_1$	$r_3$			$\Big\} \coan_1$
$C_1(b)$	$r_2$	$r_4$			$\Big\} \coan_1$
$C_2(a)$			$r_{5}$	$r_{7}$
$C_2(b)$			$r_{6}$	$r_{8}$

Table 9.4: Combined Common Antecedence Table of Example 55

From the above combined common antecedence table we can derive the combined vf truth table as shown in table 9.6.

Table 9.5
	$\Theta_1$		$\Theta_2$		$\Rcac_3$				$\Rcac_4$
	$C_1(a)$	$C_1(b)$	$C_2(a)$	$C_2(b)$		$A(a)$	$A(b)$	$\Sigma$		$A(a)$	$A(b)$	$\Sigma$
$\Theta_1\timesM\Theta_2$					$\omega^{a}_{()}$	$\omega^{a}_{(1)}$	$\omega^{a}_{(2)}$	$\omega^{a}_{(1,2)}$	$\omega^{a}_{()}$	$\omega^{a}_{(1)}$	$\omega^{a}_{(2)}$	$\omega^{a}_{(1,2)}$
$\theta^{\|0\|}_1 \timesM \theta^{\|0\|}_2$	$0$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$
$\theta^{\|0\|}_1 \timesM \theta^{\|1\|}_2$	$0$	$0$	$0$	$1$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$
$\theta^{\|0\|}_1 \timesM \theta^{\|2\|}_2$	$0$	$0$	$1$	$1$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$
$\theta^{\|1\|}_1 \timesM \theta^{\|0\|}_2$	$0$	$1$	$0$	$0$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$
$\theta^{\|1\|}_1 \timesM \theta^{\|1\|}_2$	$0$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$
$\theta^{\|1\|}_1 \timesM \theta^{\|2\|}_2$	$0$	$1$	$1$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$
$\theta^{\|2\|}_1 \timesM \theta^{\|0\|}_2$	$1$	$1$	$0$	$0$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$
$\theta^{\|2\|}_1 \timesM \theta^{\|1\|}_2$	$1$	$1$	$0$	$1$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$
$\theta^{\|2\|}_1 \timesM \theta^{\|2\|}_2$	$1$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$
					$[0]$	$[2]$	$[2]$	$2[2]$	$[0]$	$[2]$	$[2]$	$2[2]$

Table 9.5: Combined vf Truth Table of Example 55

From the combined vf truth table we can now derive the CA-table, as shown in table 9.6.

In this example the two antecedence predicates are identical and therefore a common appearance does only occur for those vf-pairs which are contributed by the same a-segments of $\Rcac_3$ and $\Rcac_4$ .

Table 9.6
		$[0]$	$[2]$			$2[2]$
$CA(\Rcac_4,\Rcac_5)$		$(0,0)$	$(0,2)$	$(1,1)$	$(2,0)$	$(0,4)$	$(2,2)$	$(4,0)$
$[0]$	$(0,0)$	$*$
$[2]$	$(0,2)$		$*$	$*$	$*$
	$(1,1)$		$*$	$*$	$*$
	$(2,0)$		$*$	$*$	$*$
$2[2]$	$(0,4)$					$*$	$*$	$*$
	$(2,2)$					$*$	$*$	$*$
	$(4,0)$					$*$	$*$	$*$

Table 9.6: CA-table of Example 55

Creation of CA-Tables of Atomic Conditionals

We can now state how CA-tables can be constructed out of combined vf truth tables.

Assumption 3 (Creation of CA-Tables for Atomic Conditionals).

Let $\Ra_1$ and $\Ra_2$ be two atomic conditionals.

Let $vf_1\in\gamma(\Ra_1)$ and $vf_2\in\gamma(\Ra_2)$ be two vf-pairs, one which occurs in the conditional contribution of $\Rcac_1$ and the other in the conditional contribution of $\Rcac_2$ .

If $\Ra_1$ and $\Ra_2$ share the same antecedence predicate then the a-segments of both $\Ra_1$ and $\Ra_2$ need to be derived from the unified set of a-atoms of $\Ra_1$ and $\Ra_2$ , i.e. from $\aat(\Ra_1)\cup\aat(\Ra_2)$ . Then $vf_1$ and $vf_2$ share a common appearance iff both are contributed by the same a-segment in the same (reduced) c-segment $\omega^c$ .

If $\Ra_1$ and $\Ra_2$ don’t share the same antecedence predicate then every vf-pair $vf_1$ of $\Ra_1$ shares a common appearance with every vf-pair $vf_2$ of $\Ra_2$ iff both $vf_1$ and $vf_2$ are contributed in the same (reduced) c-segment $\omega^c$ .

We give a rough outline of a possible proof for this assumption. The following only applies for a given c-segment $\omega^c$ , i.e. there is never any correlation between the vf-pairs contributed within two different c-segments.

