Abstract machine and compilation
Mechanizing mathematical theory of programming
languages: Operational semantics of WHILE in COQ
Lecture 8 on 24.06.2013
Provably correct implementation
Why is a formal specification of the semantics of
a programming language useful?
Provably correct implementation
Why is a formal specification of the semantics of
a programming language useful?
⇒ We can use it to argue about the correctness of its
implementation.
Provably correct implementation
Why is a formal specification of the semantics of
a programming language useful?
⇒ We can use it to argue about the correctness of its
implementation.
We will show the correctness of a translation from While to a
structured assembler code for an abstract machine.
Provably correct translation
from While to an abstract machine
The idea is:
• define the meaning of the abstract machine instructions by
an operational semantics
Provably correct translation
from While to an abstract machine
The idea is:
• define the meaning of the abstract machine instructions by
an operational semantics
• define translation functions that map expressions and
statements of While into instruction sequences
Provably correct translation
from While to an abstract machine
The idea is:
• define the meaning of the abstract machine instructions by
an operational semantics
• define translation functions that map expressions and
statements of While into instruction sequences
• show that if we
• translate a program into code, and
• execute the code on the abstract machine,
then we get the same result as is specified by the semantic
functions SNat and SSOS .
The abstract machine
The abstract machine AM has configurations of the form
hc, e, si ∈ Code × Stack × State where
• c is the sequence of instructions (or, code) to be executed,
• e is the evaluation stack, and
• s is the storage.
The abstract machine
The abstract machine AM has configurations of the form
hc, e, si ∈ Code × Stack × State where
• c is the sequence of instructions (or, code) to be executed,
• e is the evaluation stack, and
• s is the storage.
An evaluation stack is a list of values, i.e.,
e ∈ Stack = (Z ∪ T)∗
For simplicity, we assume the storage is similar to the state, so
s ∈ State = Var → (Z ∪ T)
Machine instructions
inst
::=
|
|
|
c ::=
PUSH-n | ADD | MULT | SUB
TRUE | FALSE | EQ | LE | AND | NEG
FETCH-x | STORE-x
NOOP | BRANCH(c, c) | LOOP(c, c)
| inst : c
where is the empty sequence.
Code represents the syntactic category of sequences of
instructions (c ∈ Code).
A configuration hc, e, si ∈ Code × Stack × State is terminal
when c is the empty sequence.
The operational semantics for AM
The operational semantics for AM is given by a (small-step)
transition relation on configurations in the form of
hc, e, si . hc 0 , e0 , s0 i
Rules of the operational semantics for AM (1)
hPUSH-n : c, e, σi
. hc, N JnK : e, σi
hADD : c, z1 : z2 : e, σi . hc, (z1 + z2 ) : e, σi if z1 , z2 ∈ Z
hMULT : c, z1 : z2 : e, σi . hc, (z1 ∗ z2 ) : e, σi if z1 , z2 ∈ Z
hSUB : c, z1 : z2 : e, σi . hc, (z1 − z2 ) : e, σi if z1 , z2 ∈ Z
hTRUE : c, e, σi
. hc, tt : e, σi
hFALSE : c, e, σi
. hc, ff : e, σi
hEQ : c, z1 : z2 : e, σi
. hc, (z1 = z2 ) : e, σi if z1 , z2 ∈ Z
hLE : c, z1 : z2 : e, σi
. hc, (z1 ≤ z2 ) : e, σi if z1 , z2 ∈ Z
hAND : c, t1 : t2 :e, σi
.
hc, tt : e, σi if t1 = tt and t2 = tt
hc, ff : e, σi if t1 = ff or t2 = ff, t1 , t2 ∈ T
Rules of the operational semantics for AM (2)
hc, ff : e, σi
hc, tt : e, σi
hc, (σ x) : e, σi
hc, e, σ[x 7→ z]i
hc, e, σi
hc1 : c, e, σi
hc2 : c, e, σi
hNEG : c, t : e, σi
.
hFETCH-x : c, e, σi
hSTORE-x : c, z : e, σi
hNOOP : c, e, σi
.
.
.
hBRANCH(c1 , c2 ) : c, t : e, σi .
if t = tt
if t = ff
if t = tt
if t = ff
hLOOP(c1 , c2 ) : c, e, σi
.
hc1 : BRANCH(c2 : LOOP(c1 , c2 ), NOOP) : c, e, σi
We use the notation ’:’ both for appending two instruction
sequences and for prepending an element to a sequence.
Computation sequences for AM
Given a code c and a storage σ, a computation sequence for c
and σ is either
• a finite sequence
γ0 , γ1 , γ2 , . . . , γk
of configurations such that γ0 = hc, , σi and γi . γi+1 for
0 ≤ i < k , k ≥ 0, and there is no γ such that γk . γ, or
• an infinite sequence
γ0 , γ1 , γ2 , . . .
of configurations such that γ0 = hc, , σi and γi . γi+1 for
0 ≤ i.
Terminating/diverging computation sequences
A computation sequence is
• terminating if and only if it is finite, and
• diverging if and only if it is infinite.
A terminating computation sequence may end in a terminal
configuration or in a stuck configuration, e.g., hADD, , σi.
Example (1)
Consider the code
PUSH-1 : FETCH-x : ADD : STORE-x
Given an initial storage σ where σ x = 3, we get
hPUSH-1 : FETCH-x : ADD : STORE-x, , σi
. hFETCH-x : ADD : STORE-x, 1, σi
. hADD : STORE-x, 3 : 1, σi
. hSTORE-x, 4, σi
. h, , σ[x 7→ 4]i
Example (2)
Consider the code
LOOP(TRUE, NOOP)
We get
hLOOP(TRUE, NOOP), , σi
. hTRUE : BRANCH(NOOP : LOOP(TRUE, NOOP), NOOP), , σi
. hBRANCH(NOOP : LOOP(TRUE, NOOP), NOOP), TRUE, σi
. hNOOP : LOOP(TRUE, NOOP), , σi
. hLOOP(TRUE, NOOP), , σi
...
Properties of AM
Lemma
If γ . γ 0 and γ . γ 00 , then γ 0 = γ 00 .
Lemma
If hc1 , e1 , σi .k hc10 , e10 , σ 0 i, then, for any c2 and e2 ,
hc1 : c2 , e1 : e2 , σi .k hc10 : c2 , e10 : e2 , σ 0 i.
Lemma
If hc1 : c2 , e, σi .k h, e00 , σ 00 i, then there exists a configuration
h, e0 , σ 0 i and natural numbers k1 , k2 such that k = k1 + k2 and
hc1 , e, σi .k1 h, e0 , σ 0 i and hc2 , e0 , σ 0 i .k2 h, e00 , σ 00 i.
The execution function M
We define the meaning of code as a (partial) function from
State to State
M : Code → (State ,→ State)
given by
MJcK σ =
σ0
if hc, , σi .∗ h, e, σ 0 i
undef otherwise
Is this function well-defined?