[ARITH] Improve detect linear equation (#529)

* [ARITH] Improve detect linear equation

* fix doc

HalideIR @ a40a3e2f

-Subproject commit dbf043a8d8bf379b05c56d8aa9025db55f589d6d
+Subproject commit a40a3e2fedee88d2f7b97ba4caf8a9d0eb25886f

include/tvm/arithmetic.h

@@ -118,7 +118,7 @@ class IntSet : public NodeRef {
  * \brief Range of a linear integer function.
  *  Use to do specify the possible index values.
  *
- *  set = { base + coeff * x | x in Z }
+ *  set = { coeff * x + base | x in Z }
  *
  *  When coeff != 0, it can also be written as
  *  set = { n | n % coeff == base }
@@ -127,16 +127,17 @@ class IntSet : public NodeRef {
  *  For example, if index = 0 + 4 x, then we know it can be divided by 4.
  */
 struct ModularEntry {
-  /*! \brief The base */
-  int base{0};
   /*! \brief linear co-efficient */
   int coeff{1};
+  /*! \brief The base */
+  int base{0};
 
   /*! \return entry represent everything */
   static ModularEntry everything() {
     // always safe to set 0 + x, so it can be everything.
     ModularEntry e;
-    e.base = 0; e.coeff = 1;
+    e.coeff = 1;
+    e.base = 0;
     return e;
   }
   /*!
@@ -157,14 +158,25 @@ struct IntSetNode : public Node {
   TVM_DECLARE_BASE_NODE_INFO(IntSetNode, Node);
 };
 
-
 /*!
- * \brief Detect if e can be rewritten as e = base + var * coeff
+ * \brief Detect if e can be rewritten as e = sum_{i=0}^n var[i] * coeff[i] + coeff[n]
  *  Where coeff and base are invariant of var.
  *
- * \return [base, coeff] if it is possible, empty array if it is not.
+ * \param e The expression to be detected.
+ * \param vars List of variables to be used in detection.
+ * \return [coeff[i]] if it is possible, empty array if it is not.
+ */
+Array<Expr> DetectLinearEquation(const Expr& e, const Array<Var>& vars);
+
+/*!
+ * \brief Detect if expression corresponds to clip bound of the vars
+ *
+ * \param e The expression to be detected.
+ * \param vars List of variables to be used in detection.
+ * \return concat([min_value[i], max_value[i]]), None is returned if there is no min or max value
+ *          return empty if the e does not match the pattern.
  */
-Array<Expr> DetectLinearEquation(Expr e, Var var);
+Array<Expr> DetectClipBound(const Expr& e, const Array<Var>& vars);
 
 /*!
  * \brief Find an symbolic integer set that contains all possible values of

src/api/api_arith.cc

@@ -36,6 +36,11 @@ TVM_REGISTER_API("arith.DetectLinearEquation")
     *ret = DetectLinearEquation(args[0], args[1]);
   });
 
+TVM_REGISTER_API("arith.DetectClipBound")
+.set_body([](TVMArgs args, TVMRetValue *ret) {
+    *ret = DetectClipBound(args[0], args[1]);
+  });
+
 TVM_REGISTER_API("arith.DeduceBound")
 .set_body([](TVMArgs args, TVMRetValue *ret) {
     *ret = DeduceBound(args[0], args[1],

src/arithmetic/detect_linear_equation.cc

@@ -21,22 +21,27 @@ struct LinearEqEntry {
   Expr coeff;
 };
 
+struct IntervalEntry {
+  Expr min_value;
+  Expr max_value;
+};
+
 class LinearEqDetector
     : public ExprFunctor<LinearEqEntry(const Expr&, const Expr &)> {
  public:
   explicit LinearEqDetector(Var var)
       : var_(var) {}
 
