Glossary

Symbols

$\models$
the propositional entailment function
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$\timesM$
the semantical combination function which creates the semantical combination of two or more possible worlds
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$\OmegaA(\Rcac)$
the set of a-segments of cac-conditional $\Rcac$
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$\OmegaC(\Rcac)$
the set of c-segments of cac-conditional $\Rcac$
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$\OmegaCR(\Rca)$
the set of all sets of representatives of ca-conditional $\Rca$
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$\OmegaEC$
the set of of all equivalence classes (of possible worlds) of a knowledge base $\kb$
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$\Omega(\{~ga_1,\cdots,ga_n~\})$
the set of all possible worlds which can be derived from the ground atoms $ga_1,\cdots,ga_n$
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$\Omega(\kb)$
the set of all possible worlds which can be derived from the conditionals of a knowledge base $\kb$
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$\Sigma$
a many sorted first order signature
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$\Theta_i$
a set of common antecedence segments, i.e. the set of all possible world segments which are based on the c-atoms in a common antecedence set
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$\Theta_i^{redu}$
the set of all sets of redundant segements of $\Theta_i$ , where $\Theta_i$ is a set common antecedence segments
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$\Theta_i^R$
the set of representatives of a set of common antecedence segments $\Theta_i$
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$\Theta_i^{|t|}$
a set of redundant segments, i.e. the set of those common antecedence segments within the set of common antecedence segments $\Theta_i$ , which all verify the same amount of ground atoms $t$
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$\gamma(\R)$
the conditional contribution of a conditional $R$ , i.e. the set of $\vf$ -pairs generated by conditional $\R$
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$\gcalc$
a possible method to construct the conditional structure $\gamma(\kb)$ of knowledge base $\kb$ by calculating it from the numbers of different types of ground atoms of $\kb$
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$\gci$
a possible method to calculate immediately the parametrically uniform version of the conditional structure of a knowledge base out of the non-parametrically uniform version of that knowledge base
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$\gamma(\kb)$
the conditional structure of knowledge base $\kb$
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$\gred$
a possible method to construct the conditional structure $\gamma(\kb)$ of a knowledge base $\kb$ by generating it from a reduced set of possible worlds derived from $\kb$
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$\ggi$
a possible method to generate immediately the parametrically uniform version of the conditional structure of a knowledge base out of the non-parametrically uniform version of that knowledge base
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$\gammaci(\omega)$
the conditional impact of possible world $\omega$ , i.e. the set of $\vf$ -pairs generated by possible world $\omega$ over all atomic conditional $R \in \kb$
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$\ginv$
a possible method to transform a conditional structure back to a FOPCL knowledge base
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$\gamma(\kb,(x,y)_i)$
the partial conditional structure of FOPCL knowledge base $\kb$ and $\vf$ -pair $(x,y)_i$ is the subset of $\gamma(\kb)$ which includes all conditional impacts in which $(x,y)_i$ appears
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$\gfull$
the common method, i.e. the function which constructs the conditional structure $\gamma(\kb)$ from the full set of possible worlds derived from $\kb$
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$\omega$
a possible world, i.e. a specific assignment of truth values to a n-tuple of ground atoms
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$\omegaEC$
an equivalence class of possible worlds, i.e. the set of those possible worlds which all generate the same conditional impact $\gamma(\omega)$
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$\omega^a$
an a-segment, i.e. a possible world which occurs in the set of a-segments $\OmegaA(\Rcac)$ of cac-conditional $\Rcac$
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$\omega^a_{()}$
the specific a-segment in which all a-atoms are falsified
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$\omega^a_{(j_1,\cdots,j_k)}$
the specific a-segment which verifies the a-atoms with indexes $j_1,\cdots,j_k$
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$\omega^a_{\sum}$
the cc-composed a-segment, i.e. the composed a-segment of a cc-conditional for a specific c-segment
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$\omega^c$
a c-segment, i.e. a possible world which occurs in the set of c-segments $\Omega(\cat(\Rcac))$ of cac-conditional $\Rcac$
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$\omega^c_{()}$
the specific c-segment in which all c-atoms are falsified
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$\omega^c_{(i_1,\cdots,i_l)}$
the specific c-segment which verifies the c-atoms with indexes $i_1,\cdots,i_l$
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$\phi$
the conclusion of a conditional $\R$ , a first-order formula (over $\L$ )
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$\psi$
the antecedence of a condtional $\R$ a first-order formula (over $\L$ )
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$\rho$
the probability assigned to a FOPCL condtional $\R$
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$\theta$
a common antecedence segment, i.e. a possible world segment within a set of common antecedence segments
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$\theta^{|t|}_i$
a representative of the set of redundant segments $\Theta_i$ , i.e. a redundant segment which creates the same conditional impact as all other redundant segments within a set of redundant segments $\Theta_i$
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$\theta_{(i_1,\cdots,i_l)}$
a common antecedence segment which verifies the c-atoms $i_1,\cdots,i_l$
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$\xi$
the set of instantiation restrictions of a conditional $\R$ , i.e. a set of formulas of $\L$ which only make use of the equality predicate
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A

