Chapter 17 Omega: a solver of quantifier-free problems in
Presburger Arithmetic
Pierre Crégut
17.1 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
the type 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 meaned by ``Presburger arithmetic''.
If the tactic cannot solve the goal, it fails with an error message.
In any case, the computation eventually stops.
17.1.1 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
on nat or from the predicates
=, <, <=, >, >=
on Z. In expressions of type nat, omega recognizes
plus, minus, mult, pred, S, O
and in expressions of type Z, omega recognizes
+, -, *, Zsucc, and constants.
All expressions of type nat or Z not built on these
operators are considered abstractly as if they
were arbitrary variables of type nat or Z.
17.1.2 Messages from omega
When omega does not solve the goal, one of the following errors
is generated:
Error messages: - 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
17.2 Using omega
The omega tactic does not belong to the core system. It should be
loaded by
Coq < Require Import Omega.
Coq < Open Scope Z_scope.
Example 3:
Coq < Goal forall m n:Z, 1 + 2 * m <> 2 * n.
1 subgoal
============================
forall m n : Z, 1 + 2 * m <> 2 * n
Coq < intros; omega.
Proof completed.
Example 4:
Coq < Goal forall z:Z, z > 0 -> 2 * z + 1 > z.
1 subgoal
============================
forall z : Z, z > 0 -> 2 * z + 1 > z
Coq < intro; omega.
Proof completed.
17.3 Technical data
17.3.1 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
nat are 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.
17.3.2 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
Pierre.Cregut@cnet.francetelecom.fr