# source:src/ASM/CPP2013-policy/proof.tex@3365

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1\section{The proof}
2
3In this section, we present the correctness proof for the algorithm in more
4detail. The main correctness statement is shown, slightly simplified, in~Figure~\ref{statement}.
5
6\label{sigmapolspec}
7\begin{figure}[t]
8\begin{alignat*}{6}
9\mathtt{sigma}&\omit\rlap{$\mathtt{\_policy\_specification} \equiv 10\lambda program.\lambda sigma.\lambda policy.$} \notag\\
11  & \omit\rlap{$sigma\ 0 = 0\ \wedge$} \notag\\
12        & \mathbf{let}\ & & \omit\rlap{$instr\_list \equiv code\ program\ \mathbf{in}$} \notag\\
13        &&& \omit\rlap{$\forall ppc.ppc < |instr\_list| \rightarrow$} \notag\\
14        &&& \mathbf{let}\ && pc \equiv sigma\ ppc\ \mathbf{in} \notag\\
15        &&& \mathbf{let}\ && instruction \equiv \mathtt{fetch\_pseudo\_instruction}\ instr\_list\ ppc\ \mathbf{in} \notag\\
16        &&& \mathbf{let}\ && next\_pc \equiv sigma\ (ppc+1)\ \mathbf{in}\notag\\
17        &&&&& next\_pc = pc + \mathtt{instruction\_size}\ sigma\ policy\ ppc\ instruction\ \wedge\notag\\
18  &&&&& (pc + \mathtt{instruction\_size}\ sigma\ policy\ ppc\ instruction < 2^{16}\ \vee\notag\\
19  &&&&& (\forall ppc'.ppc' < |instr\_list| \rightarrow ppc < ppc' \rightarrow \notag\\
20        &&&&& \mathbf{let}\ instruction' \equiv \mathtt{fetch\_pseudo\_instruction}\ instr\_list\ ppc'\ ppc\_ok'\ \mathbf{in} \notag\\
21        &&&&&\ \mathtt{instruction\_size}\ sigma\ policy\ ppc'\ instruction' = 0)\ \wedge \notag\\
22        &&&&& pc + (\mathtt{instruction\_size}\ sigma\ policy\ ppc\ instruction = 2^{16}))
23\end{alignat*}
24\caption{Main correctness statement\label{statement}}
25\end{figure}
26
27Informally, this means that when fetching a pseudo-instruction at $ppc$, the
28translation by $\sigma$ of $ppc+1$ is the same as $\sigma(ppc)$ plus the size
29of the instruction at $ppc$.  That is, an instruction is placed consecutively
30after the previous one, and there are no overlaps.
31
32Instructions are also stocked in
33order: the memory address of the instruction at $ppc$ should be smaller than
34the memory address of the instruction at $ppc+1$. There is one exception to
35this rule: the instruction at the very end of the program, whose successor
36address can be zero (this is the case where the program size is exactly equal
37to the amount of memory).
38
39Finally, we enforce that the program starts at address 0, i.e. $\sigma(0) = 0$.
40
41Since our computation is a least fixed point computation, we must prove
42termination in order to prove correctness: if the algorithm is halted after
43a number of steps without reaching a fixed point, the solution is not
44guaranteed to be correct. More specifically, branch instructions might be
45encoded which do not coincide with the span between their location and their
46destination.
47
48Proof of termination rests on the fact that the encoding of branch
49instructions can only grow larger, which means that we must reach a fixed point
50after at most $2n$ iterations, with $n$ the number of branch instructions in
51the program. This worst case is reached if at every iteration, we change the
52encoding of exactly one branch instruction; since the encoding of any branch
53instruction can change first from short to absolute, and then to long, there
54can be at most $2n$ changes.
55
56The proof has been carried out using the Russell'' style from~\cite{Sozeau2006}.
57We have proven some invariants of the {\sc f} function from the previous
58section; these invariants are then used to prove properties that hold for every
59iteration of the fixed point computation; and finally, we can prove some
60properties of the fixed point.
61
62\subsection{Fold invariants}
63
64In this section, we present the invariants that hold during the fold of {\sc f}
65over the program. These will be used later on to prove the properties of the
66iteration.
67
68Note that during the fixed point computation, the $\sigma$ function is
69implemented as a trie for ease of access; computing $\sigma(x)$ is achieved by
70looking up the value of $x$ in the trie. Actually, during the fold, the value
71we pass along is a pair $\mathbb{N} \times \mathtt{ppc\_pc\_map}$. The first
72component is the number of bytes added to the program so far with respect to
73the previous iteration, and the second component, {\tt ppc\_pc\_map}, is the
74actual $\sigma$ trie.
