cse 311: foundations of computing · microsoft powerpoint - lecture02-equivalences-a.pptx author:...
TRANSCRIPT
![Page 1: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/1.jpg)
CSE 311: Foundations of Computing
Lecture 2: More Logic, Equivalence & Digital Circuits
![Page 2: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/2.jpg)
If you are worried about Mathy aspects of 311
• Associated 1-credit CR/NC workshop
– CSE 390Z
– Extra collaborative practice on 311 concepts, study skills, a small
amount of assigned work
– Meets in Loew 113
– ZA Section Thursdays 3:30-4:50 pm
– If sufficient demand will add a ZB Section Thursdays 5:00-6:20
– Full participation is required for credit
– NOT for help with 311 homework
• Anyone in 311 can sign up but enrollment is limited
– Enrollment in CSE 390Z section ZA will open up later today FCFS
– If you want to register but it is full, show up anyway at 3:30
tomorrow in Loew 113.
![Page 3: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/3.jpg)
Last class: Some Connectives & Truth Tables
p ¬p
T F
F T
p q p ∧ q
T T T
T F F
F T F
F F F
p q p ∨ q
T T T
T F T
F T T
F F F
p q p ⊕ q
T T F
T F T
F T T
F F F
Negation (not) Conjunction (and)
Disjunction (or) Exclusive Or
![Page 4: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/4.jpg)
Last class: Implication
“If it’s raining, then I have my umbrella”
It’s useful to think of implications as
promises. That is “Did I lie?”
The only lie is when:
(a) It’s raining AND
(b) I don’t have my umbrella
p q p → q
T T T
T F F
F T T
F F T
It’s raining It’s not raining
I have my
umbrellaNo No
I do not have
my umbrellaYes No
![Page 5: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/5.jpg)
Last class: � → �
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
![Page 6: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/6.jpg)
Last class: Biconditional: � ↔ �
• 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
![Page 7: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/7.jpg)
Last class: Garfield Sentence with a Truth Table
� ¬ ∨ ¬ ∧ ( ∧ ) → � ( ∧ ) → � ∧ ( ∨ ¬)
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
![Page 8: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/8.jpg)
Converse, Contrapositive
Implication:
p → q
Converse:
q → p
Consider
p: x is divisible by 2
q: x is divisible by 4
Contrapositive:
¬q → ¬p
Inverse:
¬p → ¬q
p → q
q → p
¬q → ¬p
¬p → ¬q
![Page 9: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/9.jpg)
Converse, Contrapositive
Implication:
p → q
Converse:
q → p
Consider
p: x is divisible by 2
q: x is divisible by 4 Divisible By 2 Not Divisible By 2
Divisible By 4
Not Divisible By 4
Contrapositive:
¬q → ¬p
Inverse:
¬p → ¬q
p → q
q → p
¬q → ¬p
¬p → ¬q
Numbers that are…
![Page 10: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/10.jpg)
Converse, Contrapositive
Implication:
p → q
Converse:
q → p
Consider
p: x is divisible by 2
q: x is divisible by 4 Divisible By 2 Not Divisible By 2
Divisible By 4 4,8,12,... Impossible
Not Divisible By 4 2,6,10,... 1,3,5,...
Contrapositive:
¬q → ¬p
Inverse:
¬p → ¬q
p → q
q → p
¬q → ¬p
¬p → ¬q
Numbers that are…
![Page 11: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/11.jpg)
Converse, Contrapositive
Implication:
p → q
Converse:
q → p
How do these relate to each other?
Contrapositive:
¬q → ¬p
Inverse:
¬p → ¬q
p q p → q q → p ¬p ¬q ¬p → ¬q ¬q → ¬p
T T
T F
F T
F F
![Page 12: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/12.jpg)
Converse, Contrapositive
Implication:
p → q
Converse:
q → p
An implication and it’s contrapositive
have the same truth value!
Contrapositive:
¬q → ¬p
Inverse:
¬p → ¬q
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
![Page 13: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/13.jpg)
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
![Page 14: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/14.jpg)
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
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.
This is a contradiction. It’s always false no matter what truth
value p takes on.
This is a contingency. When p=T, q=T, (T → T)∧T is true.
When p=T, q=F, (T → F)∧T is false.
![Page 15: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/15.jpg)
Logical Equivalence
A = B means A and B are identical “strings”:
– p ∧ q = p ∧ q
– p ∧ q ≠ q ∧ p
![Page 16: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/16.jpg)
Logical Equivalence
A = B means A and B are identical “strings”:
– p ∧ q = p ∧ q
– p ∧ q ≠ q ∧ p
A ≡ B means A and B have identical truth values:
– p ∧ q ≡ p ∧ q
– p ∧ q ≡ q ∧ p
– p ∧ q ≢ q ∨ p
These are equal, because they are character-for-character identical.
