[Mlir-commits] [mlir] [MLIR][Presburger] Implement function to evaluate the number of terms in a generating function. (PR #78078)

[Arjun P]

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@@ -245,3 +245,215 @@ QuasiPolynomial mlir::presburger::detail::getCoefficientInRationalFunction(
   }
   return coefficients[power].simplify();
 }
+
+/// We have a generating function of the form
+/// f_p(x) = \sum_i sign_i * (x^n_i(p)) / (\prod_j (1 - x^d_{ij})
+///
+/// where sign_i is ±1,
+/// n_i \in Q^p -> Q^d is a d-vector of affine functions on p parameters, and
+/// d_{ij} \in Q^d are vectors.
+///
+/// We need to find the number of terms of the form x^t in the expansion of
+/// this function, for which we substitute x = (1, ..., 1).
+/// However, direct substitution causes the denominator to become zero.
+///
+/// We therefore use the following procedure instead:
+/// 1. Substitute x_i = (s+1)^μ_i for some vector μ. This makes the generating
+/// a function of a scalar s.
+/// 2. Write each term in this function as P(s)/Q(s), where P and Q are
+/// polynomials. P has coefficients as quasipolynomials in d parameters, while
+/// Q has coefficients as scalars.
+/// 3. Find the constant term in the expansion of each term P(s)/Q(s). This is
+/// equivalent to substituting s = 0, which by step 1's substitution is
+/// equivalent to letting x = (1, ..., 1).
+/// In this step, we cancel the factor with root zero from the numerator and
+/// denominator, thus preventing the denominator from becoming zero.
+/// Step (1) We need to find a μ_i such that we can substitute x_i =
+/// (s+1)^μ_i. After this substitution, the exponent of (s+1) in the
+/// denominator is (μ_i • d_{ij}) in each term. Clearly, this cannot become
+/// zero. Thus we find a vector μ that is not orthogonal to any of the
+/// d_{ij} and substitute x accordingly.
+///
+/// Step (2) We need to express the terms in the function as quotients of
+/// polynomials. Each term is now of the form
+/// sign_i * (s+1)^n_i / (\prod_j (1 - (s+1)^d'_{ij}))
+/// For the i'th term, we first convert all the d'_{ij} to their
+/// absolute values by multiplying and dividing by (s+1)^(-d'_{ij}) if it is
+/// negative. We change the sign accordingly.
----------------
Superty wrote:

can you mention why sign is changed (to keep same form)

https://github.com/llvm/llvm-project/pull/78078