NP-Complete - Lehigh CSE
... What Does this Thing “Polynomial Transformation” Means? A problem A can be transformed in polynomial time into a problem B if we can write a polynomial algorithm that transforms A into B such that when we solve B, we also solve A A ...
... What Does this Thing “Polynomial Transformation” Means? A problem A can be transformed in polynomial time into a problem B if we can write a polynomial algorithm that transforms A into B such that when we solve B, we also solve A A ...
Propositional Logic: Part I - Semantics
... where m ≤ 2n and for i = 1, . . . n and j = 1, . . . m, xji is either pi or ¬pi E.g. (¬p ∨ ¬q ∨ r) ∧ (p ∨ ¬q ∨ ¬r) is in CNF ¬(p ∧ q) ∨ r is not. Each of the series of disjunctions rules out a row of the truth table where formula is false. CNF ANDs together the ORs for the false rows. One way to obt ...
... where m ≤ 2n and for i = 1, . . . n and j = 1, . . . m, xji is either pi or ¬pi E.g. (¬p ∨ ¬q ∨ r) ∧ (p ∨ ¬q ∨ ¬r) is in CNF ¬(p ∧ q) ∨ r is not. Each of the series of disjunctions rules out a row of the truth table where formula is false. CNF ANDs together the ORs for the false rows. One way to obt ...
Chapter 9
... Typically: we first do the proof for Case 1 where = and function symbols are absent, then Case 2 where identity is present, and finally Case 3 where both identity and function symbols are present. In this last case, there is often a subsidiary induction on complexity to prove that all terms have som ...
... Typically: we first do the proof for Case 1 where = and function symbols are absent, then Case 2 where identity is present, and finally Case 3 where both identity and function symbols are present. In this last case, there is often a subsidiary induction on complexity to prove that all terms have som ...
1 Circuit Complexity
... A Boolean circuit C on n inputs x1 , . . . , xn is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x1 , . . . , xn ), one node of out-degree 0 (the output), and every node of the graph except the input nodes is labeled by AND, OR, or NOT; it has in-degree 2 (for AND and OR), o ...
... A Boolean circuit C on n inputs x1 , . . . , xn is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x1 , . . . , xn ), one node of out-degree 0 (the output), and every node of the graph except the input nodes is labeled by AND, OR, or NOT; it has in-degree 2 (for AND and OR), o ...
Math Notes - Summit School District
... different forms. Although these forms look different, they are equivalent; that is, their graphs are all the same line. Standard Form: An equation in ax + by = c form, such as −6x + 3y = −18. Slope-Intercept Form: An equation in y = mx + b form, such as y = 2x −6. You can find the slope (also known ...
... different forms. Although these forms look different, they are equivalent; that is, their graphs are all the same line. Standard Form: An equation in ax + by = c form, such as −6x + 3y = −18. Slope-Intercept Form: An equation in y = mx + b form, such as y = 2x −6. You can find the slope (also known ...
Propositional and First Order Reasoning
... • When DPLL terminates, it can emit a proof • Claim: – it can always emit a resolution proof – emitting proofs is only polynomial overhead, a natural extension of the algorithm ...
... • When DPLL terminates, it can emit a proof • Claim: – it can always emit a resolution proof – emitting proofs is only polynomial overhead, a natural extension of the algorithm ...
physics 201 - La Salle University
... Recall there may have been a carry-in, so the sum of inputs A and B has to be added to the carry-in. This can be accomplished with a second half adder. What should we do with the carry-outs of these two individual half adders in order to obtain one carry-out corresponding to adding all three digits? ...
... Recall there may have been a carry-in, so the sum of inputs A and B has to be added to the carry-in. This can be accomplished with a second half adder. What should we do with the carry-outs of these two individual half adders in order to obtain one carry-out corresponding to adding all three digits? ...
Functions - Vincesplace
... number (or x-value on a graph). The input values are called the domain. • The result is the output number (or y-value on a graph). Output values are the range. • The x2 key illustrates the idea of a function. ...
... number (or x-value on a graph). The input values are called the domain. • The result is the output number (or y-value on a graph). Output values are the range. • The x2 key illustrates the idea of a function. ...
