How to specify initial 'soft' values for the model? This initial model is the result of solving a similar query, and it is likely that this model has a correct pieces or even may be true for the current query.
Currently I am simulating this with an incremental solving and hard/soft constraints:
(define-fun trans_assumed ((a Int)) Int
; an initial model, which may be (partially) true
)
(declare-fun trans_sought ((a Int)) Int)
(declare-const p Bool)
(assert (=> p (forall ((a Int)) (= (trans_assumed a) (trans_sought a)))))
(check-sat p) ; in hope that trans_assumed values will be used as initial below
; add here the main constraints for trans_sought function
(check-sat) ; Z3 will use trans_assumed as a starting point for trans_sought
Does this really specify initial values for trans_sought to be trans_assumed?
Incremental mode of solving is slow compared to sequential. Any better ways of introducing initial values?
I think this is a good approach, but you may consider using more Boolean variables. Right now, it is a "all" or "nothing" approach. In your script, when (check-sat p) is executed, Z3 will look for a model where trans_assumed and trans_sought have the same interpretation. If such model does not exist, it will return with the unsat core containing p. When (check) is executed, Z3 is free to assign p to false, and the universal quantifier is essentially a don't care. That is, trans_assumed and trans_sought can be completely different.
If you use multiple Boolean variables to control the interpretation of trans_sought, you will have more flexibility.
If the rest of your problem is quantifier free, you should consider dropping the universal quantifier. This can be done if you only care about the value of trans_sought in a finite number of points.
Suppose we have that trans_assumed(0) = 1 and trans_assumed(1) = 10. Then, we can write:
assert (=> p0 (= (trans_sought 0) 1)))
assert (=> p1 (= (trans_sought 1) 10)))
In this encoding, we can query (check-sat p0 p1), (check-sat p0), (check-sat p1)
Related
I'm trying to create a function in Z3 that is transitive but not reflexive. I.e. if (transitive a b) and (transitive b c)hold then (transitive a c) should hold, but (transitive a a) should not.
I've tried to do it the following way, with 5 "tests". The first does what I expect, but the second one fails and results in unknown.
(declare-datatypes () ((T T1 T2 T3)))
(declare-fun f (T T) Bool)
(assert(f T1 T2))
(assert(f T2 T3))
; Make sure that f is not reflexive
(assert
(forall ((x T))
(not (f x x))))
; Now we create the transitivity function ourselves
(define-fun-rec transitive ((x T) (y T)) Bool
(or
(f x y)
(exists ((z T))
(and
(f x z)
(transitive z y)))))
; This works and gives sat
(push)
(assert (not (transitive T1 T1)))
(assert (not (transitive T2 T2)))
(assert (not (transitive T3 T3)))
(check-sat)
(pop)
; This fails with "unknown" and the verbose flag gives: (smt.mbqi "max instantiations 1000 reached")
(push)
(assert
(forall ((x T))
(not (transitive x x))))
(check-sat)
(pop)
My question is: how does the second test differ from the first? Why does the last one give unknown, whereas the one before that works just fine?
The "verbose" message is a hint here. mbqi stands for model-based-quantifier-instantiation. It's a method of dealing with quantifiers in SMT solving. In the first case, MBQI manages to find a model. But your transitive function is just too complicated for MBQI to handle, and thus it gives up. Increasing the limit will not likely address the problem, nor it's a long term solution.
Short story long, recursive definitions are difficult to deal with, and recursive definitions with quantifiers are even harder. The logic becomes semi-decidable, and you're at the mercy of heuristics. Even if you found a way to make z3 compute a model for this, it would be brittle. These sorts of problems are just not suitable for SMT solving; better use a proper theorem prover like Isabelle, Hol, Coq, Lean. Agda, etc. Almost all these tools offer "tactics" to dispatch subgoals to SMT solvers, so you have the best of both worlds. (Of course you lose full automation, but with quantifiers present, you can't expect any better.)
I was working with z3 with the following example.
f=Function('f',IntSort(),IntSort())
n=Int('n')
c=Int('c')
s=Solver()
s.add(c>=0)
s.add(f(0)==0)
s.add(ForAll([n],Implies(n>=0, f(n+1)==f(n)+10/(n-c))))
The last equation is inconsistent (since n=c would make it indeterminate). But, Z3 cannot detect this kind of inconsistencies. Is there any way in which Z3 can be made to detect it, or any other tool that can detect it?
