Reduction of C-Segments

In this chapter we will show that the conditional contribution of ca-conditionals can be generated by using only a subset of all lines within their vf truth tables. Many lines generate the same vf-pairs and therefore are redundant. Although this is obvious from looking at the vf truth tables, it is not immediately clear whether there is an algorithm which would allow finding such redundant lines. In this chapter we will introduce such an algorithm for the limited case of ca-conditionals. The mechanism is called the method of reduction of c-segments or simply reduction of c-segments, in short RCS.

The related results will be applied to c-conditionals and ca-conditionals in chapter 9. We therefore will state the propositions wherever possible in a more generic way, e.g. for atomic conditionals in general or or for cac-conditionals.

Figure 7.1 Overview of the Method of Reduction of C-Segments (RCS)

Figure 7.1 gives an overview of the steps performed in this chapter in order to define RCS. Step 1 shows the segmentation of possible worlds into c-segments and a-segments, as described already in section 4.7.

In section 7.1 we will introduce the common antecedence sets, which are the sets of c-atoms which are counted by exactly the same a-atoms of a conditional. Step 2 shows that these sets will be used to further segment the c-atoms into common antecedence segments, which will be described in section 7.2.

We will then show in section 7.3 that within a set of common antecedence segments all those segments are mutually redundant, which verify (or falsify) the same number of c-atoms (step 3). This allows us to chose only one of these mutually redundant sets as a representative (step 4). All the different representatives of one set of common antecedence segments then will be put together into a set of representatives as described in section 7.4 (step 5). We then end up with the so-called a reduced set of c-segments (step 6).

In the next chapter we will show how RCS allows us to generate the same conditional contribution as the common method (as defined by section 2.3) but in many cases with less effort.

7.1 Common Antecedence Sets

We will pave the ground in order to allow for a segmentation into redundant sets, as outlined in the previous section. For this we first have to identify the a-atoms which count the same c-atoms. These a-atoms are grouped into the countedby-set. Those c-atoms which then share the same countedby-sets form a common antecedence set.

We first define the countedby-sets.

Definition 39 (Set of A-Atoms Counting a C-atom)

Let $\Rcac$ be a cac-conditional. Let $c\in\cat(\Rcac)$ be a c-atom of $\Rca$ and let $\gnd(\Rcac)$ be the set of groundings of $\Rcac$ .

The set of a-atoms counting $c$ is

\countedby(\Rcac,c) := \{~a~|~\big<\big(c\big|a\big)\big>~\in\gnd(\Rcac)~\}.

Example 37 (Set of A-Atoms Counting a C-atom)

Continuing from example 32.

Then we get the sets of a-atoms which count the same c-atoms:

$\countedby(\Rcac,C(a))$	$= \{~A(b),~A(c)~\}$
$\countedby(\Rcac,C(b))$	$= \{~A(a),~A(c)~\}$
$\countedby(\Rcac,C(c))$	$= \{~A(a),~A(b)~\}$

We now can define the common antecedence set as the set of c-atoms which share the same countedby-sets.

Definition 40 (Common Antecedence Set)

Let $\Rcac$ be a cac-conditional. Let $c\in\cat(\Rcac)$ and $a\in\aat(\Rcac)$ .

The common antecedence set of $c$ is the set of c-atoms which are all counted by exactly the same a-atoms in conditional $\Rcac$ , i.e.

\coan(\Rcac,~c) = \{~c'~|~\countedby(\Rcac,c)~=~\countedby(\Rcac,c'),~c,c'\in\cat(\Rcac)~\}

The following example introduces informally the common antecedence table.

Example 38 (Common Antecedence Set)

Let $s_1 = \{~a,~b,~c~\}$ and $s_2 = \{~x,~y~\}$ be sorts, $V_1$ and $W$ variables over $s_1$ , $V_2$ a variable over $s_2$ , $C, A$ predicates and $\Rca=\big<~\big(~C(V_1, V_2)~|~A(W)~\big),~V_1\neq W, V_1\neq c~\big>$ a ca-conditional.

Then we get the groundings of $\Rca$ , i.e.

$r_1 = ~\big<~\big(~C(a,x)~\|~A(b)~\big)~\big>$	$~~~~$	$r_5 = ~\big<~\big(~C(b,x)~\|~A(a)~\big)~\big>$
$r_2 = ~\big<~\big(~C(a,x)~\|~A(c)~\big)~\big>$		$r_6 = ~\big<~\big(~C(b,x)~\|~A(c)~\big)~\big>$
$r_3 = ~\big<~\big(~C(a,y)~\|~A(b)~\big)~\big>$		$r_7 = ~\big<~\big(~C(b,y)~\|~A(a)~\big)~\big>$
$r_4 = ~\big<~\big(~C(a,y)~\|~A(c)~\big)~\big>$		$r_8 = ~\big<~\big(~C(b,y)~\|~A(c)~\big)~\big>$

From those we can derive the sets of of a-atoms counting c-atoms:

$\countedby(\Rca,C(a,x))$	$= \{~A(b),~A(c)~\}$	$= \countedby(\Rca,C(a,y))$
$\countedby(\Rca,C(b,x))$	$= \{~A(a),~A(c)~\}$	$= \countedby(\Rca,C(b,y))$

and get the common antecedence sets:

