Lecture 2: Limiting Models of Instruction Obeying Machine

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Lecture 2Limiting Models ofInstruction Obeying
Machine

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Content

• Machine Simulation and Equivalence
• Unlimited-Register Machine

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Lecture 2Limiting Models ofInstruction Obeying
Machine

• Machine Simulation and Equivalence

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Computer as a Partial Function
M ? a machine
? ? an M-program
? an encoding function
? a decoding function
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Computer as a Partial Function
A partial function
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Machine Equivalence
Output (Y)
Input (X)
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Machine Equivalence
Output (Y)
Input (X)
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Machine Equivalence
Output (Y)
Input (X)
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Machine Equivalence
Output (Y)
Input (X)
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Machine Equivalence
Output (Y)
Input (X)
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Machine Equivalence Defined
two machines
Memory sets of M1 and M2.
M1-program
M2-program
?g, h such that
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Machine Equivalence Defined
encoding function
f can be computed on M2 using ?, with
decoding function
?g, h such that
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Machine Simulation Defined
A machine M2 simulates M1 if
such that
we can specify an algorithm which given any
program ? produces ? satisfying
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Machine Simulation Defined

1. What is the algorithm?
2. How to find g and h?

Problems
A machine M2 simulates M1 if
such that
we can specify an algorithm which given any
program ? produces ? satisfying
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Theorem
The memory encoder g has to be one to one.
M2 simulates M1
A machine M2 simulates M1 if
such that
we can specify an algorithm which given any
program ? produces ? satisfying
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Theorem
The memory encoder g has to be one to one.
M2 simulates M1
Pf)
M2 simulates M1
Consider
Suppose that g is not one to one. Then, g(m1)
g(m2) M for some m1?m2.
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Stepwise Simulation
F the set of operation functions of M1.
P the set of predicates of M1.
M2 stepwise simulates M1 if
? 1-to-1 encoding function gM1?M2 such that
1) For each F?F,
? a program ?F in M2 such that
2) For each P?P,
? a program ?P in M2 such that
and ?P doesnt change M2.
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Stepwise Simulation
F the set of operation functions of M1.
P the set of predicates of M1.
M2 stepwise simulates M1 if
? 1-to-1 encoding function gM1?M2 such that
1) For each F?F,
? a program ?F in M2 such that
2) For each P?P,
? a program ?P in M2 such that
and ?P doesnt change M2.
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Stepwise Simulation
F the set of operation functions of M1.
P the set of predicates of M1.
M2 stepwise simulates M1 if
? 1-to-1 encoding function gM1?M2 such that
1) For each F?F,
? a program ?F in M2 such that
2) For each P?P,
? a program ?P in M2 such that
and ?P doesnt change M2.
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Stepwise Simulation
M2 stepwise simulates M1
M2 simulates M1
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SR4
? 4 registers (x1, x2, x3, x4)
for i 1, 2, 3, 4.
for i 1, 2, 3, 4.
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Review PC
Does PC Simulates SR4?
Does SR4 Simulates PC?
? 2 registers (x, y)
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Prove PC Simulates SR4
Step 1
Step 2
For each F?FSR4, find a ?F on PC such that
To be shown
Step 3
For each P?PSR4, find a ?P on PC such that
Exercise
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SR4
PC
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(No Transcript)
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Exercise
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2 x?
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Prove PC Simulates SR4
In fact, SR2 also simulates SR4.
Why?
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Discussion
Is SR? more powerful than SR2?
No.
Is SR? more powerful than PC?
Not sure, now.
PC
SR4
SR?
SR2
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Lecture 2Limiting Models ofInstruction Obeying
Machine

• Unlimited-Register Machine

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The Unlimited-Register Machine

• Unlimited number of registers.
• Unbounded capacity of every register.
• Powerful instructions

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The Machine R
Memory set
That is, for some, k ? 1, ni 0 for all i ?
k (finite memory are used).
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Input Output Registers of R
k input registers
l output registers
Other registers can be working registers if
necessary.
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Running R
? a program in R
e encoder
d decoder
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SR?
? The same as R
Operations Predicates
i 1, 2,
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Machine Simulations
Of course.
SR?
R
Not sure, now.
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Prove SR? Simulates R
Step 1
w working registers
Step 2
Step 3
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Prove SR? Simulates R
Step 1
w working registers
Converted to register-mode operation by using a
working register.
Step 2
Step 3
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Prove SR? Simulates R
? ? ? ?
gt
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Prove SR? Simulates R
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Prove SR? Simulates R
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Prove SR? Simulates R
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Prove SR? Simulates R
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Exercise
Using SR? to Simulate R, at least how many
working registers are required?
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Discussion
R
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Discussion

• Machine equivalence is reflexive, symmetric, and
  transitive, i.e., an equivalence relation.
• SR2 is the same powerful as R.
• To study computation, considering PC, SR2, SR4,
  SR or R is equally well.
• The above machines are register machines.

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Register Functions
Use x1, , xk as input registers of R.
Let fi Nk ? N be the function computed by an
R-program ? using xi as the output register.
We call the k functions f1, , fk the (k-adic)
register functions of ?.
We will considered register functions (the class
of all k-adic, k?1, register functions) to be
functions that are computable by R.
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Examples