diff --git a/src/analyses/constant_propagator.cpp b/src/analyses/constant_propagator.cpp
index 452e8355a93..57b854a52ee 100644
--- a/src/analyses/constant_propagator.cpp
+++ b/src/analyses/constant_propagator.cpp
@@ -31,11 +31,15 @@ void constant_propagator_domaint::assign_rec(
   valuest &values,
   const exprt &lhs,
   const exprt &rhs,
-  const namespacet &ns)
+  const namespacet &ns,
+  const constant_propagator_ait *cp)
 {
   if(lhs.id()!=ID_symbol)
     return;
 
+  if(cp && !cp->should_track_value(lhs, ns))
+    return;
+
   const symbol_exprt &s=to_symbol_expr(lhs);
 
   exprt tmp=rhs;
@@ -89,11 +93,11 @@ void constant_propagator_domaint::transform(
     const code_assignt &assignment=to_code_assign(from->code);
     const exprt &lhs=assignment.lhs();
     const exprt &rhs=assignment.rhs();
-    assign_rec(values, lhs, rhs, ns);
+    assign_rec(values, lhs, rhs, ns, cp);
   }
   else if(from->is_assume())
   {
-    two_way_propagate_rec(from->guard, ns);
+    two_way_propagate_rec(from->guard, ns, cp);
   }
   else if(from->is_goto())
   {
@@ -110,7 +114,7 @@ void constant_propagator_domaint::transform(
      values.set_to_bottom();
     else
     {
-      two_way_propagate_rec(g, ns);
+      two_way_propagate_rec(g, ns, cp);
       // If two way propagate is enabled then it may be possible to detect
       // that the branch condition is infeasible and thus the domain should
       // be set to bottom.  Without it the domain will only be set to bottom
@@ -175,7 +179,7 @@ void constant_propagator_domaint::transform(
             break;
 
           const symbol_exprt parameter_expr(p_it->get_identifier(), arg.type());
-          assign_rec(values, parameter_expr, arg, ns);
+          assign_rec(values, parameter_expr, arg, ns, cp);
 
           ++p_it;
         }
@@ -220,7 +224,8 @@ void constant_propagator_domaint::transform(
 /// handles equalities and conjunctions containing equalities
 bool constant_propagator_domaint::two_way_propagate_rec(
   const exprt &expr,
-  const namespacet &ns)
+  const namespacet &ns,
+  const constant_propagator_ait *cp)
 {
 #ifdef DEBUG
   std::cout << "two_way_propagate_rec: " << format(expr) << '\n';
@@ -238,7 +243,7 @@ bool constant_propagator_domaint::two_way_propagate_rec(
       change = false;
 
       forall_operands(it, expr)
-        if(two_way_propagate_rec(*it, ns))
+        if(two_way_propagate_rec(*it, ns, cp))
           change=true;
     }
     while(change);
diff --git a/src/analyses/constant_propagator.h b/src/analyses/constant_propagator.h
index e32a14bc0f2..125942acfe7 100644
--- a/src/analyses/constant_propagator.h
+++ b/src/analyses/constant_propagator.h
@@ -18,6 +18,8 @@ Author: Peter Schrammel
 #include "ai.h"
 #include "dirty.h"
 
+class constant_propagator_ait;
+
 class constant_propagator_domaint:public ai_domain_baset
 {
 public:
@@ -143,12 +145,15 @@ class constant_propagator_domaint:public ai_domain_baset
 protected:
   void assign_rec(
     valuest &values,
-    const exprt &lhs, const exprt &rhs,
-    const namespacet &ns);
+    const exprt &lhs,
+    const exprt &rhs,
+    const namespacet &ns,
+    const constant_propagator_ait *cp);
 
   bool two_way_propagate_rec(
     const exprt &expr,
-    const namespacet &ns);
+    const namespacet &ns,
+    const constant_propagator_ait *cp);
 
   bool partial_evaluate_with_all_rounding_modes(
     exprt &expr,
@@ -160,13 +165,35 @@ class constant_propagator_domaint:public ai_domain_baset
 class constant_propagator_ait:public ait<constant_propagator_domaint>
 {
 public:
-  explicit constant_propagator_ait(const goto_functionst &goto_functions):
-    dirty(goto_functions)
+  typedef std::function<bool(const exprt &, const namespacet &)>
+    should_track_valuet;
+
+  static bool track_all_values(const exprt &, const namespacet &)
+  {
+    return true;
+  }
+
+  explicit constant_propagator_ait(
+    const goto_functionst &goto_functions,
+    should_track_valuet should_track_value = track_all_values):
+    dirty(goto_functions),
+    should_track_value(should_track_value)
+  {
+  }
+
+  explicit constant_propagator_ait(
+    const goto_functiont &goto_function,
+    should_track_valuet should_track_value = track_all_values):
+    dirty(goto_function),
+    should_track_value(should_track_value)
   {
   }
 
   constant_propagator_ait(
-    goto_modelt &goto_model):dirty(goto_model.goto_functions)
+    goto_modelt &goto_model,
+    should_track_valuet should_track_value = track_all_values):
+    dirty(goto_model.goto_functions),
+    should_track_value(should_track_value)
   {
     const namespacet ns(goto_model.symbol_table);
     operator()(goto_model.goto_functions, ns);
@@ -175,7 +202,10 @@ class constant_propagator_ait:public ait<constant_propagator_domaint>
 
   constant_propagator_ait(
     goto_functionst::goto_functiont &goto_function,
-    const namespacet &ns):dirty(goto_function)
+    const namespacet &ns,
+    should_track_valuet should_track_value = track_all_values):
+    dirty(goto_function),
+    should_track_value(should_track_value)
   {
     operator()(goto_function, ns);
     replace(goto_function, ns);
@@ -197,6 +227,8 @@ class constant_propagator_ait:public ait<constant_propagator_domaint>
   void replace_types_rec(
     const replace_symbolt &replace_const,
     exprt &expr);
+
+  should_track_valuet should_track_value;
 };
 