-  Array<Expr> Detect(const Expr& e) {
-    LinearEqEntry ret = VisitExpr(e, e);
-    if (fail_) return Array<Expr>();
-    if (!ret.base.defined()) {
-      ret.base = make_zero(var_.type());
+  bool Detect(const Expr& e, LinearEqEntry* ret) {
+    *ret = VisitExpr(e, e);
+    if (fail_) return false;
+    if (!ret->base.defined()) {
+      ret->base = make_zero(var_.type());
     }
-    if (!ret.coeff.defined()) {
-      ret.coeff = make_zero(var_.type());
+    if (!ret->coeff.defined()) {
+      ret->coeff = make_zero(var_.type());
     }
-    return Array<Expr>{ret.base, ret.coeff};
+    return true;
   }
 
   LinearEqEntry VisitExpr_(const Add* op, const Expr& e) final {
@@ -48,6 +53,17 @@ class LinearEqDetector
     ret.coeff = AddCombine(a.coeff, b.coeff);
     return ret;
   }
+
+  LinearEqEntry VisitExpr_(const Sub* op, const Expr& e) final {
+    if (fail_) return LinearEqEntry();
+    LinearEqEntry a = VisitExpr(op->a, op->a);
+    LinearEqEntry b = VisitExpr(op->b, op->b);
+    LinearEqEntry ret;
+    ret.base = SubCombine(a.base, b.base);
+    ret.coeff = SubCombine(a.coeff, b.coeff);
+    return ret;
+  }
+
   LinearEqEntry VisitExpr_(const Mul* op, const Expr& e) final {
     if (fail_) return LinearEqEntry();
     LinearEqEntry a = VisitExpr(op->a, op->a);
@@ -94,6 +110,11 @@ class LinearEqDetector
     if (!b.defined()) return a;
     return ComputeExpr<Add>(a, b);
   }
+  Expr SubCombine(Expr a, Expr b) {
+    if (!a.defined()) return -b;
+    if (!b.defined()) return a;
+    return ComputeExpr<Sub>(a, b);
+  }
   Expr MulCombine(Expr a, Expr b) {
     if (!a.defined()) return a;
     if (!b.defined()) return b;
@@ -101,9 +122,134 @@ class LinearEqDetector
   }
 };
 
-Array<Expr> DetectLinearEquation(Expr e, Var var) {
-  return LinearEqDetector(var).Detect(e);
+Array<Expr> DetectLinearEquation(const Expr& e, const Array<Var>& vars) {
+  CHECK_GE(vars.size(), 1U);
+  Expr base = e;
+  Array<Expr> coeff;
+
+  for (Var v : vars) {
+    LinearEqEntry ret;
+    if (!LinearEqDetector(v).Detect(base, &ret)) {
+      return Array<Expr>();
+    }
+    coeff.push_back(ret.coeff);
+    base = std::move(ret.base);
+  }
+
+  std::unordered_set<const Variable*> vset;
+  for (size_t i = vars.size(); i != 1; --i) {
+    vset.insert(vars[i - 1].get());
+    // The previous coeff contains the variable
+    if (ExprUseVar(coeff[i - 2], vset)) {
+      return Array<Expr>();
+    }
+  }
+  coeff.push_back(base);
+  return coeff;
 }
 
