I'm doing quantifier elimination on LIA using F# and Z3 3.2 API.
Z3 used to have QUANT_ARITH configuration which indicates the use of Cooper's method or the Omega test for LIA quantifier elimination. But that option was replaced by ELIM_QUANTIFIERS in Z3 2.6 (see Z3 release notes).
I want to ask internally how Z3 3.2 knows which method to use for quantifier elimination? Can users affect the choice of method like QUANT_ARITH before?
Furthermore, with the introduction of strategy specification language, will Z3 allow us to customize quantifier elimination by extending or combining these methods?
The quantifier elimination module was re-implemented. The new implementation should be faster and correct.
The latest Z3 does not have a implementation of Cooper’s method or Omega test.
You can find more details about the actual quantifier elimination procedure used in Z3 at:
“Linear Quantifier Elimination as an Abstract Decision Procedure, Nikolaj Bjørner, IJCAR 2010”.
Regarding the strategy specification language, we will eventually expose tactics for performing quantifier elimination.
We are currently working on this infrastructure, more news are coming soon.
Related
After working with νZ, an extension within the SMT solver Z3 to make use of objective functions, I was surprised to find that the used optimization primitives are not part of the SMT-LIB2 syntax. These primitives are:
(maximize t) - instruct the solver to maximize t.
(minimize t) - instruct the solver to minimize t.
(assert-soft F :weight n) - assert soft constraint F, optionally with weight n.
This extension was introduced in 2014. The latest version of the SMT-LIB standard, version 2.6, still doesn't seem to introduce any kind of syntax to support objective functions. Is this really the case? If so, is there any work being done to introduce such a standard?
I'm not aware of any work to standardize Max-SAT (i.e., optimization) in SMTLib. Your best bet is to ask at the SMTLib mailing list (https://groups.google.com/g/smt-lib), to call for attention to this matter.
is there a way to express assumptions in Z3 (I am using the Z3Py library) such that the engine does not check their validity but takes them as underlying theories, just like in theorem proving?
For example, lets say that I have two unary functions with argument of type Real. I would like to tell the Z3 engine that for all input values, f1(t) is equal to f2(t).
Encoded in Z3Py that would look something like the following:
t = Real("t")
assumption1 = ForAll(t, f1(t) = f2(t)).
The problem with the presented code is that my assertion set is quite big and I use quantifiers (I am trying to prove satisfiability of a real-time system). If I add the above assertion to the set of the other assertions the checking procedure does not terminate.
is there a way to express assumptions in Z3 (I am using the Z3Py library) such that the engine does not check their validity but takes them as underlying theories, just like in theorem proving?
In fact, all assertions you add to Z3 are treated as what you call assumptions. Z3 checks satisfiability of the assertions, it does not check validity. To check validity of a formula F, you assert (not F), and check for satisfiability of (not F). If (not F) is unsat, then F is valid. If you have background axioms, you are essentially checking validity of Background => F, so you can check satisifiability of Background & (not F).
Whether Z3 terminates on your query depends on which combination of theories and quantifiers you use. The more features your queries combine the tougher it is.
For formulas over pure linear arithmetic or polynomial real arithmetic,
these are called LRA, LIA and NRA in the SMT-LIB classification (see smtlib.org) Z3 uses specialized decision procedures that have recently been added.
Yes, that's possible just as you describe it, but you will end up with quantifiers, which does of course mean that you're solving a harder problem and Z3 will behave differently (it's possible you end up using completely different solvers that don't even share much source code).
For the particular example given, it's possible to eliminate the quantifier cheaply because it has the form of a function definition (ForAll x . f(x) = ...), i.e., we can just replace all occurrences of f with the right hand side and then the quantifier is trivially satisfied. In Z3, this is done by the macro finder, which may be applied as a tactic (with name "macro-finder"), or if you are using the "smt" tactic (implicitly via others or directly), then you can set smt.macro_finder=true.
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.
Can SMT solver efficiently find a solution (or an assignment) for the pseudo-Boolean problem as described as follows:
\sum {i..m} f_i x1 x2.. xn *w_i
where f_i x1 x2 .. xn is a Boolean function, and w_i is a weight of Int type.
For your convenience, I highlight the contents in page 1 and 3, which is enough for specifying
the pseudo-Boolean problem.
SMT solvers typically address the question: given a logical formula, optionally using functions and predicates from underlying theories (such as the theory of arithmetic, the theory of bit-vectors, arrays), is the formula satisfiable or not.
They typically don't expose a way for you specify objective functions
and typically don't have built-in optimization procedures.
Some special cases are formulas that only use Booleans or a combination of Booleans and either bit-vectors or integers. Pseudo Boolean constraints can be formulated with either integers or encoded (with some care taking overflow semantics into account) using bit-vectors, or they can be encoded directly into SAT. For some formulas using bounded integers that fall in the class of psuedo-boolean problems, Z3 will try automatic reductions into bit-vectors. This applies only to benchmkars in the SMT-LIB2 format tagged as QF_LIA or applies if you explicitly invoke a tactic that performs this reduction (the "qflia" tactic should apply).
While Z3 does not directly expose objective functions, the question of augmenting
SMT solvers with objective functions is actively pursued in the research community.
One approach suggested by Nieuwenhuis and Oliveras in SAT 2006 was to build in
solving for the "weighted max SMT" problem as a custom theory. Yices comes with built-in
features for weighted max SMT, Z3 does not, but it is possible to write a custom
theory that performs the backtracking search of a weighted max SMT solver, but nothing
out of the box.
Sometimes people try to specify objective functions using quantified formulas.
In theory one could hope that quantifier elimination procedures then can solve
for the objective.
This is generally pretty bad when it comes to performance. Quantifier elimination
is an overfit and the routines (that we have) will not be efficient.
For your problem, if you want to find an optimized (maximum or minimum) result from the sum, yes Z3 has this ability. You can use the Optimize class of Z3 library instead of Solver class. The class provides two methods for 'maximization' and 'minimization' respectively. You can pass the SMT variable that is needed to be optimized and Optimization class model will give the solution for you. It actually worked with C# API using Microsoft.Z3 library. For your inconvenience, I am attaching a snippet:
Optimize opt; // initializing object
opt.MkMaximize(*your variable*);
opt.MkMinimize(*your variable*);
opt.Assert(*anything you need to do*);
I am using Z3 to check satisfiability in a logic with linear integer arithmetic, uninterpreted functions and quantifiers over boolean variables. Z3 does not provide models for satisfiable problems, I suppose this is because of the quantifiers (or the logic that I have chosen: AUFLIA).
Is there a way to make Z3 give me models for such problems, except for instantiating all the boolean variables myself?
Z3 can, in principle, decide this fragment. I say "in principle" because the complexity of the decision problem for this fragment is very high. For example, it subsumes the Bernays–Schönfinkel fragment (aka EPR) which is NEXPTIME-complete. A list of the fragments that can be decided by Z3 can be found at: http://rise4fun.com/z3/tutorial/guide
We have to make sure that model-based quantifier instantiation (MBQI) is enabled in Z3. You can do that by using the command line option MBQI=true or the SMT2 command
(set-option :mbqi true)
Z3 also has a threshold on the number of iterations of MBQI steps. We can change the threshold by using the command line option MBQI_MAX_ITERATIONS=<value> or the command
(set-option :mbqi-max-iterations 1000000)
For each MBQI step, we can ask Z3 to display which quantifiers were not satisfied by the current candidate model. Option MBQI_TRACE=true
That being said, I recently fixed a bug (crash) exposed by the SMT2 script you sent me. The fix will be available in Z3 3.2.