CSE 311 Lecture 02: Logic, Equivalence, and Circuits. Emina Torlak and Kevin Zatloukal
|
|
- Augustine Hood
- 5 years ago
- Views:
Transcription
1 CSE 311 Lecture 02: Logic, Equivalence, and Circuits Emina Torlak and Kevin Zatloukal 1
2 Toics Proositional logic A brief review of Lecture 01. Classifying comound roositions Converse, contraositive, and inverse of imlication. Tautology, contradiction, contingency. Logical equivalence Equivalence, laws of logic, and roerties of logical connectives. Digital circuits Gates, combinational circuits, and circuit equivalence. 2
3 Proositional logic A brief review of Lecture 01. 3
4 Syntax and semantics of roositional logic Syntax Atomic roositions are words in roositional logic. Proositional variables reresent atomic roositions. Comound roositions are sentences made with logical connectives:.,,,,, Semantics A variable is either true ( T) or false ( F). Truth tables show the meaning of comound roositions. 4
5 Connectives and truth tables q q q q q q F T F F F F F F F F F T F F T F F T T F T T T F F T F T T F T T T T T T T T T F q q F F T F T T T F F T T T q q F F T F T F T F F T T T 5
6 Imlication can be tricky but truth tables don t lie q q F F T F T T T F F T T T In an imlication : q q is called the remise or antecedent. is called the conclusion or consequence. q imlies whenever is true q must be true if then q q if is sufficient for q only if q is necessary for q 6
7 Translating English sentences to logic Garfield has black stries if he is an orange cat and likes lasagna, and he is an orange cat or does not like lasagna. q r = Garfield has black stries. = Garfield is an orange cat. = Garfield likes lasagna. Ste 1: abstract ( if ( q and r)) and ( q or (not r)) Ste 2: relace English connectives with logical connectives (( q r) ) ( q ( r)) 7
8 Understanding sentences with truth tables q r r (q ( r)) (q r) ((q r) ) ((q r) ) (q ( r)) F F F T T F T T F F T F F F T F F T F T T F T T F T T F T T F F T F F T T F T T T F T F F F T F T T F T T F T T T T T F T T T T Garfield has black stries if he is an orange cat and likes lasagna, and he is an orange cat or does not like lasagna. q r = Garfield has black stries. = Garfield is an orange cat. = Garfield likes lasagna. 8
9 Classifying comound roositions Converse, contraositive, and inverse of imlication. Tautology, contradiction, contingency. 9
10 Imlication and friends Imlication q Converse q Contraositive q Inverse q How do these relate to each other? q q q q q q F F T T F T T F T F F T T T F F 10
11 Imlication and friends Imlication q Converse q Contraositive q Inverse q How do these relate to each other? q q q q q q F F T T T T F T T F T F T F F T F T T T T T F F 10
12 Imlication and friends Imlication q Converse q Contraositive q Inverse q How do these relate to each other? q q q q q q F F T T T T T F T T F T F T T F F T F T F T T T T F F T 10
13 Imlication and friends Imlication q Converse q Contraositive q Inverse q How do these relate to each other? q q q q q q F F T T T T T T F T T F T F T F T F F T F T F T T T T T F F T T 10
14 Imlication and friends Imlication q Converse q Contraositive q Inverse q How do these relate to each other? q q q q q q F F T T T T T T F T T F T F T F T F F T F T F T T T T T F F T T An imlication and its contraositive have the same truth value! 10
15 Tautology, contradiction, and contingency A comound roosition is a Tautology if it is always true; Contradiction if it is always false; Contingency if it can be either true or false. ( q) 11
16 Tautology, contradiction, and contingency A comound roosition is a Tautology if it is always true; Contradiction if it is always false; Contingency if it can be either true or false. ( q) = q = T = T, q = F This is a contingency. It s true when and false when. 11
17 Tautology, contradiction, and contingency A comound roosition is a Tautology if it is always true; Contradiction if it is always false; Contingency if it can be either true or false. ( q) This is a contingency. It s true when and false when. = q = T This is a tautology. It s true no matter what truth value takes on. = T, q = F 11
18 Tautology, contradiction, and contingency A comound roosition is a Tautology if it is always true; Contradiction if it is always false; Contingency if it can be either true or false. ( q) This is a contingency. It s true when and false when. = q = T This is a tautology. It s true no matter what truth value This is a contradiction. It s false no matter what truth value takes on. = T, q = F takes on. 11
19 Logical equivalence Equivalence, laws of logic, and roerties of logical connectives. 12
