X-Git-Url: http://wien.tomnetworks.com/gitweb/?a=blobdiff_plain;f=docs%2Fabc-removal.txt;h=5c2ce91fd5f294e7a96b347bdb5f69200022c2c9;hb=52e98abd6e5c6d10ddea91a529f7b1b2336e0696;hp=82bad22330e083c5314bb8d12b4109d53ebc1b91;hpb=98a86ecd4ba2d3b4631d98ac5a4abf23f4e6c3e3;p=mono.git diff --git a/docs/abc-removal.txt b/docs/abc-removal.txt index 82bad22330e..5c2ce91fd5f 100755 --- a/docs/abc-removal.txt +++ b/docs/abc-removal.txt @@ -1,525 +1,373 @@ -Here "abc" stays for "array bounds check", or "array bound checks", or -some combination of the two... - ------------------------------------------------------------------------------- -USAGE - -Simply use the "abcrem" optimization invoking mono. - -To see if bound checks are actually removed, use "mono -v" and grep for -"ARRAY-ACCESS" in the output, there should be a message for each check -that has been removed. - -To trace the algorithm execution, use "-v -v -v", and be prepared to be -totally submersed by debugging messages... - ------------------------------------------------------------------------------- -EFFECTIVENESS - -The abc removal code can now always remove bound checks from "clean" -array scans in loops, and generally anyway there are clear conditions that -already state that the index is "safe". - -To be clearer, and give an idea of what the algorithm can and cannot do -without describing it in detail... keep in mind that only "redunant" checks -cannot be removed. By "redundant", I mean "already explicitly checked" in -the method code. + Arrays Bounds Check Elimination (ABC) + in the Mono Runtime -Unfortunately, analyzing complex expressions is not so easy (see below for -details), so the capabilities of this "abc remover" are limited. + Massimiliano Mantione (mass@ximian.com) -These are simple guidelines: -- Only simple comparisons between variables are understood. -- Only increment-decrement operations are handled. -- "switch" statements are not yet handled. - -This means that code like this will be handled well: - -for (int i = 0; i < a.Length; i++) { - a [i] = ..... -} - -The "i" variable could be declared out of the "for", the "for" could be a -"while", and maybe even implemented with "goto", the array could be scanned -in reverse order, and everything would still work. -I could have a temporary variable storing the array length, and check on -it inside the loop, and the abc removal would still occurr, like this: - -int i = 0; -int l = a.Length; -while ( i < l ) { - a [i] ...... -} - -or this: - -int l = a.Length; -for (int i = l; i > 0; i--) { - a [i] = ..... -} - - -But just something like this - -for (int i = 0; i < (a.Length -1); i++) ..... - -or like this - -for (int i = 0; i < a.Length; i += 2) ..... - -would not work, the check would stay there, sorry :-( - -Just to make you understand how things are tricky... this would work! - -int limit = a.Length - 1; -for (int i = 0; i < limit; i++) { - a [i] = ..... -} - - -A detailed explanation of the reason why things are done like this is given -below... - -All in all, the compiling phase is generally non so fast, and the abc removed -are so few, that there is hardly a performance gain in using the "abcrem" -flag in the compiling options for short programs like the ones I tried. - -Anyway, various bound checks *are* removed, and this is good :-) - ------------------------------------------------------------------------------- -THE ALGORITHM - -Unfortunately, this time google has not been my friend, and I did not -find an algorithm I could reasonably implement in the time frame I had, -so I had to invent one. The two cleasses of algorithms I found were either -simply an analisys of variable ranges (which is not very different from -what I'm doing), or they were based on type systems richer than the .NET -one (particularly, they required integer types to have an explicit range, -which could be related to array ranges). By the way, it seems that the -gcj people do not have an array bound check removal optimization (they -just have a compiler flag that unconditionally removes all the checks, -but this is not what we want). - -This array bound check removal (abc removal) algorithm is based on -symbolic execution concepts. I handle the removal like an "unreachable -code elimination" (and in fact the optimization could be extended to remove -also other unreachable sections of code, due to branches that "always go the -same way"). - -In symbolic execution, variables do not have actual (numeric) values, but -instead symbolic expressions (like "a", or "x+y"). -Also, branch conditions are handled like symbolic conditions (like "ivars []->def" should contain exactly -those store instructions, and the "get_variable_value_from_ssa_store" -function extracts the variable value from there. - -On the other hand, the conditions under which each basic block is -executed are not made fully explicit. - -However, it is not difficult to make them so. -Each BB that has more than one exit BB, in practice must end either with -a conditional branch instruction or with a switch instruction. -In the first case, the BB has exactly two exit BBs, and their execution -conditions are easy to get from the condition of the branch (see the -"get_relation_from_branch_instruction" function, and expecially the end of -"analyze_block" in abcremoval.c. -If there is a switch, the jump condition of every exit BB is the equality -of the switch argument with the