User:IssaRice/Logical induction notation: Difference between revisions

Revision as of 02:50, 3 August 2018

This is in user space because it's not really about machine learning.

Term	Notation	Type	Notes
${\mathcal {F}}$ -combination	$A$	${\mathcal {S}}\cup \{0,1\}\to {\mathcal {F}}_{n}$	Function application of an ${\mathcal {F}}$ -combination uses square brackets instead of parentheses. Why? As far as I can tell, this is because each coefficient is in ${\mathcal {F}}$ so is itself a function. This means we have two senses of "application": we can pick out the specific coefficient we want (square brackets), or we can apply each coefficient to return something (parentheses).
Holdings from $T$ against ${\overline {\mathbb {P} }}$ (a $\mathbb {Q}$ -combination)	$T({\overline {\mathbb {P} }})$	${\mathcal {S}}\cup \{0,1\}\to \mathbb {Q}$
Trading strategy	$T$	${\mathcal {S}}\cup \{1\}\to {\mathcal {E\!F}}$

Example of a 5-strategy given on p. 18 of the paper:

\underbrace {\left[(\neg \neg \phi )^{*5}-\phi ^{*5}\right]} _{\xi _{1}}\cdot (\phi -\phi ^{*5})+\underbrace {\left[\phi ^{*5}-(\neg \neg \phi )^{*5}\right]} _{\xi _{2}}\cdot \left(\neg \neg \phi -(\neg \neg \phi )^{*5}\right)

Since the coefficients ( $\xi _{1}$ and $\xi _{2}$ ) are in ${\mathcal {F}}_{5}$ , this is an ${\mathcal {E\!F}}_{5}$ -combination. Let's call this 5-strategy $T_{5}$ . We can pick out the coefficient for the $\phi$ term like $T_{5}[\phi ]=(\neg \neg \phi )^{*5}-\phi ^{*5}$ . But since each coefficient is a feature (which is a function), we can also apply each coefficient to some valuation sequence ${\overline {\mathbb {V} }}$ , like this:

T_{5}({\overline {\mathbb {V} }})=\left[(\neg \neg \phi )^{*5}({\overline {\mathbb {V} }})-\phi ^{*5}({\overline {\mathbb {V} }})\right]\cdot (\phi -\phi ^{*5}({\overline {\mathbb {V} }}))+\left[\phi ^{*5}({\overline {\mathbb {V} }})-(\neg \neg \phi )^{*5}({\overline {\mathbb {V} }})\right]\cdot \left(\neg \neg \phi -(\neg \neg \phi )^{*5}({\overline {\mathbb {V} }})\right)

Now each coefficient is a real number, so $T_{5}({\overline {\mathbb {V} }})$ is an $\mathbb {R}$ -combination. Note that since $T_{5}\colon {\mathcal {S}}\cup \{1\}\to {\mathcal {E\!F}}_{5}$ is a function that takes a sentence or the number $1$ and ${\overline {\mathbb {V} }}$ is a valuation sequence (not a sentence or number), there appears to be a type error in writing $T_{5}({\overline {\mathbb {V} }})$ . What is going on is that we aren't evaluating $T_{5}$ at ${\overline {\mathbb {V} }}$ ; rather, we are evaluating each coefficient of $T_{5}$ , to convert the range of $T_{5}$ from ${\mathcal {F}}_{5}$ to $\mathbb {R}$ .

@@ Line 9: / Line 9: @@
 | Holdings from <math>T</math> against <math>\overline{\mathbb P}</math> (a <math>\mathbb Q</math>-combination)|| <math>T(\overline{\mathbb P})</math> || <math>\mathcal S \cup \{0,1\} \to \mathbb Q</math> || ||
 |-
-| Trading strategy || <math>T</math> || <math>\mathcal S \cup \{1\} \to \mathcal{EF}</math> || ||
+| Trading strategy || <math>T</math> || <math>\mathcal S \cup \{1\} \to \mathcal{E\!F}</math> || ||
 |}
@@ Line 16: / Line 16: @@
 :<math>\underbrace{\left[(\neg\neg\phi)^{*5} -\phi^{*5}\right]}_{\xi_1} \cdot (\phi - \phi^{*5}) + \underbrace{\left[\phi^{*5} - (\neg \neg \phi)^{*5}\right]}_{\xi_2} \cdot \left(\neg\neg\phi - (\neg\neg\phi)^{*5}\right)</math>
-Since the coefficients (<math>\xi_1</math> and <math>\xi_2</math>) are in <math>\mathcal F_5</math>, this is an <math>\mathcal{EF}_5</math>-combination. Let's call this 5-strategy <math>T_5</math>. We can pick out the coefficient for the <math>\phi</math> term like <math>T_5[\phi] = (\neg\neg\phi)^{*5} -\phi^{*5}</math>. But since each coefficient is a feature (which is a function), we can also apply each coefficient to some valuation sequence <math>\overline{\mathbb V}</math>, like this:
+Since the coefficients (<math>\xi_1</math> and <math>\xi_2</math>) are in <math>\mathcal F_5</math>, this is an <math>\mathcal{E\!F}_5</math>-combination. Let's call this 5-strategy <math>T_5</math>. We can pick out the coefficient for the <math>\phi</math> term like <math>T_5[\phi] = (\neg\neg\phi)^{*5} -\phi^{*5}</math>. But since each coefficient is a feature (which is a function), we can also apply each coefficient to some valuation sequence <math>\overline{\mathbb V}</math>, like this:
 :<math>T_5(\overline{\mathbb V}) = \left[(\neg\neg\phi)^{*5}(\overline{\mathbb V}) -\phi^{*5}(\overline{\mathbb V})\right] \cdot (\phi - \phi^{*5}(\overline{\mathbb V})) + \left[\phi^{*5}(\overline{\mathbb V}) - (\neg \neg \phi)^{*5}(\overline{\mathbb V})\right] \cdot \left(\neg\neg\phi - (\neg\neg\phi)^{*5}(\overline{\mathbb V})\right)</math>
-Now each coefficient is a real number, so <math>T_5(\overline{\mathbb V})</math> is an <math>\mathbb R</math>-combination. Note that since <math>T_5\colon \mathcal S \cup \{1\} \to \mathcal{EF}_5</math> is a function that takes a sentence or the number <math>1</math> and <math>\overline{\mathbb V}</math> is a valuation sequence (''not'' a sentence or number), there appears to be a type error in writing <math>T_5(\overline{\mathbb V})</math>. What is going on is that we aren't evaluating <math>T_5</math> at <math>\overline{\mathbb V}</math>; rather, we are evaluating ''each coefficient'' of <math>T_5</math>, to convert the range of <math>T_5</math> from <math>\mathcal F_5</math> to <math>\mathbb R</math>.
+Now each coefficient is a real number, so <math>T_5(\overline{\mathbb V})</math> is an <math>\mathbb R</math>-combination. Note that since <math>T_5\colon \mathcal S \cup \{1\} \to \mathcal{E\!F}_5</math> is a function that takes a sentence or the number <math>1</math> and <math>\overline{\mathbb V}</math> is a valuation sequence (''not'' a sentence or number), there appears to be a type error in writing <math>T_5(\overline{\mathbb V})</math>. What is going on is that we aren't evaluating <math>T_5</math> at <math>\overline{\mathbb V}</math>; rather, we are evaluating ''each coefficient'' of <math>T_5</math>, to convert the range of <math>T_5</math> from <math>\mathcal F_5</math> to <math>\mathbb R</math>.
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