# source:src/ASM/TACAS2013-policy/proof.tex@3415

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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\begin{figure}[t]
7\small
8\begin{alignat*}{6}
9\mathtt{sigma}&\omit\rlap{$\mathtt{\_policy\_specification} \equiv 10\lambda program.\lambda sigma.$} \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\ ppc\ instruction\ \wedge\notag\\
18  &&&&& (pc + \mathtt{instruction\_size}\ sigma\ 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'\ \mathbf{in} \notag\\
21        &&&&&\ \mathtt{instruction\_size}\ sigma\ ppc'\ instruction' = 0)\ \wedge \notag\\
22        &&&&& pc + \mathtt{instruction\_size}\ sigma\ ppc\ instruction = 2^{16})
23\end{alignat*}
24\caption{Main correctness statement\label{statement}}
25\label{sigmapolspec}
26\end{figure}
27%
28Informally, this means that when fetching a pseudo-instruction at $ppc$, the
29translation by $\sigma$ of $ppc+1$ is the same as $\sigma(ppc)$ plus the size
30of the instruction at $ppc$.  That is, an instruction is placed consecutively
31after the previous one, and there are no overlaps. The rest of the statement deals with memory size: either the next instruction fits within memory ($next\_pc < 2^{16}$) or it ends exactly at the limit memory,
32in which case it must be the last translated instruction in the program (enforced by specfiying that the size of all subsequent instructions is 0: there may be comments or cost annotations that are not translated).
33
34Finally, we enforce that the program starts at address 0, i.e. $\sigma(0) = 0$. It may seem strange that we do not explicitly include a safety property stating that every jump instruction is of the right type with respect to its target (akin to the lemma from Figure~\ref{sigmasafe}), but this is not necessary. The distance is recalculated according to the instruction addresses from $\sigma$, which implicitly expresses safety.
35
36Since our computation is a least fixed point computation, we must prove
37termination in order to prove correctness: if the algorithm is halted after
38a number of steps without reaching a fixed point, the solution is not
39guaranteed to be correct. More specifically, branch instructions might be
40encoded which do not coincide with the span between their location and their
41destination.
42
43Proof of termination rests on the fact that the encoding of branch
44instructions can only grow larger, which means that we must reach a fixed point
45after at most $2n$ iterations, with $n$ the number of branch instructions in
46the program. This worst case is reached if at every iteration, we change the
47encoding of exactly one branch instruction; since the encoding of any branch
48instruction can change first from short to absolute, and then to long, there
49can be at most $2n$ changes.
50
51%The proof has been carried out using the Russell'' style from~\cite{Sozeau2006}.
52%We have proven some invariants of the {\sc f} function from the previous
53%section; these invariants are then used to prove properties that hold for every
54%iteration of the fixed point computation; and finally, we can prove some
55%properties of the fixed point.
56
57\subsection{Fold invariants}
58
59In this section, we present the invariants that hold during the fold of {\sc f}
60over the program. These will be used later on to prove the properties of the
61iteration. During the fixed point computation, the $\sigma$ function is
62implemented as a trie for ease of access; computing $\sigma(x)$ is achieved by
63looking up the value of $x$ in the trie. Actually, during the fold, the value
64we pass along is a pair $\mathbb{N} \times \mathtt{ppc\_pc\_map}$. The first
65component is the number of bytes added to the program so far with respect to
66the previous iteration, and the second component, {\tt ppc\_pc\_map}, is the
67actual $\sigma$ trie (which we'll call $strie$ to avoid confusion).
68%
69{\small
70\begin{alignat*}{2}
71\mathtt{out} & \mathtt{\_of\_program\_none} \equiv \lambda prefix.\lambda strie. \notag\\
72& \forall i.i < 2^{16} \rightarrow (i > |prefix| \leftrightarrow
73 \mathtt{lookup\_opt}\ i\ (\mathtt{snd}\ strie) = \mathtt{None})
74\end{alignat*}}
75%
76The first invariant states that any pseudo-address not yet examined is not
77present in the lookup trie.
