CPSC 121: Models of Computation Module 8: Sequential Circuits - - PowerPoint PPT Presentation

CPSC 121: Models of Computation

Module 8: Sequential Circuits

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Module 8: Sequential Circuits

By the start of class, you should be able to

Trace the operation of a DFA (deterministic finite- state automaton) represented as a diagram on an input, and indicate whether the DFA accepts or rejects the input. Deduce the language accepted by a simple DFA after working through multiple example inputs.

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Module 8: Sequential Circuits

Quiz 8 feedback:

Well done. Many fine answers to the push-button light question. We will revisit this problem soon.

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? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

CPSC 121: the BIG questions:

• 1. How can we build a computer that is able to

execute a user-defined program?

a) Computers execute instructions one at a time. b) They need to remember values, unlike the circuits you designed in labs 1, 2, 3 and 4. c) That is, a computer is a very large and very complicated sequential circuit.

Module 8: Sequential Circuits

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Module 8: Sequential Circuits

By the end of this module, you should be able to:

Translate a DFA into a sequential circuit that implements the DFA. Explain how and why each part of the resulting circuit works.

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Module 8: Sequential Circuits

Announcements:

Pre-class quiz #9 is due Monday November 12th at 19:00. Textbook sections:

Epp, 4th edition: 5.1 to 5.4 Epp, 3rd edition: 4.1 to 4.4 Rosen, 6th edition: 4.1, 4.2 Rosen, 7th edition: 5.1, 5.2

Assignment #4 is due tomorrow at 19:00.

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Module 8: Sequential Circuits

Announcements (continued):

Midterm #2:

Friday at 17:30 Same locations as for midterm #1. No notes, textbook, calculator or other electronic equipment is allowed. We will supply the parts of Dave’s excellent formula sheet that you need for the midterm. Covers from predicate logic (with both types of quantifiers) up to the end of today’s class, as well as labs 4, 5, 6, 7.

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Module 8: Sequential Circuits

Announcements (continued):

Pre-class quiz #10 is tentatively due Monday November 26th at 19:00. Textbook sections:

Epp, 4th edition: 6.1, 7.1 Epp, 3rd edition: 5.1, 6.1 Rosen, 6th edition: 2.1, 2.3 up to the top of page 136. Rosen, 7th edition: 2.1, 2.3 down to the bottom of page 141.

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Module 8: Sequential Circuits

Module Summary

Latches, toggles and flip-flops. Using a DFA for branch prediction. Other problems and exercises.

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Module 8.1: Latches, toggles and flip-flops

There are two types of Finite-State Automata:

Those whose output is determined solely by the final state (Moore machines).

Used to match a string to a pattern.

Input validation. Searching text for contents. Lexical Analysis: the first step in a compiler or an interpreter.

(define (fun x) (if (<= x 0) 1 (* x (fun (- x 1)))))

( define ( fun x ) ( if ( <= x 0 ) 1 ( * x ( fun ( - x 1 ) ) ) ) )

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Module 8.1: Latches, toggles and flip-flops

Those that produce output every time the state changes (Mealy machines).

Examples:

Traffic lights controller. The push button example from quiz 8b. Predicting branching in machine-language programs

Light OFF Light ON Button pressed Button pressed

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Module 8.1: Latches, toggles and flip-flops

A circuit that implements a finite state machine

• f either type needs to remember the current

state:

It needs memory.

A latch A flip-flop A register (multiple side by side flip-flops with a common clock)

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Module 8.1: Latches, toggles and flip-flops

Recall the latch from lab #5: When en is low, the MUX retains its current value. When en is high, it changes its value to d instead.

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Module 8.1: Latches, toggles and flip-flops

Problem: Design a circuit that changes state every time a button is pushed.

? ?

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Module 8.1: Latches, toggles and flip-flops

What signal does the button generate?

low high

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Module 8.1: Latches, toggles and flip-flops

Complete the circuit...

Circuit to calculate the next state

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Module 8.1: Latches, toggles and flip-flops

What is wrong with our solution?

a) We should have used XOR instead of NOT. b) The light will be in a random, unpredictable state. c) The delay introduced by the NOT gate is too long. d) There is some other problem with the circuit. e) Nothing is wrong.

▷

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Module 8.1: Latches, toggles and flip-flops

This toll booth has a similar problem.

From MIT 6.004, Fall 2002

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Module 8.1: Latches, toggles and flip-flops

Instead use this:

From MIT 6.004, Fall 2002 P.S. Call this a “bar”, not a “gate”,

• r we'll tie ourselves in (k)nots.

