Logica fuzzy - Cronologia

Mattiie: /* Decidability */

2013-01-04T15:46:39Z

‎Decidability

Mattiie: /* Last concepts (Devo ancora decidire che titolo dargli) */

2012-12-19T08:18:36Z

‎Last concepts (Devo ancora decidire che titolo dargli)

Mattiie il 15:50, 13 dic 2012

2012-12-13T15:50:48Z

Mattiie: /* Decidability */

2012-12-13T11:48:04Z

‎Decidability

Mattiie: /* Decidability */

2012-12-13T11:31:31Z

‎Decidability

Mattiie: corretto errore di scrittura

2012-12-13T10:59:53Z

corretto errore di scrittura

Mattiie: /* Decidability */

2012-12-13T10:55:32Z

‎Decidability

Mattiie: Lemma di riduzione dei mondi possibili

2012-12-13T10:48:45Z

Lemma di riduzione dei mondi possibili

Mattiie: Fn is a finite set, example on equiv 1

2012-12-13T10:37:25Z

Fn is a finite set, example on equiv 1

Mattiie: correzione errore grammaticale

2012-12-13T10:30:30Z

correzione errore grammaticale

← Versione meno recente		Versione delle 15:46, 4 gen 2013
Riga 379:		Riga 379:


−	'''Lemma''' (~~of reduction of all~~ possible worlds). Let be <math>\mu, \nu : FORM \to [0,1]</math> two assignments. If	+	'''Lemma''' (possible worlds reduction). Let be <math>\mu, \nu : FORM \to [0,1]</math> two assignments. If

	<math>\mu\equiv_n\nu</math>		<math>\mu\equiv_n\nu</math>

← Versione meno recente		Versione delle 08:18, 19 dic 2012
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	The last concepts' notes (of the first part of the course) are written directly in latex because are full of plots and Hasse diagrams.		The last concepts' notes (of the first part of the course) are written directly in latex because are full of plots and Hasse diagrams.
−	~~Soon here will be a link~~.	+	[[file:Posets Hesse.pdf]]

← Versione meno recente		Versione delle 15:50, 13 dic 2012
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	it cannot exist a <math>\mu</math> such that <math>0=\mu(X_1)=1</math>.		it cannot exist a <math>\mu</math> such that <math>0=\mu(X_1)=1</math>.
		+
		+	====Last concepts (Devo ancora decidire che titolo dargli)====
		+
		+	The last concepts' notes (of the first part of the course) are written directly in latex because are full of plots and Hasse diagrams.
		+	Soon here will be a link.

← Versione meno recente		Versione delle 11:48, 13 dic 2012
Riga 336:		Riga 336:

	Nevertheless Godel Logic is decidable. The reason is because it is possible to reduce, as we will see later, the infinity number of assignment <math>\mu_g \to [0,1]</math> to a finite number of cases.		Nevertheless Godel Logic is decidable. The reason is because it is possible to reduce, as we will see later, the infinity number of assignment <math>\mu_g \to [0,1]</math> to a finite number of cases.
		+
		+	Take a look at all the possible plots of <math>\mu(\alpha(X_1)), \alpha \in FORM_1</math> [[file:Plot.pdf]].
		+
		+	Note that, everytime we have <math>\mu(X_1)=a</math> where <math>0 < a < 1</math> and <math>\mu(\alpha) = 1</math> we have also <math>\mu(\alpha)=1</math> for every assignment <math>0 < \mu(X_1) < 1</math>. This consideration is defined in the two following theorems.

	'''Definition'''(Equivalent assignments). Let <math>\mu, \nu : FORM \to [0,1]</math> be two assignments. We can say the <math>\mu, \nu</math> are equivalent (on the first n variables) if exists a permutation <math>\sigma : \{1, ... , n\} \to \{1, ... , n\}</math> and a function <math> s : {0,1, ... , n} \to \{<,=\}</math> such that <br><br>		'''Definition'''(Equivalent assignments). Let <math>\mu, \nu : FORM \to [0,1]</math> be two assignments. We can say the <math>\mu, \nu</math> are equivalent (on the first n variables) if exists a permutation <math>\sigma : \{1, ... , n\} \to \{1, ... , n\}</math> and a function <math> s : {0,1, ... , n} \to \{<,=\}</math> such that <br><br>

