1. Chapter 2: Simplification, Optimization and Implication
Where we learn more fun things we can do with constraints.
CSC Simplification, Optimization and Implication

2. CSC5240 - Simplification, Optimization and Implication
Chapter Overview
Constraint Simplification
Projection
Constraint Simplifiers
Optimization
Implication and Equivalence
CSC Simplification, Optimization and Implication

3. Constraint Simplification - 1
Simplification: a transformation to make information implicit in the original system becomes apparent
Projection: a form of simplification to express a constraint only in terms of the variables that are of interest
Difficult and expensive but needed for presenting answer constraints in a form understandable to the user
CSC Simplification, Optimization and Implication

4. Constraint Simplification - 2
Two equivalent constraints represent the same information, but one may be “simpler” than the other
Removing redundant constraints, rewriting a primitive constraint, changing order, substituting using an equation all preserve equivalence
CSC Simplification, Optimization and Implication

5. Redundant Constraints - 1
One constraint C1 implies another C2 if the solutions of C1 are a subset of those of C2
C2 is said to be redundant wrt C1
It is written
CSC Simplification, Optimization and Implication

6. Redundant Constraints - 2
We can remove a primitive constraint which is redundant with respect to the rest of the constraint C without changing C’s meaning
This definitely produces a simpler constraint
CSC Simplification, Optimization and Implication

7. CSC5240 - Simplification, Optimization and Implication
Solved Form Solvers - 1
Since a solved form (normalizing) solver creates equivalent constraints it can be a simplifier
For example using the term constraint solver
CSC Simplification, Optimization and Implication

8. CSC5240 - Simplification, Optimization and Implication
Solved Form Solvers - 2
Or using the Gauss-Jordan solver
CSC Simplification, Optimization and Implication

9. CSC5240 - Simplification, Optimization and Implication
Projection - 1
It becomes even more important to simplify when we are only interested in some variables in the constraint
Simplified wrt to V and I
CSC Simplification, Optimization and Implication

10. CSC5240 - Simplification, Optimization and Implication
Projection - 2
The projection of a constraint C onto a set of variables V is a constraint C1 such that
Every solution of C is a solution of C1
A solution of C1 can be extended (Defn 2.2) to give a solution of C
CSC Simplification, Optimization and Implication

11. CSC5240 - Simplification, Optimization and Implication
Fouriers Algorithm
Eliminates variable y from linear ineqs in C
Write each ineq with y on one side of ineq
For each pair
produce a new ineq
The result is all new ineqs and those ineqs in C which do not involve y
CSC Simplification, Optimization and Implication

12. Fouriers Algorithm Example
Projecting out Y
Result only involving X
CSC Simplification, Optimization and Implication

13. Projecting Boolean Constraints
Any Boolean constraint C is equivalent to (x  Cx)  (x  Cx)
Cx: C with all occurrences of x replaced by 1
Cx: C with all occurrences of x replaced by 0
The projection of C onto vars(C){x} is just Cx  Cx
CSC Simplification, Optimization and Implication

14. Projecting Term Constraints
We can also project term constraints
For example
projected onto {X,Z} is
But what is X = cons(Y,Z) projected onto X?
Answer: there is no such constraint!
Must generalize to allow simplification to contain extra “local” variables
CSC Simplification, Optimization and Implication

15. Constraint Simplifiers
Constraints C1 and C2 are equivalent wrt a set of variables V if
taking any solution of one and restricting it to the variables in V, the restricted solution can be extended to be a solution of the other
E.g. X=succ(Y) and X=succ(Z) wrt {X}
CSC Simplification, Optimization and Implication

16. Simplifier Definition
A constraint simplifier is a function simpl which takes a constraint C and a set of variables V, and returns a constraint C1 that is equivalent to C wrt V
We can construct a simplifier for real inequalities from the Fouriers algorithm
CSC Simplification, Optimization and Implication

17. Tree Constraint Simplifier
apply the term solver to C obtaining C1
if C1 is false then return false
foreach equation x=t in C1
if x is in V then
if t is a variable not in V then
substitute x for t throughout C1 and result
else add x=t to result
return result
CSC Simplification, Optimization and Implication

18. Tree Simplification Example - 1
Tree constraint to be simplified wrt {Y,T}
Equivalent constraint from tree solver
Discard the first two equations, keep the third and use the last to substitute for U by T
CSC Simplification, Optimization and Implication

