Approximate Bayesian computation: Difference between revisions

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In [[logic]], '''converse nonimplication'''<ref>Lehtonen, Eero, and Poikonen, J.H.</ref> is a [[logical connective]] which is the [[negation]] of the [[conversion (logic)|converse]] of [[material conditional|implication]].
 
==Definition==
<math>_{p\not\subset q}\!</math> which is the same as <math>_{\sim(p\subset q)}\!</math>
 
===Truth table===
The [[truth table]] of <math>_{p\not\subset q}\!</math>.<ref name="Knuth">{{harvnb|Knuth|2011|p=49}}</ref>
 
{| border="1" cellpadding="1" cellspacing="0" style="text-align:center;"
|+
! style="width:35px;background:#aaaaaa;" | p
! style="width:35px;background:#aaaaaa;" | q
! style="width:35px" | <math>_{\not\subset}\!</math>
|-
| T  || T  || F
|-
| T  || F  || F
|-
| F  || T  || T
|-
| F  || F  || F
|}
 
===Venn diagram===
The [[Venn Diagram]] of "It is not the case that B implies A" (the red area is true)
 
[[File:Venn0010.svg|150px]]
 
==Properties==
 
'''falsehood-preserving''': The interpretation under which all variables are assigned a [[truth value]] of 'false' produces a truth value of 'false' as a result of converse nonimplication
 
==Symbol==
Alternatives for <math>_{p\not\subset q}\!</math> are
*<math>_{p\tilde{\leftarrow}q}\!</math>: <math>_{\tilde{\leftarrow}}\!</math> combines [[Converse implication|Converse implication's]] left arrow(<math>_{\leftarrow}\!</math>) with [[Negation|Negation's]] tilde(<math>_{\sim}\!</math>).
*<math>_{Mpq}\!</math>:  uses prefixed capital letter. 
*<math>_{p\nleftarrow q}\!</math>: <math>_{\nleftarrow }\!</math> combines ''Converse implication's'' left arrow(<math>_{\leftarrow}\!</math>) denied by means of a stroke(<math>_{/}\!</math>).
 
==Natural language==
===Grammatical===
{{Empty section|date=February 2011}}
 
===Rhetorical===
"not A but B"
 
===Colloquial===
{{Empty section|date=February 2011}}
 
==Boolean algebra==
<div id="Definition">
Converse Nonimplication in a general [[Boolean algebra (structure)|Boolean algebra]] is defined as <math>_{q \nleftarrow p=q'p}\!</math>.
 