particular index associated with its case -(but the current implementation does not handle switch statements yet). - -These individual conditions are in practice associated to each arc that -connects BBs in the CFG (with the simple case that unconditional branches -have a "TRUE" condition, because they always happen). - -So, for each BB, its *proper* entry condition is the union of all the -conditions associated to arcs that enter the BB. The "union" is like a -logical "or", in the sense that either of the condition could be true, -they are not necessarily all true. This means that if I can enter a BB -in two ways, and in one case I know that "x>0", and in the other that -"x==0", actually in the BB I know that "x>=0", which is a weaker -condition (the union of the two). - -Also, the *complete* entry condition for a BB is the "intersection" of all -the entry conditions of its dominators. This is true because each dominator -is the only way to reach the BB, so the entry condition of each dominator -must be true if the control flow reached the BB. This translates to the -logical "and" of all the "proper" conditions of the BBs met walking up in the -dominator tree. So, if one says "x>0", and another "x==1", then I know -that "x==1", which is a stronger condition (the intersection of the two). -Note that, if the two conditions were "x>0" and "x==0", then the block would -be unreachable (the intersection is empty), because some branch is impossible. - -Another observation is that, inside each BB, every variable is subject to the -complete entry condition of that very same BB, and not the one in which it is -defined (with the "complete entry condition" being the thing I defined before, -sorry if these terms "proper" and "complete" are strange, I found nothing -better). -This happens because the branch conditions are related to the control flow. -I can define "i=a", and if I am in a BB where "a>0", then "i>0", but not -otherwise. - -So, intuitively, if the available conditions say "i>=0", and i is used as an -index in an array access, then the lower bound check can be omitted. -If the condition also says "(i>=0)&&(i=0" and "index < array.length") -If the system is valid for *each possible* variable value, then the goal -functions are always true, and the abc can be removed. - -All this discussion is useful to give a precise specification to the problem -we are trying to solve. -The trouble is that, in the general case, the resulting system of equations -is like a predicate in first order logic, which is semi-decidable, and its -general solution is anyway too complex to be attempted in a JIT compiler -(which should not contain a full fledged theorem prover). - -Therefore, we must cut some corner. - - -By the way, there is also another big problem, which is caused by "recursive" -symbolic definitions. These definition can (and generally do) happen every -time there is a loop. For instance, in the following piece of code - -for ( int i = 0; i < array.length; i++ ) { - Console.WriteLine( "array [i] = " + array [i] ); -} - -one of the definitions of i is a PHI that can be either 0 or "i + 1". - -Now, we know that mathematically "i = i + 1" does not make sense, and in -fact symbolic values are not "equations", they are "symbolic definitions". - -The symbolic value of i is a generic "n", where "n" is the number of -iterations of the loop, but this is terrible to handle (and in more complex -examples the symbolic definition of i simply cannot be written, because i is -calculated in an iterative way). - -However, the definition "i = i + 1" tells us something about i: it tells us -that i "grows". So (from the PHI definition) we know that i is either 0, or -"grows". This is enough to tell that "i>=0", which is what we want! -It is important to note that recursive definitions can only occurr inside -PHI definitions, because actually a variable cannot be defined *only* in terms -of itself! - - -At this point, I can explain which corners I want to cut to make the -problem solvable. It will not remove all the abc that could theoretically -be removed, but at least it will work. - -The easiest way to cut corners is to only handle expressions which are -"reasonably simple", and ignore the rest. -Keep in mind that ignoring an expression is not harmful in itself. -The algorithm will be simply "less powerful", because it will ignore -conditions that could have caused to the removal of an abc, but will -not remove checks "by mistake" (so the resulting code will be in any case -correct). - -In a first attempt, we can consider only conditions that have the simple -possible form, which is "(compare simpleExpression simpleExpression)", -where "simpleExpression" is either "(ldind.