Changeset 2096

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Timestamp:
Jun 15, 2012, 3:25:21 PM (9 years ago)
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Changes to the English for Jaap, and some tidying up and making consistent with the other CPP submission.

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src/ASM/CPP2012-policy
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4 edited

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 r2091 Given the NP-completeness of the problem, to arrive at an optimal solution within a short space of time (using, for example, a constraint solver) will potentially take a great amount of time. (using, for example, a constraint solver) will potentially take a great amount of time. The SDCC compiler~\cite{SDCC2011}, which has the MCS-51 among its target instruction sets, simply encodes every branch instruction as a long jump The SDCC compiler~\cite{SDCC2011}, which has a backend targetting the MCS-51 instruction set, simply encodes every branch instruction as a long jump without taking the distance into account. While certainly correct (the long jump can reach any destination in memory) and rapid, it does result in a less than optimal solution. jump can reach any destination in memory) and a very fast solution to compute, it results in a less than optimal solution. The {\tt gcc} compiler suite~\cite{GCC2012}, while compiling C on the x86 architecture, uses a greatest fix point algorithm. In other words, it starts off with all branch instructions encoded as the largest jumps available, and then tries to reduce branch instructions as much as possible. On the other hand, the {\tt gcc} compiler suite~\cite{GCC2012}, while compiling C on the x86 architecture, uses a greatest fix point algorithm. In other words, it starts with all branch instructions encoded as the largest jumps available, and then tries to reduce the size of branch instructions as much as possible. Such an algorithm has the advantage that any intermediate result it returns jump is always possible, and the algorithm only reduces those branch instructions whose destination address is in range for a shorter jump. The algorithm can thus be stopped after a determined amount of steps without losing correctness. The algorithm can thus be stopped after a determined number of steps without sacrificing correctness. The result, however, is not necessarily optimal, even if the algorithm is run until it terminates naturally: the fixed point reached is the {\em greatest} The result, however, is not necessarily optimal. Even if the algorithm is run until it terminates naturally, the fixed point reached is the {\em greatest} fixed point, not the least fixed point. Furthermore, {\tt gcc} (at least for the x86 architecture) only uses short and long jumps. This makes the algorithm more rapid, as shown in the previous section, but also results in a less more efficient, as shown in the previous section, but also results in a less optimal solution. encoding can only switch from short to long, but never in the other direction. When we add absolute jumps, however, it is theoretically possible for a branch instruction to switch from absolute to long and back, as shown in the previous section. instruction to switch from absolute to long and back, as previously explained. Proving termination then becomes difficult, because there is nothing that precludes a branch instruction from switching back and forth between absolute and long indefinitely. precludes a branch instruction from oscillating back and forth between absolute and long jumps indefinitely. In order to keep the algorithm in the same complexity class and more easily from absolute to long. There is one complicating factor: suppose that a branch instruction is encoded There is one complicating factor. Suppose that a branch instruction is encoded in step $n$ as an absolute jump, but in step $n+1$ it is determined that (because of changes elsewhere) it can now be encoded as a short jump. Due to $n+1$ as well. This is not necessarily correct, however: a branch instruction that can be encoded as a short jump cannot always also be encoded as an absolute jump (a short jump can bridge segments, whereas an absolute jump cannot). Therefore, in this situation we decide to encode the branch instruction as a long jump, which is always correct. This is not necessarily correct. A branch instruction