Does the input ( boolean and arithmetic equations ) order matters to the constraint solvers like Gecode and SMT solvers like microsoft Z3 ?? If yes, which one of these two will perform better provided that I can take advantage of some pre-known heuristics using branching function in Gecode ??
(Note : I dont know if function similar to branch() in Gecode, exists in Z3)
In theory, no; order should not matter. The order of assertions should not make a difference. But in practice, they can have an impact as heuristics might end up spending a lot of time in dead-ends. SMT solvers usually work as black-boxes, i.e., it's hard to see how they are progressing unless you know their exact internals. You can, however, turn up verbosity (use z3's -v flag) and might look at the output to see if you spot any divergent behavior.
As with any general "performance of SMT solver" question, it is impossible to answer in the abstract. Each problem instance has specific characteristics, and there might be different ways of coding it to make it easier for the solver. If you post specific problems, you can get better suggestions.
Related
While reading Extending Sledgehammer with SMT solvers I read the following:
In the original Sledgehammer architecture, the available lemmas were rewritten
to clause normal form using a naive application of distributive laws before the relevance filter was invoked. To avoid clausifying thousands of lemmas on each invocation, the clauses were kept in a cache. This design was technically incompatible with the (cache-unaware) smt method, and it was already unsatisfactory for ATPs, which include custom polynomial-time clausifiers.
My understanding of SMT so far is as follows: SMTs don't work over clauses. Instead, they try to build a model for the quantifier-free part of a problem. The search is refined by instantiating quantifiers according to some set of active terms. Thus, indeed no clausal form is needed for SMT solvers.
We rewrote the relevance filter so that it operates on arbitrary HOL formulas, trying to simulate the old behavior. To mimic the penalty associated with Skolem functions in the clause-based code, we keep track of polarities and detect quantifiers that give rise to Skolem functions.
What's the penalty associated with Skolem functions? I could understand they are not good for SMTs, but here it seems that they are bad for ATPs too...
First, SMT solvers do work over clauses and there is definitely some (non-naive) normalization internally (e.g., miniscoping). But you do not need to do the normalization before calling the SMT solver (especially, since it will be more naive and generate a larger number of clauses).
Anyway, Section 6.6.7 explains why skolemization was done on the Isabelle side. To summarize: it is not possible to introduce polymorphic constants in a proof in Isabelle; hence it must be done before starting the proof.
It seems likely that, when writing the paper, not changing the filtering lead to worse performance and, hence, the penalty was added. However, I tried to find the relevant code simulating clausification in Sledgehammer, so I don't believe that this happens anymore.
How can I use Z3 to simplify an linear inequality ?
The inequality is as follows：
(x+k−1<n)∧(z>x+k−1)∧(x+k<n)∧(z<=x+k)
and the ideal result should be as following.
z-x<=k < z−x+1
But how to solve it using z3?
Z3 can perform some simplifications indeed, but expecting it to do well on symbolic computation would be too much to ask for. Since your question has no details, it's impossible to give you any guidance. If your interest is symbolic mathematics, perhaps using a symbolic-algebra tool is the better choice. For instance, Wolfram Alpha can handle your example quite well: http://www.wolframalpha.com/input/?i=(x%2Bk%E2%88%921%3Cn)%E2%88%A7(z%3Ex%2Bk%E2%88%921)%E2%88%A7(x%2Bk%3Cn)%E2%88%A7(z%3C%3Dx%2Bk)
My team has been using the Z3 solver to perform passive learning. Passive learning entails obtaining from a set of observations a model consistent with all observations in the set. We consider models of different formalisms, the simplest being Deterministic Finite Automata (DFA) and Mealy machines. For DFAs, observations are just positive or negative samples.
The approach is very simplistic. Given the formalism and observations, we encode each observation into a Z3 constraint over (uninterpreted) functions which correspond to functions in the formalism definition. For DFAs for example, this definition includes a transition function (trans: States X Inputs -> States) and an output function (out: States -> Boolean).
Encoding say the observation (aa, +) would be done as follows:
out(trans(trans(start,a),a)) == True
Where start is the initial state. To construct a model, we add all the observation constraints to the solver. We also add a constraint which limits the number of states in the model. We solve the constraints for a limit of 1, 2, 3... states until the solver can find a solution. The solution is a minimum state-model that is consistent with the observations.
I posted a code snippet using Z3Py which does just this. Predictably, our approach is not scalable (the problem is NP-complete). I was wondering if there were any (small) tweaks we could perform to improve scalability? (in the way of trying out different sorts, strategies...)
We have already tried arranging all observations into a Prefix Tree and using this tree in encoding, but scalability was only marginally improved. I am well aware that there are much more scalable SAT-based approaches to this problem (reducing it to a graph coloring problem). We would like to see how far a simple SMT-based approach can take us.
So far, what I have found is that the best Sorts for defining inputs and states are DeclareSort. It also helps if we eliminate quantifiers from the state-size constraint. Interestingly enough, incremental solving did not really help. But it could be that I am not using it properly (I am an utter novice in SMT theory).
