diff --git a/.gitignore b/.gitignore
index 71e7b25e0..766d882ea 100644
--- a/.gitignore
+++ b/.gitignore
@@ -37,3 +37,12 @@
 
 # emacs save files
 *.~
+
+# Files generated by ctags
+CTAGS
+GTAGS
+GRTAGS
+GSYMS
+GPATH
+tags
+TAGS
diff --git a/k2/csrc/fsa.h b/k2/csrc/fsa.h
index e4dfaaaee..f6d591983 100644
--- a/k2/csrc/fsa.h
+++ b/k2/csrc/fsa.h
@@ -12,6 +12,10 @@
 
 namespace k2 {
 
+using Label = int32_t;
+using StateId = int32_t;
+using Weight = float;
+
 enum {
   kFinalSymbol = -1,  // final-costs are represented as arcs with
                       // kFinalSymbol as their label, to the final
@@ -31,9 +35,9 @@ struct Range {
 };
 
 struct Arc {
-  int32_t src_state;
-  int32_t dest_state;
-  int32_t label;  // 'label' as in a finite state acceptor.
+  StateId src_state;
+  StateId dest_state;
+  Label label;  // 'label' as in a finite state acceptor.
                   // For FSTs, the other label will be present in the
                   // aux_label array.  Which of the two represents the input
                   // vs. the output can be decided by the user; in general,
@@ -91,7 +95,7 @@ struct Fsa {
 
  */
 struct DenseFsa {
-  float* weights;  // Would typically be a log-prob or unnormalized log-prob
+  Weight* weights;  // Would typically be a log-prob or unnormalized log-prob
   int32_t T;       // The number of time steps == rows in the matrix `weights`;
                    // this FSA has T + 2 states, see explanation above.
   int32_t num_symbols;  // The number of symbols == columns in the matrix
@@ -105,7 +109,7 @@ struct DenseFsa {
       CAUTION: we may later enforce that stride == num_symbols, in order to
       be able to know the layout of a phantom matrix of arcs.  (?)
    */
-  DenseFsa(float* data, int32_t T, int32_t num_symbols, int32_t stride);
+  DenseFsa(Weight* data, int32_t T, int32_t num_symbols, int32_t stride);
 };
 
 /*
diff --git a/k2/csrc/properties.cc b/k2/csrc/properties.cc
index b3fab9dc2..0821416da 100644
--- a/k2/csrc/properties.cc
+++ b/k2/csrc/properties.cc
@@ -1,36 +1,57 @@
 // k2/csrc/properties.cc
 
-// Copyright (c)  2020  Daniel Povey
+// Copyright (c)  2020 Haowen Qiu
+//                     Daniel Povey
 
 // See ../../LICENSE for clarification regarding multiple authors
 
-#include "k2/csrc/properties.h"
+#include <unordered_set>
 
+#include "k2/csrc/properties.h"
 #include "k2/csrc/fsa.h"
 
 namespace k2 {
 
-bool IsTopSorted(const Fsa& fsa) {
-  for (const auto& range : fsa.leaving_arcs) {
-    for (auto arc_idx = range.begin; arc_idx < range.end; ++arc_idx) {
-      const Arc& arc = fsa.arcs[arc_idx];
-      if (arc.dest_state < arc.src_state) {
-        return false;
-      }
+bool IsTopSorted(const Fsa &fsa) {
+  for (auto &arc : fsa.arcs) {
+    if (arc.dest_state < arc.src_state) {
+      return false;
     }
   }
   return true;
 }
 
-bool HasSelfLoops(const Fsa& fsa) {
-  // TODO(haowen): refactor code below as we have
-  // so many for-for-loop structures
-  for (const auto& range : fsa.leaving_arcs) {
-    for (auto arc_idx = range.begin; arc_idx < range.end; ++arc_idx) {
-      const Arc& arc = fsa.arcs[arc_idx];
-      if (arc.dest_state == arc.src_state) {
+bool HasSelfLoops(const Fsa &fsa) {
+  for (auto &arc : fsa.arcs) {
+    if (arc.dest_state == arc.src_state) {
+      return true;
+    }
+  }
+  return false;
+}
+
+bool IsDeterministic(const Fsa &fsa) {
+  std::unordered_set<Label> labels;
+  StateId state = 0;
+  for (auto &arc : fsa.arcs) {
+    if (arc.src_state == state) {
+      if (labels.find(arc.label) != labels.end()) {
         return false;
       }
+      labels.insert(arc.label);
+    } else {
+      state = arc.src_state;
+      labels.clear();
+      labels.insert(arc.label);
+    }
+  }
+  return true;
+}
+
+bool IsEpsilonFree(const Fsa &fsa) {
+  for (auto &arc : fsa.arcs) {
+    if (arc.label == kEpsilon) {
+      return false;
     }
   }
   return true;
diff --git a/k2/csrc/properties_test.cc b/k2/csrc/properties_test.cc
index b4e4ded6e..8597c798c 100644
--- a/k2/csrc/properties_test.cc
+++ b/k2/csrc/properties_test.cc
@@ -1,6 +1,6 @@
 // k2/csrc/properties_test.cc
 
