Refactored BVGauss preprocessing pass. (#1766)

2 files changed, 876 insertions, 0 deletions

diff --git a/src/preprocessing/passes/bv_gauss.cpp b/src/preprocessing/passes/bv_gauss.cpp
new file mode 100644
index 000000000..d1f7d09ee
--- /dev/null
+++ b/src/preprocessing/passes/bv_gauss.cpp
@@ -0,0 +1,823 @@
+/*********************                                                        */
+/*! \file bv_gauss.cpp
+ ** \verbatim
+ ** Top contributors (to current version):
+ **   Aina Niemetz
+ ** This file is part of the CVC4 project.
+ ** Copyright (c) 2009-2018 by the authors listed in the file AUTHORS
+ ** in the top-level source directory) and their institutional affiliations.
+ ** All rights reserved.  See the file COPYING in the top-level source
+ ** directory for licensing information.\endverbatim
+ **
+ ** \brief Gaussian Elimination preprocessing pass.
+ **
+ ** Simplify a given equation system modulo a (prime) number via Gaussian
+ ** Elimination if possible.
+ **/
+
+#include "preprocessing/passes/bv_gauss.h"
+
+#include "expr/node.h"
+#include "theory/rewriter.h"
+#include "theory/bv/theory_bv_utils.h"
+#include "theory/bv/theory_bv_rewrite_rules_normalization.h"
+#include "util/bitvector.h"
+
+#include <unordered_map>
+#include <vector>
+
+
+using namespace CVC4;
+using namespace CVC4::theory;
+using namespace CVC4::theory::bv;
+
+namespace CVC4 {
+namespace preprocessing {
+namespace passes {
+
+namespace {
+
+/**
+ *  Represents the result of Gaussian Elimination where the solution
+ *  of the given equation system is
+ *
+ *   INVALID ... i.e., NOT of the form c1*x1 + c2*x2 + ... % p = b,
+ *               where ci, b and p are
+ *                 - bit-vector constants
+ *                 - extracts or zero extensions on bit-vector constants
+ *                 - of arbitrary nesting level
+ *               and p is co-prime to all bit-vector constants for which
+ *               a multiplicative inverse has to be computed.
+ *
+ *   UNIQUE  ... determined for all unknowns, e.g., x = 4
+ *
+ *   PARTIAL ... e.g., x = 4 - 2z
+ *
+ *   NONE    ... no solution
+ *
+ *   Given a matrix A representing an equation system, the resulting
+ *   matrix B after applying GE represents, e.g.:
+ *
+ *   B = 1 0 0 2 <-    UNIQUE
+ *       0 1 0 3 <-
+ *       0 0 1 1 <-
+ *
+ *   B = 1 0 2 4 <-    PARTIAL
+ *       0 1 3 2 <-
+ *       0 0 1 1
+ *
+ *   B = 1 0 0 1       NONE
+ *       0 1 1 2
+ *       0 0 0 2 <-
+ */
+enum class Result
+{
+  INVALID,
+  UNIQUE,
+  PARTIAL,
+  NONE
+};
+
+bool is_bv_const(Node n)
+{
+  if (n.isConst()) { return true; }
+  return Rewriter::rewrite(n).getKind() == kind::CONST_BITVECTOR;
+}
+
+Node get_bv_const(Node n)
+{
+  Assert(is_bv_const(n));
+  return Rewriter::rewrite(n);
+}
+
+Integer get_bv_const_value(Node n)
+{
+  Assert(is_bv_const(n));
+  return get_bv_const(n).getConst<BitVector>().getValue();
+}
+
+/**
+ * Determines if an overflow may occur in given 'expr'.
+ *
+ * Returns 0 if an overflow may occur, and the minimum required
+ * bit-width such that no overflow occurs, otherwise.
+ *
+ * Note that it would suffice for this function to be Boolean.
+ * However, it is handy to determine the minimum required bit-width for
+ * debugging purposes.
+ *
+ * Note: getMinBwExpr assumes that 'expr' is rewritten.
+ *
+ * If not, all operators that are removed via rewriting (e.g., ror, rol, ...)
+ * will be handled via the default case, which is not incorrect but also not
+ * necessarily the minimum.