CPS130, Lecture 1: Introduction to Algorithms
... Suppose there exists a set S = {k | p´(k) }. Then S has a least element m0 >1 since p(1) is true. By construction of S, we have p(k) is true for all k < m0, including m0-1. Therefore by the second part of the hypothesis for either (I1) or (I2), p(m0) must be true. This is a contradiction ( p(m0 ) ...
... Suppose there exists a set S = {k | p´(k) }. Then S has a least element m0 >1 since p(1) is true. By construction of S, we have p(k) is true for all k < m0, including m0-1. Therefore by the second part of the hypothesis for either (I1) or (I2), p(m0) must be true. This is a contradiction ( p(m0 ) ...
LCD_5
... In a positive logic system, a high voltage is used to represent logical true (1), and a low voltage for a logical false (0). • Negative Logic In a negative logic system, a low voltage is used to represent logical true (1), and a high voltage for a logical false (0). ...
... In a positive logic system, a high voltage is used to represent logical true (1), and a low voltage for a logical false (0). • Negative Logic In a negative logic system, a low voltage is used to represent logical true (1), and a high voltage for a logical false (0). ...
Algebra 2 Notes 2.1 Relations and Functions Definitions: Relation: A
... Algebra 2 Notes 2.1 Relations and Functions Definitions: Relation: A pairing of two sets of numbers. The first value is the input or domain. The second value is the output or range. Since the order of the pairings is important, the relation is frequently written as a set of ordered pairs, a table, a ...
... Algebra 2 Notes 2.1 Relations and Functions Definitions: Relation: A pairing of two sets of numbers. The first value is the input or domain. The second value is the output or range. Since the order of the pairings is important, the relation is frequently written as a set of ordered pairs, a table, a ...
Algebra and Representation – Sample Exam Problems
... a. Find the Fahrenheit temperature if the temperature is 35 degrees Celsius. b. Find the Celsius temperature if the temperature is 18 degrees Fahrenheit. c. Use the formula above to find the formula for converting Fahrenheit temperatures to Celsius. Show your work. Patterns, Recursive Formulas, and ...
... a. Find the Fahrenheit temperature if the temperature is 35 degrees Celsius. b. Find the Celsius temperature if the temperature is 18 degrees Fahrenheit. c. Use the formula above to find the formula for converting Fahrenheit temperatures to Celsius. Show your work. Patterns, Recursive Formulas, and ...
study guide.
... rather than writing it twice. To make a transition write Boolean formulas as trees and reuse parts that are repeating. The connectives become circuit gates. It is possible to have more than 2 inputs into an AND or OR circuit, but not a NOT circuit. It is possible to construct arithmetic circuits (e. ...
... rather than writing it twice. To make a transition write Boolean formulas as trees and reuse parts that are repeating. The connectives become circuit gates. It is possible to have more than 2 inputs into an AND or OR circuit, but not a NOT circuit. It is possible to construct arithmetic circuits (e. ...
27FCS147LECTURE3 - Department of Computer Science
... Now let’s go the other way; let’s take a Boolean expression and draw • We have therefore just demonstrated circuit equivalence – That is, both circuits produce the exact same output for each input value combination ...
... Now let’s go the other way; let’s take a Boolean expression and draw • We have therefore just demonstrated circuit equivalence – That is, both circuits produce the exact same output for each input value combination ...
x, y, x
... A formula is in Prenex Normal Form (PNF) if all quantifiers precede a quantifier-free sub-formula. Q: Is the following formula valid? ∀x∃y(L(x, y)) ↔ ∃y∀x(L(x, y)) no. Consider L(x, y) : x < y and D = N, then lhs is true and rhs is false This suggests that formulas in PNF are very sensitive to the o ...
... A formula is in Prenex Normal Form (PNF) if all quantifiers precede a quantifier-free sub-formula. Q: Is the following formula valid? ∀x∃y(L(x, y)) ↔ ∃y∀x(L(x, y)) no. Consider L(x, y) : x < y and D = N, then lhs is true and rhs is false This suggests that formulas in PNF are very sensitive to the o ...