+// Detect clip condition as min max value
+bool DetectClipBound(
+    const Expr& cond,
+    std::unordered_map<const Variable*, IntervalEntry>* bmap) {
+  int flag = 0;
+  Var var;
+  auto fvisit = [&bmap, &flag, &var](const NodeRef& n) {
+    if (const Variable* v = n.as<Variable>()) {
+      if (bmap->count(v)) {
+        if (flag == 0) {
+          var = Var(n.node_);
+          flag = 1;
+        } else if (flag == 1) {
+          if (!var.same_as(n)) {
+            flag = -1;
+          }
+        }
+      }
+    }
+  };
+  PostOrderVisit(cond, fvisit);
+  if (flag != 1) return false;
+  // canonical form: exp >= 0
+  Expr canonical;
+  if (const LT* op = cond.as<LT>()) {
+    if (!op->a.type().is_int()) return false;
+    canonical = op->b - op->a - make_const(op->a.type(), 1);
+  } else if (const LE* op = cond.as<LE>()) {
+    if (!op->a.type().is_int()) return false;
+    canonical = op->b - op->a;
+  } else if (const GT* op = cond.as<GT>()) {
+    if (!op->a.type().is_int()) return false;
+    canonical = op->a - op->b - make_const(op->a.type(), 1);
+  } else if (const GE* op = cond.as<GE>()) {
+    if (!op->a.type().is_int()) return false;
+    canonical = op->a - op->b;
+  } else {
+    return false;
+  }
+  LinearEqEntry ret;
+  if (!LinearEqDetector(var).Detect(canonical, &ret)) return false;
+  ret.coeff = Simplify(ret.coeff);
+  IntervalEntry& p = (*bmap)[var.get()];
+  if (is_one(ret.coeff)) {
+    // var + shift >=0 -> var >= -shift
+    if (p.min_value.defined()) {
+      p.min_value = ir::Max::make(p.min_value, -ret.base);
+    } else {
+      p.min_value = -ret.base;
+    }
+    return true;
+  }
+  if (is_const(ret.coeff, -1)) {
+    // -var + shift >=0 -> var <= shift
+    if (p.max_value.defined()) {
+      p.max_value = ir::Min::make(p.max_value, ret.base);
+    } else {
+      p.max_value = ret.base;
+    }
+    return true;
+  }
+  return false;
+}
+
+
+template<typename OP>
+void SplitCommExpr(const Expr& e, std::vector<Expr>* ret) {
+  if (const OP* op = e.as<OP>()) {
+    SplitCommExpr<OP>(op->a, ret);
+    SplitCommExpr<OP>(op->b, ret);
+  } else {
+    ret->push_back(e);
+  }
+}
+
+// Detect the lower and upper bound from the expression.
+// e must be connected by and.
+Array<Expr> DetectClipBound(const Expr& e, const Array<Var>& vars) {
+  std::vector<Expr> splits;
+  SplitCommExpr<ir::And>(e, &splits);
+  std::unordered_map<const Variable*, IntervalEntry> rmap;
+  for (Var v : vars) {
+    rmap[v.get()] = IntervalEntry();
+  }
+  for (Expr cond : splits) {
+    if (!DetectClipBound(cond, &rmap)) return Array<Expr>();
+  }
+  Array<Expr> ret;
+  for (Var v : vars) {
+    IntervalEntry e = rmap[v.get()];
+    if (e.min_value.defined()) {
+      e.min_value = Simplify(e.min_value);
+    }
+    if (e.max_value.defined()) {
+      e.max_value = Simplify(e.max_value);
+    }
+    ret.push_back(e.min_value);
+    ret.push_back(e.max_value);
+  }
+  return ret;
+}
+
+
 }  // namespace arith
 }  // namespace tvm

src/pass/narrow_channel_access.cc

@@ -175,10 +175,10 @@ class ChannelAccessRewriter : public IRMutator {
     r = Range::make_by_min_extent(
         ir::Simplify(r->min), ir::Simplify(r->extent));
     if (ExprUseVar(r->extent, var)) return body;
-    Array<Expr> linear_eq = DetectLinearEquation(r->min, var);
+    Array<Expr> linear_eq = DetectLinearEquation(r->min, {var});
     if (linear_eq.size() == 0) return body;
-    Expr base = linear_eq[0];
-    Expr coeff = linear_eq[1];
+    Expr coeff = linear_eq[0];
+    Expr base = linear_eq[1];
     if (!is_zero(base)) return body;
     Expr left = ir::Simplify(adv_op->value - coeff * for_op->extent);
     if (!can_prove(left >= 0)) return body;