$A$
the predicate symbol in the antecedence of an atomic conditional
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$\aat(\Rcac)$
the a-atoms of cac-conditional $\Rcac$ , i.e. the ground atoms which occur in the antecedence of an admissible ground instance of $\Rcac$
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$\acountsc(\Rca,~a)$
the set of c-atoms counted by a-atom a of conditional $\Rca$
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$\aR$
set of all atomic conditionals
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$\at(r)$
the set of ground atoms occurring in a specific grounding $r$ of conditional $R$
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$\at_R(P_1,\cdots,P_n)$
the set of ground atoms of predicate symbols $P_1,~\cdots,~P_n$ , whereby the ground atoms occur within the admissible ground instances of conditional $\R$
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C

$C$
the predicate symbol in the conclusion of an atomic conditional
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$CA(\R_1,\R_2)$
the set of common appearances (i.e. the common appearance table) of conditionals $\R_1$ and $\R_2$
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$\cat(\Rcac)$
the c-atoms of cac-conditional $\Rcac$ , i.e. the ground atoms which occur in the conclusion of an admissible ground instance of $\Rcac$
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$\coan(\Rcac,~c)$
the common antecedence set, i.e. the set of c-atoms which are all counted by exactly the same a-atoms in cac-conditional $\Rcac$
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$Const$
a finite set of constants
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$\ccountedby(\Rcac_1,\Rcac_2,c)$
the set of a-atoms of cac-conditionals $\Rcac_1$ and $\Rcac_2$ of which the related canonical a-segments all count the c-atom $c$
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$\countedby(\Rcac,c)$
the set of a-atoms of which the related canonical a-segments count the c-atom $c$ in cac-conditional $\Rcac$
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$\allCS$
the set of all conditional structures
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D

$\diff(\Rcac)$
the difference of cac-conditional $\Rcac$ , i.e. the ground atoms which occur either in the set of c-atoms of $\Rcac$ or in the set of a-atoms of $\Rcac$ but not in both
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F

$\falij$
the counting function which counts the number of ground instances of conditional $R_i$ for which the related ground atoms in the conclusion are falsified by possible world $\omega_j$ and the related ground atoms in the antecedence are verified by a possible world $\omega_j$
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$\fala_{\Rcac}(\omega^c,~\omega^a)$
the atomic counting function which counts the number of ground instances of conditional $\Rcac$ of which the related c-atoms are falsified by c-segment $\omega^c$ and the related a-atoms are verified by a-segment $\omega^a$
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G

$\gnd(R)$
the set of admissible ground instances of a conditional $R$ - this set is also called groundings of $R$
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H

$\herbrand(\kb)$
the Herbrand base of $\kb$ , i.e. the set of ground atoms which are instantiated by the admissible ground instances of the conditionals forming the FOPCL knowledge base $\kb$
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I

