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## Basics of z3
The exercises you will solve in the practicals are only going to cover a small subset of the possibilities of solving problems with z3. If you are interested in more background or need to look into some details we suggest you to take a look [here](https://theory.stanford.edu/~nikolaj/programmingz3.html).
#### A Simple Example
A typical workflow that integrates z3 into a python script follows these steps:
- import `z3`,
- declare needed variables of specific `Sort` (this is the word we use for types in z3),
- declare a solver: `solver = Solver()` and
- add constraints for the declared variables to the solver.
- After adding all the constraints we tell the solver to try to `check()` for satisfiability and if the solver tells us that the model is satisfiable we may
- print the model.
Consider the following simple example:
``` python
# coding: utf-8
import os, sys
from z3 import *
# v-- internal z3 representation
x = Bool('x')
#^-- python variable
# v-- internal z3 representation
gamma = Bool('g') # possible, but not advisable
#^-- python variable
# Declare a solver with which we can do some work
solver = Solver()
p = Bool('p')
qu = Bool('q')
r = Bool('r')
# p -> q, r = ~q, ~p or r
# Add constraints
solver.add(Implies(p,qu))
solver.add(r == Not(qu))
solver.add(Or(Not(p), r))
# solver.add(r == q)
res = solver.check()
if res != sat:
print("unsat")
sys.exit(1)
m = solver.model()
for d in m.decls():
print("%s -> %s" % (d, m[d]))
```
At first we import z3 `from z3 import *`.
We then need to declare variables:
``` python
# v-- internal z3 representation
x = Bool('x')
#^-- python variable
# v-- internal z3 representation
gamma = Bool('g') # possible, but not advisable
#^-- python variable
```
Lets have a closer look: In z3 we declare variables of a `Sort`. In z3, you may use `BoolSort`, `IntSort`, `RealSort` and others. The example above only covers propositional logic, hence only uses variables declared as `Bool`.
In order to check for satisfiability we are going to need a solver: `solver = Solver()`.
In the next step we will add some constraints to the solver:
``` python
p = Bool('p')
qu = Bool('qu')
r = Bool('r')
# p -> q, r = ~q, ~p or r
# Add constraints
solver.add(Implies(p,qu))
solver.add(r == Not(qu))
solver.add(Or(Not(p), r))
```
Adding constraints is done with the solvers `add()` method. These constraints are added to the solver as one conjunction.
Finally, we can as z3 to check for satisfiability:
``` python
res = solver.check()
if res != sat:
print("unsat")
sys.exit(1)
```
Our example is satisfiable so in the end we print the model by asking the solver for the created model and print all of the variables which have an associated value in the created model:
``` python
m = solver.model()
for d in m.decls():
print("%s -> %s" % (d, m[d]))
> q -> True
> p -> False
> r -> False
```
This can be done with `solver.model().decls()` which lists the assignments as a dictionary. You can also evaluate individual variables in your model:
``` python
m = solver.model()
print("qu: " + str(m.eval(qu)))
print("p: " + str(m.eval(p)))
print("r: " + str(m.eval(r)))
> qu: True
> p: False
> r: False
```
### First Order Logic Types and Constraints
So far we have only touched propositional logic, but z3 is an SMT-solver so lets expand our knowledge to use these funtionalities.
From the example above, you can see that creating z3 integer variables follows the same principle as for booleans.
Python expressions are valid in constraints too, for example using a built-in function: `solver.add(sum([x,y]) == 42)`.
### Custom Datatypes and Sorts
So far we have used z3's capabilities by using boolean or integer valued variables. This already gives us quite a powerful tool, but we want to extend this to be able to use our own custom structures and datatypes. A first approach is to use the `DataType` functionality.
``` python
Colour = DataType("Colour")
```
This will create a placeholder that contains constructors and accessors for our custom `Colour` variables.
``` python
Colour.declare("green")
Colour.declare("yellow")
ColourSort = Colour.create()
```
We have now defined two constructors for possible values of our `Colour` variable type and finalized the definition of `Colour`. `.create()` returns a sort that we can now work with. z3 will now internally work with these possible values for `Colour`. You may think of `Colour` in the same way as of the `IntSort` mentioned above. Let's consider this once more. We have used `Int(...)` to tell z3 that we want it to create an internal representation of an integer variable. This could be refactored as such:
This means that `Int("x")` is only syntactic sugar to make our code more legible. But this also tells us how to use our `Colour` datatype:
``` python
x = Const("cell", ColourSort)
```
We have used the `DataType` functionality solely to model an enum-type variable. A constructor for such a datatype but might also have some accessor associated with it, allowing us to create algebraic structures like lists or trees.
Another type of a custom structures are uninterpreted sorts. These can be created using `DeclareSort(...)`:
```python
A = DeclareSort('A')
x, y = Consts('x y', A)
```
An uninterpreted sort may be used similarly as the above discussed `DataType`s. z3 will see `x` and `y` as of type `A`. Since these sorts are uninterpreted there are no semantics related to the variables. , e.g. we have no means to compare `x` and `y`.
Note that you do not have to `create()` your custom sort, it will be handled like a set of its declared variables.
### Uninterpreted Functions
Lastly, we cover uninterpreted functions that give us a way to model relationships, or mappings between variables. A function maps from a set of sorts to a sort.
Consider this example (taken from [here](https://ece.uwaterloo.ca/~agurfink/ece653w17/z3py-advanced)):
```python
from z3 import *
A = DeclareSort('A')
x, y = Consts('x y', A)
f = Function('f', A, A)
s = Solver()
s.add(f(x) == y, f(f(x)) == x, x != y)
s.check()
m = s.model()
print(m)
print("interpretation assigned to A:")
print("f(x) = " + m.evaluate(f(x)).decl().name())
print("f(y) = " + m.evaluate(f(y)).decl().name())
```
We use an uninterpreted sort `A` with values `x` and `y`. `f` is declared as a `Function(...)` mapping `A` to `A`. The function `f` is then constrainted, such that `f(x)` maps to `y`, `f(f(x))` maps to `x` again and that `x` and `y` need to be different values. Checking for satisfiability will now check whether such a function can exist. If z3 can find such a function, it will represent the look-up table for `f` in the satisfying model:
```python
[x = A!val!0,
y = A!val!1,
f = [A!val!1 -> A!val!0, else -> A!val!1]]
interpretation assigned to A:
f(x) = A!val!1
f(y) = A!val!0
```
This function does not need to be fully defined, as z3 will only check if it can exist with respect to our expressed constraints. In order to get an assignment for all possible values in our sort, we can evaluate the model using the `model_completion=True` flag. This is taken from the `seating-arrangement` example:
```python
arrangement = ["" for guest in range(len(guests))]