These are NOT equal, because they are different sequences of
characters. They “mean” the same thing though.
![Page 17: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/17.jpg)
Logical Equivalence
A = B means A and B are identical “strings”:
– p ∧ q = p ∧ q
– p ∧ q ≠ q ∧ p
A ≡ B means A and B have identical truth values:
– p ∧ q ≡ p ∧ q
– p ∧ q ≡ q ∧ p
– p ∧ q ≢ q ∨ p
These are equal, because they are character-for-character identical.
These are NOT equal, because they are different sequences of
characters. They “mean” the same thing though.
Two formulas that are equal also are equivalent.
These two formulas have the same truth table!
When p=T and q=F, p ∧ q is false, but p ∨ q is true!
![Page 18: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/18.jpg)
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.
![Page 19: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/19.jpg)
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”.
![Page 20: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/20.jpg)
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.
![Page 21: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/21.jpg)
De Morgan’s Laws
p q ¬p ¬q ¬p ∨ ¬q p ∧ q ¬(p ∧ q) ¬(p ∧ q) ↔ (¬p ∨ ¬q)
T T
T F
F T
F F
Example: ¬(p ∧ q) ≡ (¬p ∨ ¬q)
![Page 22: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/22.jpg)
De Morgan’s Laws
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
Example: ¬(p ∧ q) ≡ (¬p ∨ ¬q)
![Page 23: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/23.jpg)
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);
}
![Page 24: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/24.jpg)
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!
![Page 25: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/25.jpg)
Law of Implication
p q p → q ¬p ¬p ∨ q p → q ↔ ¬p ∨ q
T T
T F
F T
F F
p → q ≡ ¬p ∨ q
![Page 26: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/26.jpg)
Law of Implication
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
p → q ≡ ¬p ∨ q
![Page 27: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/27.jpg)
Some Equivalences Related to Implication
p → q ≡ ¬p ∨ q
p → q ≡ ¬q → ¬p
p ↔ q ≡ (p→ q) ∧ (q → p)
p ↔ q ≡ ¬p ↔ ¬q
![Page 28: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/28.jpg)
Properties of Logical ConnectivesWe will always give
you this list!
![Page 29: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/29.jpg)
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 2n entries in the column for n variables.
![Page 30: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/30.jpg)
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
![Page 31: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/31.jpg)
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)
![Page 32: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/32.jpg)
And Gate
p q p ∧ q
T T T
T F F
F T F
F F F
p q OUT
1 1 1
1 0 0
0 1 0
0 0 0
AND Connective AND Gate
q
pOUTAND
“block looks like D of AND”
pOUTAND
qp ∧ q
vs.
![Page 33: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/33.jpg)
Or Gate
p q p ∨ q
T T T
T F T
F T T
F F F
p q OUT
1 1 1
1 0 1
0 1 1
0 0 0
OR Connective OR Gate
pOUTOR
qp ∨ q
vs.
p
qOR
“arrowhead block looks like V”
OUT
![Page 34: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/34.jpg)
Not Gates
¬p
NOT Gate
p ¬ p
T F
F T
p OUT
1 0
0 1
vs.NOT Connective
Also called
inverter
p OUTNOT
p OUTNOT
![Page 35: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/35.jpg)
Blobs are Okay!
p OUTNOT
pq
OUTAND
pq
OUTOR
You may write gates using blobs instead of shapes!
![Page 36: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/36.jpg)
Combinational Logic Circuits
Values get sent along wires connecting gates
NOT
OR
AND
AND
NOT
p
q
rs
OUT
![Page 37: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/37.jpg)
Combinational Logic Circuits
Values get sent along wires connecting gates
NOT
OR
AND
AND
NOT
p
q
rs
OUT
![Page 38: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/38.jpg)
Combinational Logic Circuits
Wires can send one value to multiple gates!
OR
AND
NOT
ANDp
q
r
OUT
![Page 39: CSE 311: Foundations of Computing · Microsoft PowerPoint - lecture02-equivalences-a.pptx Author: Paul B Created Date: 1/8/2020 2:53:02 PM](https://reader035.vdocuments.us/reader035/viewer/2022071219/6055e3019f78381e5a6ce250/html5/thumbnails/39.jpg)
Combinational Logic Circuits
Wires can send one value to multiple gates!
OR
AND
NOT
ANDp
q
r
OUT
� ∧ ¬� ∨ (¬� ∧ �)