75
76\begin{alignat*}{2}
77\mathtt{out} & \mathtt{\_of\_program\_none} \equiv \lambda prefix.\lambda sigma. \notag\\
78& \forall i.i < 2^{16} \rightarrow (i > |prefix| \leftrightarrow
79 \mathtt{lookup\_opt}\ i\ (\mathtt{snd}\ sigma) = \mathtt{None})
80\end{alignat*}
81
82The first invariant states that any pseudo-address not yet examined is not
83present in the lookup trie.
84
85\begin{alignat*}{2}
86\mathtt{not} & \mathtt{\_jump\_default} \equiv \lambda prefix.\lambda sigma.\notag\\
87& \forall i.i < |prefix| \rightarrow\notag\\
88& \neg\mathtt{is\_jump}\ (\mathtt{nth}\ i\ prefix) \rightarrow\notag\\
89& \mathtt{lookup}\ i\ (\mathtt{snd}\ sigma) = \mathtt{short\_jump}
90\end{alignat*}
91
92This invariant states that when we try to look up the jump length of a
93pseudo-address where there is no branch instruction, we will get the default
94value, a short jump.
95
96\begin{alignat*}{4}
97\mathtt{jump} & \omit\rlap{$\mathtt{\_increase} \equiv \lambda pc.\lambda op.\lambda p.$} \notag\\
98        & \omit\rlap{$\forall i.i < |prefix| \rightarrow$} \notag\\
99&       \mathbf{let}\  && oj \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ op)\ \mathbf{in} \notag\\
100&       \mathbf{let}\ && j \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ p)\ \mathbf{in} \notag\\
101&&&     \mathtt{jmpleq}\ oj\ j
102\end{alignat*}
103
104This invariant states that between iterations (with $op$ being the previous
105iteration, and $p$ the current one), jump lengths either remain equal or
106increase. It is needed for proving termination.
107
108\begin{figure}[ht]
109\begin{alignat*}{6}
110\mathtt{sigma} & \omit\rlap{$\mathtt{\_compact\_unsafe} \equiv \lambda prefix.\lambda sigma.$}\notag\\
111& \omit\rlap{$\forall n.n < |prefix| \rightarrow$}\notag\\
112& \mathbf{match}\ && \omit\rlap{$\mathtt{lookup\_opt}\ n\ (\mathtt{snd}\ sigma)\ \mathbf{with}$}\notag\\
113&&& \omit\rlap{$\mathtt{None} \Rightarrow \mathrm{False}$} \notag\\
114&&& \omit\rlap{$\mathtt{Some}\ \langle pc, j \rangle \Rightarrow$} \notag\\
115&&& \mathbf{match}\ && \mathtt{lookup\_opt}\ (n+1)\ (\mathtt{snd}\ sigma)\ \mathbf{with}\notag\\
116&&&&& \mathtt{None} \Rightarrow \mathrm{False} \notag\\
117&&&&& \mathtt{Some}\ \langle pc_1, j_1 \rangle \Rightarrow
118                pc_1 = pc + \notag\\
119&&&&& \ \ \mathtt{instruction\_size\_jmplen}\ j\ (\mathtt{nth}\ n\ prefix)
120\end{alignat*}
121\caption{Temporary safety property}
122\label{sigmacompactunsafe}
123\end{figure}
124
125We now proceed with the safety lemmas. The lemma in
126Figure~\ref{sigmacompactunsafe} is a temporary formulation of the main
127property {\tt sigma\_policy\_specification}. Its main difference from the
128final version is that it uses {\tt instruction\_size\_jmplen} to compute the
129instruction size. This function uses $j$ to compute the span of branch
130instructions  (i.e. it uses the $\sigma$ under construction), instead
131of looking at the distance between source and destination. This is because
132$\sigma$ is still under construction; we will prove below that after the
133final iteration, {\tt sigma\_compact\_unsafe} is equivalent to the main
134property.