that can be encoded as a short jump cannot always also be encoded as an absolute jump, as a short jump can bridge segments, whereas an absolute jump cannot. Therefore, in this situation we have decided to encode the branch instruction as a long jump, which is always correct. The resulting algorithm, while not optimal, is at least as good as the ones The branch displacement algorithm forms part of the translation from pseudo-code to assembler. More specifically, it is used by the function that pseudocode to assembler. More specifically, it is used by the function that translates pseudo-addresses (natural numbers indicating the position of the instruction in the program) to actual addresses in memory. The original intention was to have two different functions, one function Our original intention was to have two different functions, one function $\mathtt{policy}: \mathbb{N} \rightarrow \{\mathtt{short\_jump}, \mathtt{absolute\_jump}, \mathtt{long\_jump}\}$ to associate jumps to their intended encoding, and a function $\sigma: \mathbb{N} \rightarrow \mathtt{Word}$ to associate pseudo-addresses to actual addresses. $\sigma$ \mathtt{Word}$to associate pseudo-addresses to machine addresses.$\sigma$would use$\mathtt{policy}$to determine the size of jump instructions. From the algorithmic point of view, in order to create the$\mathtt{policy}$function, we must necessarily have a translation from pseudo-addresses to actual addresses (i.e. a$\sigma$function): in order to judge the distance to machine addresses (i.e. a$\sigma$function): in order to judge the distance between a jump and its destination, we must know their memory locations. Conversely, in order to create the$\sigma$function, we need to have the algorithms. We now have a function$\sigma: \mathbb{N} \rightarrow \mathtt{Word} \times \mathtt{bool}$which associates a pseudo-address to an actual address. The boolean denotes a forced associates a pseudo-address to a machine address. The boolean denotes a forced long jump; as noted in the previous section, if during the fixed point computation an absolute jump changes to be potentially re-encoded as a short \end{figure} The algorithm, shown in figure~\ref{f:jump_expansion_step}, works by folding the The algorithm, shown in Figure~\ref{f:jump_expansion_step}, works by folding the function {\sc f} over the entire program, thus gradually constructing$sigma$. This constitutes one step in the fixed point calculation; successive steps The first two are parameters that remain the same through one iteration, the last three are standard parameters for a fold function (including$ppc$, final three are standard parameters for a fold function (including$ppc$, which is simply the number of instructions of the program already processed).$\sigma: \mathbb{N} \rightarrow \mathbb{N} \times \{\mathtt{short\_jump}, \mathtt{absolute\_jump},\mathtt{long\_jump}\}$; a function that associates a pseudo-address with an memory address and a jump length. We do this to be able to more easily compare the jump lengths between iterations. In the algorithm, pseudo-address with a memory address and a jump length. We do this to be able to more ease the comparison of jump lengths between iterations. In the algorithm, we use the notation$sigma_1(x)$to denote the memory address corresponding to$x$, and$sigma_2(x)$to denote the jump length corresponding to$x$. We cannot use$old\_sigma$without change: it might be the case that we have already increased the size some branch instructions before, making the program already increased the size of some branch instructions before, making the program longer and moving every instruction forward. We must compensate for this by adding the size increase of the program to the label's memory address according Note also that we add the pc to$sigma$at location$ppc+1$, whereas we add the jump length at location$ppc$. We do this so that$sigma(ppc)$will always return a couple with the start address of the instruction at$ppc$and the return a pair with the start address of the instruction at$ppc$and the length of its branch instruction (if any); the end address of the program can be found at$sigma(n+1)$, where$n$is the number of instructions in the • src/ASM/CPP2012-policy/conclusion.tex  r2093 \section{Conclusion} In the previous sections, we have discussed the branch displacement optimisation In the previous sections we have discussed the branch displacement