+ */
+unsigned getMinBwExpr(Node expr)
+{
+  std::vector<Node> visit;
+  /* Maps visited nodes to the determined minimum bit-width required. */
+  std::unordered_map<Node, unsigned, NodeHashFunction> visited;
+  std::unordered_map<Node, unsigned, NodeHashFunction>::iterator it;
+
+  visit.push_back(expr);
+  while (!visit.empty())
+  {
+    Node n = visit.back();
+    visit.pop_back();
+    it = visited.find(n);
+    if (it == visited.end())
+    {
+      if (is_bv_const(n))
+      {
+        /* Rewrite const expr, overflows in consts are irrelevant. */
+        visited[n] = get_bv_const(n).getConst<BitVector>().getValue().length();
+      }
+      else
+      {
+        visited[n] = 0;
+        visit.push_back(n);
+        for (const Node &nn : n) { visit.push_back(nn); }
+      }
+    }
+    else if (it->second == 0)
+    {
+      Kind k = n.getKind();
+      Assert(k != kind::CONST_BITVECTOR);
+      Assert(!is_bv_const(n));
+      switch (k)
+      {
+        case kind::BITVECTOR_EXTRACT:
+        {
+          const unsigned size = bv::utils::getSize(n);
+          const unsigned low = bv::utils::getExtractLow(n);
+          const unsigned child_min_width = visited[n[0]];
+          visited[n] = std::min(
+              size, child_min_width >= low ? child_min_width - low : 0u);
+          Assert(visited[n] <= visited[n[0]]);
+          break;
+        }
+
+        case kind::BITVECTOR_ZERO_EXTEND:
+        {
+          visited[n] = visited[n[0]];
+          break;
+        }
+
+        case kind::BITVECTOR_MULT:
+        {
+          Integer maxval = Integer(1);
+          for (const Node& nn : n)
+          {
+            if (is_bv_const(nn))
+            {
+              maxval *= get_bv_const_value(nn);
+            }
+            else
+            {
+              maxval *= BitVector::mkOnes(visited[nn]).getValue();
+            }
+          }
+          unsigned w = maxval.length();
+          if (w > bv::utils::getSize(n)) { return 0; } /* overflow */
+          visited[n] = w;
+          break;
+        }
+
+        case kind::BITVECTOR_CONCAT:
+        {
+          unsigned i, wnz, nc;
+          for (i = 0, wnz = 0, nc = n.getNumChildren() - 1; i < nc; ++i)
+          {
+            unsigned wni = bv::utils::getSize(n[i]);
+            if (n[i] != bv::utils::mkZero(wni)) { break; }
+            /* sum of all bit-widths of leading zero concats */
+            wnz += wni;
+          }
+          /* Do not consider leading zero concats, i.e.,
+           * min bw of current concat is determined as
+           *   min bw of first non-zero term
+           *   plus actual bw of all subsequent terms */
+          visited[n] = bv::utils::getSize(n) + visited[n[i]]
+                       - bv::utils::getSize(n[i]) - wnz;
+          break;
+        }
+
+        case kind::BITVECTOR_UREM_TOTAL:
+        case kind::BITVECTOR_LSHR:
+        case kind::BITVECTOR_ASHR:
+        {
+          visited[n] = visited[n[0]];
+          break;
+        }
+
+        case kind::BITVECTOR_OR:
+        case kind::BITVECTOR_NOR:
+        case kind::BITVECTOR_XOR:
+        case kind::BITVECTOR_XNOR:
+        case kind::BITVECTOR_AND:
+        case kind::BITVECTOR_NAND:
+        {
+          unsigned wmax = 0;
+          for (const Node &nn : n)
+          {
+            if (visited[nn] > wmax)
+            {
+              wmax = visited[nn];
+            }
+          }
+          visited[n] = wmax;
+          break;
+        }
+
+        case kind::BITVECTOR_PLUS:
+        {
+          Integer maxval = Integer(0);
+          for (const Node& nn : n)
+          {
+            if (is_bv_const(nn))
+            {
+              maxval += get_bv_const_value(nn);
+            }
+            else
+            {
+              maxval += BitVector::mkOnes(visited[nn]).getValue();
+            }
+          }
+          unsigned w = maxval.length();
+          if (w > bv::utils::getSize(n)) { return 0; } /* overflow */
+          visited[n] = w;
+          break;
+        }
+
+        default:
+        {
+          /* BITVECTOR_UDIV_TOTAL (since x / 0 = -1)
+           * BITVECTOR_NOT
+           * BITVECTOR_NEG
+           * BITVECTOR_SHL */
+          visited[n] = bv::utils::getSize(n);
+        }
+      }
+    }
+  }
+  Assert(visited.find(expr) != visited.end());
+  return visited[expr];
+}
+
+/**
+ * Apply Gaussian Elimination modulo a (prime) number.