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Module 8.1: Latches, toggles and flip-flops

The circuit version of this improved tollbooth is called a flip-flop:

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Module 8.1: Latches, toggles and flip-flops

Assume the value stored in the flip-flop is 1 and d = 0. As long as the clock remains low:

1 1

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Module 8.1: Latches, toggles and flip-flops

Observe that the two select input are never the same.

1 1

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Module 8.1: Latches, toggles and flip-flops

Now the clock goes high:

1 1

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Module 8.1: Latches, toggles and flip-flops

Now the clock goes low again:

1 1

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Module 8.1: Latches, toggles and flip-flops

Finally we set d = 1:

1 1 1 1

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Module 8.1: Latches, toggles and flip-flops

And we get the following improved circuit for

• ur button and light problem:

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Module 8: Sequential Circuits

Module Summary

Latches, toggles and flip-flops. Using a DFA for branch prediction. Other problems and exercises.

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Module 8.2: Using a DFA for branch prediction

How do computers really execute programs?

Programs written in a high-level language (Racket, Java) are translated into machine language. A machine-language program is a sequence of very simple instructions.

Each instruction is a sequence of 0s and 1s. Each instruction also has a human-readable version

Humans don't like looking at long sequences of 0s and 1s. The human-readable version is not actually part of the program.

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Module 8.2: Using a DFA for branch prediction

Example (modified to make it easier to understand):

(1) sum ← 0 (2) is n = 0? (3) if true go to 7 (4) sum ← sum + n (5) n ← n – 1 (6) goto 2

Some instructions like instruction 3 may tell the computer that the next instruction to execute is not the next in the sequence (4), but elsewhere (7).

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Module 8.2: Using a DFA for branch prediction

To speed things up, a modern computer starts executing an instruction before the previous one is finished. This means that when it is executing

if true go to 7

it does not yet know if the condition is true, and hence does not know if the next instruction is

sum ← sum + n

• r instruction number 7.

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Module 8.2: Using a DFA for branch prediction

We want to use a DFA to predict the branch

• utcome.

If we guess wrong, then we will ignore some of the work that was done.

We will keep track of two pieces of information:

What we will predict (F = not branch, T = branch). How confident we are that we are correct (F = not very, T = very).

• nce we know if the branch was taken, we

update this information.

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Module 8.2: Using a DFA for branch prediction

How many states will the Finite State Automaton have?

a) 2 b) 4 c) 8 d) Another value less than 8. e) Another value larger than 8.

▷

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Module 8.2: Using a DFA for branch prediction

Our prediction algorithm works as follows:

We are confident if the last two outcomes were the same, but not if they were different. We keep predicting one outcome until we have been wrong twice in a row. Being wrong twice in a row is the same as being wrong when we’re not confident about our prediction In this case we change prediction.

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Module 8.2: Using a DFA for branch prediction

Hence we get the following DFA:

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Module 8.2: Using a DFA for branch prediction

We can then turn this DFA into a sequential circuit, as described in Lab 8.

Number the states, starting with 0, and figure out how many bits you need to store the state number. Number the inputs, starting with 0, and figure out how many bits you need to represent the input. Layout enough D flip-flops to store the state (one per bit).

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Module 8.2: Using a DFA for branch prediction

We can then turn this DFA into a sequential circuit, as described in Lab 8 (continued).

For each state, build a combinational circuit that computes the next state (and the output, if needed) given the input. Send all those into multiplexers, and use the current state as the control signal (so you only keep the correct one). Store the next state back into the D flip-flops.

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Module 8: Sequential Circuits

Module Summary

Latches, toggles and flip-flops. Using a DFA for branch prediction. Other problems and exercises.

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Module 8.3: Other problems and exercises

Real numbers:

We can write numbers in decimal using the format

(-)? d+ (.d+)?

where the ( )? mean that the part in parentheses is

• ptional, and d+ stands for “1 or more digits”.

Design a DFA that will accept input strings that are valid real numbers using this format.

You can use else as a label on an edge instead of listing every character that does not appear on another edge leaving from a state.

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Module 8.3: Other problems and exercises

Real numbers (continued)

Then design a circuit that turns a LED on if the input is a valid real number, and off otherwise.

Hint: Logisim has a keyboard component you can use. Hint: my DFA for this problem has 6 states.

Design a DFA for a vending machine that sells

• ne of three items (lemon juice, whiteboard

markers, and corn flour) for 35¢ each. It should accept 5¢, 10¢ and 25¢ coins, and does not need to return change.