optimisation problem, presented an optimised solution, and discussed the proof of termination and correctness for this algorithm, as formalised in Matita. optimal solution would need techniques like constraint solvers. While outside the scope of the present research, it would be interesting to see if enough heuristics could be find to make such a solution practical for implementing in an existent compiler; this would be especially useful for embedded systems, where it is important to have a small solution as possible. heuristics could be found to make such a solution practical for implementing in an existing compiler; this would be especially useful for embedded systems, where it is important to have as small solution as possible. This algorithm is part of a greater whole, the CerCo project, which aims at the complete formalisation and verification of a compiler. More information This algorithm is part of a greater whole, the CerCo project, which aims to complete formalise and verify a concrete cost preserving compiler for a large subset of the C programming language. More information on the formalisation of the assembler, of which the present work is a part, can be found in a companion publication~\cite{DC2012}. displacement optimisation algorithm. The CompCert project is another project aimed at formally verifying a compiler. Their backend~\cite{Leroy2009} generates assembly code for (amongst others) the The CompCert project is another verified compiler project. Their backend~\cite{Leroy2009} generates assembly code for (amongst others) subsets of the PowerPC and x86 (32-bit) architectures. At the assembly code stage, there is no distinction between the span-dependent jump instructions, so a branch An offshoot of the CompCert project is the CompCertTSO project, who add thread concurrency and synchronisation to the CompCert compiler~\cite{TSO2011}. This compiler also generate assembly code and therefore does not include a branch compiler also generates assembly code and therefore does not include a branch displacement algorithm. There is also Piton~\cite{Moore1996}, which not only includes the Finally, there is also the Piton stack~\cite{Moore1996}, which not only includes the formal verification of a compiler, but also of the machine architecture targeted by that compiler. However, this architecture does not have different targeted by that compiler, a bespoke microprocessor called the FM9001. However, this architecture does not have different jump sizes (branching is simulated by assigning values to the program counter), so the branch displacement problem does not occur. so the branch displacement problem is irrelevant. \subsection{Formal development} • src/ASM/CPP2012-policy/problem.tex  r2091 fact that in many architecture sets, the encoding (and therefore size) of some instructions depends on the distance to their operand (the instruction 'span'). The branch displacement optimisation problem consists in encoding these The branch displacement optimisation problem consists of encoding these span-dependent instructions in such a way that the resulting program is as small as possible. This problem is the subject of the present paper. After introducing the problem in more detail, we will discuss the solutions used by other compilers, present the algorithm used by us in the CerCo assembler, and discuss its verification, that is the proofs of termination and correctness using the Matita theorem prover~\cite{Asperti2007}. the algorithm we use in the CerCo assembler, and discuss its verification, that is the proofs of termination and correctness using the Matita proof assistant~\cite{Asperti2007}. The research presented in this paper has been executed within the CerCo project target architecture for this project is the MCS-51, whose instruction set contains span-dependent instructions. Furthermore, its maximum addressable memory size is very small (65 Kbytes), which makes it important to generate memory size is very small (64 Kb), which makes it important to generate programs that are as small as possible. In most modern instruction sets that have them, the only span-dependent instructions are branch instructions. Taking the ubiquitous x86-64 instruction set as an example, we find that it contains are eleven different forms of the set as an example, we find that it contains eleven different forms of the unconditional branch instruction, all with