For the case of a shared antecedence predicate we first create the set of all a-atoms of both $\Ra_1$ and $\Ra_2$ and from that then create all a-segments $\omega^a_{(j_1,\cdots,j_k)}$ . We know that the entirety of these a-segments covers all possible combinations of truth values assigned to the a-atoms. This means that every a-segment represents a different truth value combination assigned to the a-atoms. As both $\Ra_1$ and $\Ra_2$ share the same antecedence predicate it follows that the a-segments of both conditionals are identical and therefore only those vf-pairs share a common appearance within $\omega^c$ which are contributed by the same a-segment.

For the case of different antecedence predicates the two sets of a atoms are mutually exclusive, i.e. it is $\aat(\Ra_1)\cap\aat(\Ra_2)=\emptyset$ and therefore in the vf truth table every combination of truth values assigned to $\aat(\Ra_1)\cap\aat(\Ra_2)$ occurs. Therefore each a-segment of $\Ra_1$ needs to be combined with every a-segment of $\Ra_2$ and therefore within $\omega^c$ every vf-pair of $\Ra_1$ shares a common appearance with every vf-pair of $\Ra_2$ .

9.2 Generation of Conditional Structures

In this chapter we will describe a method to derive the CS of a given knowledge base out of the CA-tables of the related conditionals. The described method is unfortunately quite demanding when it comes to resources and therefore seems not to be applicable in practice. Nevertheless, the short outline given here might server as a base for further investigations how a CS can be created out of a given set of conditional contributions.

We will not proof the findings in this chapter and focus on a more description of the mechanism of combination of conditional contributions.

The following definition allows us to extract those conditional impacts from a CS which all show the same specific vf-pair, which is generated by a specific conditional.

Definition 47 (Partial Conditional Structure)

Let $\kb = \{~R_1,\cdots,\R_n~\}$ be a FOPCL knowledge base with $\gamma(\kb)$ being the related condtional structure. Let $(x,y)_i$ with $x,y\in\IN_0$ and $1\leq i \leq n$ be a vf-pair which appears within $\gamma(\kb)$ as part of the conditional contribution of conditional $\R_i\in\kb$ .

The partial conditional structure of $\kb$ and $(x,y)_i$ is the subset of $\gamma(\kb)$ which includes all conditional impacts in which $(x,y)_i$ appears and is denoted by $\gamma(\kb,(x,y)_i)$ .

We now show the example of a knowledge base which includes three atomic conditionals. We will step-by-step explain how the different partial conditional structures can be created. After this example we will try to generalize the used steps.

Example 56 (Combining Common Appearances to Conditional Structures)

Let $s=\{~a,~b~c~\}$ be a sort with $C_1,~C_2,~A_1,~A_2$ being unary predicate symbols over $s$ . Let $\kba=\{~\Ra_1,~\Ra_2,~\Ra_3~\}$ with

$\Ra_1 = \big<\big(~C_1(V_1)~|~A_1(W_1)~\big), V_1\neq c~\big>$
$\Ra_2 = \big<\big(~C_1(V_1)~|~A_2(W_1)~\big)~\big>$
$\Ra_3 = \big<\big(~C_2(V_1)~|~A_1(W_1)~\big)~\big>$

The three CA-tables for the three conditionals of $\kba$ are shown in table 9.7, table 9.8 and table 9.9.

Table 9.7
		$[0]$	$[3]$				$2[3]$
$CA(\Ra_1,\Ra_2)$		$( 0, 0)_2$	$( 0, 3)_2$	$( 1, 2)_2$	$( 2, 1)_2$	$( 3, 0)_2$	$( 0, 6)_2$	$( 2, 4)_2$	$( 4, 2)_2$	$( 6, 0)_2$
$[0]$	$( 0, 0)_1$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[2]$	$( 0, 2)_1$	$*$	$*$	$*$			$*$	$*$
	$( 1, 1)_1$	$*$		$*$	$*$			$*$	$*$
	$( 2, 0)_1$	$*$			$*$	$*$			$*$	$*$
$2[2]$	$( 0, 4)_1$	$*$	$*$	$*$			$*$	$*$
	$( 2, 2)_1$	$*$		$*$	$*$			$*$	$*$
	$( 4, 0)_1$	$*$			$*$	$*$			$*$	$*$