<div id="TwoElements">
Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators <math>_{\sim}\!</math> as complement operator, <math>_{_\vee}\!</math> as join operator and <math>_{_\wedge}\!</math> as meet operator, build the Boolean algebra of [[propositional logic]].
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;"
| <math>_{\sim x}\!</math>
| style="background-color:#DDFFDD"| <math>_{1}\!</math>
| style="background-color:#DDFFDD"| <math>_{0}\!</math>
|-
| <math>_{x}\!</math>
! <math>_{0}\!</math>
! <math>_{1}\!</math>
|}
| style="border:none;" |and
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|<math>_{y}\!</math>
|style="border:none;" |
|-
!<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
|-
!<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
|-
| style="text-align:center;" |<math>_{y_\vee x}\!</math>
!<math>_{0}\!</math>
!<math>_{1}\!</math>
|<math>_{x}\!</math>
|}
| style="border:none;" |and
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|<math>_{y}\!</math>
|style="border:none;" |
|-
!<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
|-
!<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
|-
| style="text-align:center;" |<math>_{y_\wedge x}\!</math>
!<math>_{0}\!</math>
!<math>_{1}\!</math>
|<math>_{x}\!</math>
|}
| style="border:none;" |then <math>_{y \nleftarrow x}\!</math> means
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|<math>_{y}\!</math>
|style="border:none;" |
|-
!<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
|-
!<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{0}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
|-
| style="text-align:center;" |<math>_{y \nleftarrow x}\!</math>
!<math>_{0}\!</math>
!<math>_{1}\!</math>
|<math>_{x}\!</math>
|}
|-
| style="border:none;" |''(Negation)''
| style="border:none;" |
| style="border:none;" |''(Inclusive Or)''
| style="border:none;" |
| style="border:none;" |''(And)''
| style="border:none;" |
| style="border:none;" |''(Converse Nonimplication)''
|}
</div>
<div id="DivisorsOfSix">
<sup>[4]</sup>
Example of a 4-element Boolean algebra: the 4 divisors {1,2,3,6} of 6 with 1 as zero and 6 as unity element, operators <math>_{ ^{c}}\!</math> (codivisor of 6) as complement operator, <math>_{_\vee}\!</math> (least common multiple) as join operator and <math>_{_\wedge}\!</math> (greatest common divisor) as meet operator, build a Boolean algebra.
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;"
| <math>_{x^c}\!</math>
| style="background-color:#DDFFDD"| <math>_{6}\!</math>
| style="background-color:#DDFFDD"| <math>_{3}\!</math>
| style="background-color:#DDFFDD"| <math>_{2}\!</math>
| style="background-color:#DDFFDD"| <math>_{1}\!</math>
|-
| <math>_{x}\!</math>
! <math>_{1}\!</math>
! <math>_{2}\!</math>
! <math>_{3}\!</math>
! <math>_{6}\!</math>
|}
| style="border:none;" |and
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|<math>_{y}\!</math>
|style="border:none;" |
|-
!<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
|-
!<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
|-
!<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
|-
!<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
|-
| style="text-align:center;" |<math>_{y_\vee x}\!</math>
!<math>_{1}\!</math>
!<math>_{2}\!</math>
!<math>_{3}\!</math>
!<math>_{6}\!</math>
|<math>_{x}\!</math>
|}
| style="border:none;" |and
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|<math>_{y}\!</math>
|style="border:none;" |
|-
!<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
|-
!<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
|-
!<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
|-
!<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
|-
| style="text-align:center;" |<math>_{y_\wedge x}\!</math>
!<math>_{1}\!</math>
!<math>_{2}\!</math>
!<math>_{3}\!</math>
!<math>_{6}\!</math>
|<math>_{x}\!</math>
|}
| style="border:none;" |then <math>_{y \nleftarrow x}\!</math> means
|style="border:none;" |
{| class="wikitable" style="border:none; background:transparent;text-align:center;"
|<math>_{y}\!</math>
|style="border:none;" |
|-
!<math>_{6}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
|-
!<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
|-
!<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
|-
!<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{1}\!</math>
| style="background-color:#DDFFDD"|<math>_{2}\!</math>
| style="background-color:#DDFFDD"|<math>_{3}\!</math>
| style="background-color:#DDFFDD"|<math>_{6}\!</math>
|-
| style="text-align:center;" |<math>_{y \nleftarrow x}\!</math>
!<math>_{1}\!</math>
!<math>_{2}\!</math>
!<math>_{3}\!</math>
!<math>_{6}\!</math>
|<math>_{x}\!</math>
|}
|-
| style="border:none;" |''(Codivisor 6)''
| style="border:none;" |
| style="border:none;" |''(Least Common Multiple)''
| style="border:none;" |
| style="border:none;" |''(Greatest Common Divisor)''
| style="border:none;" |
| style="border:none;" |''(x's greatest Divisor [[coprime]] with y)''
|}
</div>
 
=== Properties ===
==== Non-associative ====
<math>_{r \nleftarrow (q \nleftarrow p)=(r \nleftarrow q) \nleftarrow p}\!</math> iff <math>_{rp=0}\!</math> [[#NonAssociative|<sup>[5]</sup>]] (In a [[two-element Boolean algebra]] the latter condition is reduced to <math>_{r=0}\!</math> or <math>_{p=0}\!</math>). Hence in a nontrivial Boolean algebra Converse Nonimplication is '''nonassociative'''.
 
