RCS and Conditional Contributions

In the first three sections of this chapter we will show that RCS not only allows the efficient generation of the CS of ca-conditionals (as seen in the last chapter), but also the generation of the CS of c-conditionals and in many cases also those of cc-conditionals. Section 8.4 describes how equivalence classes of possible worlds can be created based on RCS. In section 8.5 we will discuss the efficiency of RCS and section 8.6 will summarize our findings on RCS so far. Finally section 8.7 is an excursion which introduces the grounding tables which are a useful tool when it comes to determining whether there is a imbalanced usage of ground atoms within a conditional.

In the previous chapters we have created a rather massive theoretical apparatus which enabled us to proof the findings for calculation and generation related to atomic conditionals. Extending these definitions to a point where we could proof all the following findings would go far beyond what this thesis has set out to achieve. We nevertheless don’t want to skip the findings of this chapter and therefore decided to show these results but loosen the theoretical rigor. We will therefore in most of the cases not state propositions and proof them, but rather show examples and will sometimes roughly outline straight forward proofs, if possible.

In any case, the material shown here, as long as it not references strict proofs in previous chapters, should be regarded as a starting point for possible further investigations.

8.1 Generation of Conditional Contribution for CA-Conditionals

We saw already in chapter 7 how the CS of a ca-conditional can be generated by RCS.

8.2 Generation of Conditional Contribution for C-Conditionals

In order to generate the conditional contribution for a c-conditional we just need to allow for the " $\top$ " symbol in the antecedence to be treated as a single a-atom. Example 46 shows how the reduced vf truth table needs to be modified in order to do so.

Example 46 (RCS Generates Conditional Contribution of C-Conditionals)

Let $\Rc=\acond{C(V)}{\top}{V\neq c}$ be a c-conditional with the unary predicate $C$ and the variable $V$ ranging over sort $s=\{~a,~b,~c~\}$ .

The common antecedence table of $\Rc$ is shown in table 8.1.

Table 8.1
$\Rc$	$\top$
$C(a)$	$r_1$	$\coan_{1}$
$C(b)$	$r_2$	$\coan_{1}$

Table 8.1: Common Antecedence Sets of

R^c

of Example 46

From the common antecedence table we can derive the reduced vf truth table as shown in table 8.2.

Table 8.2
	$\Theta_1$		$\Rc$
	$C(a)$	$C(b)$	$\top$
	$\omega^c_{(1)}$	$\omega^c_{(2)}$	$\omega^{a_1}_{()}$
$\theta^{\|0\|}_1$	$0$	$0$	$(0,2)$
$\theta^{\|1\|}_1$	$0$	$1$	$(1,1)$
$\theta^{\|2\|}_1$	$1$	$1$	$(2,0)$
			$[2]$

Table 8.2: vf Truth Table of

R^c

of Example 46

We can read the conditional contribution of $\Rc$ directly from 8.2 and see that $\gamma(\Rc) = \{~[2]~\}$ .

RCS for c-conditonals can therefore be regarded as a simplified case of what we have shown for RCS in the chapter 7.

8.3 Generation of Conditional Contribution for CC-Conditionals - ccRCS

The application of RCS to cc-conditionals is a bit more tricky than what we have seen in the last section. We will first come up with a general assumption on how RCS needs to be extended in order to allow also the handling of cc-conditionals. We will call modified RCS method ccRCS and will not formally proof it. Afterwards we will look at an example where the sets of c-atoms and a-atoms are identical, which will give us a hint how to handle overlapping c-atoms and a-atoms. Then we will see an example where c-atoms and a-atoms are only partially overlapping.

The following paragraphs give an outline of how ccRCS works and why the related modifications are necessary.

For ccRCS the idea of redundant segments and their representatives, as defined in section 7.3 and section 7.4 still holds, as the values of the counting functions are only influenced by the sets of c-atoms and a-atoms and by the instantiation restrictions of a ca-conditional. We will therefore re-use concepts like canonical a-segments, common antecedence sets, redundant segments and representatives as we described them in earlier chapters for ca-conditionals. We will do so without formal definitions, propositions and proofs as those should be easily derived from the material in the previous chapters.