Table 9.7: CA-Table of

R^a_1

and

R^a_2

of Example 56

Table 9.8
		$[0]$	$[3]$				$2[3]$
$CA(\Ra_1,\Ra_3)$		$( 0, 0)_3$	$( 0, 3)_3$	$( 1, 2)_3$	$( 2, 1)_3$	$( 3, 0)_3$	$( 0, 6)_3$	$( 2, 4)_3$	$( 4, 2)_3$	$( 6, 0)_3$
$[0]$	$( 0, 0)_1$	$*$
$[2]$	$( 0, 2)_1$		$*$	$*$	$*$	$*$
	$( 1, 1)_1$		$*$	$*$	$*$	$*$
	$( 2, 0)_1$		$*$	$*$	$*$	$*$
$2[2]$	$( 0, 4)_1$						$*$	$*$	$*$	$*$
	$( 2, 2)_1$						$*$	$*$	$*$	$*$
	$( 4, 0)_1$						$*$	$*$	$*$	$*$

Table 9.8: CA-Table of

R^a_1

and

R^a_3

of Example 56

Table 9.9
		$[0]$	$[3]$				$2[3]$
$CA(\Ra_2,\Ra_3)$		$( 0, 0)_3$	$( 0, 3)_3$	$( 1, 2)_3$	$( 2, 1)_3$	$( 3, 0)_3$	$( 0, 6)_3$	$( 2, 4)_3$	$( 4, 2)_3$	$( 6, 0)_3$
$[0]$	$( 0, 0)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$[3]$	$( 0, 3)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 1, 2)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 1)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 3, 0)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
$2[3]$	$( 0, 6)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 2, 4)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 4, 2)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$
	$( 6, 0)_2$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$	$*$

Table 9.9: CA-Table of

R^a_2

and

R^a_3

of Example 56

We will now step-by-step generate the elements of the CS $\gamma(\kba)$ .

For $(0,0)_1$ we see

from table 9.7 that $(0,0)_1$ commonly appears with all vf-pairs of $\Ra_2$ ;
from table 9.8 that $(0,0)_1$ only commonly appears with $(0,0)_3$ , but no other vf-pair of $\Ra_3$ ; and
from table 9.9 that $(0,0)_3$ commonly appears with all vf-pairs of $\Ra_2$ .

The appearance of $(0,0)_1$ is therefore mostly restricted by bullet 2 above, i.e. if $(0,0)_1$ appears within the CS of $\kba$ then it cannot appear commonly with any other element of $\Ra_3$ than $(0,0)_3$ . As it appears commonly with all vf-pairs of $\Ra_2$ and all of those appear commonly with all vf-pairs of $\Ra_3$ , we have to combine $(0,0)_1$ with all vf-pairs of $\Ra_2$ and $(0,0)_3$ . Due to the restriction in bullet 2 we must not create those conditional impacts which

It is therefore

$\gamma(\kba,(0,0)_1) = \{$	$(~(0,0)_1,~(0,0)_2,~(0,0)_3~),$	$(~(0,0)_1,~(0,3)_2,~(0,0)_3~),$
	$(~(0,0)_1,~(1,2)_2,~(0,0)_3~),$	$(~(0,0)_1,~(2,1)_2,~(0,0)_3~),$
	$(~(0,0)_1,~(3,0)_2,~(0,0)_3~),$	$(~(0,0)_1,~(0,6)_2,~(0,0)_3~),$
	$(~(0,0)_1,~(2,4)_2,~(0,0)_3~),$	$(~(0,0)_1,~(4,2)_2,~(0,0)_3~),$
	$(~(0,0)_1,~(6,0)_2,~(0,0)_3~)~$		$\}$

We take $(0,2)_1$ next, which appears (according to table 9.7) commonly with $(0,0)_2,~(0,3)_2,~(1,2)_2,~(0,6)_2,~(2,4)_2$ as well as (according to table 9.8) with $(0,3)_3,~(1,2)_3,~(2,1)_3$ and $(3,0)_3$ . We already know from 9.9 that all combinations between $\R_2$ and $R_3$ will be included in $\gamma(\kb)$ , as long as they are not further restricted. In this case they are further restricted by table 9.7. We therefore get

$\gamma(\kba,(0,2)_1) = \{$	$(~(0,2)_1,~(0,0)_2,~(0,3)_3~),$	$(~(0,2)_1,~(0,0)_2,~(1,2)_3~),$
	$(~(0,2)_1,~(0,0)_2,~(2,1)_3~),$	$(~(0,2)_1,~(0,0)_2,~(3,0)_3~),$
	$(~(0,2)_1,~(0,3)_2,~(0,3)_3~),$	$(~(0,2)_1,~(0,3)_2,~(1,2)_3~),$
	$(~(0,2)_1,~(0,3)_2,~(2,1)_3~),$	$(~(0,2)_1,~(0,3)_2,~(3,0)_3~),$
	$(~(0,2)_1,~(1,2)_2,~(0,3)_3~),$	$(~(0,2)_1,~(1,2)_2,~(1,2)_3~),$
	$(~(0,2)_1,~(1,2)_2,~(2,1)_3~),$	$(~(0,2)_1,~(1,2)_2,~(3,0)_3~),$
	$(~(0,2)_1,~(0,6)_2,~(0,3)_3~),$	$(~(0,2)_1,~(0,6)_2,~(1,2)_3~),$
	$(~(0,2)_1,~(0,6)_2,~(2,1)_3~),$	$(~(0,2)_1,~(0,6)_2,~(3,0)_3~),$
	$(~(0,2)_1,~(2,4)_2,~(0,3)_3~),$	$(~(0,2)_1,~(2,4)_2,~(1,2)_3~),$
	$(~(0,2)_1,~(2,4)_2,~(2,1)_3~),$	$(~(0,2)_1,~(2,4)_2,~(3,0)_3~)~$	$\}$

The other elements of the CS $\gamma(\kb)$ can be generated in the same way.