+ * The given equation system is represented as a matrix of Integers.
+ *
+ * Note that given 'prime' does not have to be prime but can be any
+ * arbitrary number. However, if 'prime' is indeed prime, GE is guaranteed
+ * to succeed, which is not the case, otherwise.
+ *
+ * Returns INVALID if GE can not be applied, UNIQUE and PARTIAL if GE was
+ * successful, and NONE, otherwise.
+ *
+ * Vectors 'rhs' and 'lhs' represent the right hand side and left hand side
+ * of the given matrix, respectively. The resulting matrix (in row echelon
+ * form) is stored in 'rhs' and 'lhs', i.e., the given matrix is overwritten
+ * with the resulting matrix.
+ */
+Result gaussElim(
+    Integer prime,
+    std::vector<Integer>& rhs,
+    std::vector<std::vector<Integer>>& lhs)
+{
+  Assert(prime > 0);
+  Assert(lhs.size());
+  Assert(lhs.size() == rhs.size());
+  Assert(lhs.size() <= lhs[0].size());
+
+  /* special case: zero ring */
+  if (prime == 1)
+  {
+    rhs = std::vector<Integer>(rhs.size(), Integer(0));
+    lhs = std::vector<std::vector<Integer>>(
+        lhs.size(), std::vector<Integer>(lhs[0].size(), Integer(0)));
+    return Result::UNIQUE;
+  }
+
+  size_t nrows = lhs.size();
+  size_t ncols = lhs[0].size();
+
+  #ifdef CVC4_ASSERTIONS
+  for (size_t i = 1; i < nrows; ++i) Assert(lhs[i].size() == ncols);
+  #endif
+  /* (1) if element in pivot column is non-zero and != 1, divide row elements
+   *     by element in pivot column modulo prime, i.e., multiply row with
+   *     multiplicative inverse of element in pivot column modulo prime
+   *
+   * (2) subtract pivot row from all rows below pivot row
+   *
+   * (3) subtract (multiple of) current row from all rows above s.t. all
+   *     elements in current pivot column above current row become equal to one
+   *
+   * Note: we do not normalize the given matrix to values modulo prime
+   *       beforehand but on-the-fly. */
+
+  /* pivot = lhs[pcol][pcol] */
+  for (size_t pcol = 0, prow = 0; pcol < ncols && prow < nrows; ++pcol, ++prow)
+  {
+    /* lhs[j][pcol]: element in pivot column */
+    for (size_t j = prow; j < nrows; ++j)
+    {
+#ifdef CVC4_ASSERTIONS
+      for (size_t k = 0; k < pcol; ++k) { Assert(lhs[j][k] == 0); }
+#endif
+      /* normalize element in pivot column to modulo prime */
+      lhs[j][pcol] = lhs[j][pcol].euclidianDivideRemainder(prime);
+      /* exchange rows if pivot elem is 0 */
+      if (j == prow)
+      {
+        while (lhs[j][pcol] == 0)
+        {
+          for (size_t k = prow + 1; k < nrows; ++k)
+          {
+            lhs[k][pcol] = lhs[k][pcol].euclidianDivideRemainder(prime);
+            if (lhs[k][pcol] != 0)
+            {
+              std::swap(rhs[j], rhs[k]);
+              std::swap(lhs[j], lhs[k]);
+              break;
+            }
+          }
+          if (pcol >= ncols - 1) break;
+          if (lhs[j][pcol] == 0)
+          {
+            pcol += 1;
+            if (lhs[j][pcol] != 0)
+              lhs[j][pcol] = lhs[j][pcol].euclidianDivideRemainder(prime);
+          }
+        }
+      }
+
+      if (lhs[j][pcol] != 0)
+      {
+        /* (1) */
+        if (lhs[j][pcol] != 1)
+        {
+          Integer inv = lhs[j][pcol].modInverse(prime);
+          if (inv == -1)
+          {
+            return Result::INVALID; /* not coprime */
+          }
+          for (size_t k = pcol; k < ncols; ++k)
+          {
+            lhs[j][k] = lhs[j][k].modMultiply(inv, prime);
+            if (j <= prow) continue; /* pivot */
+            lhs[j][k] = lhs[j][k].modAdd(-lhs[prow][k], prime);
+          }
+          rhs[j] = rhs[j].modMultiply(inv, prime);
