From 519e848e4f0f3f0712df3ea964d9a4e963157560 Mon Sep 17 00:00:00 2001 From: phawaderamchandra Date: Tue, 26 Sep 2023 11:42:41 +0530 Subject: [PATCH] class 18 and 19 slides, summary added --- class_12_04092023/summary.txt | 10 +++++++++ class_13_05092023/summary.txt | 18 ++++++++++++++++ class_14_07092023/summary.txt | 18 ++++++++++++++++ class_15_11092023/summary.txt | 11 ++++++++++ class_16_12092023/summary.txt | 13 ++++++++++++ class_17_14092023/summary.txt | 20 ++++++++++++++++++ .../class_18_19_25_26092023.pdf | Bin 0 -> 368673 bytes class_18_and_19_25and26092023/summary.txt | 9 ++++++++ 8 files changed, 99 insertions(+) create mode 100644 class_12_04092023/summary.txt create mode 100644 class_13_05092023/summary.txt create mode 100644 class_14_07092023/summary.txt create mode 100644 class_15_11092023/summary.txt create mode 100644 class_16_12092023/summary.txt create mode 100644 class_17_14092023/summary.txt create mode 100644 class_18_and_19_25and26092023/class_18_19_25_26092023.pdf create mode 100644 class_18_and_19_25and26092023/summary.txt diff --git a/class_12_04092023/summary.txt b/class_12_04092023/summary.txt new file mode 100644 index 0000000..dd44d1a --- /dev/null +++ b/class_12_04092023/summary.txt @@ -0,0 +1,10 @@ +Class 12 : Normal Forms +-- literals, positive literals, negative literals. +-- CNF, DNF +-- existence of equivalent formulae in CNF and DNF for a given formula using truth tables +-- time complexity of SAT -- NP complete +-- quest for some fragment which has polytime complexity for SAT +-- Horn formulae : CNF with each clause having at most one literal +-- Algorithm for checking satisfiability of Horn Formulae + + diff --git a/class_13_05092023/summary.txt b/class_13_05092023/summary.txt new file mode 100644 index 0000000..22438f2 --- /dev/null +++ b/class_13_05092023/summary.txt @@ -0,0 +1,18 @@ +Class 13 : Horn Formulae + +Horn Formulae : CNF with each clause having at most one positive literal. +Algorithm for satisfiability-greedy algo +Termination proof +Time complexity-polytime +Correctness. +Soundness : If alogH says formula \alpha is satisfiable then \alpha is satisfiable. +Proof idea : algoH gives an assignment, we prove that this satisfies \alpha. + +Completeness: If the given formula \alpha is satisfiable then algoH prints "satisfiable". +Proof idea: Prove the contrapositive of above statement, using the following claim. + +Claim : if a proposition is marked true then it remains true in all +possible satisfying assignments of the given Horn Formulae + +-- Horn formulae satisfying algorithm and Prolog. + diff --git a/class_14_07092023/summary.txt b/class_14_07092023/summary.txt new file mode 100644 index 0000000..94649a0 --- /dev/null +++ b/class_14_07092023/summary.txt @@ -0,0 +1,18 @@ +Class 14 : *** Proof system for logic + +** Desirable properties of a proof system +** Proof systems has a set of axioms (AXIOMS) and rules (R). +** Set of formulas provable in the system = I(AXIOMS, R). + Alternatively, provable formulas have legal construction sequences over AXIOMS and R. + +** Hilbert's proof system for Propositional Logic: +* Axioms : +* Ax1 : (\alpha --> (\beta --> \alpha)) +* Ax2 : ( (\alpha --> (\beta --> \gamma)) --> ((\alpha --> \beta) --> (\alpha --> \gamma)) ) +* Ax3 : ( (\neg \beta --> \neg \alpha) --> (\alpha --> \beta) ) +* Rules : Modus Ponens + If we have proofs of \alpha, \(alpha --> \beta) + then we can derive \beta + +** Example : proved (\alpha --> \alpha) in Hilbert's proof system. + diff --git a/class_15_11092023/summary.txt b/class_15_11092023/summary.txt new file mode 100644 index 0000000..56450c5 --- /dev/null +++ b/class_15_11092023/summary.txt @@ -0,0 +1,11 @@ +Class 15 : Proofs with assumptions--inductive definition of provable formulas +example : deriving \beta from assumptions {\alpha, (\neg \alpha))}. + +Monotonicity +Strong monotonicity + +Deduction theorem +Example : { (\alpha => \beta), (\beta => \gamma)} \derives (\alpha => \gamma) + +Proof of one direction of deduction theorem. + diff --git a/class_16_12092023/summary.txt b/class_16_12092023/summary.txt new file mode 100644 index 0000000..38ca0ae --- /dev/null +++ b/class_16_12092023/summary.txt @@ -0,0 +1,13 @@ +Class 16 : DT, Soundness, consistency + +Completion of proof of deduction theorem +Example proof of \derives (\neg \neg \alpha) --> \alpha + +Soundness theoreom : \Sigma \derives \alpha \implies \Sigma \entails \alpha + +Consistency defn 1 : There does not exists a formula \alpha such that + \Sigma \derives \alpha and \Sigma \derives (\neg \alpha)) +Consistency defn 2 : There exists an alpha which is not derivable assuming \Sigma. + +Thm: Equivalence of two notions of consistency. + diff --git a/class_17_14092023/summary.txt b/class_17_14092023/summary.txt new file mode 100644 index 0000000..5d127cb --- /dev/null +++ b/class_17_14092023/summary.txt @@ -0,0 +1,20 @@ +Class 17 : Consistency, Satisfiability and Maximal consistency + +Consistency of \emptyset using soundness + +Theorem : \Sigma is satisfiable ==> \Sigma is consistent + +Example (1) \Sigma_1=\{p\} +Example (2) \Sigma_2=\{p_1,p_2,...\} +Example (3) \Sigma_3=\{ p_i -> p_j | for all i,j\} +Consistency of \Sigma іn the above examples using Theorem. + +Maximally Consistent Set (\Sigma): +(1) \Sigma is Consistent +(2) \Sigma \derives \alpha OR \Sigma \union \{\alpha} is inconsistent + +Example (4) \Sigma_1 is consistent but not MCS +Example (5) \Sigma_1 is consistent. Discussion of difficulty in +proving it's maximal consistency. 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implies satisfiability + -- Completeness Theorem for PL + -- compactness theorem using completeness + -- maximal satisfiability Iff maximal consistency +