different ranges, instruction sizes and semantics (only six are valid in 64-bit mode, for example). Some examples are shown in figure~\ref{f:x86jumps}. are shown in Figure~\ref{f:x86jumps}. \begin{figure}[h] The chosen target architecture of the CerCo project is the Intel MCS-51, which features three types of branch instructions (or jump instructions; the two terms are used interchangeably), as shown in figure~\ref{f:mcs51jumps}. are used interchangeably), as shown in Figure~\ref{f:mcs51jumps}. \begin{figure}[h] encoded using three branch instructions (for instructions whose logical negation is available, it can be done with two branch instructions, but for some instructions, this opposite is not available); the call instruction is some instructions this is not available); the call instruction is only available in absolute and long forms. Note that even though the MCS-51 architecture is much less advanced and more simple than the x86-64 architecture, the basic types of branch instruction Note that even though the MCS-51 architecture is much less advanced and simpler than the x86-64 architecture, the basic types of branch instruction remain the same: a short jump with a limited range, an intra-segment jump and a jump that can reach the entire available memory. Generally, in the code that is sent to the assembler as input, the only difference made between branch instructions is by semantics, not by span. This Generally, in code fed to the assembler as input, the only difference between branch instructions is semantics, not span. This means that a distinction is made between an unconditional branch and the several kinds of conditional branch, but not between their short, absolute or The algorithm used by the assembler to encode these branch instructions into the different machine instructions is known as the {\em branch displacement algorithm}. The optimisation problem consists in using as small an encoding as algorithm}. The optimisation problem consists of finding as small an encoding as possible, thus minimising program length and execution time. The canonical solution, as shown by Szymanski~\cite{Szymanski1978} or more recently by Dickson~\cite{Dickson2008} for the x86 instruction set, is to use a fixed point algorithm that starts out with the shortest possible encoding (all branch instruction encoded as short jumps, which is very probably not a correct fixed point algorithm that starts with the shortest possible encoding (all branch instruction encoded as short jumps, which is likely not a correct solution) and then iterates over the program to re-encode those branch instructions whose target is outside their range. situation where the span of$j$is so small that it can be encoded as a short jump. This argument continues to hold throughout the subsequent iterations of the algorithm: short jumps can change into long jumps, but not vice versa (spans only increase). Hence, the algorithm either terminates when a fixed the algorithm: short jumps can change into long jumps, but not \emph{vice versa}, as spans only increase. Hence, the algorithm either terminates early when a fixed point is reached or when all short jumps have been changed into long jumps. However, neither of these claims (termination nor optimality) hold when we add the absolute jump. The reason for this is that with absolute jumps, the encoding of a branch the absolute jump, as with absolute jumps, the encoding of a branch instruction no longer depends only on the distance between the branch instruction and its target: in order for an absolute jump to be possible, they long if this is not the case). This invalidates the termination argument: a branch instruction, once encoded \begin{figure}[ht] \begin{alltt} jmp X \vdots L$$\sb{0}$$: \vdots jmp L$$\sb{0}$$ \end{alltt} \caption{Example of a program where a long jump becomes absolute} \label{f:term_example} \end{figure} This invalidates our earlier termination argument: a branch instruction, once encoded as a long jump, can be re-encoded during a later iteration as an absolute jump. Consider the program shown in figure~\ref{f:term_example}. At the start of the Consider the program shown in Figure~\ref{f:term_example}. At the start of the first iteration, both the branch to {\tt X} and the branch to$\mathtt{L}_{0}$are encoded as small jumps. Let us assume that in this case, the placement of Hence, in the third iteration, the branch