Changeset 3350
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 Jun 14, 2013, 11:46:10 AM (5 years ago)
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Papers/itp2013/ccexec2.tex
r3349 r3350 255 255 associated to $L_1$ is the number of cycles required to execute the block 256 256 $I_3$ and \verb+COND l_2+, while the cost $k_2$ associated to $L_2$ counts the 257 cycles required by the block $I_4$ and \verb+GOTO l_1+. The compiler also guarantees257 cycles required by the block $I_4$, \verb+GOTO l_1+ and \verb+COND l_2+. The compiler also guarantees 258 258 that every executed instruction is in the scope of some code emission label, 259 259 that each scope does not contain loops (to associate a finite cost), and that … … 431 431 runs are weakly similar to the source code runs. 432 432 433 The notion of weak bisimulation for structured traces is a global property433 The notion of weak simulation for structured traces is a global property 434 434 which is hard to prove formally and much more demanding than the simple forward 435 435 simulation required for proofs of preservation of functional properties. … … 693 693 Let's consider a generic unstructured language already equipped with a 694 694 small step structured operational semantics (SOS). We introduce a 695 deterministic labelled transition system~\cite{LTS} $(S, s_{\mathrm{init}},\Lambda,\to)$695 deterministic labelled transition system~\cite{LTS} $(S,\Lambda,\to)$ 696 696 that refines the 697 697 SOS by observing function calls and the beginning of basic blocks. 698 $S$ is the set of states of the program , $s_\mathrm{init}$ the initial stateand698 $S$ is the set of states of the program and 699 699 $\Lambda = \{ \tau, RET \} \cup \Labels \cup \Functions$ 700 700 where $\Functions$ is the set of names of functions that can occur in the … … 705 705 denotes the image of this function. 706 706 The transition function is defined as $s_1 \to[o] s_2$ if 707 $s_1$ moves to $s_2$ according to the SOS ; moreover$o = f \in \Functions$ if707 $s_1$ moves to $s_2$ according to the SOS and $o = f \in \Functions$ if 708 708 the function $f$ is called, $o = RET$ if a \verb+RETURN+ is executed, 709 709 $o = L \in \Labels$ if an \verb+EMIT $L$+ is executed to signal the … … 711 711 Because we assume the language to be deterministic, the label emitted can 712 712 actually be computed simply observing $s_1$. Finally, $S$ is also endowed with 713 a relation $s\ar s'$ ($s'$ \emph{follows} $s$) when the instruction to be executed 714 in $s'$ is just after the one in $s$. 715 713 a relation $s\ar s'$ ($s'$ \emph{follows} $s$) that holds when the instruction 714 to be executed in $s'$ follows syntactically the one in $s$ in the source program. 715 716 In the rest of the paper we write $s_0 \to^{*} s_n$ for the finite execution 717 fragment $T = s_0 \to[o_0] s_1 \to[o_1] \ldots \to[o_{n1}] s_n$ 718 and, we call \emph{weak trace} of $T$ (denoted as $T$) the 719 subsequence $o_{i_0} \ldots o_{i_m}$ of $o_0 \ldots o_{n1}$ obtained dropping 720 every internal action $\tau$. 721 722 %Let $k$ be a cost model for observables actions that maps elements of 723 %$\Lambda \setminus \{\tau\}$ to any commutative cost monoid 724 %(e.g. natural numbers). We extend the domain of $k$ to executable fragments 725 %by posing $k(T) = \Sigma_{o \in T} k(o)$. 726 727 \paragraph{Structured execution fragments} 716 728 Among all possible finite execution fragments we want to 717 729 identify the ones that satisfy the requirements we sketched in the previous 718 730 section. We say that an execution fragment 719 $s_0 \to [o_0] s_1 \to[o_1] \ldots \to[o_n]s_n$720 is \emph{structured} ( markingit as $s_0 \To s_n$) iff the following conditions731 $s_0 \to^{*} s_n$ 732 is \emph{structured} (and we denote it as $s_0 \To s_n$) iff the following conditions 721 733 are met. 722 734 \begin{enumerate} … … 726 738 $s_i \ar s_{k+1}$. 727 739 In other words, $s_{i+1}$ must start execution with \verb+EMIT $\ell(f)$+ 740  so that no instruction falls outside the scope of every label  728 741 and then continue with a structured fragment returning control 729 742 to the instruction immediately following the call. 730 This captures the requirements that the body of function calls always start 731 with a label emission statement, and every function 732 call must converge yielding back control just after it. 733 \item For every $i$ and $f$, if $s_{i+1}\to[\ell(f)]s_{i+2}$ then 734 $s_i\to[f]s_{i+1}$. This is a technical condition needed to ensure that 735 labels associated with functions always follow a call. 743 744 The condition also enforces convergence of every function call, which is 745 necessary to bound the cost of the fragment. Note that 746 non convergent programs may still have structured execution fragments 747 that are worth measuring. For example, we can measure the reaction time 748 of a server implemented as an unbounded loop whose body waits for an 749 input, process it and performs an output before looping: the processing 750 steps form a structured execution fragment. 