Thanks! BTW, I am unsure how viable/useful this test is as a benchmark for SMT solvers.
As far as I understand, Z3, when encountering quantified linear real/rational arithmetic, applies a form of quantifier elimination described in Bjørner, IJCAR 2010 and more recent work by Bjørner and Monniaux (that's what qe_sat_tactic.cpp says, at least).
I was wondering
Whether it still works if the formula is multilinear, in the sense that the "constants" are symbolic. E.g. ∀x, ax≤b ⇒ ax ≤ 0 can be dealt with by separating the cases a<0, a=0 and a>0. This is possible using Weispfenning's virtual substitution approach, but I don't know what ended up being implemented in Z3 (that is, whether it implements the general approach or the one restricted to constant coefficients).
Whether it is possible, in Z3, to output the result of elimination instead of just solving for one model. There might be a Z3 tactic to do so but I don't know how this is supposed to be requested.
Whether it is possible, in Z3, to perform elimination as described above, then use the new nonlinear solver to obtain a model. Again, a succession of tactics might do the trick, but I don't know how this is supposed to be requested.
Thanks.
After long travels (including a travel where I met David at a conference), here is a short summary to answer the questions as they are posed.
There is no specific support for multi-linear forms.
The 'qe' tactic produces results of elimination, but may as a side-effect decide satisfiablity.
This is a very interesting problem to investigate, but it is not supported out of the box.
Has anyone tried proving Z3 with Z3 itself?
Is it even possible, to prove that Z3 is correct, using Z3?
More theoretical, is it possible to prove that tool X is correct, using X itself?
The short answer is: “no, nobody tried to prove Z3 using Z3 itself” :-)
The sentence “we proved program X to be correct” is very misleading.
The main problem is: what does it mean to be correct.
In the case of Z3, one could say that Z3 is correct if, at least, it never returns “sat” for an unsatisfiable problem, and “unsat” for a satisfiable one.
This definition may be improved by also including additional properties such as: Z3 should not crash; the function X in the Z3 API has property Y, etc.
After we agree on what we are supposed to prove, we have to create models of the runtime, programming language semantics (C++ in the case of Z3), etc.
Then, a tool (aka verifier) is used to convert the actual code into a set of formulas that we should check using a theorem prover such as Z3.
We need the verifier because Z3 does not “understand” C++.
The Verifying C Compiler (VCC) is an example of this kind of tool.
Note that, proving Z3 to be correct using this approach does not provide a definitive guarantee that Z3 is really correct since our models may be incorrect, the verifier may be incorrect, Z3 may be incorrect, etc.
To use verifiers, such as VCC, we need to annotate the program with the properties we want to verify, loop invariants, etc. Some annotations are used to specify what code fragments are supposed to do. Other annotations are used to "help/guide" the theorem prover. In some cases, the amount of annotations is bigger than the program being verified. So, the process is not completely automatic.
Another problem is cost, the process would be very expensive. It would be much more time consuming than implementing Z3.
Z3 has 300k lines of code, some of this code is based on very subtle algorithms and implementation tricks.
Another problem is maintenance, we are regularly adding new features and improving performance. These modifications would affect the proof.
Although the cost may be very high, VCC has been used to verify nontrivial pieces of code such as the Microsoft Hyper-V hypervisor.
In theory, any verifier for programming language X can be used to prove itself if it is also implemented in language X.
The Spec# verifier is an example of such tool.
Spec# is implemented in Spec#, and several parts of Spec# were verified using Spec#.
Note that, Spec# uses Z3 and assumes it is correct. Of course, this is a big assumption.
You can find more information about these issues and Z3 applications on the paper:
http://research.microsoft.com/en-us/um/people/leonardo/ijcar10.pdf
No, it is not possible to prove that a nontrivial tool is correct using the tool itself. This was basically stated in Gödel's second incompleteness theorem:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.
Since Z3 includes arithmetic, it cannot prove its own consistency.
Because it was mentioned in a comment above: Even if the user provides invariants, Gödels's theorem still applies. This is not a question of computability. The theorem states that no such prove can exist in a consistent system.
However you could verify parts of Z3 with Z3.
Edit after 5 years:
Actually the argument is easier than Gödel's incompleteness theorem.
Let's say Z3 is correct if it only returns UNSAT for unsatisfiable formulas.
Assume we find a formula A, such that if A is unsatisfiable then Z3 is correct (and we somehow have proven this relation).
We can give this formula to Z3, but
if Z3 returns UNSAT it could be because Z3 is correct or because of a bug in Z3. So we have not verified anything.
if Z3 returns SAT and a countermodel, we might be able to find a bug in Z3 by analyzing the model
otherwise we don't know anything.
So we can use Z3 to find bugs in Z3 and to improve confidence about Z3 (to an extremely high level), but not to formally verify it.