19. Tree Simplification Example - 2
In this example, the simplifier has managed to return the projection of the original constraint onto the variables of interest
Finding exact projection is not always possible, unfortunately
The answer will sometimes contain extra “local” variables
Try X = succ(Y)
CSC Simplification, Optimization and Implication

20. Simplifier Properties - 1
Difficult to quantify what “simple” means since there is tradeoff between different notions of simplicity
In some cases, the result of a simplifier may not even appear simpler to a human
Individuals may argue which of many textual forms of a constraint is the simplest, but there are some guiding principles
CSC Simplification, Optimization and Implication

21. Simplifier Properties - 2
Desirable properties of a simplifier are:
projecting:
weakly projecting: for all constraints C2 that are equivalent to C1 wrt V
a weakly projecting simplifier never uses more variables than is required
Both properties allow a simplifier to be used as a solver (how?)
CSC Simplification, Optimization and Implication

22. CSC5240 - Simplification, Optimization and Implication
Often given some problem which is modelled by constraints we don’t want just any solution, but a “best” solution
This is an optimization problem
We need an objective function, a mapping from solutions to reals, so that we can rank solutions
By convention, we always try to minimize
CSC Simplification, Optimization and Implication

23. CSC5240 - Simplification, Optimization and Implication
An optimization problem (C,f) consists of a constraint C and objective function f
A valuation v1 is preferred to valuation v2 if f(v1) < f(v2)
An optimal solution is a solution of C such that no other solution of C is preferred to it
Optimization problems may have more than one optimal solution
CSC Simplification, Optimization and Implication

24. CSC5240 - Simplification, Optimization and Implication
Optimization Example
An optimization problem
Find the closest point to the origin satisfying C. Some solutions and f values
Optimal solution
CSC Simplification, Optimization and Implication

25. Optimization Peculiarities
Some optimization problems have no solution.
Constraint has no solution
For any solution, there is a more preferable one. In other words, problem has no optimum. Can you give an example?
CSC Simplification, Optimization and Implication

26. Optimization as Simplification
Optimization is closely related to simplification
Project the constraint onto the variable that we want to optimize
Conjoin the projected constraint with the original constraint to compute an optimal solution to the original constraint
Very expensive in general
CSC Simplification, Optimization and Implication

27. The Simplex Algorithm - 1
The most widely used optimization algorithm
Optimizes a linear function wrt to linear constraints
Related to Gauss-Jordan elimination
Based on the following observations
Linear constraints form convex polygons
Optimal solution always occur at a vertex
CSC Simplification, Optimization and Implication

28. CSC5240 - Simplification, Optimization and Implication
The Simplex Form
An optimization problem (C, f) is in simplex form:
C is the conjunction of CE and CI
CE is a conjunction of linear equations
CI constrains all variables in C to be non-negative
f is a linear expression over variables in C
CSC Simplification, Optimization and Implication

29. CSC5240 - Simplification, Optimization and Implication
Simplex Form Example
An optimization problem in simplex form
An arbitrary problem can be put in simplex form by replacing
unconstrained var X by new vars and
inequality by new vars and
CSC Simplification, Optimization and Implication

30. CSC5240 - Simplification, Optimization and Implication
Simplex Solved Form - 1
The simplex method works by repeatedly transforming an optimization problem in “basic feasible solved form” and to another
A simplex optimization problem is in basic feasible solved (bfs) form if:
The equations are in solved form
Each constant on the right hand side is non-negative
Only parameters occur in the objective
CSC Simplification, Optimization and Implication

31. CSC5240 - Simplification, Optimization and Implication
Simplex Solved Form - 2
Every problem in bfs form has a corresponding “basic feasible solution” which corresponds to a vertex of the polygon defined by the constraint
A basic feasible solution is obtained by setting each parameter to 0 and each non-parameter to the constant in its equation
CSC Simplification, Optimization and Implication

32. Simplex Solved Form Example
An equivalent problem to the previous example in the bfs form
We can read off a solution and its objective value
CSC Simplification, Optimization and Implication

33. The Simplex Algorithm - 2
The simplex algorithm finds the optimum by repeatedly looking for an “adjacent” basic feasible solved form whose basic feasible solution decreases the value of the objective function until no such neighbour can be found
By adjacent, we mean that the new bfs form can be reached by a single “pivoting” operation
CSC Simplification, Optimization and Implication

