CSE 311: Foundations of Computing. Lecture 2: More Logic, Equivalence & Digital Circuits

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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 F F F Conjunction (and) p q p q T T T T F F F T F F F F Exclusive Or p q p q T T F T F T F T T F F F

Last Class: p q Implication: p implies q whenever p is true q must be true if p then q q if p p is sufficient for q p only if q q is necessary for p p q p q T T T T F F F T T F F T

Biconditional: p q p iff q p is equivalent to q p implies q and q implies p p is necessary and sufficient for q p q p q

Biconditional: p q p iff q p is equivalent to q p implies q and q implies p p is necessary and sufficient for q p q p q T T T T F F F T F F F T

Last class: Using Logical Connectives Measles: You can get measles Mumps: You can get mumps MMR: You had the MMR vaccine You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine then you can t get either. ((Measles and Mumps) if not MMR) and (if MMR then not (Measles or Mumps)) ((Measles Mumps) if MMR) (if MMR then (Measles Mumps))

Understanding the Vaccine Sentence You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine you can t get either. ((Measles Mumps) if MMR) (if MMR then (Measles Mumps)) ( MMR (Measles Mumps)) (MMR (Measles Mumps))

Understanding the Vaccine Sentence You can get measles and mumps if you didn t have the MMR vaccine, but if you had the MMR vaccine you can t get either. ((Measles Mumps) if MMR) (if MMR then (Measles Mumps)) ( MMR (Measles Mumps)) (MMR (Measles Mumps)) Define shorthand p : MMR q : Measles r : Mumps ( p q r ) (p q r )

Analyzing the Vaccine Sentence with a Truth Table p q r p q r p (q r) q r (q r) p (q r) p q r (p (q r)) T T T T T F T F T T F F F T T F T F F F T F F F

Analyzing the Vaccine Sentence with a Truth Table p q r p q r p (q r) q r (q r) p (q r) p q r (p (q r)) T T T F T T T F F F T T F F F T T F F F T F T F F T T F F F T F F F F T F T T T F T T T T T T F T T F T F T F F T F T F F F T T F F T F T F F F F T F F F T T F

Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q Consider p: x is divisible by 2 q: x is divisible by 4 p q q p q p p q

Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q Consider p: x is divisible by 2 q: x is divisible by 4 p q q p q p p q Divisible By 4 Not Divisible By 4 Divisible By 2 Not Divisible By 2

Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q Consider p: x is divisible by 2 q: x is divisible by 4 Divisible By 2 Not Divisible By 2 p q q p q p p q Divisible By 4 4,8,12,... Impossible Not Divisible By 4 2,6,10,... 1,3,5,...

Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q How do these relate to each other? p q p q q p p q p q q p T T F F T F T F

Converse, Contrapositive Implication: p q Converse: q p Contrapositive: q p Inverse: p q An implication and it s contrapositive have the same truth value! p q p q q p p q p q q p T T T T F F T T T F F T F T T F F T T F T F F T F F T T T T T T

Tautologies! Terminology: A compound proposition is a Tautology if it is always true Contradiction if it is always false Contingency if it can be either true or false p p p p (p q) p

Tautologies! Terminology: A compound proposition is a Tautology if it is always true Contradiction if it is always false Contingency if it can be either true or false p p This is a tautology. It s called the law of the excluded middle. If p is true, then p p is true. If p is false, then p p is true. p p This is a contradiction. It s always false no matter what truth value p takes on. (p q) p This is a contingency. When p=t, q=t, (T T) T is true. When p=t, q=f, (T F) T is false.

Logical Equivalence A = B means A and B are identical strings : p q = p q p q q p

Logical Equivalence A = B means A and B are identical strings : p q = p q These are equal, because they are character-for-character identical. p q q p These are NOT equal, because they are different sequences of characters. They mean the same thing though. A B means A and B have identical truth values: p q p q p q q p p q q p

Logical Equivalence A = B means A and B are identical strings : p q = p q These are equal, because they are character-for-character identical. p q q p These are NOT equal, because they are different sequences of characters. They mean the same thing though. A B means A and B have identical truth values: p q p q Two formulas that are equal also are equivalent. p q q p These two formulas have the same truth table! p q q p When p=t and q=f, p q is false, but p q is true!