+          if (j > prow) { rhs[j] = rhs[j].modAdd(-rhs[prow], prime); }
+        }
+        /* (2) */
+        else if (j != prow)
+        {
+          for (size_t k = pcol; k < ncols; ++k)
+          {
+            lhs[j][k] = lhs[j][k].modAdd(-lhs[prow][k], prime);
+          }
+          rhs[j] = rhs[j].modAdd(-rhs[prow], prime);
+        }
+      }
+    }
+    /* (3) */
+    for (size_t j = 0; j < prow; ++j)
+    {
+      Integer mul = lhs[j][pcol];
+      if (mul != 0)
+      {
+        for (size_t k = pcol; k < ncols; ++k)
+        {
+          lhs[j][k] = lhs[j][k].modAdd(-lhs[prow][k] * mul, prime);
+        }
+        rhs[j] = rhs[j].modAdd(-rhs[prow] * mul, prime);
+      }
+    }
+  }
+
+  bool ispart = false;
+  for (size_t i = 0; i < nrows; ++i)
+  {
+    size_t pcol = i;
+    while (pcol < ncols && lhs[i][pcol] == 0) ++pcol;
+    if (pcol >= ncols)
+    {
+      rhs[i] = rhs[i].euclidianDivideRemainder(prime);
+      if (rhs[i] != 0)
+      {
+        /* no solution */
+        return Result::NONE;
+      }
+      continue;
+    }
+    for (size_t j = i; j < ncols; ++j)
+    {
+      if (lhs[i][j] >= prime || lhs[i][j] <= -prime)
+      {
+        lhs[i][j] = lhs[i][j].euclidianDivideRemainder(prime);
+      }
+      if (j > pcol && lhs[i][j] != 0)
+      {
+        ispart = true;
+      }
+    }
+  }
+
+  if (ispart) { return Result::PARTIAL; }
+
+  return Result::UNIQUE;
+}
+
+/**
+ * Apply Gaussian Elimination on a set of equations modulo some (prime)
+ * number given as bit-vector equations.
+ *
+ * IMPORTANT: Applying GE modulo some number (rather than modulo 2^bw)
+ * on a set of bit-vector equations is only sound if this set of equations
+ * has a solution that does not produce overflows. Consequently, we only
+ * apply GE if the given bit-width guarantees that no overflows can occur
+ * in the given set of equations.
+ *
+ * Note that the given set of equations does not have to be modulo a prime
+ * but can be modulo any arbitrary number. However, if it is indeed modulo
+ * prime, GE is guaranteed to succeed, which is not the case, otherwise.
+ *
+ * Returns INVALID if GE can not be applied, UNIQUE and PARTIAL if GE was
+ * successful, and NONE, otherwise.
+ *
+ * The resulting constraints are stored in 'res' as a mapping of unknown
+ * to result (modulo prime). These mapped results are added as constraints
+ * of the form 'unknown = mapped result' in applyInternal.
+ */
+Result gaussElimRewriteForUrem(
+    const std::vector<Node>& equations,
+    std::unordered_map<Node, Node, NodeHashFunction>& res)
+{
+  Assert(res.empty());
+
+  Node prime;
+  Integer iprime;
+  std::unordered_map<Node, std::vector<Integer>, NodeHashFunction> vars;
+  size_t neqs = equations.size();
+  std::vector<Integer> rhs;
+  std::vector<std::vector<Integer>> lhs =
+      std::vector<std::vector<Integer>>(neqs, std::vector<Integer>());
+
+  res = std::unordered_map<Node, Node, NodeHashFunction>();
+
+  for (size_t i = 0; i < neqs; ++i)
+  {
+    Node eq = equations[i];
+    Assert(eq.getKind() == kind::EQUAL);
+    Node urem, eqrhs;
+
+    if (eq[0].getKind() == kind::BITVECTOR_UREM)
+    {
+      urem = eq[0];
+      Assert(is_bv_const(eq[1]));
+      eqrhs = eq[1];
+    }
+    else
+    {
+      Assert(eq[1].getKind() == kind::BITVECTOR_UREM);
+      urem = eq[1];
+      Assert(is_bv_const(eq[0]));
+      eqrhs = eq[0];
+    }
+    if (getMinBwExpr(Rewriter::rewrite(urem[0])) == 0)
+    {
+      Trace("bv-gauss-elim")
+          << "Minimum required bit-width exceeds given bit-width, "
+             "will not apply Gaussian Elimination."