* local[*])" or -"(ldlen (ldind.ref local[*]))" (or a constant, of course). - -We can also "summarize" variable definitions, keeping only what is relevant -to know: if they are greater or smaller than zero or other variables. - -One particular note on PHI functions: they work (obviously) like the logical -"or" of their definitions, and therefore are equivalent to the "logical or" -of the summarization of their definitions. - -About recursive definitions (which, believe me, are the worst thing in all -this mess), we handle only "monotonic" ones. That is, we try to understand -if the recursive definition (which, as we said above, must happen because -of a loop) always "grows" or "gets smaller". In all other cases, we decide -we cannot handle it. - -So, the algorithm to optimize one method could be something like: - -[1] Build the SSA representation. -[2] Analyze each BB. Particularly, do the following: - [2a] summarize its exit conditions - [2b] summarize all the variable definitions occurring in the BB - [2c] check if it contains array accesses (only as optimization, so that - step 3 will be performed only when necessary) -[3] Perform the removal. For each BB that contains array accesses, do: - [3a] build an evaluation area where for each variable we keep track - of its relations with zero and with each other variable. - [3b] populate the evaluation area with the initial data (the variable - definitions and the complete entry condition of the BB) - [3c] propagate the relations between variables (so that from "a b+1", because we -are interested in what value b has, and not a. And maybe the software should -be able to do this without modifying the original condition, because it was -used elsewhere (or could be reused). -Then, the software should also be able to apply logical relations to -conditions (like "(a>0)||(c<1)"), and manipulate them, too. - -In my opinion, an optimizer written in this way risks to be too complex, and -its performance not acceptable in a JIT compiler. -Therefore, I decided to represent values in a "summarized" way. -For instance, "x = a + 1" becomes "x > a" (and, by the way, "x = x + 1" -would look like "x > x", which means that "x grows"). -So, each variable value is simply a "relation" of the variable with other -variables or constants. Anything more complex is ignored. -Relations are represented with flags (EQ, LT and GT), so that it is easy -to combine them logically (it is enough to perform the bitwise operations -on the flags). The condition (EQ|LT|GT) means we know nothing special, any -value is possible. The empty condition would mean an unreachable statement -(because no value is acceptable). - -There is still the problem of identifying variables, and where to store all -these relations. After some thinking, I decided that a "brute force" approach -is the easier, and probably also the fastest. In practice, I keep in memory -an array of "variable relations", where for each variable I record its -relation with the constants 0 and 1 and an array of its relations with all -other variables. The evaluation area, therefore, looks like a square matrix -as large as the number of variables. I *know* that the matrix is rather sparse -(in the sense that only few of its cells are significant), but I decided that -this data structure was the simplest and more efficient anyway. After all, -each cell takes just one byte, and any other data structure must be some -kind of associative array (like a g_hash_table). Such a data structue is -difficult to use with a MonoMemPool, and in the end is slower than a plain -array, and has the storage overhead of all the pointers... in the end, maybe -the full matrix is not a waste at all. - -After a few benchmarks, I clearly see that the matrix works well if the -variables are not too much, otherwise it gets "too sparse", and much time -is wasted in the propagation phase. Maybe the data structure could be changed -after a more careful evaluation of the problem... - - -With these assumptions, all the memory used to represent values, conditions -and the evaluation area can be allocated in advance (from a MonoMemPool), and -used for all the analysis of one method. Moreover, the performance of logical -operations is high (they are simple bitwise operations on bytes). - -For the propagation phase, there was a function I could not easily code -with bitwise operations. As that phase is a real pain for the performance -(the complexity is quadratic, and it must be executed iteratively until -nothing changes), I decided to code it in perl, and generate a precomputed -table then used by the C code (just 64 bytes). I think this is faster than -all those branches... Really, I could think again to phases 3c and 3d (see -above), maybe there is a better way to do them. - - -Of course not all expressions can be handled by the summarization, so the -optimizer is not as effective as a full blown theorem prover... - -Another thing related to design... the implementation is as less intrusive -as possible. -I "patched" the main files to add an option for abc removal, and handle it -just like any other option, and then wrote all the relevant code in two -separated files (one header for data structures, and a C file for the code). -If the patch turns out to be useful also for other pourposes (like a more -generic "unreachable code elimination", that detects branches that will never -be taken using the analysis of variable values), it will not be difficult -to integrate the branch summarizations into the BasicBlock data structures, -and so on. - -One last thing... in some places, I knowingly coded in an inefficient way. -For instance, "remove_abc_from_block" does something only if the flag -"has_array_access_instructions" is true, but I call it anyway, letting it -do nothing (calling it conditionally would be better). There are several -things like this in the code, but I was in a hurry, and preferred to make -the code more readable (it is already difficult to understand as it is). - ------------------------------------------------------------------------------- -WHEN IS ABC REMOVAL