User:IssaRice/Logical induction notation: Difference between revisions

Revision as of 02:32, 3 August 2018

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

Term	Notation	Type	Definition	Notes
$F$ -combination	$A$	$S \cup {0, 1} \to F_{n}$		Function application of an $F$ -combination uses square brackets instead of parentheses. Why? As far as I can tell, this is because each coefficient is in $F$ 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 $\bar{P}$ (a $Q$ -combination)	$T (\bar{P})$	$S \cup {0, 1} \to Q$

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

[(\neg \neg ϕ)^{* 5} - ϕ^{* 5}] \cdot (ϕ - ϕ^{* 5}) + [ϕ^{* 5} - (\neg \neg ϕ)^{* 5}] \cdot (\neg \neg ϕ - (\neg \neg ϕ)^{* 5})

Since the coefficients are in $F_{5}$ , this is an $F_{5}$ -combination. Let's call this 5-strategy $T_{5}$ . We can pick out the coefficient for the $ϕ$ term like $T_{5} [ϕ] = (\neg \neg ϕ)^{* 5} - ϕ^{* 5}$ . But since each coefficient is a feature (which is a function), we can also apply each coefficient to some valuation sequence $\bar{V}$ , like this:

T_{5} (\bar{V}) = [(\neg \neg ϕ)^{* 5} (\bar{V}) - ϕ^{* 5} (\bar{V})] \cdot (ϕ - ϕ^{* 5} (\bar{V})) + [ϕ^{* 5} (\bar{V}) - (\neg \neg ϕ)^{* 5} (\bar{V})] \cdot (\neg \neg ϕ - (\neg \neg ϕ)^{* 5} (\bar{V}))

Now each coefficient is a real number, so $T_{5} (\bar{V})$ is an $R$ -combination.

@@ 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> || ||
 |}
+Example of a 5-strategy given on p. 18 of the paper:
+:<math>\left[(\neg\neg\phi)^{*5} -\phi^{*5}\right] \cdot (\phi - \phi^{*5}) + \left[\phi^{*5} - (\neg \neg \phi)^{*5}\right] \cdot \left(\neg\neg\phi - (\neg\neg\phi)^{*5}\right)</math>
+Since the coefficients are in <math>\mathcal F_5</math>, this is an <math>\mathcal 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.
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