135
136\begin{figure}[ht]
137\begin{alignat*}{6}
138\mathtt{sigma} & \omit\rlap{$\mathtt{\_safe} \equiv \lambda prefix.\lambda labels.\lambda old\_sigma.\lambda sigma$}\notag\\
139& \omit\rlap{$\forall i.i < |prefix| \rightarrow$} \notag\\
140& \omit\rlap{$\forall dest\_label.\mathtt{is\_jump\_to\ (\mathtt{nth}\ i\ prefix})\ dest\_label \rightarrow$} \notag\\
141& \mathbf{let} && \omit\rlap{$\ paddr \equiv \mathtt{lookup}\ labels\ dest\_label\ \mathbf{in}$} \notag\\
142& \mathbf{let} && \omit\rlap{$\ \langle j, src, dest \rangle \equiv$}\notag\\
143&&& \omit\rlap{$\mathbf{if} \ paddr\ \leq\ i\ \mathbf{then}$}\notag\\
144&&&&& \mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ sigma)\ \mathbf{in} \notag\\
145&&&&& \mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ (\mathtt{snd}\ sigma)\ \mathbf{in}\notag\\
146&&&&& \mathbf{let}\ \langle addr, \_ \rangle \equiv \mathtt{lookup}\ paddr\ (\mathtt{snd}\ sigma)\ \mathbf{in}\notag\\
147&&&&&   \langle j, pc\_plus\_jl, addr \rangle\notag\\
148&&&\mathbf{else} \notag\\
149&&&&&\mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ sigma)\ \mathbf{in} \notag\\
150&&&&&\mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ (\mathtt{snd}\ old\_sigma)\ \mathbf{in}\notag\\
153&&&\omit\rlap{$\mathbf{in}$}\notag\\
154&&&\mathbf{match} && \ j\ \mathbf{with} \notag\\
155&&&&&\mathrm{short\_jump} \Rightarrow \mathtt{short\_jump\_valid}\ src\ dest\notag\\
156&&&&&\mathrm{absolute\_jump} \Rightarrow \mathtt{absolute\_jump\_valid}\ src\ dest\notag\\
157&&&&&\mathrm{long\_jump} \Rightarrow \mathrm{True}
158\end{alignat*}
159\caption{Safety property}
160\label{sigmasafe}
161\end{figure}
162
163The lemma in figure~\ref{sigmasafe} is a safety property. It states
164that branch instructions are encoded properly, and that no wrong branch
165instructions are chosen.
166
167Note that we compute the distance using the memory address of the instruction
168plus its size. This follows the behaviour of the MCS-51 microprocessor, which
169increases the program counter directly after fetching, and only then executes
170the branch instruction (by changing the program counter again).
171
172We now encode some final, simple, properties to make sure that our policy
173remains consistent, and to keep track of whether the fixed point has been
174reached.
175
176\begin{align*}
177& \mathtt{lookup}\ 0\ (\mathtt{snd}\ policy) = 0 \notag\\
178& \mathtt{lookup}\ |prefix|\ (\mathtt{snd}\ policy) = \mathtt{fst}\ policy
179\end{align*}
180
181These two properties give the values of $\sigma$ for the start and end of the
182program; $\sigma(0) = 0$ and $\sigma(n)$, where $n$ is the number of
183instructions up until now, is equal to the maximum memory address so far.
184
185\begin{align*}
186& added = 0\ \rightarrow\ \mathtt{policy\_pc\_equal}\ prefix\ old\_sigma\ policy \notag\\
187& \mathtt{policy\_jump\_equal}\ prefix\ old\_sigma\ policy\ \rightarrow\ added = 0
188\end{align*}
189
190And finally, two properties that deal with what happens when the previous
191iteration does not change with respect to the current one. $added$ is a
192variable that keeps track of the number of bytes we have added to the program
193size by changing the encoding of branch instructions. If $added$ is 0, the program
194has not changed and vice versa.
195
196We need to use two different formulations, because the fact that $added$ is 0
197does not guarantee that no branch instructions have changed.  For instance,
198it is possible that we have replaced a short jump with an absolute jump, which
199does not change the size of the branch instruction.
200
201Therefore {\tt policy\_pc\_equal} states that $old\_sigma_1(x) = sigma_1(x)$,
202whereas {\tt policy\_jump\_equal} states that $old\_sigma_2(x) = sigma_2(x)$.
203This formulation is sufficient to prove termination and compactness.
204
205Proving these invariants is simple, usually by induction on the prefix length.
206
207\subsection{Iteration invariants}
208
209These are invariants that hold after the completion of an iteration. The main
210difference between these invariants and the fold invariants is that after the
211completion of the fold, we check whether the program size does not supersede
21264 Kb, the maximum memory size the MCS-51 can address.
213
214The type of an iteration therefore becomes an option type: {\tt None} in case
215the program becomes larger than 64 Kb, or $\mathtt{Some}\ \sigma$
216otherwise. We also no longer pass along the number of bytes added to the
217program size, but a boolean that indicates whether we have changed something
218during the iteration or not.
219
220If an iteration returns {\tt None}, we have the following invariant:
221
222\begin{alignat*}{2}
223\mathtt{nec} & \mathtt{\_plus\_ultra} \equiv \lambda program.\lambda p. \notag\\
224&\neg(\forall i.i < |program|\ \rightarrow \notag\\
225& \mathtt{is\_jump}\ (\mathtt{nth}\ i\ program)\ \rightarrow \notag\\
226& \mathtt{lookup}\ i\ (\mathtt{snd}\ p) = \mathrm{long\_jump}).