\coan_1 = \{~C(a,x),~C(a,y)~\}

~~~

\coan_2 = \{~C(b,x),~C(b,y)~\}

Finding these identical sets is a bit complicated, at least when done manually. We therefore introduce the common antecedence table, which indicates the a-atoms as column headings and the c-segments as row headings. The cells of which the related c-atom and a-atom occur both in an admissible grounding of $\Rca$ are marked (e.g. with the indicator of the admissible grounding or simply a " $*$ "). Table 7.1 shows such a table for this example. We see that the c-atoms of those lines, which have indicators in exactly the same columns, form a common antecedence set.

Table 7.1
	$A(a)$	$A(b)$	$A(c)$
$C(a,x)$		$r_1$	$r_2$	$\Big\} \coan_1$
$C(a,y)$		$r_3$	$r_4$	$\Big\} \coan_1$
$C(b,x)$	$r_5$		$r_6$	$\Big\} \coan_2$
$C(b,y)$	$r_7$		$r_8$	$\Big\} \coan_2$

Table 7.1: Common Antecedence Table of

R^{cac}

of Example 38

Table 7.2 shows the reduced vf truth table of $\Rca$ .

Table 7.2
	$\coan_1$		$\coan_2$
	$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$		$A(a)$	$A(b)$	$A(c)$	$\sum$
					$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(3)}$	$\omega^a_{(1,2)}$	$\omega^a_{(1,3)}$	$\omega^a_{(2,3)}$	$\omega^a_{(1,2,3)}$
$\omega^c_{()}$	$0$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,4)$	$(0,6)$	$(0,6)$	$(0,8)$
$\omega^c_{(4)}$	$0$	$0$	$0$	$1$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(1,3)$	$(2,4)$	$(1,5)$	$(2,6)$
$\omega^c_{(3)}$	$0$	$0$	$1$	$0$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(1,3)$	$(2,4)$	$(1,5)$	$(2,6)$
$\omega^c_{(3,4)}$	$0$	$0$	$1$	$1$	$(0,0)$	$(2,0)$	$(0,2)$	$(2,2)$	$(2,2)$	$(4,2)$	$(2,4)$	$(4,4)$

$\omega^c_{(2)}$	$0$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(1,1)$	$(1,3)$	$(1,3)$	$(1,5)$	$(2,4)$	$(2,6)$
$\omega^c_{(2,4)}$	$0$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(2,3)}$	$0$	$1$	$1$	$0$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(2,3,4)}$	$0$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(1,1)$	$(3,1)$	$(3,1)$	$(5,1)$	$(4,2)$	$(6,2)$

$\omega^c_{(1)}$	$1$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(1,1)$	$(1,3)$	$(1,3)$	$(1,5)$	$(2,4)$	$(2,6)$
$\omega^c_{(1,4)}$	$1$	$0$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(1,3)}$	$1$	$0$	$1$	$0$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(1,3,4)}$	$1$	$0$	$1$	$1$	$(0,0)$	$(2,0)$	$(1,1)$	$(3,1)$	$(3,1)$	$(5,1)$	$(4,2)$	$(6,2)$

$\omega^c_{(1,2)}$	$1$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(2,0)$	$(2,2)$	$(2,2)$	$(2,4)$	$(4,2)$	$(4,4)$
$\omega^c_{(1,2,4)}$	$1$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(3,1)$	$(4,2)$	$(5,1)$	$(6,2)$
$\omega^c_{(1,2,3)}$	$1$	$1$	$1$	$0$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(3,1)$	$(4,2)$	$(5,1)$	$(6,2)$
$\omega^c_{(1,2,3,4)}$	$1$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(4,0)$	$(6,0)$	$(6,0)$	$(8,0)$
					$[0]$	$[2]$	$[2]$	$[4]$	$[4]$	$[6]$	$[6]$	$2[4]$

Table 7.2: vf Truth Table and Common Antecedence Sets of Example 38

From this table we see that there are many redundant lines, i.e.

line 1 and 2, as well as 5 and 6 as well as 9 and 10 as well as 13 and 14 generate the same vf-pairs respectively;
the block of lines 4,5,6,7 and the the block of lines 8,9,10,11 generate the same vf-pairs.

Before we move on to the next section we state the rather obvious fact that the common antecedence sets are mutually exclusive and that they cover all c-atoms.

Proposition 17 (Common Antecedence Sets Cover Exactly All C-Atoms)

Let $\Rca$ be a ca-conditional and $\coan_1,\cdots,\coan_k$ all common antecedence sets of $\Rca$ .

It holds that $\cat(\Rca) = \bigcup_{i=1}^k \coan_i$ and for any two $\coan_i$ and $\coan_k$ of $\Rca$ with $i \neq k$ it holds that $\coan_i \cap coan_k = \emptyset$ .

Proof

Based on definition 39 it is ${\countedby(\Rcac,c) = \{~a~|~\big<(c|a)\big>~\in\gnd(\Rcac)~\}}$ , so that for all $c\in\cat(\Rca)$ there exists exactly one $\countedby$ set.

Based on definition 40 it is

\coan(\Rcac,c) = \{~c'~|~\countedby(\Rcac,c)~=~\countedby(\Rcac,c'),~c,c'\in\cat(\Rcac)~\}.

As all $\countedby$ sets are included in one of the $\coan$ sets of $\Rca$ it follows that $\cat(\Rca) = \bigcup_{i=1}^k \coan_i$ .

