Discrete Mathematics and its Applications, Lecture notes of Discrete Mathematics

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Discrete Mathematics

and its Applications

Ngày 14 tháng 9 năm 2011

Satisfiability

Definition A compound proposition is satisfiable if there is an assignment of truth values to its variables that renders the proposition true.

Satisfiability

Definition A compound proposition is satisfiable if there is an assignment of truth values to its variables that renders the proposition true.

Example Let us examine which of the following propositions is satisfiable: (^1) ( p ∨ ¬ q ) ∧ ( q ∨ ¬ r ) ∧ ( r ∨ ¬ p )

Satisfiability

Definition A compound proposition is satisfiable if there is an assignment of truth values to its variables that renders the proposition true.

Example Let us examine which of the following propositions is satisfiable: (^1) ( p ∨ ¬ q ) ∧ ( q ∨ ¬ r ) ∧ ( r ∨ ¬ p ) (^2) ( p ∨ q ∨ r ) ∧ (¬ p ∨ ¬ q ∨ ¬ r )

Satisfiability

Definition A compound proposition is satisfiable if there is an assignment of truth values to its variables that renders the proposition true.

Example Let us examine which of the following propositions is satisfiable: (^1) ( p ∨ ¬ q ) ∧ ( q ∨ ¬ r ) ∧ ( r ∨ ¬ p ) (^2) ( p ∨ q ∨ r ) ∧ (¬ p ∨ ¬ q ∨ ¬ r ) (^3) ( p ∨ ¬ q ) ∧ ( q ∨ ¬ r ) ∧ ( r ∨ ¬ p ) ∧ ( p ∨ q ∨ r ) ∧ (¬ p ∨ ¬ q ∨ ¬ r )

Comment To show that a proposition is satisfiable we need to find only one assignment of truth values that renders it true while to show that it is not satisfiable we need to check all possible truth assignments. Thus for a proposition of “only” 50 variables we need to perform 250 checks. ()^ Discrete Mathematicsand its Applications^ Ngày 14 tháng 9 năm 2011^ 1 / 1

Reductions

Reductions play an important role in mathematics. Reducing one problem to another whose solution is known, is commonly used to solve many problems even in real life.

Reductions

Reductions play an important role in mathematics. Reducing one problem to another whose solution is known, is commonly used to solve many problems even in real life.

Example (^1) You need to go to HMC. You can grab a map and start walking, or reduce the problem by riding your xe may or reduce it further by taking the train or flying, in which case you no longer need a map.

Reductions

Reductions play an important role in mathematics. Reducing one problem to another whose solution is known, is commonly used to solve many problems even in real life.

Example (^1) You need to go to HMC. You can grab a map and start walking, or reduce the problem by riding your xe may or reduce it further by taking the train or flying, in which case you no longer need a map. (^2) Here is another, a mathematical example:

Reductions

Reductions play an important role in mathematics. Reducing one problem to another whose solution is known, is commonly used to solve many problems even in real life.

Example (^1) You need to go to HMC. You can grab a map and start walking, or reduce the problem by riding your xe may or reduce it further by taking the train or flying, in which case you no longer need a map. (^2) Here is another, a mathematical example: You need to show that the integer 4854548878092130357971100435107311 is composite. You can try to factor it.

Reductions

Reductions play an important role in mathematics. Reducing one problem to another whose solution is known, is commonly used to solve many problems even in real life.

Example (^1) You need to go to HMC. You can grab a map and start walking, or reduce the problem by riding your xe may or reduce it further by taking the train or flying, in which case you no longer need a map. (^2) Here is another, a mathematical example: You need to show that the integer 4854548878092130357971100435107311 is composite. You can try to factor it. You can calculate 2 n −^1 mod n 6 = 1 which calculates much faster.

SAT and 3-SAT

Recall that for every truth table there is an equivalent conjunction of disjunctions proposition. Stephen Cook from the University of Toronto proved that every decision problem is “efficiently” reducible to deciding whether a conjunction of disjnuncions proposition (also called conjunctive normal form) is satisfiable.

SAT and 3-SAT

Recall that for every truth table there is an equivalent conjunction of disjunctions proposition. Stephen Cook from the University of Toronto proved that every decision problem is “efficiently” reducible to deciding whether a conjunction of disjnuncions proposition (also called conjunctive normal form) is satisfiable. In other words, if we knew how to “efficietly” decide whether a conjunction of disjunctions is satisfiable, we could “solve” almost every mathematical problem!

SAT and 3-SAT

Recall that for every truth table there is an equivalent conjunction of disjunctions proposition. Stephen Cook from the University of Toronto proved that every decision problem is “efficiently” reducible to deciding whether a conjunction of disjnuncions proposition (also called conjunctive normal form) is satisfiable. In other words, if we knew how to “efficietly” decide whether a conjunction of disjunctions is satisfiable, we could “solve” almost every mathematical problem! “Efficiently:” If you need to determine whether a proposition with “only” 50 boolean variables is satisfiable, you may need to perform upto 2^50 tests. But if there was a clever way to perform say only 504 tests we will consider it as efficient. Theorem SAT is reducible to 3-SAT

The classes P and NP

(^1) The class P consists of all decision problems for which we can construct the answer efficiently.