78%
79{\small
80\begin{alignat*}{2}
81\mathtt{not} & \mathtt{\_jump\_default} \equiv \lambda prefix.\lambda strie.\forall i.i < |prefix| \rightarrow\notag\\
82& \neg\mathtt{is\_jump}\ (\mathtt{nth}\ i\ prefix) \rightarrow \mathtt{lookup}\ i\ (\mathtt{snd}\ strie) = \mathtt{short\_jump}
83\end{alignat*}}
84%
85This invariant states that when we try to look up the jump length of a
86pseudo-address where there is no branch instruction, we will get the default
87value, a short jump.
88%
89{\small
90\begin{alignat*}{4}
91\mathtt{jump} & \mathtt{\_increase} \equiv \lambda pc.\lambda op.\lambda p.\forall i.i < |prefix| \rightarrow \notag\\
92&       \mathbf{let}\  oj \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ op)\ \mathbf{in} \notag\\
93&       \mathbf{let}\ j \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ p)\ \mathbf{in}\ \mathtt{jmpleq}\ oj\ j
94\end{alignat*}}
95%
96This invariant states that between iterations (with $op$ being the previous
97iteration, and $p$ the current one), jump lengths either remain equal or
98increase. It is needed for proving termination.
99%
100\begin{figure}[h]
101\small
102\begin{alignat*}{6}
103\mathtt{sigma} & \omit\rlap{$\mathtt{\_compact\_unsafe} \equiv \lambda prefix.\lambda strie.\forall n.n < |prefix| \rightarrow$}\notag\\
104& \mathbf{match}\ && \omit\rlap{$\mathtt{lookup\_opt}\ n\ (\mathtt{snd}\ strie)\ \mathbf{with}$}\notag\\
105&&& \omit\rlap{$\mathtt{None} \Rightarrow \mathrm{False}$} \notag\\
106&&& \omit\rlap{$\mathtt{Some}\ \langle pc, j \rangle \Rightarrow$} \notag\\
107&&& \mathbf{match}\ && \mathtt{lookup\_opt}\ (n+1)\ (\mathtt{snd}\ strie)\ \mathbf{with}\notag\\
108&&&&& \mathtt{None} \Rightarrow \mathrm{False} \notag\\
109&&&&& \mathtt{Some}\ \langle pc_1, j_1 \rangle \Rightarrow
110                pc_1 = pc + \notag\\
111&&&&& \ \ \mathtt{instruction\_size\_jmplen}\ j\ (\mathtt{nth}\ n\ prefix)
112\end{alignat*}
113\caption{Temporary safety property}
114\label{sigmacompactunsafe}
115\end{figure}
116%
117We now proceed with the safety lemmas. The lemma in
118Figure~\ref{sigmacompactunsafe} is a temporary formulation of the main
119property {\tt sigma\_policy\_specification}. Its main difference from the
120final version is that it uses {\tt instruction\_size\_jmplen} to compute the
121instruction size. This function uses $j$ to compute the span of branch
122instructions  (i.e. it uses the $\sigma$ under construction), instead
123of looking at the distance between source and destination. This is because
124$\sigma$ is still under construction; we will prove below that after the
125final iteration, {\tt sigma\_compact\_unsafe} is equivalent to the main
126property in Figure~\ref{sigmasafe} which holds at the end of the computation.