$\isBS(\gamma(\kb))$
the function which indicates, whether the conditional structure $\gamma(\kb)$ of the FOPCL knowledge base $\kb$ is balanced with respect to the sharing of ground atoms
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$\isBS(\kb)$
the function which indicates, whether the FOPCL knowledge base $\kb$ is balanced with respect to the sharing of ground atoms
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$\isBS(\R_1,\cdots,\R_n)$
the function which indicates, whether the set of conditionals $\R_1,\cdots,\Re_n$ is balanced with respect to the sharing of ground atoms
Page 9
$\isBU(\gamma(\kb))$
the function which indicates whether the conditional structure $\gamma(\kb)$ of the FOPCL knowledge base $\kb$ is balanced with respect to the usage of ground atoms
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$\isBU(\kb)$
the function which indicates whether the FOPCL knowledge base $\kb$ is balanced with respect to the usage of ground atoms
Page 9
$\isBU(\R_1,\cdots,\R_n)$
the function which indicates whether the set of conditionals $\R_1,\cdots,\R_n$ is balanced with respect to the usage of ground atoms
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$\isPU(\kb)$
the function which indicates whether the FOPCL knowledge base $\kb$ is parametrically uniform
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K

$\kb$
a FOPCL knowledge base, i.e. a set of FOPCL conditionals. In this thesis we refer to FOPCL knowledge bases simply as knowledge bases
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$\kba$
an atomic knowledge base
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L

$\L$
a quantifier-free first-order language defined over a many-sorted first-order signature $\Sigma$
Page 5

O

$\ovl(\Rcac)$
the overlap of cac-conditional $\Rcac$ , i.e. the ground atoms which occur in both the set of c-atoms and the set of a-atoms of $\Rcac$
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P

$|p|_{gnd(\R)}$
the grounding sum of ground atom $p$ , i.e. the number of groundings of conditional $R$ in which the grounded atom $p$ occurs
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$|(p,q)|_{gnd(\R)}$
the common grounding sum of ground atoms $p$ and $q$ , i.e. the number of groundings of conditional $R$ in which the grounded atoms $p$ and $q$ both occur
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$Pred$
a set of predicates, each having a specific arity
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$\PU(\kb)$
the parametrically uniform version of knowledge base $\kb$ after all applicably transformation rules as shown in table 2.1 have been exhaustively applied to $\kb$
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R

$\R$
a FOPCL conditional. In this thesis we refer to FOPCL conditionals simply as conditionals
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$\allR$
the set of all probabilistic conditionals
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S

$Sort$
a set of sorts
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T

$tr_{CS}$
a possible method to transform a conditional structure of a non-parametrically uniform knowledge base directly into the conditional structure of the related parametrically uniform version of that knowledge base, i.e. the transformation is performed without further knowledge of the conditionals of the knowledge base
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$tr_R$
the transformation rules which transform $\kb$ into its parametrically uniform version $\mathcal{PU}(\kb)$ by manipulating the conditionals
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V

$\verij$
the counting function which counts the number of ground instances of conditional $R_i$ of which the related ground atoms are verified by a possible world $\omega_j$
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$\vera_{\Rcac}(\omega^c,~\omega^a)$
the atomic counting function which counts the number of ground instances of conditional $\Rcac$ of which the related c-atoms are verified by c-segment $\omega^c$ and the related a-atoms are verified by a-segment $\omega^a$
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$\VF$
the set of all $\vf$ -pairs
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$\vf_{i,j}$
a $\vf$ -pair, i.e. the tuple consisting of the numbers counted by the counting functions
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$\vfa_{\Rcac}(\omega^c,~\omega^a)$
an atomic $\vf$ -pair, i.e. the tuple consisting of the numbers counted by the atomic counting functions
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$\vf_{\R_i}(\omega_j)$
a $\vf$ -pair, i.e. the tuple consisting of the numbers counted by the counting functions
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