736 751 \item The number of $RET$'s in the fragment is equal to the number of 737 calls, i.e.\ the number of observables in $\Functions$. This, together with 738 the above condition, captures the wellbracketing of the fragment with 739 respect to function calls. 752 calls performed. In combination with the previous condition, this ensures 753 wellbacketing of function calls. 754 \item 755 \label{req3} 756 For every $i$ and $f$, if $s_{i+1}\to[\ell(f)]s_{i+2}$ then 757 $s_i\to[f]s_{i+1}$. This is a technical condition needed to ensure that a 758 label associated with a function is only used at the beginning of its 759 body. Its use will become clear in~\autoref{simulation}. 740 760 \item For every $i$, if the instruction to be executed in $s_i$ is a 741 761 conditional branch, then there is an $L$ such that $s_{i+1} \to[L] s_{i+2}$ or, equivalently, that $s_{i+1}$ must start execution with an 742 762 \verb+EMIT $L$+. This captures the requirement that every branch which is 743 live code must start with a label emission. 763 live code must start with a label emission. Otherwise, it would be possible 764 to have conditional instructions whose branches are assigned different 765 costs, making impossible to assign a single cost to the label whose scope 766 contains the jump. 744 767 \end{enumerate} 745 768 One might wonder why $f$ and $\ell(f)$, that aways appear in this order, are not 746 collapsed into a single observable. This would indeed simplify some aspects of 747 the formalisation, but has the problem of misassagning the cost of calls, which would 748 fall under the associated label. As different call instructions with different costs 749 are possible, this is not acceptable. 750 751 Let $T = s_0 \to[o_0] s_1 \ldots \to[o_n] s_{n+1}$ be an 752 execution fragment. The \emph{weak trace} $T$ associated to $T$ is the 753 subsequence $o_{i_0} \ldots o_{i_m}$ of $o_0 \ldots o_n$ obtained dropping 754 every internal action $\tau$. 755 756 Let $k$ be a cost model that maps observables actions to any commutative cost 757 monoid (e.g. natural numbers). We extend the domain of $k$ to fragments 758 by posing $k(T) = \Sigma_{o \in T} k(o)$. 759 760 The labelling approach is based on the idea that the execution cost of 769 collapsed into a single observable. This would simplify some aspects of the 770 formalisation at the price of others. For example, we should add special 771 cases when the fragment starts at the beginning of a function body 772 (e.g. the one of \texttt{main}) because in that case nobody would have emitted 773 the observable $\ell(f)$. 774 775 \paragraph{Measurable execution fragments and their cost prediction.} 776 The first main theorem of CerCo deals with programs written in object code. 777 It states that the execution cost of 761 778 certain execution fragments, that we call \emph{measurable fragments}, can be 762 computed from their weak trace by choosing a $k$ that assigns to any label 763 the cost of the instructions in its scope. 764 A structured fragment 765 $T = s_0 \To s_n$ is 766 measurable if it does not start or end in the middle of a basic block. 767 Ending in the middle of a block would mean having prepaid more instructions 768 than the ones executed, and starting in the middle would mean not paying any 769 instruction up to the first label emission. 770 Formally we require $o_0 \in \Labels$ (or equivalently 779 computed from their weak trace by choosing the cost model $k$ that assigns to 780 any label the cost (in clock cycles) of the instructions in its scope, and 781 $0$ to function calls and $RET$ observables. 782 783 \begin{theorem} 784 \label{static} 785 for all measurable fragment $T = s_0 \to^{*} s_n$,\\ 786 $$\Delta_t := \verb+clock+_{s_n}  \verb+clock+_{s_0} = \Sigma_{o \in T} k(o)$$ 787 \end{theorem} 788 789 An execution fragment $s_0 \to^{*} s_n$ is 790 measurable if it is structured (up to a possible final \texttt{RETURN}) and 791 if it does not start or end in the middle of a basic block. 792 Ending in the middle of a block would mean that the last label encountered 793 would have prepaid more instructions than the ones executed; starting in the 794 middle would mean not paying any instruction up to the first label emission. 795 796 Formally, $s_0 \to^{*} s_n$ is measurable iff $o_0 \in \Labels$ (or equivalently 771 797 in $s_0$ the program must emit a label) and either 772 $s_{n1}\to[RET]s_n$ or $s_n$ must be a label emission statement 773 (i.e.\ $s_n \to[L] s_{n+1}$). 774 798 $s_0 \To s_{n1}$ and $s_{n1}\to[RET]s_n$ or 799 $s_0 \To s_n$ and $s_n$ must be a label emission statement. 