127%
128\begin{figure}[h]
129\small
130\begin{alignat*}{6}
131\mathtt{sigma} & \omit\rlap{$\mathtt{\_safe} \equiv \lambda prefix.\lambda labels.\lambda old\_strie.\lambda strie.\forall i.i < |prefix| \rightarrow$} \notag\\
132& \omit\rlap{$\forall dest\_label.\mathtt{is\_jump\_to\ (\mathtt{nth}\ i\ prefix})\ dest\_label \rightarrow$} \notag\\
133& \mathbf{let} && \omit\rlap{$\ paddr \equiv \mathtt{lookup}\ labels\ dest\_label\ \mathbf{in}$} \notag\\
134& \mathbf{let} && \omit\rlap{$\ \langle j, src, dest \rangle \equiv \mathbf{if} \ paddr\ \leq\ i\ \mathbf{then}$}\notag\\
135&&&&& \mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ strie)\ \mathbf{in} \notag\\
136&&&&& \mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ (\mathtt{snd}\ strie)\ \mathbf{in}\notag\\
137&&&&& \mathbf{let}\ \langle addr, \_ \rangle \equiv \mathtt{lookup}\ paddr\ (\mathtt{snd}\ strie)\ \mathbf{in}\notag\\
138&&&&&   \langle j, pc\_plus\_jl, addr \rangle\notag\\
139&&&\mathbf{else} \notag\\
140&&&&&\mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ strie)\ \mathbf{in} \notag\\
141&&&&&\mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ (\mathtt{snd}\ old\_strie)\ \mathbf{in}\notag\\
143&&&&&\langle j, pc\_plus\_jl, addr \rangle \mathbf{in}\ \notag\\
144&&&\mathbf{match} && \ j\ \mathbf{with} \notag\\
145&&&&&\mathrm{short\_jump} \Rightarrow \mathtt{short\_jump\_valid}\ src\ dest\notag\\
146&&&&&\mathrm{absolute\_jump} \Rightarrow \mathtt{absolute\_jump\_valid}\ src\ dest\notag\\
147&&&&&\mathrm{long\_jump} \Rightarrow \mathrm{True}
148\end{alignat*}
149\caption{Safety property}
150\label{sigmasafe}
151\end{figure}
152%
153We compute the distance using the memory address of the instruction
154plus its size. This follows the behaviour of the MCS-51 microprocessor, which
155increases the program counter directly after fetching, and only then executes
156the branch instruction (by changing the program counter again).
157
158There are also some simple, properties to make sure that our policy
159remains consistent, and to keep track of whether the fixed point has been
160reached. We do not include them here in detail. Two of these properties give the values of $\sigma$ for the start and end of the program; $\sigma(0) = 0$ and $\sigma(n)$, where $n$ is the number of instructions up until now, is equal to the maximum memory address so far. There are also two properties that deal with what happens when the previous
161iteration does not change with respect to the current one. $added$ is a
162variable that keeps track of the number of bytes we have added to the program
163size by changing the encoding of branch instructions. If $added$ is 0, the program
164has not changed and vice versa.
165
166%{\small
167%\begin{align*}
168%& \mathtt{lookup}\ 0\ (\mathtt{snd}\ strie) = 0 \notag\\
169%& \mathtt{lookup}\ |prefix|\ (\mathtt{snd}\ strie) = \mathtt{fst}\ strie
170%\end{align*}}
171
172
173%{\small
174%\begin{align*}
175%& added = 0\ \rightarrow\ \mathtt{policy\_pc\_equal}\ prefix\ old\_strie\ strie \notag\\
176%& \mathtt{policy\_jump\_equal}\ prefix\ old\_strie\ strie\ \rightarrow\ added = 0
177%\end{align*}}
178
179We need to use two different formulations, because the fact that $added$ is 0
180does not guarantee that no branch instructions have changed.  For instance,
181it is possible that we have replaced a short jump with an absolute jump, which
182does not change the size of the branch instruction. Therefore {\tt policy\_pc\_equal} states that $old\_sigma_1(x) = sigma_1(x)$, whereas {\tt policy\_jump\_equal} states that $old\_sigma_2(x) = sigma_2(x)$. This formulation is sufficient to prove termination and compactness.
183
184Proving these invariants is simple, usually by induction on the prefix length.
185
186\subsection{Iteration invariants}
187
188These are invariants that hold after the completion of an iteration. The main
189difference between these invariants and the fold invariants is that after the
190completion of the fold, we check whether the program size does not supersede
19164 Kb, the maximum memory size the MCS-51 can address. The type of an iteration therefore becomes an option type: {\tt None} in case
192the program becomes larger than 64 Kb, or $\mathtt{Some}\ \sigma$
193otherwise. We also no longer pass along the number of bytes added to the
194program size, but a boolean that indicates whether we have changed something
195during the iteration or not.
196
197If the iteration returns {\tt None}, which means that it has become too large for memory, there is an invariant that states that the previous iteration cannot
198have every branch instruction encoded as a long jump. This is needed later in the proof of termination. If the iteration returns $\mathtt{Some}\ \sigma$, the fold invariants are retained without change.
