Chapter 21 Omega: a solver of quantifier-free problems in Presburger Arithmetic
21.1 Description of omega
omega solves a goal in Presburger arithmetic, i.e. a universally
quantified formula made of equations and inequations. Equations may
be specified either on the type
nat of natural numbers or on
Z of binary-encoded integer numbers. Formulas on
nat are automatically injected into
Z. The procedure
may use any hypothesis of the current proof session to solve the goal.
Multiplication is handled by omega but only goals where at least one of the two multiplicands of products is a constant are solvable. This is the restriction meant by “Presburger arithmetic”.
If the tactic cannot solve the goal, it fails with an error message. In any case, the computation eventually stops.
21.1.1 Arithmetical goals recognized by omega
omega applied only to quantifier-free formulas built from the connectors
/\, \/, ~, ->
on atomic formulas. Atomic formulas are built from the predicates
=, le, lt, gt, ge
nat or from the predicates
=, <, <=, >, >=
Z. In expressions of type
nat, omega recognizes
plus, minus, mult, pred, S, O
and in expressions of type
Z, omega recognizes
+, -, *, Z.succ, and constants.
All expressions of type
Z not built on these
operators are considered abstractly as if they
were arbitrary variables of type
21.1.2 Messages from omega
When omega does not solve the goal, one of the following errors is generated:
- omega can’t solve this system
This may happen if your goal is not quantifier-free (if it is universally quantified, try intros first; if it contains existentials quantifiers too, omega is not strong enough to solve your goal). This may happen also if your goal contains arithmetical operators unknown from omega. Finally, your goal may be really wrong!
- omega: Not a quantifier-free goal
If your goal is universally quantified, you should first apply intro as many time as needed.
- omega: Unrecognized predicate or connective: ident
- omega: Unrecognized atomic proposition: prop
- omega: Can’t solve a goal with proposition variables
- omega: Unrecognized proposition
- omega: Can’t solve a goal with non-linear products
- omega: Can’t solve a goal with equality on type
21.2 Using omega
The omega tactic does not belong to the core system. It should be loaded by
Coq < Open Scope Z_scope.
forall m n : Z, 1 + 2 * m <> 2 * n
Coq < intros; omega.
No more subgoals.
forall z : Z, z > 0 -> 2 * z + 1 > z
Coq < intro; omega.
No more subgoals.
21.3 Technical data
21.3.1 Overview of the tactic
- The goal is negated twice and the first negation is introduced as an hypothesis.
- Hypothesis are decomposed in simple equations or inequations. Multiple goals may result from this phase.
- Equations and inequations over
natare translated over
Z, multiple goals may result from the translation of substraction.
- Equations and inequations are normalized.
- Goals are solved by the OMEGA decision procedure.
- The script of the solution is replayed.
21.3.2 Overview of the OMEGA decision procedure
The OMEGA decision procedure involved in the omega tactic uses a small subset of the decision procedure presented in
"The Omega Test: a fast and practical integer programming algorithm for dependence analysis", William Pugh, Communication of the ACM , 1992, p 102-114.
Here is an overview, look at the original paper for more information.
- Equations and inequations are normalized by division by the GCD of their coefficients.
- Equations are eliminated, using the Banerjee test to get a coefficient equal to one.
- Note that each inequation defines a half space in the space of real value of the variables.
- Inequations are solved by projecting on the hyperspace defined by cancelling one of the variable. They are partitioned according to the sign of the coefficient of the eliminated variable. Pairs of inequations from different classes define a new edge in the projection.
- Redundant inequations are eliminated or merged in new equations that can be eliminated by the Banerjee test.
- The last two steps are iterated until a contradiction is reached (success) or there is no more variable to eliminate (failure).
It may happen that there is a real solution and no integer one. The last steps of the Omega procedure (dark shadow) are not implemented, so the decision procedure is only partial.
- The simplification procedure is very dumb and this results in many redundant cases to explore.
- Much too slow.
- Certainly other bugs! You can report them to https://coq.inria.fr/bugs/.