CPSC 121: Models of Computation 2010/11 Winter Term 2

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CPSC 121: Models of Computation2010/11 Winter Term 2

Propositional Logic: Conditionals and Logical Equivalence

Steve Wolfman, based on notes by Patrice Belleville and others

1

Outline

• Prereqs, Learning Goals, and Quiz Notes

• Problems and Discussion– Logical equivalence practice– How to write a logical equivalence proof– A flaw in our model

• Next Lecture Notes

2

Quiz 2 Review:Circuits?

Our most common meaning for circuit will be “the set of gates (and other electronics components) that collectively map a set of inputs to the desired output(s)”.

When we ask if multiple different circuits can correctly map input numbers to an output that controls an LED in the display, we mean “are there alternative designs to solve the problem?”

3

Quiz 2 Review:“Meaning” of

When I say “if it’s raining, I’ll bring my umbrella”, do I mean:

In other words, I might bring my umbrella even if it’s not raining?

Probably not, in English.

BUT, in logic always means exactly that truth table! It just isn’t a great match for “normal” English. Better for legalistic English…

4

r u r u

T T T

T F F

F T T

F F T


Consider the following truth table for p ~q

And, consider the following statement: “p cannot be true unless q is false.”

Does this mean the same thing as p ~q?5

p q ~q p ~q

T T F F

T F T T

F T F T

F F T T



And, consider the following statement: “p cannot be true unless q is false.”


p q ~q p ~q

T T F F

T F T T

F T F T

F F T T



And, consider the following statement: “If p is true, it guarantees q is false.”


p q ~q p ~q

T T F F

T F T T

F T F T

F F T T


Consider the following truth table for p q

And, consider the following statement: “Either p and q, or ~p and ~q.”

Does this mean the same thing as p q?8

p q p q

T T T

T F F

F T F

F F T

Quiz 2 Review:“F” vs. “c”

I think “c” is a terrible symbol to use for a contradiction because “a” and “b” are natural variables to use in propositional logic expressions, and “c” is a natural one to use next.

We will use “F” instead and avoid capital letters as names for propositional logic variables.

9

Learning Goals: Pre-Class

By the start of class, you should be able to:– Translate back and forth between simple natural

language statements and propositional logic, now with conditionals and biconditionals.

– Evaluate the truth of propositional logical statements that include conditionals and biconditionals using truth tables.

– Given a propositional logic statement and an equivalence rule, apply the rule to create an equivalent statement.

Example: given (u s) s, apply p q ~p q.Note: p maps to (u s) and q maps to s.Result: ~(u s) s 10

Learning Goals: In-Class

By the end of this unit, you should be able to:– Explore alternate forms of propositional logic

statements by application of equivalence rules, especially in order to simplify complex statements or massage statements into a desired form.

– Evaluate propositional logic as a “model of computation” for combinational circuits, including at least one explicit shortfall (e.g., referencing gate delays, fan-out, transistor count, wire length, instabilities, shared sub-circuits, etc.).

11

Where We Are inThe Big Stories

Theory

How do we model computational systems?

Now: practicing our second technique for formally establishing the truth of a statement (logical equivalence proofs).

(The first technique was truth tables.)

Hardware

How do we build devices to compute?

Now: learning to modify circuit designs using our logical model, gaining more practice designing circuits, and identifying a flaw in our model for circuits.

12

Outline




13

Problem: 4-Segment LED Display

Problem: Build a circuit that displays the numbers 1 through 9 represented by four Boolean values a, b, c, and d on a 4-segment Boolean display.

1 2 3 4 5

6 7 8 9

14

Problem: 4-Segment LED Display

Problem: Build a circuit that displays the numbers 1 through 9 represented by four Boolean values a, b, c, and d on a 4-segment Boolean display.

1 2 3 4 5

6 7 8 9

15

321

654

987

RECALL: Representing Positive Integers

16

This is the convention we (and computers) use for the positive integers 0-9, which requires 4 variables:

# a b c d

0 F F F F

1 F F F T

2 F F T F

3 F F T T

4 F T F F

5 F T F T

6 F T T F

7 F T T T

8 T F F F

9 T F F T

...