to$\mathtt{L}_{0}$can be encoded as an absolute jump. At first glance, there is nothing that prevents us from making a construction where two branch instructions interact in such a way as to keep switching between long and absolute encodings for an indefinite amount of iterations. \begin{figure}[h] \begin{alltt} jmp X \vdots L$$\sb{0}$$: \vdots jmp L$$\sb{0}$$ \end{alltt} \caption{Example of a program where a long jump becomes absolute} \label{f:term_example} \end{figure} In fact, this situation mirrors the explanation by Szymanski~\cite{Szymanski1978} of why the branch displacement optimisation problem is NP-complete. In this explanation, a condition for NP-completeness is the fact that programs be allowed to contain {\em pathological} jumps. These are branch instructions that can normally not be encoded as a short(er) jump, but gain this property when some other branch instructions are encoded as a long(er) jump. This is exactly what happens in figure~\ref{f:term_example}: by encoding the first branch instruction as a long jump, another branch instructions switches from long to absolute (which is shorter). The optimality argument no longer holds either. Let us consider the program shown in figure~\ref{f:opt_example}. Suppose that the distance between constructing a configuration where two branch instructions interact in such a way as to iterate indefinitely between long and absolute encodings. This situation mirrors the explanation by Szymanski~\cite{Szymanski1978} of why the branch displacement optimisation problem is NP-complete. In this explanation, a condition for NP-completeness is the fact that programs be allowed to contain {\em pathological} jumps. These are branch instructions that can normally not be encoded as a short(er) jump, but gain this property when some other branch instructions are encoded as a long(er) jump. This is exactly what happens in Figure~\ref{f:term_example}. By encoding the first branch instruction as a long jump, another branch instruction switches from long to absolute (which is shorter). In addition, our previous optimality argument no longer holds. Consider the program shown in Figure~\ref{f:opt_example}. Suppose that the distance between$\mathtt{L}_{0}$and$\mathtt{L}_{1}$is such that if {\tt jmp X} is encoded as a short jump, there is a segment border just after$\mathtt{L}_{1}$. Let segment, but far enough away from$\mathtt{L}_{1}$that they cannot be encoded as short jumps. \begin{figure}[ht] \begin{alltt} L$$\sb{0}$$: jmp X X: \vdots L$$\sb{1}$$: \vdots jmp L$$\sb{1}$$ \vdots jmp L$$\sb{1}$$ \vdots jmp L$$\sb{1}$$ \vdots \end{alltt} \caption{Example of a program where the fixed-point algorithm is not optimal} \label{f:opt_example} \end{figure} Then, if {\tt jmp X} were to be encoded as a short jump, which is clearly this solution might actually be smaller than the one reached by the smallest fixed point algorithm. \begin{figure}[h] \begin{alltt} L$$\sb{0}$$: jmp X X: \vdots L$$\sb{1}$$: \vdots jmp L$$\sb{1}$$ \vdots jmp L$$\sb{1}$$ \vdots jmp L$$\sb{1}$$ \vdots \end{alltt} \caption{Example of a program where the fixed-point algorithm is not optimal} \label{f:opt_example} \end{figure} • src/ASM/CPP2012-policy/proof.tex  r2091 \section{The proof} In this section, we will present the correctness proof of the algorithm in more detail. The main correctness statement is as follows: In this section, we present the correctness proof for the algorithm in more detail. The main correctness statement is as follows (slightly simplified, here): \clearpage \begin{lstlisting} definition sigma_policy_specification :=:$\lambda$program: pseudo_assembly_program.$\lambda$sigma: Word → Word.$\lambda$policy: Word → bool. sigma (zero$\ldots$) = zero$\ldots\wedge\forall$ppc: Word.$\forall$ppc_ok. let instr_list := \snd program in let pc ≝ sigma ppc in let labels := \fst (create_label_cost_map (\snd program)) in let lookup_labels :=$\lambda$x.bitvector_of_nat ? (lookup_def ?? labels x 0) in let instruction := \fst (fetch_pseudo_instruction (\snd program) ppc ppc_ok) in let next_pc := \fst (sigma (add ? ppc (bitvector_of_nat ? 1))) in (nat_of_bitvector$\ldots$ppc ≤ |instr_list| → next_pc = add ? pc (bitvector_of_nat$\ldots$(instruction_size lookup_labels sigma policy ppc instruction)))$\wedge$((nat_of_bitvector$\ldots$ppc < |instr_list| → nat_of_bitvector$\ldots$pc < nat_of_bitvector$\ldots$next_pc)$\vee$(nat_of_bitvector$\ldots$ppc = |instr_list| → next_pc = (zero$\ldots$))). definition sigma_policy_specification :=$\lambda$program: pseudo_assembly_program.