+          << std::endl;
+      return Result::INVALID;
+    }
+    rhs.push_back(get_bv_const_value(eqrhs));
+
+    Assert(is_bv_const(urem[1]));
+    Assert(i == 0 || get_bv_const_value(urem[1]) == iprime);
+    if (i == 0)
+    {
+      prime = urem[1];
+      iprime = get_bv_const_value(prime);
+    }
+
+    std::unordered_map<Node, Integer, NodeHashFunction> tmp;
+    std::vector<Node> stack;
+    stack.push_back(urem[0]);
+    while (!stack.empty())
+    {
+      Node n = stack.back();
+      stack.pop_back();
+
+      /* Subtract from rhs if const */
+      if (is_bv_const(n))
+      {
+        Integer val = get_bv_const_value(n);
+        if (val > 0) rhs.back() -= val;
+        continue;
+      }
+
+      /* Split into matrix columns */
+      Kind k = n.getKind();
+      if (k == kind::BITVECTOR_PLUS)
+      {
+        for (const Node& nn : n) { stack.push_back(nn); }
+      }
+      else if (k == kind::BITVECTOR_MULT)
+      {
+        Node n0, n1;
+        /* Flatten mult expression. */
+        n = RewriteRule<FlattenAssocCommut>::run<true>(n);
+        /* Split operands into consts and non-consts */
+        NodeBuilder<> nb_consts(NodeManager::currentNM(), k);
+        NodeBuilder<> nb_nonconsts(NodeManager::currentNM(), k);
+        for (const Node& nn : n)
+        {
+          Node nnrw = Rewriter::rewrite(nn);
+          if (is_bv_const(nnrw))
+          {
+            nb_consts << nnrw;
+          }
+          else
+          {
+            nb_nonconsts << nnrw;
+          }
+        }
+        Assert(nb_nonconsts.getNumChildren() > 0);
+        /* n0 is const */
+        unsigned nc = nb_consts.getNumChildren();
+        if (nc > 1)
+        {
+          n0 = Rewriter::rewrite(nb_consts.constructNode());
+        }
+        else if (nc == 1)
+        {
+          n0 = nb_consts[0];
+        }
+        else
+        {
+          n0 = bv::utils::mkOne(bv::utils::getSize(n));
+        }
+        /* n1 is a mult with non-const operands */
+        if (nb_nonconsts.getNumChildren() > 1)
+        {
+          n1 = Rewriter::rewrite(nb_nonconsts.constructNode());
+        }
+        else
+        {
+          n1 = nb_nonconsts[0];
+        }
+        Assert(is_bv_const(n0));
+        Assert(!is_bv_const(n1));
+        tmp[n1] += get_bv_const_value(n0);
+      }
+      else
+      {
+        tmp[n] += Integer(1);
+      }
+    }
+
+    /* Note: "var" is not necessarily a VARIABLE but can be an arbitrary expr */
+
+    for (const auto& p : tmp)
+    {
+      Node var = p.first;
+      Integer val = p.second;
+      if (i > 0 && vars.find(var) == vars.end())
+      {
+        /* Add column and fill column elements of rows above with 0. */
+        vars[var].insert(vars[var].end(), i, Integer(0));
+      }
+      vars[var].push_back(val);
+    }
+
+    for (const auto& p : vars)
+    {
+      if (tmp.find(p.first) == tmp.end())
+      {
+        vars[p.first].push_back(Integer(0));
+      }
+    }
+  }
+
+  size_t nvars = vars.size();
+  Assert(nvars);
+  size_t nrows = vars.begin()->second.size();
+#ifdef CVC4_ASSERTIONS
+  for (const auto& p : vars) { Assert(p.second.size() == nrows); }
+#endif
+
+  if (nrows < 1)
+  {
+    return Result::INVALID;
+  }
+
+  for (size_t i = 0; i < nrows; ++i)
+  {
+    for (const auto& p : vars)
+    {
+      lhs[i].push_back(p.second[i]);
+    }
+  }
+
+#ifdef CVC4_ASSERTIONS
+  for (const auto& row : lhs) { Assert(row.size() == nvars); }
+  Assert(lhs.size() == rhs.size());
+#endif
+
+  if (lhs.size() > lhs[0].size())
+  {
+    return Result::INVALID;
+  }
+
+  Trace("bv-gauss-elim") << "Applying Gaussian Elimination..." << std::endl;