 #endif // CPROVER_ANALYSES_CONSTANT_PROPAGATOR_H
diff --git a/unit/Makefile b/unit/Makefile
index ca8920e0c9d..496f4cb0143 100644
--- a/unit/Makefile
+++ b/unit/Makefile
@@ -9,6 +9,7 @@ SRC = unit_tests.cpp \
 SRC += unit_tests.cpp \
        analyses/ai/ai_simplify_lhs.cpp \
        analyses/call_graph.cpp \
+       analyses/constant_propagator.cpp \
        analyses/disconnect_unreachable_nodes_in_graph.cpp \
        analyses/does_remove_const/does_expr_lose_const.cpp \
        analyses/does_remove_const/does_type_preserve_const_correctness.cpp \
diff --git a/unit/analyses/constant_propagator.cpp b/unit/analyses/constant_propagator.cpp
new file mode 100644
index 00000000000..8c591e1a63c
--- /dev/null
+++ b/unit/analyses/constant_propagator.cpp
@@ -0,0 +1,114 @@
+/*******************************************************************\
+
+Module: Unit test for constant propagation
+
+Author: Diffblue Ltd
+
+\*******************************************************************/
+
+#include <testing-utils/catch.hpp>
+
+#include <analyses/constant_propagator.h>
+
+#include <ansi-c/ansi_c_language.h>
+
+#include <goto-programs/goto_convert_functions.h>
+
+#include <util/message.h>
+
+static bool starts_with_x(const exprt &e, const namespacet &)
+{
+  if(e.id() != ID_symbol)
+    return false;
+  return has_prefix(id2string(to_symbol_expr(e).get_identifier()), "x");
+}
+
+SCENARIO("constant_propagator", "[core][analyses][constant_propagator]")
+{
+  GIVEN("A simple GOTO program")
+  {
+    null_message_handlert null_out;
+
+    goto_modelt goto_model;
+    namespacet ns(goto_model.symbol_table);
+
+    // Create the program:
+    // int x = 1;
+    // int y = 2;
+
+    symbolt local_x;
+    symbolt local_y;
+    local_x.name = "x";
+    local_x.type = integer_typet();
+    local_x.mode = ID_C;
+    local_y.name = "y";
+    local_y.type = integer_typet();
+    local_y.mode = ID_C;
+
+    code_blockt code;
+    code.copy_to_operands(code_declt(local_x.symbol_expr()));
+    code.copy_to_operands(code_declt(local_y.symbol_expr()));
+    code.copy_to_operands(
+      code_assignt(
+        local_x.symbol_expr(), constant_exprt("1", integer_typet())));
+    code.copy_to_operands(
+      code_assignt(
+        local_y.symbol_expr(), constant_exprt("2", integer_typet())));
+
+    symbolt main_function_symbol;
+    main_function_symbol.name = "main";
+    main_function_symbol.type = code_typet();
+    main_function_symbol.value = code;
+    main_function_symbol.mode = ID_C;
+
+    goto_model.symbol_table.add(local_x);
+    goto_model.symbol_table.add(local_y);
+    goto_model.symbol_table.add(main_function_symbol);
+
+    goto_convert(goto_model, null_out);
+
+    const goto_functiont &main_function = goto_model.get_goto_function("main");
+
+    // Find the instruction after "y = 2;"
+    goto_programt::const_targett test_instruction =
+      main_function.body.instructions.begin();
+    while(
+      test_instruction != main_function.body.instructions.end() &&
+      (!test_instruction->is_assign() ||
+       to_code_assign(test_instruction->code).lhs() != local_y.symbol_expr()))
+    {
+      ++test_instruction;
+    }
+
+    REQUIRE(test_instruction != main_function.body.instructions.end());
+    ++test_instruction;
+
+    WHEN("We apply conventional constant propagation")
+    {
+      constant_propagator_ait constant_propagator(main_function);
+      constant_propagator(main_function, ns);
+
+      THEN("The propagator should discover values for both 'x' and 'y'")
+      {
+        const auto &final_domain = constant_propagator[test_instruction];
+
+        REQUIRE(final_domain.values.is_constant(local_x.symbol_expr()));
+        REQUIRE(final_domain.values.is_constant(local_y.symbol_expr()));
+      }
+    }
+
+    WHEN("We apply constant propagation for symbols beginning with 'x'")
+    {
+      constant_propagator_ait constant_propagator(main_function, starts_with_x);
+      constant_propagator(main_function, ns);
+
+      THEN("The propagator should discover a value for 'x' but not 'y'")
+      {
+        const auto &final_domain = constant_propagator[test_instruction];
+
+        REQUIRE(final_domain.values.is_constant(local_x.symbol_expr()));
+        REQUIRE(!final_domain.values.is_constant(local_y.symbol_expr()));
+      }
+    }
+  }
+}