As each $c\in\cat(\Rca)$ can only be included in exactly one set $\countedby(\Rca,c)$ and as all equal $\countedby$ sets contribute their c-atoms into exactly one $\coan$ -set it follows that all $\coan$ sets are mutually exclusive, i.e. for any tow $\coan_i$ and $\coan_k$ of $\Rca$ with $i \neq k$ it holds that $\coan_i \cap \coan_k = \emptyset$ .□

Example 39 (Common Antecedence Sets Cover Exactly All C-Atoms)

Continuing from example 38.

The reduced vf truth table in table 7.2 shows that $\coan_1$ and $\coan_2$ are not overlapping and that they cover all c-atoms.

7.2 Common Antecedence Segments

A common antecedence set is the set of those c-atoms of a ca-conditional which are all counted by exactly the same a-segment. In this section we introduce the common antecedence segments, which are based on the c-atoms in the common antecedence sets. We will show that these common antecedence segments cover all c-atoms and that they are mutually exclusive.

Definition 41 (Common Antecedence Segment)

Let $\Rcac$ be a cac-conditional. Let $\coan_i$ be a common antecedence set of $\Rcac$ .

The set of common antecedence segments of $\coan_i$ is the set of possible world segments which are based on the c-atoms of $\coan_i$ and is denoted by $\Theta_i$ so that $\Theta_i = \Omega(\coan_i)$ . It is $\theta\in\Omega(\coan_i)$ a common antecedence segment.

The numbering of the indexes of the ground atoms within the common antecedence segments $\theta_{(i_1,\cdots,i_l)}\in\Theta_i$ is the same as for the related c-segment.

Note that with this definition we further segmented the c-segments into common antecedence segments, based on the (mutually exclusive) common antecedence sets.

Example 40 (Common Antecedence Segment)

Continuing from example 38.

We modify the reduced vf truth table of $\Rca$ , indicating the common antecedence segments, as shown in table 7.3.

Table 7.3
		$\Theta_1$		$\Theta_2$
		$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$		$A(a)$	$A(b)$	$A(c)$	$\sum$
						$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(3)}$	$\omega^a_{(1,2)}$	$\omega^a_{(1,3)}$	$\omega^a_{(2,3)}$	$\omega^a_{(1,2,3)}$
$\omega^c_{()} =$	$\theta_{()}\timesM\theta_{()}$	$0$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,4)$	$(0,6)$	$(0,6)$	$(0,8)$
$\omega^c_{(4)} =$	$\theta_{()}\timesM\theta_{(4)}$	$0$	$0$	$0$	$1$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(1,3)$	$(2,4)$	$(1,5)$	$(2,6)$
$\omega^c_{(3)} =$	$\theta_{()}\timesM\theta_{(3)}$	$0$	$0$	$1$	$0$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(1,3)$	$(2,4)$	$(1,5)$	$(2,6)$
$\omega^c_{(3,4)} =$	$\theta_{()}\timesM\theta_{(3,4)}$	$0$	$0$	$1$	$1$	$(0,0)$	$(2,0)$	$(0,2)$	$(2,2)$	$(2,2)$	$(4,2)$	$(2,4)$	$(4,4)$

$\omega^c_{(2)} =$	$\theta_{(2)}\timesM\theta_{()}$	$0$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(1,1)$	$(1,3)$	$(1,3)$	$(1,5)$	$(2,4)$	$(2,6)$
$\omega^c_{(2,4)} =$	$\theta_{(2)}\timesM\theta_{(4)}$	$0$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(2,3)} =$	$\theta_{(2)}\timesM\theta_{(3)}$	$0$	$1$	$1$	$0$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(2,3,4)} =$	$\theta_{(2)}\timesM\theta_{(3,4)}$	$0$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(1,1)$	$(3,1)$	$(3,1)$	$(5,1)$	$(4,2)$	$(6,2)$

$\omega^c_{(1)} =$	$\theta_{(1)}\timesM\theta_{()}$	$1$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(1,1)$	$(1,3)$	$(1,3)$	$(1,5)$	$(2,4)$	$(2,6)$
$\omega^c_{(1,4)} =$	$\theta_{(1)}\timesM\theta_{(4)}$	$1$	$0$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(1,3)} =$	$\theta_{(1)}\timesM\theta_{(3)}$	$1$	$0$	$1$	$0$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\omega^c_{(1,3,4)} =$	$\theta_{(1)}\timesM\theta_{(3,4)}$	$1$	$0$	$1$	$1$	$(0,0)$	$(2,0)$	$(1,1)$	$(3,1)$	$(3,1)$	$(5,1)$	$(4,2)$	$(6,2)$

$\omega^c_{(1,2)} =$	$\theta_{(1,2)}\timesM\theta_{()}$	$1$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(2,0)$	$(2,2)$	$(2,2)$	$(2,4)$	$(4,2)$	$(4,4)$
$\omega^c_{(1,2,4)} =$	$\theta_{(1,2)}\timesM\theta_{(4)}$	$1$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(3,1)$	$(4,2)$	$(5,1)$	$(6,2)$
$\omega^c_{(1,2,3)} =$	$\theta_{(1,2)}\timesM\theta_{(3)}$	$1$	$1$	$1$	$0$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(3,1)$	$(4,2)$	$(5,1)$	$(6,2)$
$\omega^c_{(1,2,3,4)} =$	$\theta_{(1,2)}\timesM\theta_{(3,4)}$	$1$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(4,0)$	$(6,0)$	$(6,0)$	$(8,0)$