$\lambda$sigma: Word$\rightarrow$Word.$\lambda$policy: Word$\rightarrow$bool. sigma (zero$\ldots$) = zero$\ldots\wedge\forall$ppc: Word.$\forall$ppc_ok. let$\langle$preamble, instr_list$\rangle$:= program in let pc := sigma ppc in let instruction := \fst (fetch_pseudo_instruction instr_list ppc ppc_ok) in let next_pc := \fst (sigma (add ? ppc (bitvector_of_nat ? 1))) in (nat_of_bitvector$\ldots$ppc$\leq$|instr_list|$\rightarrow$next_pc = add ? pc (bitvector_of_nat$\ldots$(instruction_size$\ldots$sigma policy ppc instruction)))$\wedge$((nat_of_bitvector$\ldots$ppc < |instr_list|$\rightarrow$nat_of_bitvector$\ldots$pc < nat_of_bitvector$\ldots$next_pc)$\vee$(nat_of_bitvector$\ldots$ppc = |instr_list|$\rightarrow$next_pc = (zero$\ldots$))). \end{lstlisting} Informally, this means that when fetching a pseudo-instruction at$ppc$, the translation by$\sigma$of$ppc+1$is the same as$\sigma(ppc)$plus the size of the instruction at$ppc$; i.e. an instruction is placed immediately of the instruction at$ppc$. That is, an instruction is placed consecutively after the previous one, and there are no overlaps. The other condition enforced is the fact that instructions are stocked in Instructions are also stocked in order: the memory address of the instruction at$ppc$should be smaller than the memory address of the instruction at$ppc+1$. There is one exeception to to the amount of memory). And finally, we enforce that the program starts at address 0, i.e.$\sigma(0) = 0$. Finally, we enforce that the program starts at address 0, i.e.$\sigma(0) = 0$. Since our computation is a least fixed point computation, we must prove a number of steps without reaching a fixed point, the solution is not guaranteed to be correct. More specifically, branch instructions might be encoded who do not coincide with the span between their location and their encoded which do not coincide with the span between their location and their destination. long, there can be at most$2n$changes. This proof has been executed in the Russell'' style from~\cite{Sozeau2006}. The proof has been carried out using the Russell'' style from~\cite{Sozeau2006}. We have proven some invariants of the {\sc f} function from the previous section; these invariants are then used to prove properties that hold for every Note that during the fixed point computation, the$\sigma$function is implemented as a trie for ease access; computing$\sigma(x)$is done by looking implemented as a trie for ease of access; computing$\sigma(x)$is achieved by looking up the value of$x$in the trie. Actually, during the fold, the value we pass along is a couple$\mathbb{N} \times \mathtt{ppc_pc_map}$. The natural number is the number of bytes added to the program so far with respect to the previous iteration, and {\tt ppc\_pc\_map} is a couple of the current size of the program and our$\sigma$function. pass along is a pair$\mathbb{N} \times \mathtt{ppc_pc_map}$. The first component is the number of bytes added to the program so far with respect to the previous iteration, and the second component, {\tt ppc\_pc\_map}, is a pair consisting of the current size of the program and our$\sigma$function. \begin{lstlisting} definition out_of_program_none :=$\lambda$prefix:list labelled_instruction.$\lambda$sigma:ppc_pc_map.$\forall$i.i < 2^16 → (i > |prefix|$\leftrightarrow\forall$i.i < 2^16$\rightarrow$(i > |prefix|$\leftrightarrow$bvt_lookup_opt$\ldots$(bitvector_of_nat ? i) (\snd sigma) = None ?). \end{lstlisting} This invariant states that every pseudo-address not yet treated cannot be found in the lookup trie. \begin{lstlisting} definition not_jump_default ≝ This invariant states that any pseudo-address not yet examined is not present in the lookup trie. \begin{lstlisting} definition not_jump_default :=$\lambda$prefix:list labelled_instruction.$\lambda$sigma:ppc_pc_map.$\forall$i.i < |prefix| → ¬is_jump (\snd (nth i ? prefix$\langle$None ?, Comment []$\rangle$)) →$\forall$i.i < |prefix|$\rightarrow$¬is_jump (\snd (nth i ? prefix$\langle$None ?, Comment []$\rangle$))$\rightarrow$\snd (bvt_lookup$\ldots$(bitvector_of_nat ? i) (\snd sigma)$\langle$0,short_jump$\rangle$) = short_jump. \begin{lstlisting} definition jump_increase := λprefix:list labelled_instruction.λop:ppc_pc_map.λp:ppc_pc_map. ∀i.i ≤ |prefix| →$\lambda$prefix:list labelled_instruction.