POSSIBLE? WHEN IS IT USEFUL? - -Warning! Random ideas ahead! - -In general, managed code should be safe, so abc removal is possible only if -there already is some branch in the method that does the check. -So, one check on the index is "mandatory", the abc removal can only avoid a -second one (which is useless *because* of the first one). -If in the code there is not a conditional branch that "avoids" to access -an array out of its bounds, in general the abc removal cannot occur. - -It could happen, however, that one method is called from another one, and -the check on the index is performed in the caller. - -For instance: - -void -caller( int[] a, int n ) { - if ( n <= a.Length ) { - for ( int i = 0; i < n; i++ ) { - callee( a, i ); +Here "abc" stays for "array bounds check", or "array bound checks", or +some combination of the two. + +* Usage + + Simply use the "abcrem" optimization invoking mono. + + To see if bound checks are actually removed, use "mono -v" and + grep for "ARRAY-ACCESS" in the output, there should be a + message for each check that has been removed. + + To trace the algorithm execution, use "-v -v -v", and be + prepared to be totally submersed by debugging messages... + +* Effectiveness + + The abc removal code can now always remove bound checks from + "clean" array scans in loops, and generally anyway there are + clear conditions that already state that the index is "safe". + + To be clearer, and give an idea of what the algorithm can and + cannot do without describing it in detail... keep in mind that + only "redundant" checks cannot be removed. By "redundant", I + mean "already explicitly checked" in the method code. + + Unfortunately, analyzing complex expressions is not so easy + (see below for details), so the capabilities of this "abc + remover" are limited. + + These are simple guidelines: + + - Only expressions of the following kinds are handled: + - constant + - variable [+/- constant] + - Only comparisons between "handled" expressions are understood. + - "switch" statements are not yet handled. + + This means that code like this will be handled well: + + for (int i = 0; i < a.Length; i++) { + a [i] = ..... + } + + The "i" variable could be declared out of the "for", the "for" + could be a "while", and maybe even implemented with "goto", + the array could be scanned in reverse order, and everything + would still work. + + I could have a temporary variable storing the array length, + and check on it inside the loop, and the abc removal would + still occurr, like this: + + int i = 0; + int l = a.Length; + while ( i < l ) { + a [i] ...... + } + + or this: + + int l = a.Length; + for (int i = l; i > 0; i--) { + a [i] = ..... } - } -} - -void callee( int[] a, int i ) { - Console.WriteLine( "a [i] = " + a [i] ); -} - - -In this case, it should be possible to have two versions of the callee, one -without checks, to be called from methods that do the check themselves, and -another to be called otherwise. -This kind of optimization could be profile driven: if one method is executed -several times, wastes CPU time for bound checks, and is mostly called from -another one, it could make sense to analyze the situation in detail. - -However, one simple way to perform this optimization would be to inline -the callee (so that the abc removal algorithm can see both methods at once). - -The biggest benefits of abc removal can be seen for array accesses that -occur "often", like the ones inside inner loops. This is why I tried to -handle "recursive" variable definitions, because without them you cannot -optimize inside loops, which is also where you need it most. - -Another possible optimization (alwais related to loops) is the following: - -void method( int[] a, int n ) { - for ( int i = 0; i < n; i++ ) { - Console.WriteLine( "a [i] = " + a [i] ); - } -} - -In this case, the check on n is missing, so the abc could not be removed. -However, this would mean that i will be checked twice at each loop iteration, -one against n, and the other against a.Length. The code could be transformed -like this: - -void method( int[] a, int n ) { - if ( n < a.Length ) { - for ( int i = 0; i < n; i++ ) { - Console.WriteLine( "a [i] = " + a [i] ); // Remove abc + + The following two examples would work: + + for (int i = 0; i < (a.Length -1); i++) ..... + for (int i = 0; i < a.Length; i += 2) ..... + + But just something like this + + int delta = -1; + for (int i = 0; i < (a.Length + delta); i++) ..... + + or like this + + int delta = +2; + for (int i = 0; i < a.Length; i += delta) ..... + + would not work, the check would stay there. (unless + a combination of cfold, consprop and copyprop is used, too, + which would make the constant value of "delta" explicit). + + Just to make you understand how things are tricky... this would work! + + int limit = a.Length - 1; + for (int i = 0; i < limit; i++) { + a [i] = ..... } - } else { - for ( int i = 0; i < n; i++ ) { - Console.WriteLine( "a [i] = " + a [i] ); // Leave abc + + A detailed explanation of the reason why things are done like + this is given below. + +* The Algorithm + + This array bound check removal (abc removal) algorithm is + based on symbolic execution concepts. I handle the removal + like an "unreachable code elimination" (and in fact the + optimization could be extended to remove also other + unreachable sections of code, due to branches that "always go + the same way"). + + In