						$[0]$	$[2]$	$[2]$	$[4]$	$[4]$	$[6]$	$[6]$	$2[4]$

Table 7.3: vf Truth Table and Common Antecedence Segments of Example 40

Each of the two common antecedence sets forms a common antecedence segment, i.e. it is $\Theta_1=\Omega(\coan_1)$ and $\Theta_2=\Omega(\coan_2)$ .

Each c-segment $\omega^c\in\OmegaC(\Rcac)$ is now represented as the semantical combination of two antecedence segments, one from $\Theta_1$ , the other from $\Theta_2$ .

The common antecedence segments in $\Theta_1$ are $\theta_{()},~\theta_{(1)},~\theta_{(2)}$ and $\theta_{(1,2)}$ .

The common antecedence segments in $\Theta_2$ are $\theta_{()},~\theta_{(3)},~\theta_{(4)}$ and $\theta_{(3,4)}$ .

7.3 Redundant Segments

The examples given so far show that certain common antecedence segments are mutually redundant. We now put those common antecedence segments which verify the same amount of c-atoms (and thereby also falsify the same amount of c-atoms). In the next section we then will only take out one of the representatives of these segments.

Definition 42 (Redundant Segments)

Let $\Rcac$ be a cac-conditional. Let $\Theta_i$ be a set of common antecedence segments of $\Rcac$ .

Then a set of redundant segment $\Theta_i^{|t|}$ is the set of those common antecedence segments $\theta\in\Theta_i$ which all verify exactly $t$ ground atoms.

The set of all sets of redundant segments of $\Theta_i$ is denoted as $\Theta_i^{redu}$ .

Example 41 (Redundant Segments)

Continuing from example 40.

Within $\Theta_1$ the sets of redundant segments are

$\Theta_1^{|0|} = \{~\theta_{()}~\}$ ;
$\Theta_1^{|1|} = \{~\theta_{(1)},~\theta_{(2)}~\}$ , as both these common antecedence segments verify the same amount ( $t=1$ ) of c-atoms, i.e. both verify either $C(a,x)$ or $C(a,y)$ ;
$\Theta_1^{|2|} = \{~\theta_{(1,2)}~\}$ .

It is $\Theta_1^{redu} = \{~\Theta_1^{|0|},~\Theta_1^{|1|},~\Theta_1^{|2|}~\}$

Within $\Theta_2$ the sets of redundant segments are

$\Theta_2^{|0|} = \{~\theta_{()}~\}$ ;
$\Theta_2^{|1|} = \{~\theta_{(3)},~\theta_{(4)}~\}$ , as both these common antecedence segments verify the same amount ( $t=1$ ) of c-atoms, i.e. both verify either $C(b,x)$ or $C(b,y)$ ;
$\Theta_2^{|2|} = \{~\theta_{(3,4)}~\}$ .

It is $\Theta_2^{redu} = \{~\Theta_2^{|0|},~\Theta_2^{|1|},~\Theta_2^{|2|}~\}$

The next proposition shows that all the segments in a set of redundant segments generate the same vf-pairs. This means that they are truly mutually redundant when looking at the generated vf-pairs.

Proposition 18 (Sequence of Truth Values in a Set of Redundant Segments)

Let $\Rcac$ be a ca-conditional, with $\Theta$ being a set of common antecedence segments of $\Rcac$ , which is derived from a subset $gnd_{\Theta}$ of the c-atoms of $\Rcac$ .

Let $t\in\IN$ with $0 \leq t \leq |gnd_{\Theta}|$ and $\Theta^{|t|}$ be a set of redundant segments of $\Rcac$ . Let $\theta_1, \theta_2 \in \Theta^{|t|}$ .

Let $Y$ be the set of c-atoms of $\Rcac$ for which it holds that $Y \cup gnd_{\Theta} = \cat(\Rcac)$ and $Y \cap gnd_{\Theta} = \emptyset$ .

For any given $\omega^a \in\OmegaA(\Rcac)$ it holds for all $\omega^Y\in\Omega(Y)$ that

\vfa_{\Rcac}(\omega^Y\timesM\theta_1,~\omega^a) =\vfa_{\Rcac}(\omega^Y\timesM\theta_2,~\omega^a)

Proof

If $\omega^a \not\models a$ then all counting functions result in $0$ and the proposition holds.

We assume that $\omega^a \models a$ . We first transform the atomic counting functions and start with $\vera_{\Rcac}(\omega^Y\timesM\theta_1,~\omega^a)$ .

	$\big\|\big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\omega^Y \timesM \theta_1 \models c,~\omega^a \models a~\big\}\big\|$	$(1)$
$=$	$\big\|\big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\omega^Y \timesM \theta_1 \models c~\big\}\big\|$	$(2)$
$=$	$\big\|\big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\omega^Y \models c \vee \theta_1 \models c~\big\}\big\|$	$(3)$
$=$	$\big\|\big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\omega^Y \models c~\big\} \cup \big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\theta_1 \models c~\big\}\big\|$	$(4)$
$=$	$\big\|\big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\omega^Y \models c~\big\}\big\| + \big\|\big\{~\big<(c\|a)\big>~\in\gnd(\Rcac)~\big\|~\theta_1 \models c~\big\}\big\|$	$(5)$

