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sav08:propositional_logic_informally [2008/02/19 18:46] vkuncak |
sav08:propositional_logic_informally [2012/02/21 15:09] vkuncak |
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| ====== Propositional Logic ====== | ====== Propositional Logic ====== | ||
| - | Propositional logic is the core part of most logical formalisms (such as first-order and higher-order logic). It also describes well digital circuits that are the basis of most modern computing devices. In this course we will | + | Propositional logic is the core part of most logical formalisms (such as first-order and higher-order logic). It also describes well digital circuits that are the basis of most modern computing devices. |
| - | * review propositional logic informally to make sure everyone is comfortable with it (today) | + | |
| - | * define propositional logic formula syntax and semantics formally | + | ===== Importance of Propositional Logic ===== |
| - | * learn how to write programs that build, transform, and prove validity of such formulas (SAT solvers) | + | |
| - | * use these techniques to build verification tools | + | * Many search problems can be encoded using propositional formulas |
| + | * Checking equivalence of logical combinatorial circuits in hardware directly reduces to SAT | ||
| + | * Can encode integer arithmetic with fixed bitwidth (e.g. operations on 16-bit integers) | ||
| + | * An important class of logics can be obtained by giving meaning to propositional variables | ||
| ===== Propositional Formulas ===== | ===== Propositional Formulas ===== | ||
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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. | ||
| + | |||
| ===== 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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| Suggest another tautology. | Suggest another tautology. | ||
| - | ===== Importance of Propositional Logic ===== | + | ===== More information ===== |
| - | * Many search problems can be encoded using propositional formulas | ||
| - | * Checking equivalence of logical combinatorial circuits in hardware directly reduces to SAT | ||
| - | * Can encode integer arithmetic with fixed bitwidth (e.g. operations on 16-bit integers) | ||
| - | * An important class of logics can be obtained by giving meaning to propositional variables | ||
| - | |||
| - | ===== 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 | ||
| + | |||