<math>
::\begin{align}
(r \nleftarrow q) \nleftarrow p &= r'q \nleftarrow p \qquad \qquad \qquad ~~~~ \text{(by definition)} \\
&= (r'q)'p \qquad \qquad \qquad ~~~~~~ \text{(by definition)} \\
&= (r + q')p \qquad \qquad ~~~~~~~~~ \text{(De Morgan's laws)} \\
&= (r + r'q')p \qquad \qquad ~~~~~~~ \text{(Absorption law)} \\
&= rp + r'q'p \\
&= rp + r'(q \nleftarrow p) \qquad ~~~~~~~~ \text{(by definition)} \\
&= rp + r \nleftarrow (q \nleftarrow p) \qquad ~~~~ \text{(by definition)} \\
\end{align}
</math>
 
Clearly, it is associative iff <math>_{rp=0}\!</math>.
 
==== Non-commutative ====
 
* <math>_{q \nleftarrow p=p \nleftarrow q\,}\!</math> iff <math>_{q=p\,}\!</math> [[#NonCommutative|<sup>[6]</sup>]]. Hence Converse Nonimplication is '''noncommutative'''.
 
==== Neutral and absorbing elements ====
 
* <math>_{0}\!</math> is a left [[neutral element]] (<math>_{0 \nleftarrow p=p}\!</math>) and a right [[absorbing element]] (<math>_{p \nleftarrow 0=0}\!</math>).
* <math>_{1 \nleftarrow p=0}\!</math>, <math>_{p \nleftarrow 1=p'}\!</math>, and <math>_{p \nleftarrow p=0}\!</math>.
* Implication <math>_{q \rightarrow p}\!</math> is the dual of Converse Nonimplication <math>_{q \nleftarrow p}\!</math> [[#Dual|<sup>[7]</sup>]].
</div>
 
 
 
