What is static program analysis?
matt.might.net
matt.might.net
The distinction between algorithms and heuristics is, in my opinion, insufficiently appreciated. For example, the very phrase "genetic algorithms" is a contradiction in terms -- these are clearly heuristics, not algorithms.
Anyway, back to your point, we who are still selling static analysis -- I'm not sure why you used the past tense there; it's a growing business -- find that people who don't understand the halting problem at all are perhaps as much a source of difficulty as those who are aware of it but don't understand its implications. The latter group are skeptical that our products can work at all, but the former are often baffled and indignant that the products don't understand properties of their code that seem obvious. (The fact that analysis often takes a long time and needs large amounts of RAM also surprises some people.)
But I think the world is gradually coming to understand what static analysis can and can't do. It's an education process and will take time.
Matt is talking about situations where you can make guarantees. That is, if his analyzer says that a register can never take on a negative value, that is a guarantee. It is not based on a heuristic, it is based on a proof. How this guarantee is possible, despite the halting problem, is because he designs an abstract representation of the program being analyzed. It loses a lot of information, but by using it, one can actually prove things about the program.
If I understand you correctly, you're talking about heuristics which allow one to say "It is very likely that this program will exhibit that behavior." It's not a guarantee, because it's not based on a proof, but rather on heuristics which try to match patterns in the code in question to patterns in code that was known to exhibit such behavior. Carmack talks about using such static analysis in his projects in a popular blog post (HN discussions: http://news.ycombinator.com/item?id=4543553 and http://news.ycombinator.com/item?id=3388290).
In simpler terms, Matt's approach avoids the halting problem by only providing guarantees on a limited set of behaviors. The approach you're talking about avoids the halting problem by not providing guarantees, but by providing likelihoods.
How this guarantee is possible, despite the halting problem, is because he designs an abstract representation of the program being analyzed. It loses a lot of information, but by using it, one can actually prove things about the program.
I would rather say: the halting problem shows that there exists at least one program for which it is impossible to prove termination - not that every termination problem is undecidable, as the trivial example "return 5;" shows.
Well, something like "this condition is likely to be true" could help give an estimate for whether a loop will run many times, which leads to things like a better estimate for the cost of spilling to the stack values used in that loop instead of keeping them in registers.
the halting problem shows that there exists at least one program for which it is impossible to prove termination
I'm not sure that "every halting checker has some input where it gives the wrong answer" permits the conclusion "there is some input where they all give the wrong answer." Is "the machine must generate a proof of its answer" what lets you make that extra leap?
The reason that everyone brings up the halting problem in this context is not that we want to test programs if they halt. (Well, we may, but we're generally looking for other behaviors.) It's because for some behaviors, if you want to be able to handle all cases, then that problem can be reduced to the halting problem. Matt provides an excellent demonstration of this with array indices.
I'm confused by your restatement, because it does not restate what I said. You're talking about the halting problem. I'm talking about proving general behaviors in programs, despite the theoretical result that solving the halting problem is impossible, and that many behaviors can be reduced to the halting problem. The solution, as Matt explains, is to find ways to describe behaviors so they do not reduce to the halting problem. Using this technique, you still can have false negatives ("I don't know"), but you can guarantee no false positives.
Indeed! I hadn't quite made out what you were trying to say in that post - my statement was not relevant.
if (p == NULL)
fatal_error();
*p = 0;
The art of static analysis, then, is to design heuristics that provide useful information in enough cases to be of practical value.
Cousot's Analysis of Signs is sound. It can fail by returning {-,0,+}. However, when it succeeds, it does make guarantees. If it returns {0} for an integer, then it has proven that that integer will always and only have the value 0. Furthermore, if that integer can be 0, it is guaranteed the analysis will return a set containing 0.
Pragmatic analyses have no such guarantees; if a pragmatic analysis of signs returns {0}, then anything could happen. They are "pragmatic" in that their value is determined by how correlated their output is with the answer on real programs; there are no theorems to prove about them.
If you can provide a guarantee based on a proof, then I do not consider that a heuristic. I consider there to be two differences between proof-based analysis and heuristic-based analysis:
1. Proof-based analysis may provide guarantees. In other words, no false-positives, but false-negatives are possible. Heuristics cannot provide guarantees, only probabilities, and they can have both false-positives and false-negatives.
2. Heuristic-based analysis require data to create. For example, let's say we analyzed some code and saw:
struct thing* = alloc_thing();
process_thing(thing);
thing = alloc_thing();
Note that most analyses do no actual automated theorem proving; they just, conceptually at least, instantiate existing proofs. If, by "find a proof," you mean "show the presence of bugs if there are any," then that's part of the same soundness proof. Conversely, showing the absence of bugs if there aren't any is completeness; you can't have both due to the halting problem.
Whether an analyzer correctly implements an analysis is a totally separate beast, and is an example of verification. Xavier Leroy has done some excellent work in verifying static analyzers.
Example http://stackoverflow.com/questions/12729110/strange-integer-...
It works for the register language with infinite precision integers.
If you want it to work for C, you'll have to redefine the abstractions of addition and multiplication (and every other operation) appropriately.
You'll also have to parameterize it by word size, endianness, signedness and type of arithmetic.
That is, sound static analysis for C is almost certainly going to be platform-specific.
But, you can still do it.
Indeed. Arguably, compilers do it all the time to be able to (safely) do optimizations. Hence static analyzers are usually an extension to an compiler. Frama-C kind of being the exception, using CIL instead of the intermediate representation of a production compiler.