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Lecture 2:Limiting Models ofInstruction Obeying Machine
虞台文大同大學資工所智慧型多媒體研究室
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Content Machine Simulation and Equivalence Unlimited-Register Machine
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Lecture 2:Limiting Models ofInstruction Obeying Machine
Machine Simulation and Equivalence
大同大學資工所智慧型多媒體研究室
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Computer as a Partial Function
M a machine an M-program:e X M:d M Y
an encoding function a decoding function
Me d
input output
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Computer as a Partial Function
d eM
Me d
input output
A partial function
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Machine Equivalence
Input(X)
Output(Y)
1Me d:f X Y 1d e M
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Machine Equivalence
Input(X)
Output(Y)
1Me d:f X Y
2 M
g h
1 2h g M M
1d e M
2d eh g M :f X Y
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Machine Equivalence
Input(X)
Output(Y)
1Me d:f X Y
2 M
g h
1 2h g M M
1d e M
2d eh g M :f X Y
e d
2d e M
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Machine Equivalence
Input(X)
Output(Y)
e d:f X Y
2 M
g h
1 2h g M M
2d e M
e d
d d h e g e
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Machine Equivalence
Input(X)
Output(Y)
:f X Y
2 M
1 2h g M M
e d
d d h e g e
2d e M
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Machine Equivalence Defined
1 2, :M M
::
two machines
M1-programM2-program
1 2, :M M Memory sets of M1 and M2.
11 1: M MM
22 2: M M M g h
g, h such that1 2h g M M
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Machine Equivalence Defined
11 1: M MM
22 2: M M M g h
g, h such that1 2h g M M
:f X YLet can be computed on M1 using , i.e., 1f d e M
2( ) ( )f d h g e M
f can be computed on M2 using ’, with
encoding function
decoding function2:e g e X M
2:d d h M Y
1f d e M
2f d e M
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Machine Simulation Defined
A machine M2 simulates M1 if
1 2
2 1
:
:gh
M MM M
such that
we can specify an algorithm which given any program produces ’ satisfying
1 2h g M M
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Machine Simulation Defined
A machine M2 simulates M1 if
1 2
2 1
:
:gh
M MM M
such that
we can specify an algorithm which given any program produces ’ satisfying
1 2h g M M
Problems: 1. What is the algorithm?2. How to find g and h?
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TheoremA machine M2 simulates M1 if
1 2
2 1
:
:gh
M MM M
such that
we can specify an algorithm which given any program produces ’ satisfying
1 2h g M M
The memory encoder g has to be one to one.
M2 simulates M1
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Theorem
ConsiderPf)
START HALT: Identity
1 ( ) ( )m m m M I
The memory encoder g has to be one to one.
M2 simulates M1
Suppose that g is not one to one.Then, g(m1) = g(m2) = M for some m1m2.
1 2h g M M M2 simulates M1
1 21 1( ) ( )m h g m M M 2 ( )h M M
1 1 21 12 , ( ) ( )m m m m M M
1 22 2( ) ( )m h g m M M 2 ( )h M M
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Stepwise SimulationM2 stepwise simulates M1 if
1-to-1 encoding function g:M1M2 such that1) For each FF,
2) For each PP,
F: the set of operation functions of M1.P: the set of predicates of M1.
a program F in M2 such that 1 2 FFg gM M
a program P in M2 such that
1
2 1
1
( )P
P
P
P
true m trueg m false m false
m
MM M
M
and P doesn’t change M2.
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Stepwise SimulationM2 stepwise simulates M1 if
1-to-1 encoding function g:M1M2 such that1) For each FF,
2) For each PP,
F: the set of operation functions of M1.P: the set of predicates of M1.
a program F in M2 such that 1 2 FFg gM M
a program P in M2 such that
1
2 1
1
( )P
P
P
P
true m trueg m false m false
m
MM M
M
and P doesn’t change M2.
FF 1 2 FFg gM M
g
gF F
m m
1 ( )F mM2 ( )
Fm M
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Stepwise SimulationM2 stepwise simulates M1 if
1-to-1 encoding function g:M1M2 such that1) For each FF,
2) For each PP,
F: the set of operation functions of M1.P: the set of predicates of M1.
a program F in M2 such that 1 2 FFg gM M
a program P in M2 such that
1
2 1
1
( )P
P
P
P
true m trueg m false m false
m
MM M
M
and P doesn’t change M2.
PP P truefalse truefalse P
m ( )g m
m ( )g mm ( )g m
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Stepwise Simulation
M2 stepwise simulates M1
M2 simulates M1
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SR4
Memory set4SRM
4SRM N N N N
4 registers (x1, x2, x3, x4)
Operations4SRF
Predicates4SRP
1i ix x 1i ix x
for i = 1, 2, 3, 4.
0?ix for i = 1, 2, 3, 4.
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Review PC
Memory set PCM
PCM N N
2 registers (x, y)
Operations PCF
Predicates PCP
1x x 1x x
0?x
1y y 1y y
0?y ?x y
y x y y x y
Does PC Simulates SR4?Does SR4 Simulates PC?