$\lambda$op:ppc_pc_map.$\lambda$p:ppc_pc_map.$\forall$i.i$\leq$|prefix|$\rightarrow$let$\langle$opc,oj$\rangle$:= bvt_lookup$\ldots$(bitvector_of_nat ? i) (\snd op)$\langle$0,short_jump$\rangle$in \end{lstlisting} This invariant states that between iterations ($op$being the previous This invariant states that between iterations (with$op$being the previous iteration, and$p$the current one), jump lengths either remain equal or increase. It is needed for proving termination. \begin{lstlisting} definition sigma_compact_unsafe := λprogram:list labelled_instruction.λlabels:label_map.λsigma:ppc_pc_map. ∀n.n < |program| →$\lambda$program:list labelled_instruction.$\lambda$labels:label_map.$\lambda$sigma:ppc_pc_map.$\forall$n.n < |program|$\rightarrow$match bvt_lookup_opt$\ldots$(bitvector_of_nat ? n) (\snd sigma) with [ None ⇒ False | Some x ⇒ let$\langle$pc,j$\rangle$:= x in [ None$\Rightarrow$False | Some x$\Rightarrow$let$\langle$pc,j$\rangle$:= x in match bvt_lookup_opt$\ldots$(bitvector_of_nat ? (S n)) (\snd sigma) with [ None ⇒ False | Some x1 ⇒ let$\langle$pc1,j1$\rangle$≝ x1 in [ None$\Rightarrow$False | Some x1$\Rightarrow$let$\langle$pc1,j1$\rangle$:= x1 in pc1 = pc + instruction_size_jmplen j (\snd (nth n ? program$\langle$None ?, Comment []$\rangle$))) This is a temporary formulation of the main property ({\tt sigma\_policy\_specification}); its main difference with the final version is that it uses {\tt instruction\_size\_jmplen} to from the final version is that it uses {\tt instruction\_size\_jmplen} to compute the instruction size. This function uses$j$to compute the span of branch instructions (i.e. it uses the$\sigma$function under construction), \begin{lstlisting} definition sigma_safe := λprefix:list labelled_instruction.λlabels:label_map.λadded:$\mathbb{N}$. λold_sigma:ppc_pc_map.λsigma:ppc_pc_map. ∀i.i < |prefix| → let$\langle$pc,j$\rangle$:=$\lambda$prefix:list labelled_instruction.$\lambda$labels:label_map.$\lambda$added:$\mathbb{N}$.$\lambda$old_sigma:ppc_pc_map.$\lambda$sigma:ppc_pc_map.$\forall$i.i < |prefix|$\rightarrow$let$\langle$pc,j$\rangle$:= bvt_lookup$\ldots$(bitvector_of_nat ? i) (\snd sigma)$\langle$0,short_jump$\rangle$in let pc_plus_jmp_length := bitvector_of_nat ? (\fst (bvt_lookup$\ldots$Note that we compute the distance using the memory address of the instruction plus its size: this is due to the behaviour of the MCS-51 architecture, which plus its size: this follows the behaviour of the MCS-51 microprocessor, which increases the program counter directly after fetching, and only then executes the branch instruction (by changing the program counter again). \begin{lstlisting} (added = 0 → policy_pc_equal prefix old_sigma policy)) (policy_jump_equal prefix old_sigma policy → added = 0)) (added = 0$\rightarrow$policy_pc_equal prefix old_sigma policy)) (policy_jump_equal prefix old_sigma policy$\rightarrow$added = 0)) \end{lstlisting} And finally, two properties that deal with what happens when the previous iteration does not change with respect to the current one.$added$is the iteration does not change with respect to the current one.$added$is a variable that keeps track of the number of bytes we have added to the program size by changing the encoding of branch instructions; if this is 0, the program size by changing the encoding of branch instructions. If$added$is 0, the program has not changed and vice versa. We need to use two different formulations, because the fact that$added$is 0 does not guarantee that no branch instructions have changed: it is possible that we have replaced a short jump with a absolute jump, which does not change the size of the branch instruction. does not guarantee that no branch instructions have changed. For instance, it is possible that we have replaced a short jump with a absolute jump, which does not change the size of the branch instruction. Therefore {\tt policy\_pc\_equal} states that$old\_sigma_1(x) = sigma_1(x)$, difference between these invariants and the fold invariants is that after the completion of the fold, we check whether the program size does not supersede 65 Kbytes (the maximum memory size the MCS-51 can address). 