227\end{alignat*}
228
229This invariant is applied to $old\_sigma$; if our program becomes too large
230for memory, the previous iteration cannot have every branch instruction encoded
231as a long jump. This is needed later in the proof of termination.
232
233If the iteration returns $\mathtt{Some}\ \sigma$, the invariants
234{\tt out\_of\_program\_none},\\
235{\tt not\_jump\_default}, {\tt jump\_increase},
236and the two invariants that deal with $\sigma(0)$ and $\sigma(n)$ are
237retained without change.
238
239Instead of using {\tt sigma\_compact\_unsafe}, we can now use the proper
240invariant:
241
242\begin{alignat*}{6}
243\mathtt{sigma} & \omit\rlap{$\mathtt{\_compact} \equiv \lambda program.\lambda sigma.$} \notag\\
244& \omit\rlap{$\forall n.n < |program|\ \rightarrow$} \notag\\
245& \mathbf{match}\ && \omit\rlap{$\mathtt{lookup\_opt}\ n\ (\mathtt{snd}\ sigma)\ \mathbf{with}$}\notag\\
246&&& \omit\rlap{$\mathrm{None}\ \Rightarrow\ \mathrm{False}$}\notag\\
247&&& \omit\rlap{$\mathrm{Some}\ \langle pc, j \rangle \Rightarrow$}\notag\\
248&&& \mathbf{match}\ && \mathtt{lookup\_opt}\ (n+1)\ (\mathtt{snd}\ sigma)\ \mathbf{with}\notag\\
249&&&&&   \mathrm{None}\ \Rightarrow\ \mathrm{False}\notag\\
250&&&&& \mathrm{Some} \langle pc1, j1 \rangle \Rightarrow\notag\\
251&&&&& \ \ pc1 = pc + \mathtt{instruction\_size}\ n\ (\mathtt{nth}\ n\ program)
252\end{alignat*}
253
254
255This is almost the same invariant as ${\tt sigma\_compact\_unsafe}$, but differs in that it
256computes the sizes of branch instructions by looking at the distance between
257position and destination using $\sigma$.
258
259In actual use, the invariant is qualified: $\sigma$ is compact if there have
260been no changes (i.e. the boolean passed along is {\tt true}). This is to
261reflect the fact that we are doing a least fixed point computation: the result
262is only correct when we have reached the fixed point.
263
264There is another, trivial, invariant in case the iteration returns
265$\mathtt{Some}\ \sigma$: it must hold that $\mathtt{fst}\ sigma < 2^{16}$.
266We need this invariant to make sure that addresses do not overflow.
267
268The invariants taken directly from the fold invariants are trivial to prove.
269
270The proof of {\tt nec\_plus\_ultra} goes as follows: if we return {\tt None},
271then the program size must be greater than 64 Kb. However, since the
272previous iteration did not return {\tt None} (because otherwise we would
273terminate immediately), the program size in the previous iteration must have
274been smaller than 64 Kb.
275
276Suppose that all the branch instructions in the previous iteration are
277encoded as long jumps. This means that all branch instructions in this
278iteration are long jumps as well, and therefore that both iterations are equal
279in the encoding of their branch instructions. Per the invariant, this means that
280$added = 0$, and therefore that all addresses in both iterations are equal.
281But if all addresses are equal, the program sizes must be equal too, which
282means that the program size in the current iteration must be smaller than
28364 Kb. This contradicts the earlier hypothesis, hence not all branch
284instructions in the previous iteration are encoded as long jumps.
285
286The proof of {\tt sigma\_compact} follows from {\tt sigma\_compact\_unsafe} and
287the fact that we have reached a fixed point, i.e. the previous iteration and
288the current iteration are the same. This means that the results of
289{\tt instruction\_size\_jmplen} and {\tt instruction\_size} are the same.
290
291\subsection{Final properties}
292
293These are the invariants that hold after $2n$ iterations, where $n$ is the
294program size (we use the program size for convenience; we could also use the
295number of branch instructions, but this is more complex). Here, we only
296need {\tt out\_of\_program\_none}, {\tt sigma\_compact} and the fact that
297$\sigma(0) = 0$.
298
299Termination can now be proved using the fact that there is a $k \leq 2n$, with
300$n$ the length of the program, such that iteration $k$ is equal to iteration
301$k+1$. There are two possibilities: either there is a $k < 2n$ such that this
302property holds, or every iteration up to $2n$ is different. In the latter case,
303since the only changes between the iterations can be from shorter jumps to
304longer jumps, in iteration $2n$ every branch instruction must be encoded as
305a long jump. In this case, iteration $2n$ is equal to iteration $2n+1$ and the
306fixed point is reached.
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