As far as I can tell, your assertion that the last equation is inconsistent does not match the documentation of the SMT-LIB standard. The page Theories: Reals says:
Since in SMT-LIB logic all function symbols are interpreted
as total functions, terms of the form (/ t 0) are meaningful in
every instance of Reals. However, the declaration imposes no
constraints on their value. This means in particular that
for every instance theory T and
for every value v (as defined in the :values attribute) and
closed term t of sort Real,
there is a model of T that satisfies (= v (/ t 0)).
Similarly, the page Theories: Ints says:
See note in the Reals theory declaration about terms of the form
(/ t 0).
The same observation applies here to terms of the form (div t 0) and
(mod t 0).
Therefore, it stands to reason to believe that no SMT-LIB compliant tool would ever print unsat for the given formula.
Z3 does not check for division by zero because, as Patrick Trentin mentioned, the semantics of division by zero according to SMT-LIB are that it returns an unknown value.
You can manually ask Z3 to check for division by zero, to ensure that you never depend division by zero. (This is important, for example, if you are modeling a language where division by zero has a different semantics from SMT-LIB.)
For your example, this would look as follows:
(declare-fun f (Int) Int)
(declare-const c Int)
(assert (>= c 0))
(assert (= (f 0) 0))
; check for division by zero
(push)
(declare-const n Int)
(assert (>= n 0))
(assert (= (- n c) 0))
(check-sat) ; reports sat, meaning division by zero is possible
(get-model) ; an example model where division by zero would occur
(pop)
;; Supposing the check had passed (returned unsat) instead, we could
;; continue, safely knowing that division by zero could not happen in
;; the following.
(assert (forall ((n Int))
(=> (>= n 0)
(= (f (+ n 1))
(+ (f n) (/ 10 (- n c)))))))
I am trying to prove an inductive fact in Z3, an SMT solver by Microsoft. I know that Z3 does not provide this functionality in general, as explained in the Z3 guide (section 8: Datatypes), but it looks like this is possible when we constrain the domain over which we want to prove the fact. Consider the following example:
(declare-fun p (Int) Bool)
(assert (p 0))
(assert (forall ((x Int))
(=>
(and (> x 0) (<= x 20))
(= (p (- x 1)) (p x) ))))
(assert (not (p 20)))
(check-sat)
The solver responds correctly with unsat, which means that (p 20) is valid. The problem is that when we relax this constraint any further (we replace 20 in the previous example by any integer greater than 20), the solver responds with unknown.
I find this strange because it does not take Z3 long to solve the original problem, but when we increase the upper limit by one it becomes suddenly impossible. I have tried to add a pattern to the quantifier as follows:
(declare-fun p (Int) Bool)
(assert (p 0))
(assert (forall ((x Int))
(! (=>
(and (> x 0) (<= x 40))
(= (p (- x 1)) (p x) )) :pattern ((<= x 40)))))
(assert (not (p 40)))
(check-sat)
Which seems to work better, but now the upper limit is 40. Does this mean that I can better not use Z3 to prove such facts, or am I formulating my problem incorrectly?
Z3 uses many heuristics to control quantifier instantiation. One one them is based on the "instantiation depth". Z3 tags every expression with a "depth" attribute. All user supplied assertions are tagged with depth 0. When a quantifier is instantiated, the depth of the new expressions is bumped. Z3 will not instantiate quantifiers using expressions tagged with a depth greater than a pre-defined threshold. In your problem, the threshold is reached: (p 40) is depth 0, (p 39) is depth 1, (p 38) is depth 2, etc.
To increase the threshold, you should use the option:
(set-option :qi-eager-threshold 100)
Here is the example with this option: http://rise4fun.com/Z3/ZdxO.
Of course, using this setting, Z3 will timeout, for example, for (p 110).
In the future, Z3 will have better support for "bounded quantification". In most cases, the best approach for handling this kind of quantifier is to expand it.
With the programmatic API, we can easily "instantiate" expressions before we send them to Z3.