+  Result ret = gaussElim(iprime, rhs, lhs);
+
+  if (ret != Result::NONE && ret != Result::INVALID)
+  {
+    std::vector<Node> vvars;
+    for (const auto& p : vars) { vvars.push_back(p.first); }
+    Assert(nvars == vvars.size());
+    Assert(nrows == lhs.size());
+    Assert(nrows == rhs.size());
+    NodeManager *nm = NodeManager::currentNM();
+    if (ret == Result::UNIQUE)
+    {
+      for (size_t i = 0; i < nvars; ++i)
+      {
+        res[vvars[i]] = nm->mkConst<BitVector>(
+            BitVector(bv::utils::getSize(vvars[i]), rhs[i]));
+      }
+    }
+    else
+    {
+      Assert(ret == Result::PARTIAL);
+
+      for (size_t pcol = 0, prow = 0; pcol < nvars && prow < nrows;
+           ++pcol, ++prow)
+      {
+        Assert(lhs[prow][pcol] == 0 || lhs[prow][pcol] == 1);
+        while (pcol < nvars && lhs[prow][pcol] == 0) pcol += 1;
+        if (pcol >= nvars)
+        {
+          Assert(rhs[prow] == 0);
+          break;
+        }
+        if (lhs[prow][pcol] == 0)
+        {
+          Assert(rhs[prow] == 0);
+          continue;
+        }
+        Assert(lhs[prow][pcol] == 1);
+        std::vector<Node> stack;
+        for (size_t i = pcol + 1; i < nvars; ++i)
+        {
+          if (lhs[prow][i] == 0) continue;
+          /* Normalize (no negative numbers, hence no subtraction)
+           * e.g., x = 4 - 2y  --> x = 4 + 9y (modulo 11) */
+          Integer m = iprime - lhs[prow][i];
+          Node bv = bv::utils::mkConst(bv::utils::getSize(vvars[i]), m);
+          Node mult = nm->mkNode(kind::BITVECTOR_MULT, vvars[i], bv);
+          stack.push_back(mult);
+        }
+
+        if (stack.empty())
+        {
+          res[vvars[pcol]] = nm->mkConst<BitVector>(
+              BitVector(bv::utils::getSize(vvars[pcol]), rhs[prow]));
+        }
+        else
+        {
+          Node tmp = stack.size() == 1
+                         ? stack[0]
+                         : nm->mkNode(kind::BITVECTOR_PLUS, stack);
+
+          if (rhs[prow] != 0)
+          {
+            tmp = nm->mkNode(kind::BITVECTOR_PLUS,
+                             bv::utils::mkConst(
+                                 bv::utils::getSize(vvars[pcol]), rhs[prow]),
+                             tmp);
+          }
+          Assert(!is_bv_const(tmp));
+          res[vvars[pcol]] = nm->mkNode(kind::BITVECTOR_UREM, tmp, prime);
+        }
+      }
+    }
+  }
+  return ret;
+}
+
+}  // namespace
+
+
+
+BVGauss::BVGauss(PreprocessingPassContext* preprocContext)
+    : PreprocessingPass(preprocContext, "bv-gauss")
+{
+}
+
+PreprocessingPassResult BVGauss::applyInternal(
+    AssertionPipeline* assertionsToPreprocess)
+{
+  std::vector<Node> assertions(assertionsToPreprocess->ref());
+  std::unordered_map<Node, std::vector<Node>, NodeHashFunction> equations;
+
+  while (!assertions.empty())
+  {
+    Node a = assertions.back();
+    assertions.pop_back();
+    CVC4::Kind k = a.getKind();
+
+    if (k == kind::AND)
+    {
+      for (const Node& aa : a)
+      {
+        assertions.push_back(aa);
+      }
+    }
+    else if (k == kind::EQUAL)
+    {
+      Node urem;
+
+      if (is_bv_const(a[1]) && a[0].getKind() == kind::BITVECTOR_UREM)
+      {
+        urem = a[0];
+      }
+      else if (is_bv_const(a[0]) && a[1].getKind() == kind::BITVECTOR_UREM)
+      {
+        urem = a[1];
+      }
+      else
+      {
+        continue;
+      }
+
+      if (urem[0].getKind() == kind::BITVECTOR_PLUS && is_bv_const(urem[1]))
+      {
+        equations[urem[1]].push_back(a);
+      }
+    }
+  }
+
+  std::unordered_map<Node, Node, NodeHashFunction> subst;
+  std::vector<Node>& atpp = assertionsToPreprocess->ref();