800 801 \textbf{CSC: PROVA} 802 % The theorem is proved by structural induction over the structured 803 % trace, and is based on the invariant that 804 % iff the function that computes the cost model has analysed the instruction 805 % to be executed at $s_2$ after the one to be executed at $s_1$, and if 806 % the structured trace starts with $s_1$, then eventually it will contain also 807 % $s_2$. When $s_1$ is not a function call, the result holds trivially because 808 % of the $s_1\exec s_2$ condition obtained by inversion on 809 % the trace. The only non 810 % trivial case is the one of function calls: the cost model computation function 811 % does recursion on the first instruction that follows that function call; the 812 % \verb+as_after_return+ condition of the \verb+tal_base_call+ and 813 % \verb+tal_step_call+ grants exactly that the execution will eventually reach 814 % this state. 815 816 \paragraph{Weak similarity and cost invariance.} 775 817 Given two deterministic unstructured programming languages with their own 776 operational semantics, we say that a state $s_2$ of the second language 777 (weakly) simulates the state $s_1$ of the first iff the two unique weak traces 778 that originate from them are equal. If $s_1$ also (weakly) simulates $s_2$, 779 then the two states are weakly trace equivalent. 780 781 or, equivalently 782 because of determinism, that they are weakly bisimilar. 818 operational semantics, we say that two execution fragments are 819 \emph{weakly trace equivalent} if their weak traces are equal. 820 821 A compiler (pass) that preserves the program semantics also preserves weak 822 traces and propagates measurability iff for every measurable 823 fragment $T_1 = s_1 \to^{*} s_1'$ of the source code, the corresponding 824 execution fragment $T_2 = s_2 \to^{*} s_2'$ of the object code is measurable 825 and $T_1$ and $T_2$ are weakly trace equivalent. The very intuitive notion of 826 ``corresponding fragment'' is made clear in the forward simulation proof of 827 preservation of the semantics of the program by saying that $s_2$ and $s_1$ 828 are in a certain relation. Clearly the property holds for a compiler if it 829 holds for each compiler pass. 830 831 Having proved in~\autoref{static} that the statically computed cost model is 832 accurate for the object code, we get as a corollary that it is also accurate 833 for the source code if the compiler preserves weak traces and 834 propagates measurability. Thus it becomes possible to compute cost models 835 on the object code, transfer it to the source code and then reason comfortably 836 on the source code only. 837 838 \begin{theorem}\label{preservation} 839 Given a compiler that preserves weak traces and propagates measurability, 840 for all measurable execution fragment $T_1 = s_1 \to^{*} s_1'$ of the source 841 code such that $T_2 = s_2 \to^{*} s_2'$ is the corresponding fragment of the 842 object code, 843 844 $$\Delta_t := \verb+clock+_{s_2'}  \verb+clock+_{s_2} = \Sigma_{o \in T_2} k(o) = \Sigma_{o \in T_1} k(o)$$ 845 \end{theorem} 846 847 \section{Forward simulation} 848 \label{simulation} 849 Because of \autoref{preservation}, to certify a compiler for the labelling 850 approach we need to both prove that it respects the functional semantics of the 851 program, and that it preserves weak traces and propagates measurability. 852 The first property is standard and can be proved by means of a forward simulation argument (see for example~\cite{compcert}) that runs like this. 853 First a relation between the corresponding 854 source and target states is established. Then a lemma establishes 855 a local simulation condition: given two states in relation, if the source 856 one performs one step then the target one performs zero or more steps and 857 the two resulting states are synchronized again according to the relation. 858 Finally, the lemma is iterated over the execution trace to establish the 859 final result. 860 861 In principle, preservation of weak traces could be easily shown with the same 862 argument (and also at the same time). Surprisingly, propagation of 863 measurability cannot. What makes the standard forward 864 simulation proof work is the fact that usually a compiler pass performs some 865 kind of local or global analysis of the code followed by a compositional, order 866 preserving translation of every instruction. In order to produce structured 867 traces, however, code emission cannot be fully compositional any longer. 