20 Equivalence of comound roositions A and B are logically equivalent, written as A B, if they have the same truth values in all ossible cases. q q q q q q 13
21 Equivalence of comound roositions A and B are logically equivalent, written as A B, if they have the same truth values in all ossible cases. q q Two formulas that are syntactically identical are also equivalent. q q q q 13
22 Equivalence of comound roositions A and B are logically equivalent, written as A B, if they have the same truth values in all ossible cases. q q Two formulas that are syntactically identical are also equivalent. q q These two formulas are syntactically different but have the same truth table! q q 13
23 Equivalence of comound roositions A and B are logically equivalent, written as A B, if they have the same truth values in all ossible cases. q q Two formulas that are syntactically identical are also equivalent. q q These two formulas are syntactically different but have the same truth table! q q = T q = F q q When and, is false but is true! 13
24 A B versus A B A B A B is an assertion that and have the same truth tables. This is not a comound roosition (sentence) in roositional logic! It is also sometimes called a semantic judgement. A B is a roosition that may be true or false deending on the truth values of the variables that occur in and. A B 14
25 A B versus is an assertion that and have the same truth tables. This is not a comound roosition (sentence) in roositional logic! It is also sometimes called a semantic judgement. is a roosition that may be true or false deending on the truth values of the variables that occur in and. and A B A B A B A B A B (A B) T A B have the same meaning. A and B are equivalent when A B is a tautology. 14
26 Imortant equivalences: DeMorgan s laws ( q) q ( q) q How do we check that an equivalence A B holds? 15
27 Imortant equivalences: DeMorgan s laws ( q) q ( q) q How do we check that an equivalence Use truth tables to check that A B A B holds? is a tautology: q q q q ( q) ( q) ( q F F T T F T T F T F F T T T F F ) 15
28 Imortant equivalences: DeMorgan s laws ( q) q ( q) q How do we check that an equivalence Use truth tables to check that A B A B holds? is a tautology: q q q q ( q) ( q) ( q F F T T T F T T F T T F T F T T T F F T T F T T T T F F F T F T ) 15
29 Imortant equivalences: DeMorgan s laws ( q) q ( q) q How do we check that an equivalence Use truth tables to check that A B A B holds? is a tautology: q q q q ( q) ( q) ( q F F T T T F T T F T T F T F T T T F F T T F T T T T F F F T F T Fun fact: you can also to check that is a contradiction! use a theorem rover (A B) ) 15
30 Imortant equivalences: law of imlication q q q q q ( q) ( q) F F T T T T F T T T T T T F F F F T T T T F T T 16
31 Imortant equivalences: law of imlication q q q q q ( q) ( q) F F T T T T F T T T T T T F F F F T T T T F T T More equivalences related to imlication q q q ( q) (q ) q q 16
32 Imortant equivalences: roerties of connectives Identity T F Domination F F T T Idemotence Commutativity q q q q Associativity ( q) r (q r) ( q) r (q r) Distributivity (q r) ( q) ( r) (q r) ( q) ( r) Absortion ( q) ( q) Negation F T Double negation We will always give you this list! 17
33 Digital circuits Gates, combinational circuits, and circuit equivalence. 18
34 Comuting with logic Digital circuits imlement roositional logic: T F corresonds to 1 or high voltage. corresonds to 0 or low voltage. Digital gates are functions that take values 0/1 as inuts and roduce 0/1 as outut; corresond to logical connectives (many of them). 19
35 AND gate AND connective AND gate q q F F F F T F T F F T T T q q AND out q AND out Block looks like the D of an AND. 20
36 OR gate OR connective OR gate q q F F F F T T T F T T T T q q OR out q OR out Arrowhead block looks like. 21
37 NOT gate NOT connective NOT gate F T T F NOT out NOT out Also called an inverter. 22
38 Blobs are OK! You may write gates using blobs instead of shaes. q q AND OR NOT out out out 23
39 Combinational logic circuits: wiring u gates NOT AND out q NOT AND r s OR Values get sent along wires connecting gates. 24
40 Combinational logic circuits: wiring u gates NOT AND out q NOT AND r s OR Values get sent along wires connecting gates. ( q (r s)) 24
41 Combinational logic circuits: wiring u gates AND q NOT OR out r AND Wires can send one value to multile gates. 25
42 Combinational logic circuits: wiring u gates AND q NOT OR out r AND Wires can send one value to multile gates. ( q) ( q r) 25
43 Checking (circuit) equivalence Describe an algorithm for checking if two logical exressions (or circuits) are equivalent. What is the run time of the algorithm? Why do we care? 26