Here is an example in Python (http://rise4fun.com/Z3Py/44lE):
p = Function('p', IntSort(), BoolSort())
s = Solver()
s.add(p(0))
s.add([ p(x+1) == p(x) for x in range(40)])
s.add(Not(p(40)))
print s.check()
Finally, in Z3, patterns containing arithmetic symbols are not very effective. The problem is that Z3 preprocess the formula before solving. Then, most patterns containing arithmetic symbols will never match. For more information on how to use patterns/triggers effectively, see this article. The author also provides a slide deck.
In the following example, I tried to use uninterpreted Boolean function like "(declare-const p (Int) Bool)" rather than single Boolean constant for each assumption. But it does not work (it gives compilation error).
(set-option :produce-unsat-cores true)
(set-option :produce-models true)
(declare-fun p (Int) Bool)
;(declare-const p1 Bool)
;(declare-const p2 Bool)
; (declare-const p3 Bool)
;; We assert (=> p C) to track C using p
(declare-const x Int)
(declare-const y Int)
(assert (=> (p 1) (> x 10)))
;; An Boolean constant may track more than one formula
(assert (=> (p 1) (> y x)))
(assert (=> (p 2) (< y 5)))
(assert (=> (p 3) (> y 0)))
(check-sat (p 1) (p 2) (p 3))
(get-unsat-core)
Output
Z3(18, 16): ERROR: invalid check-sat command, 'not' expected, assumptions must be Boolean literals
Z3(19, 19): ERROR: unsat core is not available
I understand that it is not possible (unsupported) to use Boolean function. Is there any reason behind that? Is there different way to do that?
We have this restriction because Z3 applies many simplifications before it solves a problem. Some of them will rewrite formulas and terms. The problem that is actually solved by Z3 is very often quite different from the input problem. We would have trace back the simplified assumptions to the original assumptions, or introduce auxiliary variables. Restricting to Boolean literals avoids this issue, and makes the interface very clean. Note that this restriction does not limit the expressiveness. If you think it is too annoying to declare many Boolean variables to track different assertions. I suggest you take a look at the new Python front-end for Z3 called Z3Py. It is much more convenient to use than SMT 2.0. Here is your example in Z3Py: http://rise4fun.com/Z3Py/cL
In this example, instead of creating an uninterpreted predicate p, a "vector" (actually, it is a Python list) o Boolean constants is created.
The Z3Py online tutorial contains many examples.
It is also possible to implement in Z3Py the approach that creates auxiliary variables.
Here is the script that does the trick. I defined a function check_ext that does all the plumbing. http://rise4fun.com/Z3Py/B4
How to specify initial 'soft' values for the model? This initial model is the result of solving a similar query, and it is likely that this model has a correct pieces or even may be true for the current query.
Currently I am simulating this with an incremental solving and hard/soft constraints:
(define-fun trans_assumed ((a Int)) Int
; an initial model, which may be (partially) true
)
(declare-fun trans_sought ((a Int)) Int)
(declare-const p Bool)
(assert (=> p (forall ((a Int)) (= (trans_assumed a) (trans_sought a)))))
(check-sat p) ; in hope that trans_assumed values will be used as initial below
; add here the main constraints for trans_sought function
(check-sat) ; Z3 will use trans_assumed as a starting point for trans_sought
Does this really specify initial values for trans_sought to be trans_assumed?
Incremental mode of solving is slow compared to sequential. Any better ways of introducing initial values?
I think this is a good approach, but you may consider using more Boolean variables. Right now, it is a "all" or "nothing" approach. In your script, when (check-sat p) is executed, Z3 will look for a model where trans_assumed and trans_sought have the same interpretation. If such model does not exist, it will return with the unsat core containing p. When (check) is executed, Z3 is free to assign p to false, and the universal quantifier is essentially a don't care. That is, trans_assumed and trans_sought can be completely different.
If you use multiple Boolean variables to control the interpretation of trans_sought, you will have more flexibility.
If the rest of your problem is quantifier free, you should consider dropping the universal quantifier. This can be done if you only care about the value of trans_sought in a finite number of points.
Suppose we have that trans_assumed(0) = 1 and trans_assumed(1) = 10. Then, we can write:
assert (=> p0 (= (trans_sought 0) 1)))
assert (=> p1 (= (trans_sought 1) 10)))
In this encoding, we can query (check-sat p0 p1), (check-sat p0), (check-sat p1)