868 869 For example, consider~requirement \ref{req3} that asks every function body 870 to start with a label emission statement. Some compiler passes must 871 add preambles to functions, for example to take care of the parameter passing 872 convention. In order to not violate the requirement, the preamble must be 873 inserted after the label emission. In the forward simulation proof, however, 874 function call steps in the source language are simulated by the new function 875 call followed by the execution of the preamble, and only at the end of the 876 preamble the reached states are again in the expected relation. In the meantime, 877 however, the object code has already performed the label emission statement, 878 that still needs to be executed in the source code, breaking forward simulation. 879 880 Another reason why the standard argument breaks is due to the requirement that 881 function calls should yield back control after the calling point. This must be 882 enforced just after 883 884 \textbf{XXXXXXXXXXXXXXXXXX} 885 886 A compiler preserves the program semantics by suppressing or introducing 887 $\tau$ actions. 888 889 Intuitively, it is because 890 891 To understand why, consider the case of a function call 892 and the pass that fixes the parameter passing conventions. A function 893 call in the source code takes in input an arbitrary number of pseudoregisters (the actual parameters to pass) and returns an arbitrary number of pseudoregisters (where the result is stored). A function call in the target language has no 894 input nor output parameters. The pass must add explicit code before and after 895 the function call to move the pseudoregisters content from/to the hardware 896 registers or the stack in order to implement the parameter passing strategy. 897 Similarly, each function body must be augmented with a preamble and a postamble 898 to complete/initiate the parameter passing strategy for the call/return phase. 899 Therefore what used to be a call followed by the next instruction to execute 900 after the function return, now becomes a sequence of instructions, followed by 901 a call, followed by another sequence. The two states at the beginning of the 902 first sequence and at the end of the second sequence are in relation with 903 the status before/after the call in the source code, like in an usual forward 904 simulation. How can we prove however the additional condition for function calls 905 that asks that when the function returns the instruction immediately after the 906 function call is called? To grant this invariant, there must be another relation 907 between the address of the function call in the source and in the target code. 908 This additional relation is to be used in particular to relate the two stacks. 909 910 911 912 783 913 % @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ 784 914 % … … 1043 1173 % % trace_label_return \(s_1\) \(s_2\) \(\to\) list (as_cost_label S) 1044 1174 % % \end{alltt} 1045 %1046 % \paragraph{Cost prediction on structured traces.}1047 %1048 % The first main theorem of CerCo about traces1049 % (theorem \verb+compute_max_trace_label_return_cost_ok_with_trace+)1050 % holds for the1051 % instantiation1052 % of the structured traces to the concrete status of object code programs.1053 % Simplifying a bit, it states that1054 % \begin{equation}\label{th1}1055 % \begin{array}{l}\forall s_1,s_2. \forall \tau: \verb+TLR+~s_1~s_2.~1056 % \verb+clock+~s_2 = \verb+clock+~s_1 +1057 % \Sigma_{\alpha \in \tau}\;k(\alpha)1058 % \end{array}1059 % \end{equation}1060 % where the cost model $k$ is statically computed from the object code1061 % by associating to each label $\alpha$ the sum of the cost of the instructions1062 % in the basic block that starts at $\alpha$ and ends before the next labelled1063 % instruction. The theorem is proved by structural induction over the structured1064 % trace, and is based on the invariant that1065 % iff the function that computes the cost model has analysed the instruction1066 % to be executed at $s_2$ after the one to be executed at $s_1$, and if1067 % the structured trace starts with $s_1$, then eventually it will contain also1068 % $s_2$. When $s_1$ is not a function call, the result holds trivially because1069 % of the $s_1\exec s_2$ condition obtained by inversion on1070 % the trace. The only non1071 % trivial case is the one of function calls: the cost model computation function1072 % does recursion on the first instruction that follows that function call; the1073 % \verb+as_after_return+ condition of the \verb+tal_base_call+ and1074 % \verb+tal_step_call+ grants exactly that the execution will eventually reach1075 % this state.1076 1175 % 1077 1176 % \paragraph{Structured traces similarity and cost prediction invariance.} … … 1281 1380 % As should be expected, even though the rules are asymmetric $\approx$ is in fact 1282 1381 % an equivalence relation. 1283 \section{Forward simulation}1284 \label{simulation}1285 1382 1286 1383 We summarise here the results of the previous sections. Each intermediate
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