44 Checking (circuit) equivalence Describe an algorithm for checking if two logical exressions (or circuits) are equivalent. Comute the entire truth table for both of them! What is the run time of the algorithm? Why do we care? 26
45 Checking (circuit) equivalence Describe an algorithm for checking if two logical exressions (or circuits) are equivalent. Comute the entire truth table for both of them! What is the run time of the algorithm? There are entries in the column for variables. Why do we care? 2 n n 26
46 Checking (circuit) equivalence Describe an algorithm for checking if two logical exressions (or circuits) are equivalent. Comute the entire truth table for both of them! What is the run time of the algorithm? There are entries in the column for variables. 2 n n Why do we care? Program and hardware verification reduces to logical equivalence checking! 26
47 Summary Proositions can be tautologies, contradictions, or contingencies. Tautologies are always true. Contradictions are never true. Contingencies are sometimes true. Proositions are equivalent when they have the same truth values. Use truth tables or laws of logic to establish equivalence. Digital circuits imlement roositional logic! F/ T corresond to 0/1 (low/high voltage), resectively. Gates imlement logical connectives. Combinational circuits imlement comound roositions. 27
Consider the following propositions: p: lecture has started q: If I have a banana, then I have a pear r: I finished this problem
Pre-Lecture Problem Do it! Do it now! What are you waiting for?! Consider the following roositions: : lecture has started : If I have a banana, then I have a ear r: I finished this roblem (i) Which of
More informationCSE 311: Foundations of Computing. Lecture 2: More Logic, Equivalence & Digital Circuits
CSE 311: Foundations of Computing Lecture 2: More Logic, Equivalence & Digital Circuits Last class: Some Connectives & Truth Tables Negation (not) p p T F F T Disjunction (or) p q p q T T T T F T F T T
More informationCSE 311: Foundations of Computing. Lecture 3: Digital Circuits & Equivalence
CSE 311: Foundations of Computing Lecture 3: Digital Circuits & Equivalence Homework #1 You should have received An e-mail from [cse311a/cse311b] with information pointing you to look at Canvas to submit
More informationThe Logic of Compound Statements. CSE 2353 Discrete Computational Structures Spring 2018
CSE 2353 Discrete Comutational Structures Sring 2018 The Logic of Comound Statements (Chater 2, E) Note: some course slides adoted from ublisher-rovided material Outline 2.1 Logical Form and Logical Equivalence
More informationCSE 311: Foundations of Computing. Lecture 2: More Logic, Equivalence & Digital Circuits
CSE 311: Foundations of Computing Lecture 2: More Logic, Equivalence & Digital Circuits Last class: Some Connectives & Truth Tables Negation (not) p p T F F T Disjunction (or) p q p q T T T T F T F T T
More informationHW1 graded review form? HW2 released CSE 20 DISCRETE MATH. Fall
CSE 20 HW1 graded review form? HW2 released DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Translate sentences from English to propositional logic using appropriate
More informationPropositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
More information2. PROPOSITIONAL LOGIC
2. PROPOSITIONAL LOGIC Contents 2.1: Informal roositional logic 2.2: Syntax of roositional logic 2.3: Semantics of roositional logic 2.4: Logical equivalence 2.5: An examle 2.6: Adequate sets of connectives
More informationWhy Proofs? Proof Techniques. Theorems. Other True Things. Proper Proof Technique. How To Construct A Proof. By Chuck Cusack
Proof Techniques By Chuck Cusack Why Proofs? Writing roofs is not most student s favorite activity. To make matters worse, most students do not understand why it is imortant to rove things. Here are just
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationAgenda. Propositional Logic. Atomic propositions. References. Truth values. Examples of atomic propositions
Proositional Logic Andrew Simson Revised by David Lightfoot Agenda Atomic roositions Logical oerators Truth tables Precedence Tautologies, contradictions and contingencies Euational reasoning 1 2 References
More informationPROFIT MAXIMIZATION. π = p y Σ n i=1 w i x i (2)
PROFIT MAXIMIZATION DEFINITION OF A NEOCLASSICAL FIRM A neoclassical firm is an organization that controls the transformation of inuts (resources it owns or urchases into oututs or roducts (valued roducts
More informationTautologies, Contradictions, and Contingencies
Section 1.3 Tautologies, Contradictions, and Contingencies A tautology is a proposition which is always true. Example: p p A contradiction is a proposition which is always false. Example: p p A contingency