The modification to the RCS procedure is needed when it comes to summing up the canonical a-segments of the cc-conditional, which is due to two reasons:

In the case of fully overlapping c- and a-atoms we need to be aware that the a-atom only counts any c-atom iff the a-atom is verified by the related possible world. As c-atoms and a-atoms are identical it holds that the set of possible worlds $\Omega$ is identical to the set of c-segments and the set of a-segments. Therefore we can say that
1. an a-segment only counts, if the related c-atom (which is identical to the a-atom) is set; and
2. if several c-atoms are set, then all the related canonical a-segments count.
Item 2 of the above list is especially important, as it says that the counting of the a-segments is now fully dependent on the related c-atoms. This makes a difference for the composed a-segments - for ca-conditionals the truth values of a-segments are independent of the truth values assigned to the c-atoms. But for cc-conditionals we can identify a single composed a-segment for every c-segment, i.e. the composed a-segment for which all related c-atoms are set.

Therefore for cc-conditionals we can avoid to look into all different combinations of composed a-segments, as we only need one such composed a-segment which we will simply call cc-composed a-segment and we will indicate it by the symbol $\omega^a_{\sum}$ .

Example 47 will show a related case.
In example 48 we will see the case when the c-atoms and a-atoms only partially overlap, i.e. there is a difference so that the c-atoms does not include a ground atom which is part of the a-atoms. We will call the atoms in the difference {em d-atoms}. We now need to be aware that the d-atoms still do determine whether or not the related a-segment is counting, as we have shown in the item above.

Therefore, when creating the reduced vf truth table for a cc-conditional, we still need to address the d-atoms as if they were c-atoms, i.e. they get their own columns in the c-segment section of the table. As the d-atoms are not counted by the a-segments (the d-atoms only determine whether a a-segment counts or not), i.e. as they contribute the same value $0$ to all counting functions, we can comprise them within one redundant segment.

As we have seen now that ccRCS works similar to RCS we will drop the name ccRCS again and just name the whole procedure, for all types of atomic conditionals, RCS.

The above is just a high-level reasoning why ccRCS delivers the same results as the common method (as defined in definition 14) in a more efficient way. As we assume this holds we can now formulate one of our major findings.

Assumption 2 (RCS Implements $\gamma _{red}$ ).

RCS results in the same conditional contribution of an atomic conditional as $\gamma_{tab}$ , the common method.

RCS is an implementation of $\gred$ under the restriction that only conditional contributions of atomic conditionals need to be generated.

The result is less than we hoped for when we outlined the objectives in section 3.1, as both the restriction to a-conditionals and the restriction to conditional contributions are extremely limiting. Still, the mechanisms developed so far might help to better understand about CSs. A more detailed discussion of the results will be given in section 8.6.

The following example shows the case when the set of c-atoms and a-atoms of a cc-conditional are identical.

Example 47 (ccRCS Generates Conditional Contribution of CC-Conditionals - Identical C-Atoms and A-Atoms)

Let $\Rcc_1=\acond{C(V_1,V_2)}{C(W_1,W_2)}{V=W}$ be a c-conditional with predicates $C$ and $A$ and variables $V_1$ and $W_1$ ranging over sort $s_1=\{~a,~b~\}$ and variables $V_2$ and $W_2$ ranging over sort $s_2=\{~x,~y~\}$ .

We then get $\cat(\Rcc_1)=\{~C(a,x),~C(a,y),~C(b,x),~C(b,y)~\}$ and $\aat(\Rcc_1)=\{~C(a,x),~C(a,y),~C(b,x),~C(b,y)~\}$ , so that $\ovl(\Rcc_1)=\{~C(a,x),~C(a,y),~C(b,x),~C(b,y)~\}$ and $\diff(\Rcc_1) = \emptyset$ .

The common antecedence table of $\Rcc_1$ is shown in table 8.3.

Table 8.3
$\Rcc_1$	$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$
$C(a,x)$	$r_1$	$r_3$			$\coan_{1}$
$C(a,y)$	$r_2$	$r_4$			$\coan_{1}$
$C(b,x)$			$r_5$	$r_7$	$\coan_{2}$
$C(b,y)$			$r_6$	$r_8$	$\coan_{2}$

Table 8.3: Common Antecedence Sets of

R^{cc}_1

of Example 47

From the common antecedence table we can derive the reduced vf truth table as shown in table 8.4. Note that this table does not list all possible composed a-segments but only the cc-composed a-segment $\omega^a_{\sum}$ as defined in this section. We also have indicated those a-segments, which do not count any vf-pairs due to the setting of the related c-atom, with " $--$ ".