← Versione meno recente		Versione delle 11:31, 13 dic 2012
Riga 396:		Riga 396:
	It was a very strong theorem because, to prove that <math>\alpha \in TAUT</math> where <math> \alpha \in FORM</math> is now possible:		It was a very strong theorem because, to prove that <math>\alpha \in TAUT</math> where <math> \alpha \in FORM</math> is now possible:
	infact we cannot compute <math>\mu(\alpha)</math> for every possible <math>\mu</math>, because the set <math>A = \{\mu : FORM \to [0,1]\}</math> it is an infinitive set '''but we can''' compute <math>\mu(\alpha), \mu \in \mathcal{F}_n</math> because it is a finite set and, thanks the lemma above, we know that it is the same thing.		infact we cannot compute <math>\mu(\alpha)</math> for every possible <math>\mu</math>, because the set <math>A = \{\mu : FORM \to [0,1]\}</math> it is an infinitive set '''but we can''' compute <math>\mu(\alpha), \mu \in \mathcal{F}_n</math> because it is a finite set and, thanks the lemma above, we know that it is the same thing.
		+
		+	Now we can write the algorithm we talked at the beginning of the paragraph:
		+
		+	Input: <math>\alpha \in FORM_n</math>
		+
		+	#Enumerate of the possible permutations <math>\sigma : \{1, ... , n\} \to \{1, ... , n\}</math>
		+	#Enumerate of the possible <math>s : \{0, ... , n\} \to \{=, <\}</math>
		+	#For every couple <math>(\sigma, s)</math> find, if exists, an assignment <math>\mu : FORM \to [0,1]</math> such that it is true <br><math>0s(0)\mu(X_{\sigma(1)})s(1)\mu(X_{\sigma(1)})...s(n-1)\mu(X_{\sigma(n)})s(n)1</math><br>
		+	#Compute <math>\mu(\alpha)</math>
		+	#if <math>\mu(\alpha) < 1</math> exit with output NO else go on cycling
		+	#Exit with output YES
		+
		+	Note that:
		+	#finding <math>\mu</math> such that it is true that <math>0s(0)\mu(X_{\sigma(1)})s(1)\mu(X_{\sigma(1)})...s(n-1)\mu(X_{\sigma(n)})s(n)1</math> is equals to finding an representative <math>\mu</math> for its own equivalence class.
		+	#We can arrive to step 6 if only, for every representative <math>\mu</math>, we have <math>\mu(\alpha) = 1</math>
		+	#<math>\|\sigma\| = n!, \|s\| = 2^{n+1}</math>
		+
		+	<math>\|\mathcal{F}_n\| \le n!2^{n+1}</math> because there are repetitions or untrue sentences. For example:
		+
		+	it cannot exist a <math>\mu</math> such that <math>0=\mu(X_1)=1</math>.

@@ Riga 328: / Riga 328: @@
 ====Decidability====
-Godel logic is decidable is exists a program such that:
+Godel logic is decidable if exists a program such that:
 Input: <math>\alpha \in FORM</math>
 Output: Yes if <math>\alpha \in THM_g</math>, No else.

@@ Riga 383: / Riga 383: @@
 <math>\mu \models_g \alpha(X_1, ... , X_n)</math> if and only if <math>\nu \models_g \alpha(X_1, ... , X_n)</math>.
-On the other hand, if <math>\mu \not\equiv_n \nu</math> then
+on the other hand, if <math>\mu \not\equiv_n \nu</math> then
 exists a formula <math>\alpha(X_1, ... , X_n) \in FORM</math> such that it true that either
@@ Riga 392: / Riga 392: @@
 <math>\mu \not\models_g \alpha(X_1, ... , X_n)</math> but <math>\nu \models_g \alpha(X_1, ... , X_n)</math>

← Versione meno recente		Versione delle 10:48, 13 dic 2012
Riga 373:		Riga 373:
	#<math>[\nu]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ \eta(X_1) = 1 \}</math>		#<math>[\nu]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ \eta(X_1) = 1 \}</math>
	#<math>[\theta]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ 0 < \eta(X_1) < 1 \}</math>		#<math>[\theta]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ 0 < \eta(X_1) < 1 \}</math>
		+
		+
		+	'''Lemma''' (of reduction of all possible worlds). Let be <math>\mu, \nu : FORM \to [0,1]</math> two assignments. If
		+
		+	<math>\mu\equiv_n\nu</math>
		+
		+	then for every formula <math>\alpha(X_1,...,X_n) \in FORM</math>
		+
		+	<math>\mu \models_g \alpha(X_1, ... , X_n)</math> if and only if <math>\nu \models_g \alpha(X_1, ... , X_n)</math>.
		+
		+	On the other hand, if <math>\mu \not\equiv_n \nu</math> then
		+
		+	exists a formula <math>\alpha(X_1, ... , X_n) \in FORM</math> such that it true that either
		+
		+	<math>\mu \models_g \alpha(X_1, ... , X_n)</math> but <math>\nu \not\models_g \alpha(X_1, ... , X_n)</math>
		+
		+	or
		+
		+	<math>\mu \not\models_g \alpha(X_1, ... , X_n)</math> but <math>\nu \models_g \alpha(X_1, ... , X_n)</math>

← Versione meno recente		Versione delle 10:37, 13 dic 2012
Riga 366:		Riga 366:
	to refer to the quotient set of the relation <math>\equiv_n</math> on the set of all possible assignment.		to refer to the quotient set of the relation <math>\equiv_n</math> on the set of all possible assignment.
	We can call it also ''family of all equivalence classes''.		We can call it also ''family of all equivalence classes''.
		+
		+	'''Proposition'''. <math>\mathcal{F}_n</math> is a finite set.
		+
		+	Example: for <math>n = 1\ \|\mathcal{F}_n\| = 3</math> infact we have
		+	#<math>[\mu]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ \eta(X_1) = 0 \}</math>
		+	#<math>[\nu]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ \eta(X_1) = 1 \}</math>
		+	#<math>[\theta]_{\equiv 1} = \{\eta : FORM \to [0,1]\ \|\ 0 < \eta(X_1) < 1 \}</math>

← Versione meno recente		Versione delle 10:30, 13 dic 2012
Riga 365:		Riga 365:

	to refer to the quotient set of the relation <math>\equiv_n</math> on the set of all possible assignment.		to refer to the quotient set of the relation <math>\equiv_n</math> on the set of all possible assignment.
−	We can call it also ''family of all ~~equivalent~~ classes''.	+	We can call it also ''family of all equivalence classes''.