symbolic execution, variables do not have actual (numeric) + values, but instead symbolic expressions (like "a", or "x+y"). + Also, branch conditions are handled like symbolic conditions + (like "ivars + []->def" should contain exactly those store + instructions, and the "get_variable_value_from_ssa_store" + function extracts the variable value from there. + + On the other hand, the conditions under which each basic block + is executed are not made fully explicit. + + However, it is not difficult to make them so. + + Each BB that has more than one exit BB, in practice must end + either with a conditional branch instruction or with a switch + instruction. + + In the first case, the BB has exactly two exit BBs, and their + execution conditions are easy to get from the condition of the + branch (see the "get_relation_from_branch_instruction" + function, and especially the end of "analyze_block" in + abcremoval.c. + + If there is a switch, the jump condition of every exit BB is + the equality of the switch argument with the particular index + associated with its case (but the current implementation does + not handle switch statements yet). + + These individual conditions are in practice associated to each + arc that connects BBs in the CFG (with the simple case that + unconditional branches have a "TRUE" condition, because they + always happen). + + So, for each BB, its *proper* entry condition is the union of + all the conditions associated to arcs that enter the BB. The + "union" is like a logical "or", in the sense that either of + the condition could be true, they are not necessarily all + true. This means that if I can enter a BB in two ways, and in + one case I know that "x>0", and in the other that "x==0", + actually in the BB I know that "x>=0", which is a weaker + condition (the union of the two). + + Also, the *complete* entry condition for a BB is the + "intersection" of all the entry conditions of its + dominators. This is true because each dominator is the only + way to reach the BB, so the entry condition of each dominator + must be true if the control flow reached the BB. This + translates to the logical "and" of all the "proper" conditions + of the BBs met walking up in the dominator tree. So, if one + says "x>0", and another "x==1", then I know that "x==1", which + is a stronger condition (the intersection of the two). + + Note that, if the two conditions were "x>0" and "x==0", then + the block would be unreachable (the intersection is empty), + because some branch is impossible. + + Another observation is that, inside each BB, every variable is + subject to the complete entry condition of that very same BB, + and not the one in which it is defined (with the "complete + entry condition" being the thing I defined before, sorry if + these terms "proper" and "complete" are strange, I found + nothing better). + + This happens because the branch conditions are related to the + control flow. I can define "i=a", and if I am in a BB where + "a>0", then "i>0", but not otherwise. + + So, intuitively, if the available conditions say "i>=0", and i + is used as an index in an array access, then the lower bound + check can be omitted. If the condition also says + "(i>=0)&&(i=0" and "index < + array.length") + + If the system is valid for *each possible* variable value, then the goal + functions are always true, and the abc can be removed. + + All this discussion is useful to give a precise specification + to the problem we are trying to solve. + + The trouble is that, in the general case, the resulting system + of equations is like a predicate in first order logic, which + is semi-decidable, and its general solution is anyway too + complex to be attempted in a JIT compiler (which should not + contain a full fledged theorem prover). + + Therefore, we must cut some corner. + + There is also another big problem, which is caused by + "recursive" symbolic definitions. These definition can (and + generally do) happen every time there is a loop. For instance, + in the following piece of code: + + for ( int i = 0; i < array.length; i++ ) { + Console.WriteLine( "array [i] = " + array [i] ); } - } -} - -This could result in performance improvements (again, probably this -optimization should be profile driven). - ------------------------------------------------------------------------------- -OPEN ISSUES - -There are several issues that should still be addressed... - -One is related to aliasing. For now, I decided to operate only on local -variables, and ignore aliasing problems (and alias analisys has not yet -been implemented anyway). -Actually, I identified the local arrays with their length, because I -totally ignore the contents of arrays (and objects in general). - -Also, in several places in the code I only handle some cases, and ignore -other more complex ones, which could anyway work (there are comments where -this happens). Anyway, this is not harmful (the code is not as effective -as it could be). - -Another possible improvement is the explicit handling of all constants. -For now, code like - -void -method () { - int[] a = new int [16]; - for ( int i = 0; i < 16; i++ ) { - a [i] = i; - } -} - -is not handled, because the two constants "16" are lost when variable -relations are summarized (actually they are not lost, they are stored in the -summarized values, but they are not yet used correctly). - -The worst thing, anyway, is