More informationLECTURE # 3 Laws of Logic
APPLYING LAWS O LOGIC Using law of logic, simlify the statement form [~(~ )] Solution: [~(~ )] [~(~) (~)] LECURE # 3 Laws of Logic [ (~)] [ ] (~) (~) Which is the simlified statement form. EXAMPLE Using
More information2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
More informationFoundations of Computation
The Austalian National University Semester 2, 2018 Research School of Comuter Science Assignment 1 Dirk Pattinson Foundations of Comutation Released: Tue Aug 21 2018 Due: Tue Se 4 2018 (any time) Mode:
More informationAnnouncements. CS243: Discrete Structures. Propositional Logic II. Review. Operator Precedence. Operator Precedence, cont. Operator Precedence Example
Announcements CS243: Discrete Structures Propositional Logic II Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Tuesday 09/11 Weilin s Tuesday office hours are
More informationChapter Summary. Propositional Logic. Predicate Logic. Proofs. The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.
Chapter 1 Chapter Summary Propositional Logic The Language of Propositions (1.1) Applications (1.2) Logical Equivalences (1.3) Predicate Logic The Language of Quantifiers (1.4) Logical Equivalences (1.4)
More informationSection 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.
Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency
More informationCSE 311: Foundations of Computing I. Lecture 1: Propositional Logic
CSE 311: Foundations of Computing I Lecture 1: Propositional Logic About CSE 311 Some Perspective Computer Science and Engineering Programming CSE 14x Theory Hardware CSE 311 About the Course We will study
More information[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 400 lecture note #2. 1) Basics
400 lecture note #2 [Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments ( 2.3, 3.4) 1) Basics An argument is a sequence of statements ( s1, s2,, sn). All statements in an argument, excet for
More informationAnnouncements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
More informationIt is not the case that ϕ. p = It is not the case that it is snowing = It is not. r = It is not the case that Mary will go to the party =
Introduction to Propositional Logic Propositional Logic (PL) is a logical system that is built around the two values TRUE and FALSE, called the TRUTH VALUES. true = 1; false = 0 1. Syntax of Propositional
More informationSTD. XII Sci. Triumph Maths
Useful for all Engineering Entrance Examinations held across India. STD. XII Sci. Triumh Maths Based on Maharashtra Board Syllabus Salient Features Exhaustive subtoic wise coverage of MCQs. Imortant formulae
More informationPropositional Logic: Equivalence
Propositional Logic: Equivalence Alice Gao Lecture 5 Based on work by J. Buss, L. Kari, A. Lubiw, B. Bonakdarpour, D. Maftuleac, C. Roberts, R. Trefler, and P. Van Beek 1/42 Outline Propositional Logic:
More informationPropositional Logic 1
Propositional Logic 1 Section Summary Propositions Connectives Negation Conjunction Disjunction Implication; contrapositive, inverse, converse Biconditional Truth Tables 2 Propositions A proposition is
More informationIntroduction to Sets and Logic (MATH 1190)
Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition
More informationChapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary! Propositional Logic! The Language of Propositions! Applications! Logical Equivalences! Predicate Logic! The Language
More informationKnowledge Representation. Propositional logic
CS 2710 Foundations of AI Lecture 10 Knowledge Representation. Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Knowledge-based agent Knowledge base Inference engine Knowledge
More informationTruth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
More informationCSE 20: Discrete Mathematics
Spring 2018 Summary Last time: Today: Logical connectives: not, and, or, implies Using Turth Tables to define logical connectives Logical equivalences, tautologies Some applications Proofs in propositional
More informationDefinition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
More informationChapter 1, Part I: Propositional Logic. With Question/Answer Animations
Chapter 1, Part I: Propositional Logic With Question/Answer Animations Chapter Summary Propositional Logic The Language of Propositions Applications Logical Equivalences Predicate Logic The Language of
More informationSample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
More informationThe Foundations: Logic and Proofs. Part I