Problem: Equivalent Circuits

Problem: Consider these circuits for the top LED in the 4-segment display (assuming inputs abcd from the previous table):

1.(~a b) (a ~b ~c)

2.(~a b) a3.a b

4.a b

17

321

654

987


Problem: Consider these circuits for the top LED in the 4-segment display (assuming inputs abcd from the previous table).

1.(~a b) (a ~b ~c)

2.(~a b) a3.a b

4.a b

Which of 2, 3, and 4 are equivalent?a.None of them.b.2 and 3 but not 4.c.2 and 4 but not 3.d.3 and 4 but not 2.e.2, 3, and 4.

18

321

654

987



1.(~a b) (a ~b ~c)

2.(~a b) a3.a b

4.a b

How many of these are correct?a.None of them.b.One of them.c.Two of them.d.Three of them.e.All four of them.

19

321

654

987



1.(~a b) (a ~b ~c)

2.(~a b) a3.a b

4.a b

Which of these proves that circuit 1 is not equivalent to circuit 3:a.2 and 4 are equivalentb.1 mentions c but 3 does not.c.1 is false with a=T b=F c=T.d.None of these.

20

321

654

987

Outline




21

Writing an Equiv Proof: Prove (~a b) a a b

Theorem: (~a b) a a bProof:

(~a b) a

a (~a b) by commutativity

(a ~a) (a b) by distribution

...to be filled in...

a b by identity

QED22



(~a b) a




a b by identity

QED23

State your theorem.

Explicitly start the proof.

Start with one side...



(~a b) a




a b by identity

QED24


Each line starts with to indicate it’s equivalent to the previous line.

...and work to the other.

End with QED!



(~a b) a



...

a b by identity

QED25

Give the next statement. And justify how you got it.

Problem: Prove (~a b) a a b


(~a b) a



?????? by negation

a b by identity

QED

26

What’s missing?

a.(a b)b.F (a b)c.a (a b)d.None of these, but I know what it is.e.None of these, and there’s not enough information to tell.

Outline




27

“Multiplexer”

A circuit that, given three inputs a, b, and c (the “control” signal), outputs a’s value when c is F and b’s when c is T.

This circuit is called a multiplexer.

a

bout

c

0

1

28

(~a b c) ( a ~b ~c) ( a b ~c) ( a b c)

a b c out

F F F F

F F T F

F T F F

F T T T

T F F T

T F T F

T T F T

T T T T

MUX Design

Here’s one implementation of the multiplexer (MUX) circuit.

(Is this equivalentto the previousslide’s formula?Good question...prove that it is forpractice at home!)

29

a

bout

c

0

1

Truthy MUX

What is the intended output if both a and b are T?

a. T

b. F

c. Unknown... but could be answered given a value for c.

d. Unknown... and might still be unknown even given a value for c.

30

a

bout

c

0

1

Glitch in MUX Design

Imagine the circuit is in steady-state with a, b, and c all T.

Trace how changes flow when we change c to F, if each gate takes 10ns to operate.

31PRACTICE: Prove that this circuit’s model is logically

equivalent to the previous slide’s statement.

a

bout

c

0

1

Trace: -1 ns

32

T

F

T

TT

T

F

a

bout

c

0

1

Trace: 0 ns

33

T

F

T

TT

F

F

OFF

*

*

a

bout

c

0

1

Trace: 10 ns

34

T

F

T

FT

F

T

OFF

*

*

a

bout

c

0

1

Tracing a MUX

Assume the output starts at 1. We want it to end up at 1. How long does it take before it’s stable? (Choose the best answer.)

a.0 ns (it never varies from 1)

b.10 ns (it varies before 10ns, but not after)

c.20 ns

d.30 ns

e.40 ns35

a

bout

c

0

1

Trace: -1 ns

36

T

F

T

TT

T

F

a

bout

c

0

1

Trace: 0 ns

37

T

F

T

TT

F

F

OFF

*

*

a

bout

c

0

1

Trace: 10 ns

38

T

F

T

FT

F

T

OFF

*

*

a

bout

c

0

1

Trace: 20 ns

39

T

T

F

FT

F

T

OFF

*

a

bout

c

0

1

Trace: 30 ns

40

T

T

T

FT

F

T

OFF

a

bout

c

0

1

Trace: 40 ns

41

T

T

T

FT

F

T

OFF

a

bout

c

0

1

Other MUX Glitches?