Step (2) holds as we assume $\omega^a \models a$ . Step (5) holds as $Y \cap gnd_{\Theta} = \emptyset$ .

The same transformation can be done for $\vera_{\Rcac}(\omega^Y\timesM\theta_2,~\omega^a)$ , which results in $\big|\big\{~\big<(c|a)\big>~\in\gnd(\Rcac)~\big|~\omega^Y \models c~\big\}\big| + \big|\big\{~\big<(c|a)\big>~\in\gnd(\Rcac)~\big|~\theta_2 \models c~\big\}\big|$ .

We now show that $\vera_{\Rcac}(\omega^Y\timesM\theta_1,~\omega^a) = \vera_{\Rcac}(\omega^Y\timesM\theta_2,~\omega^a)$ . As $\big|\big\{~\big<(c|a)\big>~\in\gnd(\Rcac)~\big|~\omega^Y \models c~\big\}\big|$ appears in both sums it can be left out, which leaves us with:

\big|\big\{~\big<(c|a)\big>~\in\gnd(\Rcac)~\big|~\theta_1 \models c~\big\}\big| = \big|\big\{~\big<(c|a)\big>~\in\gnd(\Rcac)~\big|~\theta_2 \models c~\big\}\big|

which holds, as both $\theta_1$ and $\theta_2$ verify the same amount of $c$ atoms.

The same can be shown in a similar way for $\fala_{\Rcac}(\omega^Y\timesM\theta_1,~\omega^a)$ and $\fala_{\Rcac}(\omega^Y\timesM\theta_2,~\omega^a)$ .□

The main implication of proposition 18 is that if two lines within a set of common antecedence segments have the same amount of c-atoms verified (and falsified respectively), they contribute same amount of verified and falsified values to any vf-pair of $\Rcac$ . In other words: within a set of redundant segments the sequence of truth values is irrelevant with respect to the generated conditional contribution.

Example 42 (Sequence of Truth Values in a Set of Redundant Segments)

Continuing from example 38 and example 40.

The vf truth table shows different colors for the different sets of redundant segments. Within such a segment the sequence of the truth values is irrelevant, as they create the same vf-pairs.

We see that it holds of all $\theta_1\in\Theta_1$ and all $\omega^a\in\OmegaA(\Rcac)$ that

$\vfa_{\Rcac}(\theta_1\timesM\theta_{(4)},\omega^a) = \vfa_{\Rcac}(\theta_1\timesM\theta_{(3)},\omega^a)$

In the same way it holds for all $\theta_2\in\Theta_2$ and all $\omega^a\in\OmegaA(\Rcac)$ that

$\vfa_{\Rcac}(\theta_{(2)}\timesM\theta_2,~\omega^a) = \vfa_{\Rcac}(\theta_{(1)}\timesM\theta_2,~\omega^a)$ and

$\vfa_{\Rcac}(\theta_{(2,4)}\timesM\theta_2,~\omega^a) = \vfa_{\Rcac}(\theta_{(2,3)}\timesM\theta_2,~\omega^a)$

7.4 Set of Representatives

The representatives of the redundant segments will enable us to reduce the number of possible world lines of the vf-truth table of a ca-conditional and with that will allow for a much faster generation of the related CS. In order to get there we will first replace all the sets of redundant segments within a common antecedence segment with their respective representatives. This will result in a set of representatives which then will replace the related set of common antecedence segments in the c-segment.

Definition 43 (Set of Representative)

Let $\Rcac$ be a cac-conditional. Let ${\Theta_i^{redu} = \{~\Theta^{|1|}_i,\cdots,\Theta^{|n|}_i~\}}$ be the set of sets of redundant segments of $\Rcac$ . Let ${\Theta^{|t|}_i\in\Theta_i^{redu}}$ with ${1\leq t\leq n}$ .

A representative of $\Theta^{|t|}_i$ is a randomly picked element of $\Theta^{|t|}_i$ and is denoted by $\theta^{|t|}_i$ . Although it is of no relevance, which of the redundant segments takes the role of the representative, usually the redundant segment with the lowest binary value is chosen.

The set of representatives of $\Theta_i^{redu}$ is the set $\theta^{|t|}_i$ and is denoted by $\Theta_i^R$ .

The following example shows in detail how the above definition can be applied.

Example 43 (Set of Representative)

Continuing from example 42.

We repeat the most important findings of some of the previous examples.

We defined $s_1 = \{~a,~b,~c~\}$ and $s_2 = \{~x,~y~\}$ as sorts, $V_1$ and $W$ as variables over $s_1$ , $V_2$ as a variable over $s_2$ , $C$ and $A$ as predicates and $\Rca=\big<~\big(~C(V_1, V_2)~|~A(W)~\big),~V_1\neq W, V_1\neq c~\big>$ as a ca-conditional.

The common antecedence segments are

\Theta_1 = \{~\theta_{()},~\theta_{(1)},~\theta_{(2)},~\theta_{(1,2)},~\},

\Theta_2 = \{~\theta_{()},~\theta_{(3)},~\theta_{(4)},~\theta_{(3,4)},~\}.

The sets of redundant segments are