tests/python/unittest/test_arith_detect_clip_bound.py

+import tvm
+
+def test_basic():
+    a = tvm.var("a")
+    b = tvm.var("b")
+    c = tvm.var("c")
+    m = tvm.arith.DetectClipBound(tvm.all(a * 1 < b * 6,
+                                          a - 1 > 0), [a])
+    assert tvm.ir_pass.Simplify(m[1] - (b * 6 - 1)).value == 0
+    assert m[0].value == 2
+    m = tvm.arith.DetectClipBound(tvm.all(a * 1 < b * 6,
+                                          a - 1 > 0), [a, b])
+    assert len(m) == 0
+    m = tvm.arith.DetectClipBound(tvm.all(a + 10 * c <= 20,
+                                          b - 1 > 0), [a, b])
+    assert tvm.ir_pass.Simplify(m[1] - (20 - 10 * c)).value == 0
+    assert tvm.ir_pass.Simplify(m[2] - 2).value == 0
+
+
+if __name__ == "__main__":
+    test_basic()

tests/python/unittest/test_arith_detect_linear_equation.py

@@ -3,22 +3,41 @@ import tvm
 def test_basic():
     a = tvm.var("a")
     b = tvm.var("b")
-    m = tvm.arith.DetectLinearEquation(a * 4 + b * 6 + 7, a)
-    assert m[1].value == 4
-    assert tvm.ir_pass.Simplify(m[0] - (b * 6 + 7)).value == 0
+    m = tvm.arith.DetectLinearEquation(a * 4 + b * 6 + 7, [a])
+    assert m[0].value == 4
+    assert tvm.ir_pass.Simplify(m[1] - (b * 6 + 7)).value == 0
 
-    m = tvm.arith.DetectLinearEquation(a * 4 * (a+1) + b * 6 + 7, a)
+    m = tvm.arith.DetectLinearEquation(a * 4 * (a+1) + b * 6 + 7, [a])
     assert len(m) == 0
 
-    m = tvm.arith.DetectLinearEquation(a * 4  + (a+1) + b * 6 + 7, a)
-    assert m[1].value == 5
-    assert tvm.ir_pass.Simplify(m[0] - (b * 6 + 7 + 1)).value == 0
+    m = tvm.arith.DetectLinearEquation(a * 4  + (a+1) + b * 6 + 7, [a])
+    assert m[0].value == 5
+    assert tvm.ir_pass.Simplify(m[1] - (b * 6 + 7 + 1)).value == 0
 
-    m = tvm.arith.DetectLinearEquation(a * b + 7, a)
-    assert m[1] == b
+    m = tvm.arith.DetectLinearEquation(a * b + 7, [a])
+    assert m[0] == b
 
-    m = tvm.arith.DetectLinearEquation(b * 7, a)
-    assert m[1].value == 0
+    m = tvm.arith.DetectLinearEquation(b * 7, [a])
+    assert m[0].value == 0
+
+def test_multivariate():
+    v = [tvm.var("v%d" % i) for i in range(4)]
+    b = tvm.var("b")
+    m = tvm.arith.DetectLinearEquation(v[0] * (b + 4) + v[0] + v[1] * 8, v)
+    assert(tvm.ir_pass.Equal(tvm.ir_pass.Simplify(m[0]), b + 5))
+    assert(m[1].value == 8)
+
+    m = tvm.arith.DetectLinearEquation(v[0] * (b + 4) + v[0] + v[1] * 8 * v[2], v)
+    assert(len(m) == 0)
+
+    m = tvm.arith.DetectLinearEquation(v[0] * (b + 4) + v[0] + v[1] * 8 * v[1] + v[3], v)
+    assert(len(m) == 0)
+
+    m = tvm.arith.DetectLinearEquation(((v[0] * b + v[1]) * 8 + v[2] + 1) * 2, v)
+    assert(m[1].value == 16)
+    assert(m[2].value == 2)
+    assert(m[len(m)-1].value == 2)
 
 if __name__ == "__main__":
     test_basic()
+    test_multivariate()