34. The Simplex Algorithm - 3
starting from a problem in bfs form
repeat
Choose a variable y with negative coefficient in the obj. func.
Find the equation x = b + cy where c<0 and -b/c is minimal
Rewrite this equation with y being the subject y = -b/c + 1/c x + ...
Substitute -b/c + 1/c x for y in all other eqns and obj. func.
until no such variable y exists or no such equation exists
if no such y exists then optimum is found
else there is no optimum solution
CSC Simplification, Optimization and Implication

35. CSC5240 - Simplification, Optimization and Implication
Simplex Example
Choose variable Z, the 2nd eqn is only one with neg. coeff
Choose variable Y, the first eqn is only one with neg. coeff
No variable can be chosen, optimal value 2 is found
CSC Simplification, Optimization and Implication

36. Another Simplex Example - 1
An equivalent simplex form is:
An optimization problem showing contours of the objective function
CSC Simplification, Optimization and Implication

37. Another Simplex Example - 2
Basic feasible solved form: circle
Choose S3, replace using 2nd eq
Optimal solution: box
CSC Simplification, Optimization and Implication

38. CSC5240 - Simplification, Optimization and Implication
The Missing Part - 1
How do we get the initial bfs form and its basic feasible solution?
Solve a different simplex problem
Add artificial variables to make equations in basic feasible solved form
Minimize the sum of the artificial variables
If the sum is zero, we can construct a bfs form for the original problem
CSC Simplification, Optimization and Implication

39. The Missing Part Example - 1
Original simplex form equations
With artificial vars in bfs form:
Objective function rewritten: minimize
CSC Simplification, Optimization and Implication

40. The Missing Part Example - 2
Problem after minimization of objective function
Removing the artificial variables, the original problem is equivalent to
CSC Simplification, Optimization and Implication

41. CSC5240 - Simplification, Optimization and Implication
The Missing Part - 2
Must remember to eliminate the basic variables of the resultant bfs form from the original objective function
Finding a bfs form of the original constraint is exactly a constraint satisfaction problem, and thus the first phase of the simplex method is an efficient constraint solver for linear inequalities
CSC Simplification, Optimization and Implication

42. Canonical Form Simplifiers - 1
A simplifier simpl is a canonical form simplifier if
whenever C1 and C2 are equivalent wrt to V
The canonical form of a constraint C is
CSC Simplification, Optimization and Implication

43. Canonical Form Simplifiers - 2
Canonical form simplifiers are useful in determining if two constraints are equivalent since such simplifiers will simplify equivalent constraints to exactly the same syntactical form
Must take variable names into account
May have to do systematic variable renaming
CSC Simplification, Optimization and Implication

44. Implication and Equivalence - 1
Other important operations involving constraints are implication and equivalence tests
implication: test if C1 implies C2
impl(C1, C2) answers true, false or unknown
whether C1 has solutions which are a subset of the solutions of C2
CSC Simplification, Optimization and Implication

45. Implication and Equivalence - 2
An arbitrary implication test can be simplified to a sequence of implication tests
If C2 is , then if and only if for each i = 1,…,n
equivalence: test if C1 and C2 are equivalent
equiv(C1, C2) answers true, false or unknown
whether C1 and C2 have exactly the same set of solutions
CSC Simplification, Optimization and Implication

46. CSC5240 - Simplification, Optimization and Implication
Implication Example
For the house constraints CH, will stage B have to be reached after stage C?
For this question the answer is false, but if we require the house to be finished in 15 days the answer is true
CSC Simplification, Optimization and Implication

47. Implication and Equivalence - 3
We can use impl to define equiv and vice versa
We can use a solver to do simple impl tests
e.g.
CSC Simplification, Optimization and Implication

48. CSC5240 - Simplification, Optimization and Implication
Chapter Summary
Equivalent constraints can be written in many forms, hence we desire simplification
Particularly if we are only interested in the interaction of some of the variables
Many problems desire a optimal solution, there are algorithms (simplex) to find them
We may also be interested in asking questions involving implication
CSC Simplification, Optimization and Implication

49. CSC5240 - Simplification, Optimization and Implication
Homework
Read the entire Chapter 2 of the textbook, including Section 2.11 (very useful)
The following would take at most 2 hours
Exercises: 2.1, 2.3, 2.5
Practical exercises: 2.1, 2.2, 2.3, 2.4
CSC Simplification, Optimization and Implication

50. CSC5240 - Simplification, Optimization and Implication
Acknowledgement
With the consent of Peter Stuckey, I have shamelessly adopted and adapted a number of slides produced by him for use in this presentation
I am solely responsible, however, for any errors therein
CSC Simplification, Optimization and Implication