A B vs. A B A B is an assertion over all possible truth values that A and B always have the same truth values. A B is a proposition that may be true or false depending on the truth values of the variables in A and B. A B and (A B) T have the same meaning.

De Morgan s Laws (p q) p q (p q) p q Negate the statement: My code compiles or there is a bug. To negate the statement, ask when is the original statement false.

De Morgan s Laws (p q) p q (p q) p q Negate the statement: My code compiles or there is a bug. To negate the statement, ask when is the original statement false. It s false when not(my code compiles) AND not(there is a bug). Translating back into English, we get: My code doesn t compile and there is not a bug.

De Morgan s Laws Example: (p q) ( p q) p q p q p q p q (p q) (p q) ( p q) T T T F F T F F

De Morgan s Laws Example: (p q) ( p q) p q p q p q p q (p q) (p q) ( p q) T T F F F T F T T F F T T F T T F T T F T F T T F F T T T F T T

De Morgan s Laws (p q) p q (p q) p q if (!(front!= null && value > front.data)) front = new ListNode(value, front); else { ListNode current = front; while (current.next!= null && current.next.data < value)) current = current.next; current.next = new ListNode(value, current.next); }

De Morgan s Laws (p q) p q (p q) p q!(front!= null && value > front.data) front == null value <= front.data You ve been using these for a while!

Law of Implication p q p q p q p q p p q p q p q T T T F F T F F

Law of Implication p q p q p q p q p p q p q p q T T T F T T T F F F F T F T T T T T F F T T T T

Some Equivalences Related to Implication p q p q p q q p p q (p q) (q p) p q p q

Properties of Logical Connectives We will always give you this list!

Digital Circuits Computing With Logic T corresponds to 1 or high voltage F corresponds to 0 or low voltage Gates Take inputs and produce outputs (functions) Several kinds of gates Correspond to propositional connectives (most of them)

And Gate AND Connective p q p q p q T T T T F F F T F F F F vs. AND Gate p q AND OUT p q OUT 1 1 1 1 0 0 0 1 0 0 0 0 p q AND OUT block looks like D of AND

Or Gate OR Connective p q p q p q T T T T F T F T T F F F vs. OR Gate p q OR OUT p q OUT 1 1 1 1 0 1 0 1 1 0 0 0 p q OR OUT arrowhead block looks like V

Not Gates NOT Connective vs. NOT Gate p p NOT OUT p p p OUT Also called inverter T F 1 0 F T 0 1 p NOT OUT

Blobs are Okay! You may write gates using blobs instead of shapes! p q AND OUT p q OR OUT p NOT OUT

Combinational Logic Circuits p NOT AND OUT q NOT AND r s OR Values get sent along wires connecting gates

Combinational Logic Circuits p NOT AND OUT q NOT AND r s OR Values get sent along wires connecting gates

Combinational Logic Circuits p AND q NOT OR OUT r AND Wires can send one value to multiple gates!

Combinational Logic Circuits p AND q NOT OR OUT r AND Wires can send one value to multiple gates! p q ( q r)

Computing Equivalence Describe an algorithm for computing if two logical expressions/circuits are equivalent. What is the run time of the algorithm? Compute the entire truth table for both of them! There are 2 n entries in the column for n variables.

Some Familiar Properties of Arithmetic x + y = y + x (Commutativity) p q q p p q q p x y + z = x y + x z (Distributivity) p q r p q (p r) p q r p q (p r) x + y + z = x + (y + z) (Associativity) p q r p q r p q r p (q r)

Understanding Connectives Reflect basic rules of reasoning and logic Allow manipulation of logical formulas Simplification Testing for equivalence Applications Query optimization Search optimization and caching Artificial Intelligence Program verification

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