</div>
<div id="NonCommutative">
<sup>[6]</sup>
{| class="wikitable" style="background-color:white;"
!colspan="5"| Converse Nonimplication is noncommutative
|-
! Step
! Make use of
! colspan="3"|Resulting in
|-
| <math>_{s.1 \,}\!</math>
| [[#Definition|Definition]]
|colspan="3"|<math>_{q\tilde{\leftarrow}p=q'p\,}\!</math>
|-
| <math>_{s.2 \,}\!</math>
| [[#Definition|Definition]]
|colspan="3"|<math>_{p\tilde{\leftarrow}q=p'q\,}\!</math>
|-
| <math>_{s.3 \,}\!</math>
| <math>_{s.1\ s.2 \,}\!</math>
|colspan="3"|<math>_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\ \Leftrightarrow\ q'p=qp'\,}\!</math>
|-
| <math>_{s.4 \,}\!</math>
|
|<math>_{q\,}\!</math>
| <math>_{=\,}\!</math>
| <math>_{q.1\,}\!</math>
|-
| <math>_{s.5 \,}\!</math>
| <math>_{s.4.right\,}\!</math> - expand Unit element
|
| <math>_{=\,}\!</math>
| <math>_{q.(p+p')\,}\!</math>
|-
| <math>_{s.6 \,}\!</math>
| <math>_{s.5.right\,}\!</math> - evaluate expression
|
| <math>_{=\,}\!</math>
| <math>_{qp+qp'\,}\!</math>
|-
| <math>_{s.7 \,}\!</math>
| <math>_{s.4.left=s.6.right \,}\!</math>
|colspan="3"|<math>_{q=qp+qp'\,}\!</math>
|-
| <math>_{s.8 \,}\!</math>
|
|<math>_{q'p=qp'\,}\!</math>
|<math>_{\Rightarrow\,}\!</math>
|<math>_{qp+qp'=qp+q'p\,}\!</math>
|-
| <math>_{s.9 \,}\!</math>
| <math>_{s.8 \,}\!</math> - regroup common factors
|
|<math>_{\Rightarrow\,}\!</math>
|<math>_{q.(p+p')=(q+q').p\,}\!</math>
|-
| <math>_{s.10 \,}\!</math>
| <math>_{s.9 \,}\!</math> - join of complements equals unity
|
|<math>_{\Rightarrow\,}\!</math>
|<math>_{q.1=1.p\,}\!</math>
|-
| <math>_{s.11 \,}\!</math>
| <math>_{s.10.right \,}\!</math> - evaluate expression
|
|<math>_{\Rightarrow\,}\!</math>
|<math>_{q=p\,}\!</math>
|-
| <math>_{s.12 \,}\!</math>
| <math>_{s.8\ s.11\,}\!</math>
|colspan="3"|<math>_{q'p=qp'\ \Rightarrow\ q=p\,}\!</math>
|-
| <math>_{s.13 \,}\!</math>
|
|colspan="3"|<math>_{q=p\ \Rightarrow\ q'p=qp'\,}\!</math>
|-
| <math>_{s.14 \,}\!</math>
|<math>_{s.12\ s.13 \,}\!</math>
|colspan="3"|<math>_{q=p\ \Leftrightarrow\ q'p=qp'\,}\!</math>
|-
| <math>_{s.15 \,}\!</math>
| <math>_{s.3\ s.14 \,}\!</math>
|colspan="3"|<math>_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\ \Leftrightarrow\ q=p\,}\!</math>
|-
|}
</div>
<div id="Dual">
<sup>[7]</sup>
{| class="wikitable" style="background-color:white;"
!colspan="5"| Implication is the dual of Converse Nonimplication
|-
! Step
! Make use of
! colspan="3"|Resulting in
|-
| <math>_{s.1 \,}\!</math>
| [[#Definition|Definition]]
|<math>_{dual(q\tilde{\leftarrow}p)\,}\!</math>
|<math>_{=\,}\!</math>
|<math>_{dual(q'p)\,}\!</math>
|-
| <math>_{s.2 \,}\!</math>
|<math>_{s.1.right\,}\!</math> - .'s [[Duality (mathematics)|dual]] is +
|
| <math>_{=\,}\!</math>
| <math>_{q'+p\,}\!</math>
|-
| <math>_{s.3 \,}\!</math>
| <math>_{s.2.right\,}\!</math> - [[Involution (mathematics)|Involution]] complement
|
| <math>_{=\,}\!</math>
| <math>_{(q'+p)''\,}\!</math>
|-
| <math>_{s.4 \,}\!</math>
| <math>_{s.3.right\,}\!</math> - [[De Morgan's laws]] applied once
|
| <math>_{=\,}\!</math>
| <math>_{(qp')'\,}\!</math>
|-
| <math>_{s.5 \,}\!</math>
| <math>_{s.4.right\,}\!</math> - [[Commutativity| Commutative law]]
|
| <math>_{=\,}\!</math>
| <math>_{(p'q)'\,}\!</math>
|-
| <math>_{s.6 \,}\!</math>
| <math>_{s.5.right\,}\!</math>
|
| <math>_{=\,}\!</math>
| <math>_{(p\tilde{\leftarrow}q)'\,}\!</math>
|-
| <math>_{s.7 \,}\!</math>
| <math>_{s.6.right\,}\!</math>
|
| <math>_{=\,}\!</math>
| <math>_{p\leftarrow q\,}\!</math>
|-
| <math>_{s.8 \,}\!</math>
| <math>_{s.7.right\,}\!</math>
|
| <math>_{=\,}\!</math>
| <math>_{q\rightarrow p\,}\!</math>
|-
| <math>_{s.9 \,}\!</math>
| <math>_{s.1.left=s.8.right \,}\!</math>
|colspan="3"|<math>_{dual(q\tilde{\leftarrow}p)=q\rightarrow p\,}\!</math>
|-
|}
</div>
 
==Computer science==
An example for converse nonimplication in computer science can be found when performing a [[Join (SQL)#Right outer join|right outer join]] on a set of tables from a [[database]], if records not matching the join-condition from the "left" table are being excluded.<ref>http://www.codinghorror.com/blog/2007/10/a-visual-explanation-of-sql-joins.html</ref>
 
==Notes==
{{Reflist}}
 
==References==
*{{cite book|last=Knuth|first=Donald E.|authorlink=Donald Knuth|year=2011|title=The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1|edition=1st|publisher=Addison-Wesley Professional|isbn=0-201-03804-8|ref=harv}}
 
{{Logical connectives}}
 
{{DEFAULTSORT:Converse Nonimplication}}
[[Category:Logical connectives]]

Revision as of 22:34, 16 February 2014

Golda is what's created on my birth certification even though it is not the title on my birth certificate. For years she's been living in Kentucky but her spouse wants them to move. I am truly fond of to go to karaoke but I've been using on new issues lately. My working day occupation is a travel agent.

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