Table 8.4
	$\Theta_1$		$\Theta_2$
	$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$		$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$	$\sum$
$\Theta_2^R\timesM \Theta_2^R$					$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(3)}$	$\omega^a_{(4)}$	$\omega^a_{\sum}$
$\theta^{\|0\|}_1 \timesM \theta^{\|0\|}_2$	$0$	$0$	$0$	$0$	$(0,0)$	$--$	$--$	$--$	$--$	$(0,0)$
$\theta^{\|0\|}_1 \timesM \theta^{\|1\|}_2$	$0$	$0$	$0$	$1$	$--$	$--$	$--$	$--$	$(1,1)$	$(1,1)$
$\theta^{\|0\|}_1 \timesM \theta^{\|2\|}_2$	$0$	$0$	$1$	$1$	$--$	$--$	$--$	$(2,0)$	$(2,0)$	$(4,0)$

$\theta^{\|1\|}_1 \timesM \theta^{\|0\|}_2$	$0$	$1$	$0$	$0$	$--$	$--$	$(1,1)$	$--$	$--$	$(1,1)$
$\theta^{\|1\|}_1 \timesM \theta^{\|1\|}_2$	$0$	$1$	$0$	$1$	$--$	$--$	$(1,1)$	$--$	$(1,1)$	$(2,2)$
$\theta^{\|1\|}_1 \timesM \theta^{\|2\|}_2$	$0$	$1$	$1$	$1$	$--$	$--$	$(1,1)$	$(2,0)$	$(2,0)$	$(5,1)$

$\theta^{\|2\|}_1 \timesM \theta^{\|0\|}_2$	$1$	$1$	$0$	$0$	$--$	$(2,0)$	$(2,0)$	$--$	$--$	$(4,0)$
$\theta^{\|2\|}_1 \timesM \theta^{\|1\|}_2$	$1$	$1$	$0$	$1$	$--$	$(2,0)$	$(2,0)$	$--$	$(1,1)$	$(5,1)$
$\theta^{\|2\|}_1 \timesM \theta^{\|2\|}_2$	$1$	$1$	$1$	$1$	$--$	$(2,0)$	$(2,0)$	$(2,0)$	$(2,0)$	$(8,0)$

Table 8.4: vf Truth Table of

R^{cc}_1

of Example 47

Due to fact that $\Rcc_1$ is a cc-conditional we can read its conditional contribution from the $\omega^a_{\sum}$ column, so that $\gamma(\Rcc_1) = \{~(0,0),~(1,1),~(2,0),~(2,2),~(4,0),~(5,1),~(8,0)~\}$ .

Example 47 shows that relationship between MOS and conditional structures of ca-conditionals, which we outlined in chapter 8 does not hold for cc-conditionals.

The following example shows the case when the set of c-atoms and a-atoms of a cc-conditional are only partially overlapping.

Example 48 (ccRCS Generates Conditional Contribution of CC-Conditionals - Different C-Atoms and A-Atoms)

Let $\Rcc_2=\acond{C(V}{C(W)}{V\neq c}$ be a c-conditional with predicates $C$ and $A$ and variables $V$ and $W$ ranging over sort $s_1=\{~a,~b,~c~\}$ .

We then get $\cat(\Rcc_2)=\{~C(a),~C(b)~\}$ and $\aat(\Rcc_2)=\{~C(a),~C(b),~C(c)~\}$ , so that $\ovl(\Rcc_2) = \{~C(a),~C(b)~\}$ and $\diff(\Rcc_2) = \{~C(c)~\}$ .

The common antecedence table of $\Rcc_2$ is shown in table 8.5.

