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Lars Tveito 2019-11-19 01:59:04 +01:00
parent 48c20b294e
commit b6ac1a131f
1 changed files with 16 additions and 12 deletions
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@ -81,11 +81,12 @@ cry for help.
In addition, one axiom is added to formalize induction. Because a theory is
closed under implication, the theory consists of all true first-order
propositions that follows from these axioms, which corresponds to the true
propositions about natural numbers.
However, in Z3, we don't see such axioms; they just provide the formal
justification for implementing special solvers for problem domains like
natural numbers other commonly used theories. In z3, we could write
natural numbers other commonly used theories. In Z3, we could write
something like this:
#+BEGIN_SRC z3
@ -198,11 +199,11 @@ cry for help.
** Constraint modeling
We need to capture our intention with first-order logic formulas, and
preferably quantifier-free. In SMT-solving, quantifier-free means that we
only try to solve a set of constraints where no variable is bound by a
quantifier; these are usually much easier to solve. Rather, we use a finite
set of constant symbols, with some associated sort, and try to find an
interpretation for them.
preferably quantifier-free. In the context of SMT-solving, quantifier-free
means that we only try to solve a set of constraints where no variable is
bound by a quantifier; these are usually much easier to solve. Rather, we
use a finite set of constant symbols, with some associated sort, and try to
find an interpretation for them.
The end result needs to be a set of committees, where each committee
consists of two examiners with a number of exams to correct. An important
@ -215,12 +216,13 @@ cry for help.
A natural encoding could be modeling a committee as an integer constant,
where the value assigned to a committee corresponds to the number of exams
they correct. It is quite easy to compute all possible committees, and make
they correct. If the committee don't are not assigned any exams, we discard
it completely. It is quite easy to compute all possible committees, and make
one integer constant for each of them.
Let's write a function that takes a list of capacities, and return a
dictionary, associating (Python) committees to their corresponding integer
constant.
dictionary, associating committees to their corresponding integer constant.
Remember that we represent a committee as a set of exactly two examiners.
#+BEGIN_SRC python :tangle committees.py
def committees(C):
@ -230,11 +232,13 @@ cry for help.
return {c : Int(str(c)) for c in cs}
#+END_SRC
** Capacity constraints
Now we must ensure that no examiner receives more exams than their capacity.
Given an examiner $i$, where $0 <= i < N$, we let $c_i$ denote the set of
Given an examiner $i$, where $0 <= i < |C|$, we let $c_i$ denote the set of
all committees $i$ participates in. Then $\sum{c_i} <= C[i]$, i.e. the sum
of the committees $c_i$ does not exceed the capacity of the examiner $i$. We
write a function that encodes these constraints.
of the exams corrected by committees in $c_i$ does not exceed the capacity
of the examiner $i$. We write a function that encodes these constraints.
#+BEGIN_SRC python :tangle committees.py
def capacity_constraint(comms, C):