The approach taken by example 56 generates the CS out of the CA-tables of a given FOPCL knowledge base $\kb$ by roughly performing the following steps:

create all CA-tables of all conditionals in $\kb$ ;
select the CA-table $CA(\Ra_1,\R_2)$ ;
select a vf-pair $vf_1^1$ of $\Ra_1$ which indicates a "*" in $CA(\Ra_1,\Ra_2)$ , i.e. it shares a common appearance with $vf_2^1$ , the related vf-pair of $\Ra_2$ ;
combine $vf_1^1$ with $vf_2^1$ to a partial conditional impact;
1. select the next CA-table $CA(\Ra_1,\Ra_3)$ and select the first vf-pair $vf_3^1$ which has a common appearance with $vf_1^1$ ;
2. select the CA-table $CA(\Ra_2,\Ra_3)$ and check whether $vf_2^1$ and $vf_3^1$ share a common appearance;
3. if $vf_2^1$ and $vf_3^1$ share a common appearance then combine $vf_1^1$ with $vf_2^1$ and $vf_3^1$ to a partial conditional impact;
4. if $vf_2^1$ and $vf_3^1$ don’t share a common appearance then select the next vf-pair of $\Ra_3$ which shares a common appearance with $vf_x^1$ ;
5. repeat this for all conditionals if $\kb$ ;
add the resulting conditional impact to the CS;
repeat this for all vf-pairs of $\Ra_1$ .

We see already from this basic description that this approach is consuming a lot of resources and time as it iterates several times through all common appearance tables of all conditionals of $\kb$ . We therefore propose a quicker approach, which also seems to be easier to implement and will only require one iteration through all CA-tables.

create all CA-tables of all conditionals in $\kb$ ;
create a CS $\gamma_{full}(\kb)$ that includes all possible combinations of all vf-pairs of all conditionals;
1. select CA-table $CA(\Ra_1,\Ra_2)$ ;
2. for every two vf-pairs of $\Ra_1$ and $\Ra_2$ which don’t share a common appearance drop all conditional impacts from $\gamma_{full}$ which indicate a common appearance of these two vf-pairs
3. repeat this for all CA-tables;
the remaining lines are the CS of $\kb$ .

Unfortunately this procedure requires to first create the CS $\gamma_{full}(\kb)$ which for bigger knowledge bases might be a challenge at least to the memory resources.

9.3 Summary and Discussion

In section 9.1 we developed an assumption on how the conditional contributions of two atomic conditionals can be combined into a CA-table. We were also able to outline a possible proof for the assumption. With this we can compose all common appearances of a knowledge base.

How efficient this method is depends on the relations between the different conditionals within a knowledge base. We saw that the CA-tables can be constructed very easily if the two conditionals do not hold any common predicate. But once they have predicates in common it is also very likely that also their combined common antecedence sets differ and therefore demand for less reduction, i.e. RCS still works (as we saw in the combined vf truth tables) but in such cases we need to investigate more c-segments than when looking only at a single atomic conditional.

In section 9.2 we then investigated possibilities to re-combine the conditional contributions of atomic conditionals into a CS.

In oder to do so we first need to create all CA-tables of all conditionals in an atomic knowledge base. This on its own is already a huge effort and consumes a lot of resources. But unfortunately we even then could not find a mechanism which would combine the vf-pairs with reasonable effort into the CS. We therefore resisted to say that the function $\gred$ as defined in section 3.1 exists on a CS level - after all $\gred$ was meant to be a more efficient version of the common method $\gfull$ .

Therefore, to say it straight, the results of this chapter are rather unsatisfying. Whilst RCS gives a real advantage compared to the common method, we lose all the gained advantage when combining the conditional contributions (the output of RCS) into a CS.

As we have said before, the focus of this thesis was for a long time (most like for too long time) on the creation of conditional contributions. We therefore did not investigate the problem of their combination with the same amount of effort. It would therefore worthwhile looking into alternative possibilities which allow the combination of conditional contributions into a CS.

Notes

It is actually a combined reduced vf truth table, but for the sake of briefness we reduced the name by the word "reduced".