[ConstraintSystem] Add helpers to deal with linear constraints.

This patch introduces a new ConstraintSystem class, that maintains a set
of linear constraints and uses Fourier–Motzkin elimination to eliminate
constraints to check if there are solutions for the system.

It also adds a convert-constraint-log-to-z3.py script, which can parse
the debug output of the constraint system and convert it to a python
script that feeds the constraints into Z3 and checks if it produces the
same result as the LLVM implementation. This is for verification
purposes.

Reviewed By: spatel

Differential Revision: https://reviews.llvm.org/D84544

1 files changed, 141 insertions, 0 deletions

diff --git a/llvm/lib/Analysis/ConstraintSystem.cpp b/llvm/lib/Analysis/ConstraintSystem.cpp
new file mode 100644
index 0000000..95fe6c9
--- /dev/null
+++ b/llvm/lib/Analysis/ConstraintSystem.cpp
@@ -0,0 +1,141 @@
+//===- ConstraintSytem.cpp - A system of linear constraints. ----*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "llvm/Analysis/ConstraintSystem.h"
+#include "llvm/ADT/SmallVector.h"
+#include "llvm/ADT/StringExtras.h"
+#include "llvm/Support/Debug.h"
+
+#include <algorithm>
+#include <string>
+
+using namespace llvm;
+
+#define DEBUG_TYPE "constraint-system"
+
+bool ConstraintSystem::eliminateUsingFM() {
+  // Implementation of Fourier–Motzkin elimination, with some tricks from the
+  // paper Pugh, William. "The Omega test: a fast and practical integer
+  // programming algorithm for dependence
+  //  analysis."
+  // Supercomputing'91: Proceedings of the 1991 ACM/
+  // IEEE conference on Supercomputing. IEEE, 1991.
+  assert(!Constraints.empty() &&
+         "should only be called for non-empty constraint systems");
+  unsigned NumVariables = Constraints[0].size();
+  SmallVector<SmallVector<int64_t, 8>, 4> NewSystem;
+
+  unsigned NumConstraints = Constraints.size();
+  uint32_t NewGCD = 1;
+  // FIXME do not use copy
+  for (unsigned R1 = 0; R1 < NumConstraints; R1++) {
+    if (Constraints[R1][1] == 0) {
+      SmallVector<int64_t, 8> NR;
+      NR.push_back(Constraints[R1][0]);
+      for (unsigned i = 2; i < NumVariables; i++) {
+        NR.push_back(Constraints[R1][i]);
+      }
+      NewSystem.push_back(std::move(NR));
+      continue;
+    }
+
+    // FIXME do not use copy
+    bool EliminatedInRow = false;
+    for (unsigned R2 = R1 + 1; R2 < NumConstraints; R2++) {
+      if (R1 == R2)
+        continue;
+
+      // FIXME: can we do better than just dropping things here?
+      if (Constraints[R2][1] == 0)
+        continue;
+
+      if ((Constraints[R1][1] < 0 && Constraints[R2][1] < 0) ||
+          (Constraints[R1][1] > 0 && Constraints[R2][1] > 0))
+        continue;
+
+      unsigned LowerR = R1;
+      unsigned UpperR = R2;
+      if (Constraints[UpperR][1] < 0)
+        std::swap(LowerR, UpperR);
+
+      SmallVector<int64_t, 8> NR;
+      for (unsigned I = 0; I < NumVariables; I++) {
+        if (I == 1)
+          continue;
+
+        int64_t M1, M2, N;
+        if (__builtin_mul_overflow(Constraints[UpperR][I],
+                                   ((-1) * Constraints[LowerR][1] / GCD), &M1))
+          return false;
+        if (__builtin_mul_overflow(Constraints[LowerR][I],
+                                   (Constraints[UpperR][1] / GCD), &M2))
+          return false;
+        if (__builtin_add_overflow(M1, M2, &N))
+          return false;
+        NR.push_back(N);
+
+        NewGCD = APIntOps::GreatestCommonDivisor({32, (uint32_t)NR.back()},
+                                                 {32, NewGCD})
+                     .getZExtValue();
+      }
+      NewSystem.push_back(std::move(NR));
+      EliminatedInRow = true;
+    }
+  }
+  Constraints = std::move(NewSystem);
+  GCD = NewGCD;
+
+  return true;
+}
+
+bool ConstraintSystem::mayHaveSolutionImpl() {
+  while (!Constraints.empty() && Constraints[0].size() > 1) {
+    if (!eliminateUsingFM())
+      return true;
+  }
+
+  if (Constraints.empty() || Constraints[0].size() > 1)
+    return true;
+
+  return all_of(Constraints, [](auto &R) { return R[0] >= 0; });
+}
+
+void ConstraintSystem::dump(ArrayRef<std::string> Names) const {
+  if (Constraints.empty())
+    return;
+
+  for (auto &Row : Constraints) {
+    SmallVector<std::string, 16> Parts;
+    for (unsigned I = 1, S = Row.size(); I < S; ++I) {
+      if (Row[I] == 0)
+        continue;
+      std::string Coefficient = "";
+      if (Row[I] != 1)
+        Coefficient = std::to_string(Row[I]) + " * ";
+      Parts.push_back(Coefficient + Names[I - 1]);
+    }
+    assert(!Parts.empty() && "need to have at least some parts");
+    LLVM_DEBUG(dbgs() << join(Parts, std::string(" + "))
+                      << " <= " << std::to_string(Row[0]) << "\n");
+  }
+}
+
+void ConstraintSystem::dump() const {
+  SmallVector<std::string, 16> Names;
+  for (unsigned i = 1; i < Constraints.back().size(); ++i)
+    Names.push_back("x" + std::to_string(i));
+  LLVM_DEBUG(dbgs() << "---\n");
+  dump(Names);
+}
+
+bool ConstraintSystem::mayHaveSolution() {
+  dump();
+  bool HasSolution = mayHaveSolutionImpl();
+  LLVM_DEBUG(dbgs() << (HasSolution ? "sat" : "unsat") << "\n");
+  return HasSolution;
+}