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