1 //===- ConstraintSytem.cpp - A system of linear constraints. ----*- C++ -*-===//
2 //
3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4 // See https://llvm.org/LICENSE.txt for license information.
5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6 //
7 //===----------------------------------------------------------------------===//
8 
9 #include "llvm/Analysis/ConstraintSystem.h"
10 #include "llvm/ADT/SmallVector.h"
11 #include "llvm/Support/MathExtras.h"
12 #include "llvm/ADT/StringExtras.h"
13 #include "llvm/Support/Debug.h"
14 
15 #include <algorithm>
16 #include <string>
17 
18 using namespace llvm;
19 
20 #define DEBUG_TYPE "constraint-system"
21 
eliminateUsingFM()22 bool ConstraintSystem::eliminateUsingFM() {
23   // Implementation of Fourier–Motzkin elimination, with some tricks from the
24   // paper Pugh, William. "The Omega test: a fast and practical integer
25   // programming algorithm for dependence
26   //  analysis."
27   // Supercomputing'91: Proceedings of the 1991 ACM/
28   // IEEE conference on Supercomputing. IEEE, 1991.
29   assert(!Constraints.empty() &&
30          "should only be called for non-empty constraint systems");
31   unsigned NumVariables = Constraints[0].size();
32   SmallVector<SmallVector<int64_t, 8>, 4> NewSystem;
33 
34   unsigned NumConstraints = Constraints.size();
35   uint32_t NewGCD = 1;
36   // FIXME do not use copy
37   for (unsigned R1 = 0; R1 < NumConstraints; R1++) {
38     if (Constraints[R1][1] == 0) {
39       SmallVector<int64_t, 8> NR;
40       NR.push_back(Constraints[R1][0]);
41       for (unsigned i = 2; i < NumVariables; i++) {
42         NR.push_back(Constraints[R1][i]);
43       }
44       NewSystem.push_back(std::move(NR));
45       continue;
46     }
47 
48     // FIXME do not use copy
49     for (unsigned R2 = R1 + 1; R2 < NumConstraints; R2++) {
50       if (R1 == R2)
51         continue;
52 
53       // FIXME: can we do better than just dropping things here?
54       if (Constraints[R2][1] == 0)
55         continue;
56 
57       if ((Constraints[R1][1] < 0 && Constraints[R2][1] < 0) ||
58           (Constraints[R1][1] > 0 && Constraints[R2][1] > 0))
59         continue;
60 
61       unsigned LowerR = R1;
62       unsigned UpperR = R2;
63       if (Constraints[UpperR][1] < 0)
64         std::swap(LowerR, UpperR);
65 
66       SmallVector<int64_t, 8> NR;
67       for (unsigned I = 0; I < NumVariables; I++) {
68         if (I == 1)
69           continue;
70 
71         int64_t M1, M2, N;
72         if (MulOverflow(Constraints[UpperR][I],
73                                    ((-1) * Constraints[LowerR][1] / GCD), M1))
74           return false;
75         if (MulOverflow(Constraints[LowerR][I],
76                                    (Constraints[UpperR][1] / GCD), M2))
77           return false;
78         if (AddOverflow(M1, M2, N))
79           return false;
80         NR.push_back(N);
81 
82         NewGCD = APIntOps::GreatestCommonDivisor({32, (uint32_t)NR.back()},
83                                                  {32, NewGCD})
84                      .getZExtValue();
85       }
86       NewSystem.push_back(std::move(NR));
87       // Give up if the new system gets too big.
88       if (NewSystem.size() > 500)
89         return false;
90     }
91   }
92   Constraints = std::move(NewSystem);
93   GCD = NewGCD;
94 
95   return true;
96 }
97 
mayHaveSolutionImpl()98 bool ConstraintSystem::mayHaveSolutionImpl() {
99   while (!Constraints.empty() && Constraints[0].size() > 1) {
100     if (!eliminateUsingFM())
101       return true;
102   }
103 
104   if (Constraints.empty() || Constraints[0].size() > 1)
105     return true;
106 
107   return all_of(Constraints, [](auto &R) { return R[0] >= 0; });
108 }
109 
dump(ArrayRef<std::string> Names) const110 void ConstraintSystem::dump(ArrayRef<std::string> Names) const {
111   if (Constraints.empty())
112     return;
113 
114   for (auto &Row : Constraints) {
115     SmallVector<std::string, 16> Parts;
116     for (unsigned I = 1, S = Row.size(); I < S; ++I) {
117       if (Row[I] == 0)
118         continue;
119       std::string Coefficient;
120       if (Row[I] != 1)
121         Coefficient = std::to_string(Row[I]) + " * ";
122       Parts.push_back(Coefficient + Names[I - 1]);
123     }
124     assert(!Parts.empty() && "need to have at least some parts");
125     LLVM_DEBUG(dbgs() << join(Parts, std::string(" + "))
126                       << " <= " << std::to_string(Row[0]) << "\n");
127   }
128 }
129 
dump() const130 void ConstraintSystem::dump() const {
131   SmallVector<std::string, 16> Names;
132   for (unsigned i = 1; i < Constraints.back().size(); ++i)
133     Names.push_back("x" + std::to_string(i));
134   LLVM_DEBUG(dbgs() << "---\n");
135   dump(Names);
136 }
137 
mayHaveSolution()138 bool ConstraintSystem::mayHaveSolution() {
139   LLVM_DEBUG(dump());
140   bool HasSolution = mayHaveSolutionImpl();
141   LLVM_DEBUG(dbgs() << (HasSolution ? "sat" : "unsat") << "\n");
142   return HasSolution;
143 }
144 
isConditionImplied(SmallVector<int64_t,8> R)145 bool ConstraintSystem::isConditionImplied(SmallVector<int64_t, 8> R) {
146   // If all variable coefficients are 0, we have 'C >= 0'. If the constant is >=
147   // 0, R is always true, regardless of the system.
148   if (all_of(makeArrayRef(R).drop_front(1), [](int64_t C) { return C == 0; }))
149     return R[0] >= 0;
150 
151   // If there is no solution with the negation of R added to the system, the
152   // condition must hold based on the existing constraints.
153   R = ConstraintSystem::negate(R);
154 
155   auto NewSystem = *this;
156   NewSystem.addVariableRow(R);
157   return !NewSystem.mayHaveSolution();
158 }
159