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Prove PC Simulates SR4
Step 1:
Step 2:
Step 3:
Define a 1-to-1 encoding function 4: SR PCg M M
For each FFSR4, find a F on PC such that …
For each PPSR4, find a P on PC such that …
: , , , 2 3 5 7 ,0i j k lg i j k l
To be shown
Exercise
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FSR4 PC
F
1i ix x 1i ix x
1i ix x 1i ix x
FF
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11i ii i x xx x
FF 1 11 1 11 x xx x
START
0?x
1x x
1y y
1y y
0?y
1y y
1x x
false
truefalse
HALTtrue
20
y xx
0x y
y
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11i ii i x xx x
FF 1 11 1 11 x xx x
2 22 2 11 x xx x
3 33 3 11 x xx x
4 44 4 11 x xx x Exercise
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11i ii i x xx x
FF 1 11 1 11 x xx x
HALT
START
2 | x?yxx0
/ 2x y 0x y
y
true false
START
0?x
1x x
1y y
false
true
0?x
1x x
1y y
true
false
FALSEHALT
TRUEHALT
2 | x?
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Prove PC Simulates SR4
Step 1:
Step 2:
Step 3:
Define a 1-to-1 encoding function 4: SR PCg M M
For each FFSR4, find a F on PC such that …
For each PPSR4, find a P on PC such that …
: , , , 2 3 5 7 ,0i j k lg i j k l
To be shown
Exercise
In fact, SR2 also simulates SR4.Why?
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Discussion
PC SR4
SR2
SR
Is SR more powerful than SR2? No.Is SR more powerful than PC? Not sure, now.
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Lecture 2:Limiting Models ofInstruction Obeying Machine
Unlimited-Register Machine
大同大學資工所智慧型多媒體研究室
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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 RMemory set
Operations RF Predicates RP
10, 0R i i ii
M n n n
except fi nite many
That is, for some, k 1, ni = 0 for all i k(finite memory are used).
ix m
i jx x m
i jx x m
i jx x m
i jx x m
i jx x
i j kx x x
i j kx x x
i j kx x x
i j kx x x
?ix m
?ix m
?i jx x
?i jx x
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Input & Output Registers of R
1 1 22,, , ,, ,, ,, kR lw wu wuM u
k inputregisters
l outputregisters
Other registers can be working registers if necessary.
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Running R1 1 22,, , ,, ,, ,, kR lw wu wuM u
: a program in Re : encoder
d : decoder11 1: , , 0, ,0, , , ,0,k kkue x x xux
1 1 1: , , , , , ,l l ld y w yw y y
: -tuple -tupled e k lR
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SR
Memory set SRM
The same as R
Operations & Predicates 1i ix x
1i ix x i = 1, 2, …
0?ix
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Machine Simulations
SRRSimulates?
Simulates?
Of course.
Not sure, now.
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Prove SR Simulates R
Step 1:
Step 2:Step 3:
?g
1 1 1: , , 0, , 0, , , ,0,k w kg x x y y x x
FF PP
w working registers
ix m
i jx x m
i jx x m
i jx x m
i jx x m
i jx x
i j kx x x
i j kx x x
i j kx x x
i j kx x x
?ix m
?ix m
?i jx x
?i jx x
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Prove SR Simulates R
Step 1:
Step 2:Step 3:
?g
1 1 1: , , 0, , 0, , , ,0,k w kg x x y y x x
FF PP
w working registers
ix m
i jx x m
i jx x m
i jx x m
i jx x m
i jx x
i j kx x x
i j kx x x
i j kx x x
i j kx x x
?ix m
?ix m
?i jx x
?i jx x
Converted to register-mode operation by using a working
register.
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Prove SR Simulates R
i ji j x x mx x m
START
y m
i jx x y
0y
HALT
??ii x mx m
>=START
y m
0y
TRUEHALT
?ix y
0y
FALSEHALT
true false
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Prove SR Simulates Ri j kx x x i j kx x x
HALT
START
y xk
y>0 ?
xi 0
xi xi + xj
true
false
y y 1
HALT
START
y xj
xk>0 ?
xi 0
y y xk
true
false
xk>y ?
y 0
true
false
xi xi + 1
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Prove SR Simulates Ri j kx x x i j kx x x
HALT
START
y xk
y>0 ?
xi xj
true
false
xi xi + 1
y y 1
HALT
START
y xk
y>0 ?
xi xj
true
false
xi xi 1
y y 1
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Prove SR Simulates Ri jx x
HALT
START
y 0
xj>0?
xi 0
true
false
xj xj 1
xi xi + 1
y y + 1
y>0?true
xj xj + 1
y y 1
false
i i jx x x
HALT
START
y xi + xj
xi y
y 0
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Prove SR Simulates R
?i jx x ?i jx x
START
y xi xj
y>0 ?true
TRUEHALT
FALSEHALT
false
START
false
TRUEHALT
FALSEHALT
truexj > xi?
xi > xj?false
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Exercise
Using SR to Simulate R, at least how many working registers are required?
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Discussion
PC SR4
SR2
SR 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 FunctionsUse 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, k1, register functions) to be functions that are computable by R.
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Examples
START
x1 x3 5
x2 x1 + x3
HALT
1 2 3 4 5, , , ,x x x x x
3 2 3 4 55 , , , ,x x x x x
3 3 3 4 55 ,6 , , ,x x x x x
1 2 3 4 5 31 : , , , , 5x x x xf x x
1 2 3 4 5 32 : , , , , 6x x x xf x x
1 2 3 43 5 3: , , , ,x x x x xf x
1 2 3 44 5 4: , , , ,x x x x xf x
1 2 3 45 5 5: , , , ,x x x x xf x