-// Copyright (c)  2020  Haowen Qiue
+// Copyright (c)  2020  Haowen Qiu
 //                      Fangjun Kuang (csukuangfj@gmail.com)
 
 // See ../../LICENSE for clarification regarding multiple authors
@@ -17,35 +17,84 @@ using k2::Fsa;
 using k2::Range;
 
 // TODO(haowen): create Fsa examples in a more elegant way (add methods
-// addState, addArc, etc.)
-static Fsa CreateNonTopSortedFsaExample() {
-  std::vector<Arc> arcs = {
-      {0, 2, 0},
-      {2, 1, 0},
-      {0, 1, 0},
-  };
-  std::vector<Range> leaving_arcs = {
-      {0, 1},
-      {1, 2},
-      {2, 3},
-  };
+// addState, addArc, etc.) and use Test Fixtures by constructing
+// reusable FSA examples.
+TEST(Properties, IsNotTopSorted) {
+  std::vector<Arc> arcs = {{0, 1, 0}, {0, 2, 0}, {2, 1, 0}, };
+  std::vector<Range> leaving_arcs = {{0, 2}, {2, 3}, };
   Fsa fsa;
   fsa.leaving_arcs = leaving_arcs;
   fsa.arcs = arcs;
-  return fsa;
+  bool sorted = IsTopSorted(fsa);
+  EXPECT_FALSE(sorted);
 }
 
 TEST(Properties, IsTopSorted) {
-  Fsa fsa = CreateNonTopSortedFsaExample();
+  std::vector<Arc> arcs = {{0, 1, 0}, {0, 2, 0}, {1, 2, 0}, };
+  std::vector<Range> leaving_arcs = {{0, 2}, {2, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
+  fsa.arcs = arcs;
   bool sorted = IsTopSorted(fsa);
-  EXPECT_FALSE(sorted);
+  EXPECT_TRUE(sorted);
+}
+
+TEST(Properties, HasNotSelfLoops) {
+  std::vector<Arc> arcs = {{0, 1, 0}, {0, 2, 0}, {1, 2, 0}, };
+  std::vector<Range> leaving_arcs = {{0, 2}, {2, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
+  fsa.arcs = arcs;
+  bool hasSelfLoops = HasSelfLoops(fsa);
+  EXPECT_FALSE(hasSelfLoops);
+}
 
-  std::vector<Arc> arcs = {
-      {0, 1, 0},
-      {1, 2, 0},
-      {0, 2, 0},
-  };
+TEST(Properties, HasSelfLoops) {
+  std::vector<Arc> arcs = {{0, 1, 0}, {1, 2, 0}, {1, 1, 0}, };
+  std::vector<Range> leaving_arcs = {{0, 1}, {1, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
   fsa.arcs = arcs;
-  sorted = IsTopSorted(fsa);
-  EXPECT_TRUE(sorted);
+  bool hasSelfLoops = HasSelfLoops(fsa);
+  EXPECT_TRUE(hasSelfLoops);
+}
+
+TEST(Properties, IsNotDeterministic) {
+  std::vector<Arc> arcs = {{0, 1, 2}, {1, 2, 0}, {1, 3, 0}, };
+  std::vector<Range> leaving_arcs = {{0, 1}, {1, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
+  fsa.arcs = arcs;
+  bool isDeterministic = IsDeterministic(fsa);
+  EXPECT_FALSE(isDeterministic);
+}
+
+TEST(Properties, IsDeterministic) {
+  std::vector<Arc> arcs = {{0, 1, 2}, {1, 2, 0}, {1, 3, 2}, };
+  std::vector<Range> leaving_arcs = {{0, 1}, {1, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
+  fsa.arcs = arcs;
+  bool isDeterministic = IsDeterministic(fsa);
+  EXPECT_TRUE(isDeterministic);
+}
+
+TEST(Properties, IsNotEpsilonFree) {
+  std::vector<Arc> arcs = {{0, 1, 2}, {0, 2, 0}, {1, 2, 1}, };
+  std::vector<Range> leaving_arcs = {{0, 2}, {2, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
+  fsa.arcs = arcs;
+  bool isEpsilonFree = IsEpsilonFree(fsa);
+  EXPECT_FALSE(isEpsilonFree);
+}
+
+TEST(Properties, IsEpsilonFree) {
+  std::vector<Arc> arcs = {{0, 1, 2}, {0, 2, 1}, {1, 2, 1}, };
+  std::vector<Range> leaving_arcs = {{0, 2}, {2, 3}, };
+  Fsa fsa;
+  fsa.leaving_arcs = leaving_arcs;
+  fsa.arcs = arcs;
+  bool isEpsilonFree = IsEpsilonFree(fsa);
+  EXPECT_TRUE(isEpsilonFree);
 }