From a42072e028f651c24042189643440f45eed722e3 Mon Sep 17 00:00:00 2001 From: phawaderamchandra Date: Tue, 29 Aug 2023 14:58:24 +0530 Subject: [PATCH] logic and applications repo initiated --- class_01_01082023/plan_cs621.pdf | Bin 0 -> 873289 bytes class_02_03082023/summary.txt | 5 +++++ class_03_08082023/summary.txt | 7 +++++++ class_04_10082023/summary.txt | 7 +++++++ class_05_14082023/summary.txt | 5 +++++ class_06_17082023/summary.txt | 15 +++++++++++++++ class_07_21082023/summary.txt | 14 ++++++++++++++ class_08_22082023/summary.txt | 19 +++++++++++++++++++ class_09_24082023/summary.txt | 11 +++++++++++ class_10_29082023/summary.txt | 15 +++++++++++++++ 10 files changed, 98 insertions(+) create mode 100644 class_01_01082023/plan_cs621.pdf create mode 100644 class_02_03082023/summary.txt create mode 100644 class_03_08082023/summary.txt create mode 100644 class_04_10082023/summary.txt create mode 100644 class_05_14082023/summary.txt create mode 100644 class_06_17082023/summary.txt 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literal 0 HcmV?d00001 diff --git a/class_02_03082023/summary.txt b/class_02_03082023/summary.txt new file mode 100644 index 0000000..9ffc266 --- /dev/null +++ b/class_02_03082023/summary.txt @@ -0,0 +1,5 @@ +Inductive definition of sets +separation property : to prove that some element is not element of +this set. + + diff --git a/class_03_08082023/summary.txt b/class_03_08082023/summary.txt new file mode 100644 index 0000000..e7bc8bd --- /dev/null +++ b/class_03_08082023/summary.txt @@ -0,0 +1,7 @@ +Definition of inductive sets +Definition of wffs +Definition of T1 : last symbol from right should be a ')' or a +proposition. + +violating example : (p and q)^r + diff --git a/class_04_10082023/summary.txt b/class_04_10082023/summary.txt new file mode 100644 index 0000000..e38046c --- /dev/null +++ b/class_04_10082023/summary.txt @@ -0,0 +1,7 @@ +Property T2 : : equal number of ('s and )'s +Some example ((p)() +Satisfies T2 but not T1 + +Definition of PIS : proper initial segment +Also called proper prefix. + diff --git a/class_05_14082023/summary.txt b/class_05_14082023/summary.txt new file mode 100644 index 0000000..b41afbb --- /dev/null +++ b/class_05_14082023/summary.txt @@ -0,0 +1,5 @@ +initial proper segment contd. +It captures mismatched parenthesis +)(() +this example satisfies property T2 : equal number of ('s and )'s + diff --git a/class_06_17082023/summary.txt b/class_06_17082023/summary.txt new file mode 100644 index 0000000..78903b1 --- /dev/null +++ b/class_06_17082023/summary.txt @@ -0,0 +1,15 @@ +Semantics of propositional logic + +Valuation +Satisfiability +unÑ•atisfiable +validity +contradiction +tautology +logical implication +logical equivalence + +claim : \alpha logically implies \beta Iff (\alpha \implies \beta) is a tautology. +claim : If \alpha is a contradiction then for any \beta, \alpha logically implies \beta. + + diff --git a/class_07_21082023/summary.txt b/class_07_21082023/summary.txt new file mode 100644 index 0000000..cacf392 --- /dev/null +++ b/class_07_21082023/summary.txt @@ -0,0 +1,14 @@ +Semantic notions for a set of formulas: + +\Sigma - satisfiable, examples. + +Logical implication \Sigma \entails \alpha + +complete -- \Sigma \entails \alpha (inclusive OR) \Sigma \entails (\neg \alpha) + +\Sigma is satisfiable and complete then + \Sigma \entails \alpha (XOR) \Sigma \entails (\neg \alpha) + +Maximum Satisfiability + In addition to being satisfiable, Sigma has the following property: + \neg( \Sigma \entails \alpha) \implies \Sigma \cup \{ \alpha\} is not satisfiable. diff --git a/class_08_22082023/summary.txt b/class_08_22082023/summary.txt new file mode 100644 index 0000000..c232dcb --- /dev/null +++ b/class_08_22082023/summary.txt @@ -0,0 +1,19 @@ +Thm : \Sigma is Maximum Satisfiability + iff + \Sigma is satisfiable and complete. +Proof of above theorem. + +Claim: \Sigma \entails \alpha Iff \Sigma \cup \{ \neg \alpha\} is Not satisfiable. + + +Finite models theorem: (FMT) +A set \Sigma \entails some wff \alpha +implies +there is a finite subset of \Sigma which \entails \alpha. + +Proof of FMT using claim above. + +Claim : \Sigma is maximally satisfiable Iff + \Sigma has a unique valuation. +Left as an exercise. + diff --git a/class_09_24082023/summary.txt b/class_09_24082023/summary.txt new file mode 100644 index 0000000..084dd76 --- /dev/null +++ b/class_09_24082023/summary.txt @@ -0,0 +1,11 @@ +Finite satisfiability +Statement of Compactness theorem +Proof : +Godel Numbering +Construction of \Delta from \Sigma +Proving that \Delta is satisfiable. +Lemma : Given \Sigma is FS. Then for any formula \alpha + \Sigma \cup \{ \alpha \} is satisfiable OR + \Sigma \cup \{ \neg \alpha \} is satisfiable. +Proof of this lemma is remaining. + diff --git a/class_10_29082023/summary.txt b/class_10_29082023/summary.txt new file mode 100644 index 0000000..64f88c8 --- /dev/null +++ b/class_10_29082023/summary.txt @@ -0,0 +1,15 @@ +Applications of compactness theorem: +2-colorability of graphs + +Goal: Given a graph G=(V,E) it is 2-colorable iff every finite subset of G is 2-colorable. + +Proof outline: +Given a graph G=(V,E) construct a set \Sigma of wffs such that +G is 2-colorable +iff (step 1) +\Sigma is satisfiable. +iff +(by CT) \Sigma is finitely satisfiable +iff (step 2) +Each finite subset of G is 2-colorable. +

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