The Foundations: Logic and Proofs Part I Chapter Summary Propositional Logic n The Language of Propositions n Applications n Logical Equivalences Predicate Logic n The Language of Quantifiers n Logical
More informationCSE 599d - Quantum Computing When Quantum Computers Fall Apart
CSE 599d - Quantum Comuting When Quantum Comuters Fall Aart Dave Bacon Deartment of Comuter Science & Engineering, University of Washington In this lecture we are going to begin discussing what haens to
More informationModel checking, verification of CTL. One must verify or expel... doubts, and convert them into the certainty of YES [Thomas Carlyle]
Chater 5 Model checking, verification of CTL One must verify or exel... doubts, and convert them into the certainty of YES or NO. [Thomas Carlyle] 5. The verification setting Page 66 We introduce linear
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 12
8.3: Algebraic Combinatorics Lionel Levine Lecture date: March 7, Lecture Notes by: Lou Odette This lecture: A continuation of the last lecture: comutation of µ Πn, the Möbius function over the incidence
More informationMath.3336: Discrete Mathematics. Propositional Equivalences
Math.3336: Discrete Mathematics Propositional Equivalences Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationTopic: Lower Bounds on Randomized Algorithms Date: September 22, 2004 Scribe: Srinath Sridhar
15-859(M): Randomized Algorithms Lecturer: Anuam Guta Toic: Lower Bounds on Randomized Algorithms Date: Setember 22, 2004 Scribe: Srinath Sridhar 4.1 Introduction In this lecture, we will first consider
More informationPropositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
More informationPropositional logic ( ): Review from Mat 1348
CSI 2101 / Winter 2008: Discrete Structures. Propositional logic ( 1.1-1.2): Review from Mat 1348 Dr. Nejib Zaguia - Winter 2008 1 Propositional logic: Review Mathematical Logic is a tool for working with
More informationSets of Real Numbers
Chater 4 Sets of Real Numbers 4. The Integers Z and their Proerties In our revious discussions about sets and functions the set of integers Z served as a key examle. Its ubiquitousness comes from the fact
More informationComputer arithmetic. Intensive Computation. Annalisa Massini 2017/2018
Comuter arithmetic Intensive Comutation Annalisa Massini 7/8 Intensive Comutation - 7/8 References Comuter Architecture - A Quantitative Aroach Hennessy Patterson Aendix J Intensive Comutation - 7/8 3
More informationLogic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016 Propositions Proposition:
More informationBoolean Logic. CS 231 Dianna Xu
Boolean Logic CS 231 Dianna Xu 1 Proposition/Statement A proposition is either true or false but not both The sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 2 Boolean variables
More informationPropositional Logic. Spring Propositional Logic Spring / 32
Propositional Logic Spring 2016 Propositional Logic Spring 2016 1 / 32 Introduction Learning Outcomes for this Presentation Learning Outcomes... At the conclusion of this session, we will Define the elements
More informationLearning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
More informationUNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
More informationReview. Propositional Logic. Propositions atomic and compound. Operators: negation, and, or, xor, implies, biconditional.
Review Propositional Logic Propositions atomic and compound Operators: negation, and, or, xor, implies, biconditional Truth tables A closer look at implies Translating from/ to English Converse, inverse,
More informationCSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Reminders Exam 1 in one week One note sheet ok Review sessions Saturday / Sunday Assigned seats: seat map on Piazza shortly
More informationPropositional Logic and Semantics
Propositional Logic and Semantics English is naturally ambiguous. For example, consider the following employee (non)recommendations and their ambiguity in the English language: I can assure you that no
More informationCS 2740 Knowledge Representation. Lecture 4. Propositional logic. CS 2740 Knowledge Representation. Administration
Lecture 4 Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square dministration Homework assignment 1 is out Due next week on Wednesday, September 17 Problems: LISP programming a PL
More informationInference in Propositional Logic
Inference in Propositional Logic Deepak Kumar November 2017 Propositional Logic A language for symbolic reasoning Proposition a statement that is either True or False. E.g. Bryn Mawr College is located
More informationTopic 7: Using identity types
Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules
More informationLecture 7. Logic. Section1: Statement Logic.