Table 8.5
$\Rcc_2$	$C(a)$	$C(b)$	$C(c)$
$C(a)$	$r_1$	$r_3$	$r_5$	$\coan_{1}$
$C(b)$	$r_2$	$r_4$	$r_5$	$\coan_{1}$

Table 8.5: Common Antecedence Sets of

R^{cc}_2

of Example 48

From the common antecedence table we can derive the reduced vf truth table as shown in table 8.6. Although the common antecedence table resulted in a single common antecedence set $\Theta_1$ the reduced vf truth table lists ground atom $C(c)$ as an additional common antecedence set $\Theta_2$ , as $\diff(\Rcc_2) = \{~C(c)~\}$ . Note that the truth value of $C(c)$ is never counted and is only listed here in order to set the related a-segments accordingly.

Table 8.6
	$\Theta_1$		$\Theta_2$
	$C(a)$	$C(b)$	$C(c)$		$C(a)$	$C(b)$	$C(c)$	$\sum$
$\Theta_i^R\times\Theta_2^R$				$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(3)}$	$\omega^a_{\sum}$
$\theta^{\|0\|}_1 \timesM \theta^{\|0\|}_2$	$0$	$0$	$0$	$(0,0)$	$--$	$--$	$--$	$(0,0)$
$\theta^{\|0\|}_1 \timesM \theta^{\|1\|}_2$	$0$	$0$	$1$	$--$	$--$	$--$	$(0,2)$	$(0,2)$
$\theta^{\|1\|}_1 \timesM \theta^{\|0\|}_2$	$0$	$1$	$0$	$--$	$--$	$(1,1)$	$--$	$(1,1)$
$\theta^{\|1\|}_1 \timesM \theta^{\|1\|}_2$	$0$	$1$	$1$	$--$	$--$	$(1,1)$	$(1,1)$	$(2,2)$
$\theta^{\|2\|}_1 \timesM \theta^{\|0\|}_2$	$1$	$1$	$0$	$--$	$(2,0)$	$(2,0)$	$--$	$(4,0)$
$\theta^{\|2\|}_1 \timesM \theta^{\|1\|}_2$	$1$	$1$	$1$	$--$	$(2,0)$	$(2,0)$	$(2,0)$	$(6,0)$

Table 8.6: vf Truth Table of

R^{cc}_2

of Example 48

Due to fact that $\Rcc_2$ is a cc-conditional we can read its conditional contribution from the $\omega^a_{\sum}$ column, so that $\gamma(\Rcc_2) = \{~(0,0),~(0,2),~(1,1),~(4,0),~(6,0)~\}$ .

8.4 Equivalence Classes of Possible Worlds and RCS

We now investigate how the equivalence classes of possible worlds can be derived by only looking at the reduced set of c-segments and the a-segments of a ca-conditional. This section will only give an informal introduction on how equivalence classes of possible worlds can be generate.

Definition 15 defines the equivalence classes of possible worlds as the set of possible worlds $\omega\in\Omega(\Rca)$ which create the same conditional impact. So far we have segmented the possible worlds into c-segments and a-segments and then further split up and reduced the c-segments into redundant segments. RCS works based on the representatives of the c-segments plus the a-segments and delivers the vf-pairs every a-segment counts for specific combinations of representatives. This is obvious from e.g. table 7.4 in example 43, where only 9 lines are needed to calculate the conditional contribution, but due to 4 c-atoms and 3 a-atoms we know that $2^{(4+3)} = 2^7 = 128$ worlds exist. How can we sort these 128 worlds into equivalence classes based on the 9 lines in the table?

For every vf-pair $i_j$ in every line $i$ within the reduced vf truth table we assume a separate equivalence class $\omegaEC^i$ . Each of these lines is represented by a specific semantical combination " $\timesM$ " of representatives. We know that the related sets of redundant segments all contribute the same vf-pairs to the conditional structure. The c-segment part of the related equivalence classes therefore contains the semantical combination of the sets of redundant segments (for which the representatives in the lines of the reduced vf truth table stand for) plus the a-segment.

Once we have created this combination we then can further semantically combine each of these equivalence classes one-by-one with all a-segments which contribute the same vf-pair in line $i$ .

Example 49 (RCS and Equivalence Classes of Possible Worlds)

Continuing from example 43.

The reduced vf truth table in table 7.4 shows 9 lines, each of them with a specific conditional impact.