64 Kb, the maximum memory size the MCS-51 can address. The type of an iteration therefore becomes an option type: {\tt None} in case the program becomes larger than 65 KBytes, or$\mathtt{Some}\ \sigma$the program becomes larger than 64 Kb, or$\mathtt{Some}\ \sigma$otherwise. We also no longer use a natural number to pass along the number of bytes added to the program size, but a boolean that indicates whether we have \begin{lstlisting} definition nec_plus_ultra := λprogram:list labelled_instruction.λp:ppc_pc_map. ¬(∀i.i < |program| → is_jump (\snd (nth i ? program$\langle$None ?, Comment []$\rangle$)) →$\lambda$program:list labelled_instruction.$\lambda$p:ppc_pc_map. ¬($\forall$i.i < |program|$\rightarrow$is_jump (\snd (nth i ? program$\langle$None ?, Comment []$\rangle$))$\rightarrow$\snd (bvt_lookup$\ldots$(bitvector_of_nat 16 i) (\snd p)$\langle$0,short_jump$\rangle$) = long_jump). This invariant is applied to$old\_sigma$; if our program becomes too large for memory, the previous iteration cannot have every branch instruction encoded as a long jump. This is needed later on in the proof of termination. as a long jump. This is needed later in the proof of termination. If the iteration returns$\mathtt{Some}\ \sigma$, the invariants \begin{lstlisting} definition sigma_compact := λprogram:list labelled_instruction.λlabels:label_map.λsigma:ppc_pc_map. ∀n.n < |program| →$\lambda$program:list labelled_instruction.$\lambda$labels:label_map.$\lambda$sigma:ppc_pc_map.$\forall$n.n < |program|$\rightarrow$match bvt_lookup_opt$\ldots$(bitvector_of_nat ? n) (\snd sigma) with [ None ⇒ False | Some x ⇒ let$\langle$pc,j$\rangle$:= x in [ None$\Rightarrow$False | Some x$\Rightarrow$let$\langle$pc,j$\rangle$:= x in match bvt_lookup_opt$\ldots$(bitvector_of_nat ? (S n)) (\snd sigma) with [ None ⇒ False | Some x1 ⇒ let$\langle$pc1,j1$\rangle$:= x1 in [ None$\Rightarrow$False | Some x1$\Rightarrow$let$\langle$pc1,j1$\rangle$:= x1 in pc1 = pc + instruction_size (λid.bitvector_of_nat ? (lookup_def ?? labels id 0)) (λppc.bitvector_of_nat ? ($\lambda$id.bitvector_of_nat ? (lookup_def ?? labels id 0)) ($\lambda$ppc.bitvector_of_nat ? (\fst (bvt_lookup$\ldots$ppc (\snd sigma)$\langle$0,short_jump$\rangle$))) (λppc.jmpeqb long_jump (\snd (bvt_lookup$\ldots$ppc ($\lambda$ppc.jmpeqb long_jump (\snd (bvt_lookup$\ldots$ppc (\snd sigma)$\langle$0,short_jump$\rangle$))) (bitvector_of_nat ? n) (\snd (nth n ? program$\langle$None ?, Comment []$\rangle$)) \end{lstlisting} This is the same invariant as${\tt sigma\_compact\_unsafe}$, but instead it This is almost the same invariant as${\tt sigma\_compact\_unsafe}$, but differs in that it computes the sizes of branch instructions by looking at the distance between position and destination using$\sigma$. The proof of {\tt nec\_plus\_ultra} works as follows: if we return {\tt None}, then the program size must be greater than 65 Kbytes. However, since the then the program size must be greater than 64 Kb. However, since the previous iteration did not return {\tt None} (because otherwise we would terminate immediately), the program size in the previous iteration must have been smaller than 65 Kbytes. been smaller than 64 Kb. Suppose that all the branch instructions in the previous iteration are encodes as long jumps. This means that all branch instructions in this encoded as long jumps. This means that all branch instructions in this iteration are long jumps as well, and therefore that both iterations are equal in the encoding of their branch instructions. Per the invariant, this means that But if all addresses are equal, the program sizes must be equal too, which means that the program size in the current iteration must be smaller than 65 Kbytes. This contradicts the earlier hypothesis, hence not all branch 64 Kb. This contradicts the earlier hypothesis, hence not all branch instructions in the previous iteration are encoded as long jumps. These are the invariants that hold after$2n$iterations, where$n$is the program size (we use the program size for convenience; we could also use the number of branch instructions, but this is more complicated). Here, we only number of branch instructions, but this is more complex). Here, we only need {\tt out\_of\_program\_none}, {\tt sigma\_compact} and the fact that$\sigma(0) = 0$. Termination can now be proven through the fact that there is a$k \leq 2n$, with Termination can now be proved using the fact that there is a$k \leq 2n$, with$n$the length of the program, such that iteration$k$is equal to iteration$k+1$. There are two possibilities: either there is a$k < 2n\$ such that this
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