Ling 726: Mathematical Linguistics, Logic, Section : Statement Logic V. Borschev and B. Partee, October 5, 26 p. Lecture 7. Logic. Section: Statement Logic.. Statement Logic..... Goals..... Syntax of Statement
More informationProposition/Statement. Boolean Logic. Boolean variables. Logical operators: And. Logical operators: Not 9/3/13. Introduction to Logical Operators
Proposition/Statement Boolean Logic CS 231 Dianna Xu A proposition is either true or false but not both he sky is blue Lisa is a Math major x == y Not propositions: Are you Bob? x := 7 1 2 Boolean variables
More informationKnowledge Representation. Propositional logic.
CS 1571 Introduction to AI Lecture 10 Knowledge Representation. Propositional logic. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Announcements Homework assignment 3 due today Homework assignment
More informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More informationOutline. EECS150 - Digital Design Lecture 26 Error Correction Codes, Linear Feedback Shift Registers (LFSRs) Simple Error Detection Coding
Outline EECS150 - Digital Design Lecture 26 Error Correction Codes, Linear Feedback Shift Registers (LFSRs) Error detection using arity Hamming code for error detection/correction Linear Feedback Shift
More information1 Tautologies, contradictions and contingencies
DEDUCTION (I) TAUTOLOGIES, CONTRADICTIONS AND CONTINGENCIES & LOGICAL EQUIVALENCE AND LOGICAL CONSEQUENCE October 6, 2003 1 Tautologies, contradictions and contingencies Consider the truth table of the
More information10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference
Logic? Propositional Logic (Rosen, Chapter 1.1 1.3) TOPICS Propositional Logic Truth Tables Implication Logical Proofs 10/1/12 CS160 Fall Semester 2012 2 What is logic? Logic is a truth-preserving system
More informationChapter 4: Classical Propositional Semantics
Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical
More informationDistributed Rule-Based Inference in the Presence of Redundant Information
istribution Statement : roved for ublic release; distribution is unlimited. istributed Rule-ased Inference in the Presence of Redundant Information June 8, 004 William J. Farrell III Lockheed Martin dvanced
More informationA statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
More informationRecitation Week 3. Taylor Spangler. January 23, 2012
Recitation Week 3 Taylor Spangler January 23, 2012 Questions about Piazza, L A TEX or lecture? Questions on the homework? (Skipped in Recitation) Let s start by looking at section 1.1, problem 15 on page
More informationThe Logic of Compound Statements cont.
The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :
More informationIntroduction to Decision Sciences Lecture 2
Introduction to Decision Sciences Lecture 2 Andrew Nobel August 24, 2017 Compound Proposition A compound proposition is a combination of propositions using the basic operations. For example (p q) ( p)
More informationCSE507. Introduction. Computer-Aided Reasoning for Software. Emina Torlak courses.cs.washington.edu/courses/cse507/17wi/
Computer-Aided Reasoning for Software CSE507 courses.cs.washington.edu/courses/cse507/17wi/ Introduction Emina Torlak emina@cs.washington.edu Today What is this course about? Course logistics Review of
More informationChapter 1, Section 1.1 Propositional Logic
Discrete Structures Chapter 1, Section 1.1 Propositional Logic These class notes are based on material from our textbook, Discrete Mathematics and Its Applications, 6 th ed., by Kenneth H. Rosen, published
More informationEvaluating Circuit Reliability Under Probabilistic Gate-Level Fault Models
Evaluating Circuit Reliability Under Probabilistic Gate-Level Fault Models Ketan N. Patel, Igor L. Markov and John P. Hayes University of Michigan, Ann Arbor 48109-2122 {knatel,imarkov,jhayes}@eecs.umich.edu
More informationSec 3.3 The Conditional & Circuits
Sec 3.3 The Conditional & Circuits Conditional statement: connective if... then. a compound statement that uses the Conditional statements are also known as implications, and can be written as: p q (pronounced
More informationLogic and Proofs. Jan COT3100: Applications of Discrete Structures Jan 2007
COT3100: Propositional Equivalences 1 Logic and Proofs Jan 2007 COT3100: Propositional Equivalences 2 1 Translating from Natural Languages EXAMPLE. Translate the following sentence into a logical expression:
More informationCIRCUITS AND ELECTRONICS. The Digital Abstraction
6.002 CIRCUITS AND ELECTRONICS The Digital Abstraction Review Discretize matter by agreeing to observe the lumped matter discipline Lumped Circuit Abstraction Analysis tool kit: KVL/KCL, node method, superposition,
More informationConvex Analysis and Economic Theory Winter 2018
Division of the Humanities and Social Sciences Ec 181 KC Border Conve Analysis and Economic Theory Winter 2018 Toic 16: Fenchel conjugates 16.1 Conjugate functions Recall from Proosition 14.1.1 that is
More informationProof Nets and Boolean Circuits
Proof Nets and Boolean Circuits Kazushige Terui terui@nii.ac.j National Institute of Informatics, Tokyo 14/07/04, Turku.1/44 Motivation (1) Proofs-as-Programs (Curry-Howard) corresondence: Proofs = Programs
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
More informationPropositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173
Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical
More informationWhat is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
More informationBoolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.