+
+  for (const auto& eq : equations)
+  {
+    if (eq.second.size() <= 1) { continue; }
+
+    std::unordered_map<Node, Node, NodeHashFunction> res;
+    Result ret = gaussElimRewriteForUrem(eq.second, res);
+    Trace("bv-gauss-elim") << "result: "
+                           << (ret == Result::INVALID
+                                   ? "INVALID"
+                                   : (ret == Result::UNIQUE
+                                          ? "UNIQUE"
+                                          : (ret == Result::PARTIAL
+                                                 ? "PARTIAL"
+                                                 : "NONE")))
+                           << std::endl;
+    if (ret != Result::INVALID)
+    {
+      NodeManager *nm = NodeManager::currentNM();
+      if (ret == Result::NONE)
+      {
+        atpp.clear();
+        atpp.push_back(nm->mkConst<bool>(false));
+      }
+      else
+      {
+        for (const Node& e : eq.second)
+        {
+          subst[e] = nm->mkConst<bool>(true);
+        }
+        /* add resulting constraints */
+        for (const auto& p : res)
+        {
+          Node a = nm->mkNode(kind::EQUAL, p.first, p.second);
+          Trace("bv-gauss-elim") << "added assertion: " << a << std::endl;
+          atpp.push_back(a);
+        }
+      }
+    }
+  }
+
+  if (!subst.empty())
+  {
+    /* delete (= substitute with true) obsolete assertions */
+    for (auto& a : atpp)
+    {
+      a = a.substitute(subst.begin(), subst.end());
+    }
+  }
+  return PreprocessingPassResult::NO_CONFLICT;
+}
+
+}  // namespace passes
+}  // namespace preprocessing
+}  // namespace CVC4
diff --git a/src/preprocessing/passes/bv_gauss.h b/src/preprocessing/passes/bv_gauss.h
new file mode 100644
index 000000000..a902f391c
--- /dev/null
+++ b/src/preprocessing/passes/bv_gauss.h
@@ -0,0 +1,53 @@
+/*********************                                                        */
+/*! \file bv_gauss.h
+ ** \verbatim
+ ** Top contributors (to current version):
+ **   Aina Niemetz
+ ** This file is part of the CVC4 project.
+ ** Copyright (c) 2009-2017 by the authors listed in the file AUTHORS
+ ** in the top-level source directory) and their institutional affiliations.
+ ** All rights reserved.  See the file COPYING in the top-level source
+ ** directory for licensing information.\endverbatim
+ **
+ ** \brief Gaussian Elimination preprocessing pass.
+ **
+ ** Simplify a given equation system modulo a (prime) number via Gaussian
+ ** Elimination if possible.
+ **/
+
+#include "cvc4_private.h"
+
+#ifndef __CVC4__PREPROCESSING__PASSES__BV_GAUSS_ELIM_H
+#define __CVC4__PREPROCESSING__PASSES__BV_GAUSS_ELIM_H
+
+#include "preprocessing/preprocessing_pass.h"
+#include "preprocessing/preprocessing_pass_context.h"
+
+namespace CVC4 {
+namespace preprocessing {
+namespace passes {
+
+class BVGauss : public PreprocessingPass
+{
+ public:
+   BVGauss(PreprocessingPassContext* preprocContext);
+
+ protected:
+  /**
+   * Apply Gaussian Elimination on (possibly multiple) set(s) of equations
+   * modulo some (prime) number given as bit-vector equations.
+   *
+   * Note that these sets of equations do not have to be modulo some prime
+   * but can be modulo any arbitrary number. However, GE is guaranteed to
+   * succeed modulo a prime number, which is not necessarily the case if a
+   * given set of equations is modulo a non-prime number.
+   */
+   PreprocessingPassResult applyInternal(
+       AssertionPipeline* assertionsToPreprocess) override;
+};
+
+}  // namespace passes
+}  // namespace preprocessing
+}  // namespace CVC4
+
+#endif