We look at the second line, which represents the semantical combination $\theta_1^{|0|} \timesM \theta_2^{|1|}$ and generates the conditional impact $\gamma^1 = (0,0),~(0,2),~(0,4),~(0,6),~(0,8)$ .

Let’s start from $(0,0)$ . We know that $\omega^a_{()}$ in combination with all segments in $\Theta_1^{|0|}$ and $\Theta_2^{|1|}$ will contribute $(0,0)$ to the conditional impact. Therefore we can form the equivalence class by creating the following combination:

$\omegaEC^{0/1/0}$	$= \Theta_1^{\|0\|} \timesM \Theta_2^{\|1\|} \timesM \omega^a_{()}$
	$= \{~00~\} \timesM \{~01,~10~\} \timesM \{~000~\}$
	$= \{~0001000,~0010000~\}$

Obviously, those are not all possible worlds which create the conditional impact $(0,0)$ - it is actually contributed by all combinations of representatives, as table 7.4 clearly shows. We can look at another line, e.g. line 8:

$\omegaEC^{1/1/0}$	$= \Theta_1^{\|1\|} \timesM \Theta_2^{\|1\|} \timesM \omega^a_{()}$
	$= \{~01,~10~\} \timesM \{~01,~10~\} \timesM \{~000~\}$
	$= \{~0001000,~0101000,~0010000,~0110000~\}$

With this we have shown how equivalence classes of possible worlds can be generated by the mechanisms provided by RCS. Still there are issues to be further investigated, e.g. how equivalence classes in the case of cc-conditionals can be generated. Also, as this chapter only treats conditional contributions, we have not shown a way for the generation of equivalence classes when the conditional contributions get combined.

8.5 Effectiveness of the Method of Reduction of C-Segments

RCS is a more efficient method for the generation of conditional contributions of atomic conditionals than the common method which is described in section 2.3. We can regard RCS as a partial implementation of the function $\gred$ , which we introduced in section 3.1 as one of the objectives of this thesis in section 3.1.

One of the restrictions of RCS is that it only allows for the generation of conditional contributions. Nevertheless, every conditional contribution is a CS, as shown in section 4.3. Therefore we can measure the efficiency of RCS by comparing it with $\gfull$ as defined by definition 12 on the basis of how many possible worlds it uses for the generation of the conditional contribution. This was done e.g. in example 45, where for the particular given conditional RCS was 40 % more efficient than the common method.

Unfortunately this reduction rate is not the same for every conditional. The cases we have treated in the examples of this chapter so far worked well. We will now give two examples, where the method of reduced c-segments has lower or even no advantage compared to the common method.

Example 50 ( $V\neq W$ Type Instantiation Restriction And Common Antecedence Sets)

Let $s=\{~a,b,c~\}$ be a sort and let $C$ and $A$ be unary predicates with variables $V$ and $W$ ranging over $s$ .

Let $\R_1=\acond{C(V)}{A(W)}{V\neq W}$ be a ca-conditional.

Then we get the following common antecedence sets:

Table 8.7
	$A(a)$	$A(b)$	$A(c)$
$C(a)$		$r_1$	$r_2$	$coan_1$
$C(b)$	$r_3$		$r_4$	$coan_2$
$C(c)$	$r_5$	$r_6$		$coan_3$

Table 8.7: Common Antecedence Set of

R_1

of Example 50

Example 50 shows that a non-local instantiation restriction of the type $V\neq W$ can cause every single c-segment to result in a separate common antecedence set. RCS works by reducing the number of redundant segments within a set of common antecedence segments. This means that RCS works only in an efficient manner (compared to the common method for generation of a conditional contribution) if there are more than one c-segments within a common antecedence segment. Therefore in the case of the conditional shown in example 50 RCS does not give an advantage compared to the common method.

Example 51 ( $V=W$ Type Instantiation Restriction And Common Antecedence Sets)

Let $s=\{~a,b,c~\}$ be a sort and let $C$ and $A$ be unary predicates with variables $V$ and $W$ ranging over $s$ .

Let $\R_1=\acond{C(V)}{A(W)}{V=W}$ be a ca-conditional.