$\Theta_1^{\|0\|} = \{~\theta_{()}~\},$		$\Theta_1^{\|1\|} = \{~\theta_{(1)},~\theta_{(2)}~\},$		$\Theta_1^{\|2\|} = \{~\theta_{(1,2)}~\},$
$\Theta_2^{\|0\|} = \{~\theta_{()}~\},$		$\Theta_2^{\|1\|} = \{~\theta_{(3)},~\theta_{(4)}~\},$		$\Theta_2^{\|2\|} = \{~\theta_{(3,4)}~\}.$

With that we can form the sets of sets of redundant segments

$\Theta_1^{redu} = \{~\Theta_1^{\|0\|},~\Theta_1^{\|1\|},~\Theta_1^{\|2\|},~\},$
$\Theta_2^{redu} = \{~\Theta_2^{\|0\|},~\Theta_2^{\|1\|},~\Theta_2^{\|2\|},~\}.$

We now choose for each of the sets of redundant segments that representative, which holds the lowest binary number (i.e. the lowest binary interpretation of the numbered ground atoms), so that we get:

$\theta_1^{\|0\|} = \theta_{()} \in \Theta_1^{\|0\|},$
$\theta_1^{\|1\|} = \theta_{(1)} \in \Theta_1^{\|1\|},$
$\theta_1^{\|2\|} = \theta_{(1,2)} \in \Theta_1^{\|2\|},$
$\theta_2^{\|0\|} = \theta_{()} \in \Theta_2^{\|0\|},$
$\theta_2^{\|1\|} = \theta_{(3)} \in \Theta_2^{\|1\|},$
$\theta_2^{\|2\|} = \theta_{(3,4)} \in \Theta_2^{\|2\|},$

So we can finally form the sets of representatives, which are

\Theta_1^R = \{~\theta_1^{|0|},~\theta_1^{|1|},~\theta_1^{|2|}~\},

\Theta_2^R = \{~\theta_2^{|0|},~\theta_2^{|1|},~\theta_2^{|2|}~\}

The vf truth table in table 7.3 can now be reduced by only showing the the representatives for each common antecedence segment.

Table 7.4
	$\Theta_1^R$		$\Theta_2^R$
	$C(a,x)$	$C(a,y)$	$C(b,x)$	$C(b,y)$		$A(a)$	$A(b)$	$A(c)$	$\sum$
$\Theta^R_1\timesM\Theta^R_2$					$\omega^a_{()}$	$\omega^a_{(1)}$	$\omega^a_{(2)}$	$\omega^a_{(3)}$	$\omega^a_{(1,2)}$	$\omega^a_{(1,3)}$	$\omega^a_{(2,3)}$	$\omega^a_{(1,2,3)}$
$\theta_1^{\|0\|}\timesM\theta_2^{\|0\|}$	$0$	$0$	$0$	$0$	$(0,0)$	$(0,2)$	$(0,2)$	$(0,4)$	$(0,4)$	$(0,6)$	$(0,6)$	$(0,8)$
$\theta_1^{\|0\|}\timesM\theta_2^{\|1\|}$	$0$	$0$	$0$	$1$	$(0,0)$	$(1,1)$	$(0,2)$	$(1,3)$	$(1,3)$	$(2,4)$	$(1,5)$	$(2,6)$
$\theta_1^{\|0\|}\timesM\theta_2^{\|2\|}$	$0$	$0$	$1$	$1$	$(0,0)$	$(2,0)$	$(0,2)$	$(2,2)$	$(2,2)$	$(4,2)$	$(2,4)$	$(4,4)$

$\theta_1^{\|1\|}\timesM\theta_2^{\|0\|}$	$0$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(1,1)$	$(1,3)$	$(1,3)$	$(1,5)$	$(2,4)$	$(2,6)$
$\theta_1^{\|1\|}\timesM\theta_2^{\|1\|}$	$0$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(1,1)$	$(2,2)$	$(2,2)$	$(3,3)$	$(3,3)$	$(4,4)$
$\theta_1^{\|1\|}\timesM\theta_2^{\|2\|}$	$0$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(1,1)$	$(3,1)$	$(3,1)$	$(5,1)$	$(4,2)$	$(6,2)$

$\theta_1^{\|2\|}\timesM\theta_2^{\|0\|}$	$1$	$1$	$0$	$0$	$(0,0)$	$(0,2)$	$(2,0)$	$(2,2)$	$(2,2)$	$(2,4)$	$(4,2)$	$(4,4)$
$\theta_1^{\|2\|}\timesM\theta_2^{\|1\|}$	$1$	$1$	$0$	$1$	$(0,0)$	$(1,1)$	$(2,0)$	$(3,1)$	$(3,1)$	$(4,2)$	$(5,1)$	$(6,2)$
$\theta_1^{\|2\|}\timesM\theta_2^{\|2\|}$	$1$	$1$	$1$	$1$	$(0,0)$	$(2,0)$	$(2,0)$	$(4,0)$	$(4,0)$	$(6,0)$	$(6,0)$	$(8,0)$
					$[0]$	$[2]$	$[2]$	$[4]$	$[4]$	$[6]$	$[6]$	$2[4]$

Table 7.4: Reduced vf Truth Table of Example 43

Compared with the earlier versions of this reduced vf truth table, table 7.4 leaves out all those lines which, within a specific set of redundant segments, have the same amount of truth values verified and falsified than an already existing line. Therefore from $\Theta_1$ the combination $C(a,x) = 1 \wedge C(a,y) = 0$ has been dropped, as it is already covered by $C(a,x) = 0 \wedge C(a,y) = 1$ . For the same reason in $\Theta_2$ the combination $C(b,x) = 1 \wedge C(b,y) = 0$ has been dropped.

When writing down such reduced vf truth tables manually, we can fill the rows with the value " $1$ " from the left side. So in $\Theta_2$ in the first line we have $0~0$ , then the next line we get $0 ~1$ and after that $1~1$ . We now have to repeat this block of three lines for all remaining possible world segments in $\Theta_1$ .

By comparing the vf truth table of example 40 with the reduced vf truth table of the above example 43 we see that the results of both tables, i.e. the conditional contribution of $\Rcac$ , are identical. This leads to the assumption that the generation of the conditional contribution in general might be done in a more efficient manner for situations when there are common antecedence sets which include at least two c-atoms. The following section will show that this assumption is true.