that for now I fail completely in distinguishing -between signed and unsigned variables/operations/conditions, and between -different integer sizes (in bytes). I did this on pourpose, just for lack of -time, but this can turn out to be terribly wrong. -The problem is caused by the fact that I handle arithmetical operations and -conditions as "ideal" operations, but actually they can overflow and/or -underflow, giving "surprising" results. - -For instance, look at the following two methods: - -public static void testSignedOverflow() -{ - Console.WriteLine( "Testing signed overflow..." ); - int[] ia = new int[70000]; - int i = 1; - while ( i =0", which is what we want! It is important to note + that recursive definitions can only occurr inside PHI + definitions, because actually a variable cannot be defined + *only* in terms of itself! + + At this point, I can explain which corners I want to cut to + make the problem solvable. It will not remove all the abc that + could theoretically be removed, but at least it will work. + + The easiest way to cut corners is to only handle expressions + which are "reasonably simple", and ignore the rest. + + Keep in mind that ignoring an expression is not harmful in + itself. The algorithm will be simply "less powerful", because + it will ignore conditions that could have caused to the + removal of an abc, but will not remove checks "by mistake" (so + the resulting code will be in any case correct). + + The expressions we handle are the following (all of integer + type): + + - constant + - variable + - variable + constant + - constant + variable + - variable - constant + + And, of course, PHI definitions. + + Any other expression causes the introduction of an "any" value + in the evaluation, which makes all values that depend from it + unknown as well. + + We will call these kind of definitions "summarizable" + definitions. + + In a first attempt, we can consider only branch conditions + that have the simplest possible form (the comparison of two + summarizable expressions). + + We can also simplify the effect of variable definitions, + keeping only what is relevant to know: their value range with + respect to zero and with respect to the length of the array we + are currently handling. + + One particular note on PHI functions: they work (obviously) + like the logical "or" of their definitions, and therefore are + equivalent to the "logical or" of the summarization of their + definitions. + + About recursive definitions (which, believe me, are the worst + thing in all this mess), we handle only "monotonic" ones. That + is, we try to understand if the recursive definition (which, + as we said above, must happen because of a loop) always + "grows" or "gets smaller". In all other cases, we decide we + cannot handle it. + + One critical thing, once we have defined all these data + structures, is how the evaluation is actually performed. + + In a first attempt I coded a "brute force" approach, which for + each BB tried to examine all possible conditions between all + variables, filling a sort of "evaluation matrix". The problem + was that the complexity of this evaluation was quadratic (or + worse) on the number of variables, and that many variables + were examined even if they were not involved in any array + access. + + Following the ABCD paper: + + http://citeseer.ist.psu.edu/bodik00abcd.html + + I rewrote the algorithm in a more "sparse" way. + + Now, the main data structure is a graph of relations between + variables, and each attempt to remove a check performs a + traversal of the graph, looking for a path from the index to + the array length that satisfies the properties "index >= 0" + and "index < length". If such a path is found, the check is + removed. It is true that in theory *each* traversal has a + complexity which is exponential on the number of variables, + but in practice the graph is not very connected, so the + traversal terminates quickly. + + + Then, the algorithm to optimize one method looks like this: + + [1] Preparation: + + [1a] Build the SSA representation. + + [1b] Prepare the evaluation graph (empty) + + [1b] Summarize each variable definition, and put + the resulting relations in the evaluation + graph + + [2] Analyze each BB, starting from the entry point and + following the dominator tree: + + [2a] Summarize its entry condition, and put the resulting relations + in the evaluation graph (this is the reason + why the BBs are examined following the + dominator tree, so that conditions are added + to the graph in a "cumulative" way) + + [2b] Scan the BB instructions, and for each array + access perform step [3] + + [2c] Process children BBs following the dominator + tree (step [2]) + + [2d] Remove from the evaluation area the conditions added at step [2a] + (so that backtracking along the tree the area + is properly cleared) + + [3] Attempt the removal: + + [3a] Summarize the index expression, to see if we can handle it; there + are three cases: the index is either a + constant, or a variable (with an optional + delta) or cannot be handled (is a "any") + + [3b] If the index can be handled, traverse the evaluation area searching + a path from the index variable to the array + length (if the index is a constant, just + examine the array length to see if it has + some relation with this constant) + + [3c] Use the results of step [3b] to decide if the check is redundant +