Then we get the following common antecedence sets:

Table 8.8
	$A(a)$	$A(b)$	$A(c)$
$C(a)$	$r_1$			$coan_1$
$C(b)$		$r_2$		$coan_2$
$C(c)$			$r_3$	$coan_3$

Table 8.8: Common Antecedence Set of

R_1

of Example 51

Example 51 shows that also non-local instantiation restrictions of the type $V=W$ can cause that every the method of c-segment reduction is not more efficient than the common method.

Nevertheless, when we look again at example 38 we see that non-local instantiation restrictions do not necessarily prevent the method of c-segment reduction to be efficient at all. As the $V\neq W$ type instantiation restriction in example 38 applies only to one of two variables in predicate $C$ , the c-segments can still be grouped into common antecedence sets where each holds more than one element.

But the efficiency of RCS can hardly be judged only on the number of c-segments it makes use of for generating the conditional contribution. Another factor is e.g. the additivity of canonical a-segments, as outlined in 5.2. Whilst for the common method the vf-pairs of all possible worlds have to be counted, RCS requires the addition of the canonical vf-pairs of only the c-segments. This itself can reduce the effort for the generation of a conditional contribution immensely, depending on the amount of ground atoms in the a-segments and also in how fast the related algorithm can handle the addition of vf-pairs.

8.6 Summary and Discussion

RCS allows us to generate the conditional contribution of a ca-conditional from a reduced set of possible worlds. It reduces the number of c-segments which are required for the generation of the conditional structure, whilst it does not reduce on the number of a-segments. We have to be aware that this result only offers a very limited approach to the $\gred$ function we introduced in section 3.1.

First of all, RCS only applies to atomic conditionals, i.e. to conditional which include only a single predicate $C$ in the conclusion part as well as a single predicate $A$ in the antecedence part. This already is a major restriction, which leaves out all conditionals which include complex logic formula (such as and/or conjunctions of predicates). We also do not look at cases where negated predicates are used, as stated in section 4.2. Whilst the work on this thesis started out by investigating all types of conditionals, it became clear after a while that for an initial step only the atomic conditionals allowed for a more efficient generation of the conditional contributions. Whether such methods also exist for other types of conditionals needs to be investigated further.

Second, RCS only allows the generation of conditional contributions, i.e. it works for single conditionals but not for whole knowledge bases. In chapter 9 we show how the individual conditional contributions of a knowledge base can be combined into to related CS by a different mechanism, but the RCS itself does not allow the generation of a CS of more than a single conditional contribution.

Third, RCS only works efficiently if the number of non-local instantiation restrictions is limited, as we have seen in section 8.5. We also saw in section 8.5 that there is no straight forward way to calculate how much more efficient RCS is compared to the common method.

Fourth, RCS does currently not take into account negated predicates, as outlined in section 4.2. As said there, it seems likely that RCS can easily be extended to use negated conditionals, but this was not shown on a level that would hold for all atomic conditionals.

Finally, RCS does not directly create the relationship between a CS and its related equivalence classes of possible worlds, as shown in section 8.4. As shown there it is nevertheless a rather straight forward procedure to generate these equivalence classes from the

Within this chapter we worked out a number of tools and methods, such as the segmentation of possible worlds, the common antecedence sets and the idea of representatives of redundant sets, which all might be useful also in further investigations of CSs.

We also have created a rather massive theoretical apparatus in order to proof that RCS generates the same results as the common method. The following chapters will build on these

8.7 Excursion: Grounding Tables

In this section we introduce the grounding table, which are modified common antecedence tables. Grounding tables allow us to easily read whether there is an imbalanced use within an atomic conditional.

The two examples shown here treat the conditionals used in section 3.3, where we showed that the CS of a balanced and an imbalanced (with respect to usage of ground atoms) knowledge base can create the same conditional structure.

We define the grounding table informally in the next example.

Example 52 (Grounding Table)

Let $s=\{~a,~b,~c~\}$ be a sort. Let $C_1$ and $A$ be predicates and let $V_1, V_2$ and $W$ be variables ranging over $s$ .

Let $\R_1 = \big<\big(~C_1(V_1,V_2)~|~A(W)~\big),~V_1 \neq W,~V_2 \neq W~\big>$ be an atomic conditional.

Table 8.9 shows the grounding table of $\R_1$ .

