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sav08:propositional_logic_informally [2008/02/19 18:46] vkuncak |
sav08:propositional_logic_informally [2010/02/22 01:09] vkuncak |
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| for all values of its parameters. Let us draw formula as a tree. We introduce a column for each tree node. We obtain the value of a tree node by looking at the value of its children and applying the truth table for the operation in the node. The root gives the truth value of the formula. | for all values of its parameters. Let us draw formula as a tree. We introduce a column for each tree node. We obtain the value of a tree node by looking at the value of its children and applying the truth table for the operation in the node. The root gives the truth value of the formula. | ||
| - | ===== Validity, Satisfiability ===== | + | ===== Validity, Satisfiability of Logical Formulas ===== |
| Formula is //valid// if it evaluates to //true// for //all// values of its variables (all columns for formula say //true//). | Formula is //valid// if it evaluates to //true// for //all// values of its variables (all columns for formula say //true//). | ||
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| **SAT** is a well-known NP-complete problem. | **SAT** is a well-known NP-complete problem. | ||
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| ===== Propositional Tautologies ===== | ===== Propositional Tautologies ===== | ||
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| \[\begin{array}{l} | \[\begin{array}{l} | ||
| - | (p \rightarrow q) \leftrightarrow (p \lor (\lnot q)) \\ | + | (p \rightarrow q) \leftrightarrow ((\lnot p) \lor q) \\ |
| - | (p \leftrightarrow q) \leftrightarrow ((p \rightarrow q) \land (r \rightarrow p)) \\ | + | (p \leftrightarrow q) \leftrightarrow ((p \rightarrow q) \land (q \rightarrow p)) \\ |
| (p \land q) \leftrightarrow (q \land p) \\ | (p \land q) \leftrightarrow (q \land p) \\ | ||
| (p \land (q \land r)) \leftrightarrow ((p \land q) \land r)) \\ | (p \land (q \land r)) \leftrightarrow ((p \land q) \land r)) \\ | ||
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| ===== More information ===== | ===== More information ===== | ||
| - | * forthcoming lectures | + | |
| * [[Calculus of Computation Textbook]], Chapter 1 (for now especially Sections 1.1, 1.2, 1.3) | * [[Calculus of Computation Textbook]], Chapter 1 (for now especially Sections 1.1, 1.2, 1.3) | ||
| * [[:Gallier Logic Book]], Sections 3.1, 3.2, 3.3 | * [[:Gallier Logic Book]], Sections 3.1, 3.2, 3.3 | ||
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