7.5 Reduced Set of C-Segments

We now combine the different $\Theta$ -representatives and get a reduced set of c-segments which reflect those worlds needed to generate the whole CS of $\Rcac$ .

Definition 44 (Reduced Set of C-Segments)

Let $\Rca$ be a ca-conditional. Let $\Theta^R_1, \cdots, \Theta^R_n$ be the sets of representatives of $\Rca$ .

Then the reduced set of c-segments is

\OmegaCR(\Rca) = \Theta^R_1~\timesM~\cdots~\timesM~\Theta^R_n

Example 44 (Reduced Set of C-Segments)

Continuing from example 43.

The reduced set of c-segments is already shown in table 7.4. It can be constructed based on definition 44, i.e. it is

\OmegaCR(\Rca)

=

\Theta_1^R \timesM \Theta_2^R.

In order to form $\OmegaCR(\Rca)$ we have to semantically combine every representative of $\Theta_1^R$ with every representative of $\Theta_2^R$ . The first column of table 7.4 shows the full set of these semantical combinations.

The following proposition is the main result of this chapter. It states that the conditional contribution of a ca-conditional can be generated from the reduced set of c-segments.

Proposition 19 (Reduced Set of C-Segments and Conditional Contribution)

Let $\Rca$ be a ca-conditional.

It is

\gamma(\Rca) = \bigcup_{\omega^c_r\in\OmegaCR(\Rca)}~~\bigcup_{\omega^a\in\OmegaA(\Rca)}\vfa_{\Rca}(\omega^c_r,~\omega^a)

In order to proof this proposition we will look at a ca-conditional $\Rca$ of which the c-atoms are further split into common antecedence sets¹. We then look at two c-segments $\omega^c_i$ and $\omega^c_j$ which assign the same truth values to all c-atoms, besides in the common antecedence set $coan_r$ . Nevertheless, within $coan_r$ the two c-segments verify the same amount $t$ of c-atoms (and falsify the same amount of c-atoms respectively).

This first of all means that there is a set of common antecedence segments $\Theta_r$ , which is further split into sets of redundant segments, of which $\Theta_r^{|t|}$ is one. Now within $\Theta_r^{|t|}$ we have different combinations of truth values, but always the same amount $t$ of verified c-atoms. One of these combinations of truth values within $\Theta_r^{|t|}$ is the combination assigned by $\omega^c_i$ , which we will denote as $\theta^{|t|}_r(i)$ . Another combination is the one assigned by $\omega^c_j$ which we will denote as $\theta^{|t|}_r(j)$ . Both $\theta^{|t|}_r(i)$ and $\theta^{|t|}_r(j)$ are in the same set of redundant segments $\Theta^{|t|}_r$ .

We then look at specific a-segment $\omega^a\in\OmegaA(\Rca)$ and the vf-pairs it contributes for the two c-segments $\omega^c_i$ and $\omega^c_j$ .

From the above it is clear that for $\omega^a$ the counting functions count the same values for $\omega^c_i$ and $\omega^c_j$ for all common antecedence segments besides $coan_r$ , as the truth values set in these segments of $\omega^c_i$ and $\omega^c_j$ are identical.

It is also clear that the counting functions of both $\omega^c_i$ and $\omega^c_j$ count the same values for $coan_r$ , as $\omega^c_i$ and $\omega^c_j$ both verify $t$ ground atoms of $coan_r$ .

This means that for a given $\omega^a$ the counting functions of $\omega^c_i$ and $\omega^c_j$ result in the same values and therefore it holds that $\vera_{\Rca}(\omega^c_i,~\omega^a) = \vera_{\Rca}(\omega^c_j,~\omega^a)$ . Therefore the resulting conditional contribution of $\Rca$ will be the same, regardless whether we calculate it with or without $\omega^c_j$ .

In other words, we can pick $\theta^{|t|}_r(i)$ as the representative of $\Theta^{|t|}_r$ and forget about all other elements of $\Theta^{|t|}_r$ .

As the above is generic and holds

for all a-segments $\omega^a\in\OmegaA(\Rca)$ ,
for all sets of redundant segments $\Theta^{|t|}_r$ ,
for all common antecedence sets $\coan_r$ ,
for all c-segments $\omega^c$ ,

it follows that for the calculation of the conditional contribution of $\Rca$ it is sufficient to apply the atomic counting functions only to the reduced set of c-segments.

Proof

Let $coans(\Rca) = \{~\coan_1,~\cdots~\coan_n~\}$ be the common antecedence sets of $\Rca$ .

