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CPSC 121: Models of Computation2008/9 Winter Term 2

Propositional Logic: A First Model of Computation

Steve Wolfman, based on notes by Patrice Belleville and others

Introductions, again

Steven Wolfman (call me Steve!)wolf@cs.ubc.caICICS 239; office hours:Mon, Tue, Wed 10-12 and Thu 2-4

but I have an open door policy:If my door is open, come in and talk!Also, I will usually be available after class.And, you can make appointments with me.

Quiz 1 Notes (1 of 2)

Biggest issue: how do I start solving the problem?

• What is it asking (both informally and formally)?

• What is the answer, ignoring propositional logic?

• How do we model the problem in propositional logic?

• Where do I go from there?

Quiz 1 Notes (2 of 2)

Switch problem:• What happens when

all are on?• Do we need “state”?• Problems with XOR

PC problem• What is the intended

output (in normal numbers)?

• How many outputs are there (in propositional logic)?

• How do we get formulas for those?

• How do we simplify?

Lecture Prerequisites

Read Sections 1.1 and 1.4

Solve problems like Exercise Set 1.1, #1-18 and Exercise Set 1.4, #1-17.

You should have completed the open-book, untimed quiz on WebCT that was due before this class.

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.

– Evaluate the truth of propositional logic statements using truth tables.

– Translate back and forth between propositional logic statements and circuits that assess the truth of those statements.

Learning Goals: In-Class

By the end of this unit, you should be able to:– Build combinational computational systems using

propositional logic expressions and equivalent digital logic circuits that solve real problems, e.g., our 7- or 4-segment LED displays.

– 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.).

The second goal is really a multi-unit goal that we will start working with this unit.

Logic as Tool for Reasoning about Truth

Where should we start?

Logic as Model for Physical Computations

Input a Input b

a b

~aa

5V

0V

http://alumni.media.mit.edu/~paulo/courses/howmake/mlfabfinalproject.htm

Problem: Two-Way Switch

Problem: Build a circuit to control a light so that the light changes state any time either of the two switches that control it is flipped.

Digital Logic Challenges

If you finish early, discuss the problem with your neighbours, try a related problem, or don’t just build a circuit, build one that…• Has minimum or uniform “gate delay”• Uses the fewest transistors• Uses the fewest gates across the whole

circuit• Uses minimum area/wiring length on the

circuit

Concept Q: 2-Way Switch

In general, not just in your solution, is the light on or off when both switches are up? (Pick the best answer.)

a.On, in every correct solution.b.Off, in every correct solution.c. It depends, though any single correct solution

should always do the same thing.d.It depends, and a single correct solution

might do different things at different times.e.Neither on nor off.

Problem: Three-Way Switch

Problem: Build a circuit to control a light so that the light changes state any time any of the three switches that control it is flipped.

Concept Q: Three-Way Switch

In your circuit, which of these tells you with certainty whether the light is on or off? (Pick the best answer.)

a.Whether an odd number of switches is on.b.Whether the majority (two or more) of

switches are on.c.Whether all the switches are on.d.Whether a switch has been flipped recently.e.None of these.

Problem: 7-Segment LED Display

Problem: build a circuit that displays the numbers 0 through 9 represented by four Boolean values p, q, r, and s.

(The whole problem requires a lot of grunt work. Please pick one segment, and solve the problem for that segment.)

Truth Table and Numbers

If we agree on a convention for the rows of a truth table, we can assign a number to each row… # p q r

0 T T T

1 T T F

2 T F T

3 T F F

4 F T T

5 F T F

6 F F T

7 F F F

Of course, as Epp says, we could agree on a different convention.

Representing Positive Integers

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

Just the opposite of our previous version.

# 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

# a b c d

8 T F F F

9 T F F T

Concept Q: 7-Segment LED

Imagine we were working on an LED in a display for some other kind of number (like Roman numerals). Which of these would definitely make our problem easier?

a. More entries in the LED’s column of the truth table that are true (turn some Fs into Ts).

b. Fewer entries in the LED’s column of the truth table that are true (turn some Ts into Fs).

c. More legal inputs (e.g., 0-9 and “A” for 10).d. Fewer legal inputs (e.g., only 0-8).e. None of these would definitely make the problem

easier.

Problem: Updating the “PC”

Problem: Create logic that calculates the amount to increase the PC by, given the values NeedValC and NeedRegIDs.

Problem: Updating the “PC”

Concept Q: Fetch Logic

What’s the minimum number of Boolean outputs necessary for this circuit?

a.1

b.2

c.3

d.4

e.Cannot be determined from the information given.

Learning Goals: In-Class

By the end of this unit, you should be able to:– Build combinational computational systems using

propositional logic expressions and equivalent digital logic circuits that solve real problems, e.g., our 7- or 4-segment LED displays.

– 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.).

Bonus goal that foreshadows “induction”:Give two examples of recursive definitions.

Next Lecture Learning Goals: Pre-Class

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

language statements and proposition 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

Next Lecture Prerequisites

Reread Sections 1.1 and 1.4.

Read Section 1.2.

Solve problems like Exercise Set 1.1, #19-29, and 39-46; Set 1.2, #5-15, 20, 22, and 38-46; and Set 1.4, #26-32.

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

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More problems to solve...

(on your own or if we have time)

Problem: 4-Segment LED Display

Problem: build a circuit that displays the numbers 1 through 9 represented by four Boolean values p, q, r, and s on a 4-segment Boolean display.

1 2 3 4 5

6 7 8 9

Problem: One-Bit Addition

Problem: build a circuit that takes three one-bit numbers as input and outputs their sum as a two-bit number.

Problem: Logicians and Hats

Problem: Three logicians are each wearing a black hat or a white hat, but not all white. Nobody can see their own hat. However, A can see the hats of B and C, and B can see the hats of A and C. C is blind. You go and ask them one by one in the order A, B, C, whether they know the color of their own hat. A answers “No”. B answers “No”. Then C answers “Yes”. Explain how this is possible.

Problem: Criminals and Hats

Problem: A warden plans to line up 100 prisoners in order tomorrow. The warden will place a white or black hat on each prisoner’s head so that no prisoner can see the hat on his or her own head, but they can see the hats of the prisoners in front of them. From the back of the line, the warden will ask each prisoner “Is your hat black?” If the prisoner answers correctly, he or she is set free. Incorrect answers lead to immediate and noisy beheading. If they answer anything other than “yes” or “no”, all prisoners are beheaded. They get one hour as a group to plan their strategy.

How many prisoners can be saved?

The three-way light switch might give you a hint.