CSE 311 Lecture 02: Logic, Equivalence, and Circuits. Emina Torlak and Kevin Zatloukal
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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
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