Let $\omega^c_i,\omega^c_a\in\OmegaC(\Rca)$ be common antecedence segments of $\Rca$ , whereby $\omega^c_i$ and $\omega^c_j$ both assign the same truth values to the c-atoms in all common antecedence sets of $\Rca$ , besides $\coan_r$ , i.e. to all $coans(\Rca)\backslash \coan_r$ with $1\leq r\leq n$ . Let $\omega^c_i$ and $\omega^c_j$ verify the same number $t$ of c-atoms in $\coan_r$ . Let $ver_{rest}$ be the number of the c-atoms verified by $\omega^c_i$ as well as $\omega^c_j$ within $coans(\Rca)\backslash \coan_r$ .

Let $\Theta_r$ be the set of common antecedence segments which is derived from $\coan_r$ . Let $\Theta_r^{redu} = \{~\Theta_r^{|0|},~\cdots,~\Theta_r^{|k|}~\}$ be the set of of all sets of redundant segments of $\Omega_r$ . We denote the truth values assigned by $\omega^c_i$ to the c-atoms within $\coan_r$ as $\theta^{|t|}_r(i)$ and the truth values assigned by $\omega^c_j$ to $\coan_r$ as $\theta^{|t|}_r(j)$ . With the definition of redundant segments (cf definition 42) it holds that $\theta^{|t|}_r(i),\theta^{|t|}_r(j)\in\Theta^{|t|}_r$ .

Let $\omega^a\in\OmegaA(\Rca)$ be an a-segment of $\Rca$ . Then it holds that

\vera_{\Rca}(\omega^c_i,~\omega^a)~=~\vera_{\Rca}(\omega^c_j,~\omega^a)~=~ver_{rest} + t.

The same can be shown for the falsified c-atoms, i.e. that it holds that

\fala_{\Rca}(\omega^c_i,~\omega^a)~=~\fala_{\Rca}(\omega^c_j,~\omega^a)

and it follows that

\vfa_{\Rca}(\omega^c_i,~\omega^a)~=~\vfa_{\Rca}(\omega^c_j,~\omega^a).

We choose $\theta^{|t|}_r(i) = \theta^{|t|}_r$ as the representative of $\Theta_r^{|t|}$ , as defined in definition 43.

We now have shown that it makes no difference to the conditional contribution of $\Rca$ whether it includes the vf-pairs generated by $\omega^c_i$ or by $\omega^c_j$ or by both, as they all result in the same vf-pair. This holds for all a-segments, all sets of redundant segments as well as all common antecedence sets of $\Rca$ , i.e. it holds that any given a-segment will contribute the same vf-pair to the conditional contribution of $\Rca$ , regardless which representatives of the different redundant segments (of the different common antecedence segments) are chosen.

Therefore it holds that

\gamma(\Rca) = \bigcup_{\omega^c_r\in\OmegaCR(\Rca)}~~\bigcup_{\omega^a\in\OmegaA(\Rca)}\vfa_{\Rca}(\omega^c_r,~\omega^a)

□

The following example shows that RCS is more efficient than the common method for the generation of the conditional contribution as defined by section 2.3.

Example 45 (Reduced Set of C-Segments and Conditional Contribution)

Continuing from example 43.

We see from the last line in the reduced vf truth table in table 7.4 that $\gamma(\Rca) = \{~[0],~[2],~[4],~[6],~2[4]~\}$ which is exactly the same result as shown in the full vf truth table in table 7.2.

The reduced vf truth table in table 7.4 results in exactly the same conditional contribution as table 7.2, but only makes use of the reduced number of c-segments which are required by $\gred$ . Whilst $\gfull$ requires 16 c-segments, $\gred$ only requires 9. This is a reduction of over 40 %.

We have now a complete view of RCS - it was defined by this chapter according to the steps outlined in figure 7.1. The following chapter will further detail how RCS can be applied and will also discuss its advantages and limits.

Notes

It can be that there is only one common antecedence set, but for this informal overview we assume there are several

$r_1 = ~\big<~\big(~C(a,x)~\|~A(b)~\big)~\big>$	$~~~~$	$r_5 = ~\big<~\big(~C(b,x)~\|~A(a)~\big)~\big>$
$r_2 = ~\big<~\big(~C(a,x)~\|~A(c)~\big)~\big>$		$r_6 = ~\big<~\big(~C(b,x)~\|~A(c)~\big)~\big>$
$r_3 = ~\big<~\big(~C(a,y)~\|~A(b)~\big)~\big>$		$r_7 = ~\big<~\big(~C(b,y)~\|~A(a)~\big)~\big>$
$r_4 = ~\big<~\big(~C(a,y)~\|~A(c)~\big)~\big>$		$r_8 = ~\big<~\big(~C(b,y)~\|~A(c)~\big)~\big>$

$\Theta_1^{\|0\|} = \{~\theta_{()}~\},$		$\Theta_1^{\|1\|} = \{~\theta_{(1)},~\theta_{(2)}~\},$		$\Theta_1^{\|2\|} = \{~\theta_{(1,2)}~\},$
$\Theta_2^{\|0\|} = \{~\theta_{()}~\},$		$\Theta_2^{\|1\|} = \{~\theta_{(3)},~\theta_{(4)}~\},$		$\Theta_2^{\|2\|} = \{~\theta_{(3,4)}~\}.$

$\theta_1^{\|0\|} = \theta_{()} \in \Theta_1^{\|0\|},$
$\theta_1^{\|1\|} = \theta_{(1)} \in \Theta_1^{\|1\|},$
$\theta_1^{\|2\|} = \theta_{(1,2)} \in \Theta_1^{\|2\|},$
$\theta_2^{\|0\|} = \theta_{()} \in \Theta_2^{\|0\|},$
$\theta_2^{\|1\|} = \theta_{(3)} \in \Theta_2^{\|1\|},$
$\theta_2^{\|2\|} = \theta_{(3,4)} \in \Theta_2^{\|2\|},$