Table 8.9
	$A(a)$	$A(b)$	$A(c)$	$\|c\|_{\gnd(\R_1)}$
$C_1(a,a)$		$r_5$	$r_9$	$2$	$\coan_{1}$
$C_1(b,b)$	$r_1$		$r_{12}$	$2$	$\coan_{2}$
$C_1(c,c)$	$r_4$	$r_8$		$2$	$\coan_{3}$
$C_1(a,b)$			$r_{10}$	$1$	$\coan_{4}$
$C_1(b,a)$			$r_{11}$	$1$	$\coan_{4}$
$C_1(a,c)$		$r_6$		$1$	$\coan_{5}$
$C_1(c,a)$		$r_7$		$1$	$\coan_{5}$
$C_1(b,c)$	$r_2$			$1$	$\coan_{6}$
$C_1(c,b)$	$r_3$			$1$	$\coan_{6}$
$\|a\|_{\gnd(\R_1)}$	$4$	$4$	$4$

Table 8.9: Grounding Table of

R_1

of Example 52

The grounding table is an extended version of the common antecedence table which we introduced in section 7.1. Grounding tables allow us to determine whether there is an imbalanced use of ground atoms in $\R_1$ or not. If the numbers in the last row are all the same and it the numbers in the one-but-last column are all the same, then the conditional is balanced with respect to the usage of ground atoms.

Proposition 20 (Grounding Table Indicates Imbalanced Use)

Let $\R = \big<~\big(~C(V_1,\cdots,V_k)~|~A(W_1,\cdots,W_l)~\big),~\xi_{\R}~\big>$ be an atomic conditional.

If it holds that $C \neq A$ then $\R$ is balanced with respect to the use of ground atoms if and only if within the grounding table the grounding sums of all rows are equal and the grounding sums of all columns are equal.

If it holds that $C = A$ then $\R$ is balanced with respect to the use of ground atoms if and only if within the grounding table the grounding sums of all rows and all columns are equal.

Proof

The proposition follows immediately from item 1 in definition 5.□

The following example shows that the conditional contribution of certain conditionals can immediately be read from the grounding table.

Example 53 (Grounding Table Indicates Imbalanced Use)

Let $s_1=\{~a,~b,~c~\}$ and $s_2=\{~x,~y~\}$ be two sorts. Let $C_2$ and $A$ be predicates and let $V_1, V_2$ and $W$ be variables ranging over $s_1$ and let $X$ be a variable ranging over $s_2$ .

Let $\R_2= \big<\big(~C_2(V_1,V_2,X)~|~A(W)~\big),~V_1 \neq V_2,~V_1 \neq W, V_2 \neq W~\big>$ be an atomic conditional.

Table 8.10 shows the atomic conditional table of $\R_2$ .

Table 8.10
	$A(a)$	$A(b)$	$A(c)$	$\|c\|_{\gnd(\R_1)}$
$C_2(b,c,x)$	$r_1$			$1$	$\coan_{1}$
$C_2(b,c,y)$	$r_2$			$1$
$C_2(c,b,x)$	$r_3$			$1$
$C_2(c,b,y)$	$r_4$			$1$
$C_2(a,c,x)$		$r_5$		$1$	$\coan_{2}$
$C_2(a,c,y)$		$r_6$		$1$
$C_2(c,a,x)$		$r_7$		$1$
$C_2(c,a,y)$		$r_8$		$1$
$C_2(a,b,x)$			$r_{9}$	$1$	$\coan_{3}$
$C_2(a,b,y)$			$r_{10}$	$1$
$C_2(b,a,x)$			$r_{11}$	$1$
$C_2(b,a,y)$			$r_{12}$	$1$
$\|a\|_{\gnd(\R_1)}$	$4$	$4$	$4$

Table 8.10: Grounding Table of

R_2

of Example 53

As it is $C_2 \neq A$ and as for all rows it is $|c|_{\gnd(\R_2)} =1$ (for all c-atoms $c$ ) as well as for all columns it is $|a|_{\gnd(\R_2)} =4$ (for all a-atoms $a$ ), it holds with proposition 20 that $\R_1$ is balanced with respect to the use of ground atoms.

Grounding tables are a useful tool and come for free when creating the common antecedence table.