The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or
More informationRecall that the expression x > 3 is not a proposition. Why?
Predicates and Quantifiers Predicates and Quantifiers 1 Recall that the expression x > 3 is not a proposition. Why? Notation: We will use the propositional function notation to denote the expression "
More informationLecture 3: Boolean Algebra
Lecture 3: Boolean Algebra Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University Overview
More informationRound-off Errors and Computer Arithmetic - (1.2)
Round-off Errors and Comuter Arithmetic - (.). Round-off Errors: Round-off errors is roduced when a calculator or comuter is used to erform real number calculations. That is because the arithmetic erformed
More informationLogical equivalences 12/8/2015. S T: Two statements S and T involving predicates and quantifiers are logically equivalent
1/8/015 Logical equivalences CSE03 Discrete Computational Structures Lecture 3 1 S T: Two statements S and T involving predicates and quantifiers are logically equivalent If and only if they have the same
More informationKP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8
KP/Worksheets: Propositional Logic, Boolean Algebra and Computer Hardware Page 1 of 8 Q1. What is a Proposition? Q2. What are Simple and Compound Propositions? Q3. What is a Connective? Q4. What are Sentential
More informationDESIGN AND IMPLEMENTATION OF ENCODERS AND DECODERS. To design and implement encoders and decoders using logic gates.
DESIGN AND IMPLEMENTATION OF ENCODERS AND DECODERS AIM To design and implement encoders and decoders using logic gates. COMPONENTS REQUIRED S.No Components Specification Quantity 1. Digital IC Trainer
More informationSwitches: basic element of physical implementations
Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element
More informationPOINTS ON CONICS MODULO p
POINTS ON CONICS MODULO TEAM 2: JONGMIN BAEK, ANAND DEOPURKAR, AND KATHERINE REDFIELD Abstract. We comute the number of integer oints on conics modulo, where is an odd rime. We extend our results to conics
More information3 Properties of Dedekind domains
18.785 Number theory I Fall 2016 Lecture #3 09/15/2016 3 Proerties of Dedekind domains In the revious lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following
More informationExample. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives
Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.
More informationEXERCISE 10 SOLUTIONS
CSE541 EXERCISE 10 SOLUTIONS Covers Chapters 10, 11, 12 Read and learn all examples and exercises in the chapters as well! QUESTION 1 Let GL be the Gentzen style proof system for classical logic defined
More informationGödel s incompleteness theorem
Gödel s incomleteness theorem Bengt Ringnér Centre for Mathematical Sciences, Lund University, Lund, Sweden. Homeage: htt://www.maths.lth.se/ /bengtr December 8, 2008 1 A remarkable equation At the revious
More informationC241 Homework Assignment 4
C241 Homework Assignment 4 1. Which of the following formulas are tautologies and which are contradictions? Which of the formulas are logically equivalent to each other? (a) p (q r) (b) ( p r) (q r) (c)
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More information1 Boolean Algebra Simplification
cs281: Computer Organization Lab3 Prelab Our objective in this prelab is to lay the groundwork for simplifying boolean expressions in order to minimize the complexity of the resultant digital logic circuit.
More informationPacket #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
More informationLING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, Propositional logic
LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, 2016 Propositional logic 1 Vocabulary of propositional logic Vocabulary (1) a. Propositional letters: p, q, r, s, t, p 1, q 1,..., p 2, q 2,...
More information1.1 Language and Logic
c Oksana Shatalov, Spring 2018 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
More information2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
More information