199
200Instead of using {\tt sigma\_compact\_unsafe}, we can now use the proper
201invariant:
202%
203{\small
204\begin{alignat*}{6}
205\mathtt{sigma} & \omit\rlap{$\mathtt{\_compact} \equiv \lambda program.\lambda sigma.$} \notag\\
206& \omit\rlap{$\forall n.n < |program|\ \rightarrow$} \notag\\
207& \mathbf{match}\ && \omit\rlap{$\mathtt{lookup\_opt}\ n\ (\mathtt{snd}\ sigma)\ \mathbf{with}$}\notag\\
208&&& \omit\rlap{$\mathrm{None}\ \Rightarrow\ \mathrm{False}$}\notag\\
209&&& \omit\rlap{$\mathrm{Some}\ \langle pc, j \rangle \Rightarrow$}\notag\\
210&&& \mathbf{match}\ && \mathtt{lookup\_opt}\ (n+1)\ (\mathtt{snd}\ sigma)\ \mathbf{with}\notag\\
211&&&&&   \mathrm{None}\ \Rightarrow\ \mathrm{False}\notag\\
212&&&&& \mathrm{Some} \langle pc1, j1 \rangle \Rightarrow\notag\\
213&&&&& \ \ pc1 = pc + \mathtt{instruction\_size}\ n\ (\mathtt{nth}\ n\ program)
214\end{alignat*}}
215%
216This is almost the same invariant as ${\tt sigma\_compact\_unsafe}$, but differs in that it
217computes the sizes of branch instructions by looking at the distance between
218position and destination using $\sigma$. In actual use, the invariant is qualified: $\sigma$ is compact if there have
219been no changes (i.e. the boolean passed along is {\tt true}). This is to
220reflect the fact that we are doing a least fixed point computation: the result
221is only correct when we have reached the fixed point.
222
223There is another, trivial, invariant in case the iteration returns
224$\mathtt{Some}\ \sigma$: it must hold that $\mathtt{fst}\ sigma < 2^{16}$.
225We need this invariant to make sure that addresses do not overflow.
226
227The proof of {\tt nec\_plus\_ultra} goes as follows: if we return {\tt None},
228then the program size must be greater than 64 Kb. However, since the
229previous iteration did not return {\tt None} (because otherwise we would
230terminate immediately), the program size in the previous iteration must have
231been smaller than 64 Kb.
232
233Suppose that all the branch instructions in the previous iteration are
234encoded as long jumps. This means that all branch instructions in this
235iteration are long jumps as well, and therefore that both iterations are equal
236in the encoding of their branch instructions. Per the invariant, this means that
237$added = 0$, and therefore that all addresses in both iterations are equal.
238But if all addresses are equal, the program sizes must be equal too, which
239means that the program size in the current iteration must be smaller than
24064 Kb. This contradicts the earlier hypothesis, hence not all branch
241instructions in the previous iteration are encoded as long jumps.
242
243The proof of {\tt sigma\_compact} follows from {\tt sigma\_compact\_unsafe} and
244the fact that we have reached a fixed point, i.e. the previous iteration and
245the current iteration are the same. This means that the results of
246{\tt instruction\_size\_jmplen} and {\tt instruction\_size} are the same.
247
248\subsection{Final properties}
249
250These are the invariants that hold after $2n$ iterations, where $n$ is the
251program size (we use the program size for convenience; we could also use the
252number of branch instructions, but this is more complex). Here, we only
253need {\tt out\_of\_program\_none}, {\tt sigma\_compact} and the fact that
254$\sigma(0) = 0$.
255
256Termination can now be proved using the fact that there is a $k \leq 2n$, with
257$n$ the length of the program, such that iteration $k$ is equal to iteration
258$k+1$. There are two possibilities: either there is a $k < 2n$ such that this
259property holds, or every iteration up to $2n$ is different. In the latter case,
260since the only changes between the iterations can be from shorter jumps to
261longer jumps, in iteration $2n$ every branch instruction must be encoded as
262a long jump. In this case, iteration $2n$ is equal to iteration $2n+1$ and the
263fixed point is reached.
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