The mux glitches because information from c travels two paths with different delays. While the longer path “catches up” the circuit can be incorrect.

PRACTICE: Trace the circuit to show why none of these can cause a glitch:• Changing a (keeping b and c constant)• Changing b (keeping a and c constant)• Changing c (keeping a and b constant if at least

one of them is F)

42

a

bout

c

0

1

A Glitchless MUX

This circuit uses what we know when a = b = T.

PRACTICE: Prove that it’s logically equivalent to the original MUX.

Hint: break a b up into two cases, one where c is true and one where c is false: a b c and a b ~c.

43

a

bout

c

0

1

Outline




44

Learning Goals: In-Class

By the end of this unit, you should be able to:– Explore alternate forms of propositional logic

statements by application of equivalence rules, especially in order to simplify complex statements or massage statements into a desired form.

– Evaluate propositional logic as a “model of computation” for combinational circuits, including at least one explicit shortfall (e.g., referencing gate delays, fan-out, transistor count, wire length, instabilities, shared sub-circuits, etc.).

45

Next Lecture Learning Goals: Pre-Class

By the start of class, you should be able to:– Convert positive numbers from decimal to binary

and back.

– Convert positive numbers from hexadecimal to binary and back.

– Take the two’s complement of a binary number.

– Convert signed (either positive or negative) numbers to binary and back.

– Add binary numbers.

46

Next Lecture Prerequisites

Read Section 1.5 and the supplemental handout: http://www.ugrad.cs.ubc.ca/~cs121/2009W1/Handouts/signed-binary-decimal-conversions.html

Solve problems like Exercise Set 1.5, #1-16, 27-36, and 41-46.

NOTE: the binary numbers in problems 27-30 are signed (that is, they use the two’s complement representation scheme).

Complete the open-book, untimed quiz on Vista that’s due before the next class.

47

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snack

Some Things to Try...

(on your own if you have time, not required)

48

PRACTICE Problem: Arbitrary Logic Expression

Problem: Aliens hold the Earth hostage, and only you can save it by proving (a b) ~(b a) ~a b.

Reminder (p. 20 of Epp, 3d ed.): p q ~p qHint: usually start these problems from the more complex side.

49

Problem: String equals

Consider the following from the Java 6 API documentation for the String class’s equals method:...The [method returns] true if and only if the argument is not null and is a String object that represents the same sequence of characters as this object.

Let n1 mean “this string is null”, n2 mean “the argument is null”, r mean “the method returns true”, and s mean “the two strings are objects that represent the same sequence of characters”.

Presumably any two null strings are equal to each other. Then, equality would become something like “the method returns true if and only if the two strings both null or are objects that represent the same sequence of characters”.

Problem: Is that logically equivalent to the statement from the API? Why or why not?

50

Problem: Decisions in Code

Consider the following code, part of a “binary bounds search”:if target equals value then

if lean-left-mode is true

call the go-left routine

otherwise

call the go-right routine

otherwise if target is less than value then

call the go-left routine

otherwise

call the go-right routine

51

Problem: Decisions in Code

Let gl mean “the go-left routine is called”

Problem: complete the logical expression

gl _____________________

Problem: Use your logic to simplify the pseudocode so it requires just one “if/otherwise”.

52

Problem: Prove (~a b) a a b


(~a b) a



...

a b by identity

QED53

State your theorem.

Explicitly start the proof.


Each line starts with to indicate it’s equivalent to the previous line.

Give the next statement. And justify how you got it.

End with QED!

...and work to the other.

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CPSC 121: Models of Computation 2010/11 Winter Term 2