diff --git a/NEWS b/NEWS
index bd6265530..db61afd3b 100644
--- a/NEWS
+++ b/NEWS
@@ -24,6 +24,12 @@ New in spot 2.7.5.dev (not yet released)
   - genaut learned --m-nba=N to generate Max Michel's NBA familly.
     (NBAs with N+1 states whose determinized have at least N! states.)
 
+  - Due to some new simplification of parity acceptance, the output of
+    "ltl2tgba -P -D" is now using a minimal number of colors.  This
+    means that recurrence properties have an acceptance condition
+    amongst "Inf(0)", "t", or "f", and persistance properties have an
+    acceptance condition among "Fin(0)", "t", or "f".
+
   Library:
 
   - Add generic_accepting_run() as a variant of generic_emptiness_check() that
@@ -117,6 +123,13 @@ New in spot 2.7.5.dev (not yet released)
   - The postprocessor and translator classes are now using
     reduce_parity() for further simplifications.
 
+  - The code for checking recurrence/persistence properties can also
+    use the fact the reduce_parity() with return "Inf(0)" (or "t" or
+    "f") for deterministic automata corresponding to recurrence
+    properties, and "Fin(0)" (or "t" or "f") for persistence
+    properties.  This can be altered with the SPOT_PR_CHECK
+    environment variable.
+
 New in spot 2.7.5 (2019-06-05)
 
   Build:
diff --git a/bin/man/spot-x.x b/bin/man/spot-x.x
index d0e6e94ab..a243b34b1 100644
--- a/bin/man/spot-x.x
+++ b/bin/man/spot-x.x
@@ -132,8 +132,8 @@ ignoring them), and setting this variable will interfer with that.
 .TP
 \fBSPOT_PR_CHECK\fR
 Select the default algorithm that must be used to check the persistence
-or recurrence property of a formula f. The values it can take are 1
-or 2. Both methods work either on f or !f thanks to the duality of
+or recurrence property of a formula f. The values it can take are between
+1 and 3. All  methods work either on f or !f thanks to the duality of
 persistence and recurrence classes.  See
 .UR https://spot.lrde.epita.fr/hierarchy.html
 this page
@@ -145,8 +145,13 @@ for more details. If it is set to:
 It will try to check if f (or !f) is co-Büchi realizable in order to
 tell if f belongs to the persistence (or the recurrence) class.
 .IP 2
-It checks if f (or !f) is det-Büchi realizable to tell if f belongs
-to the recurrence (or the persistence) class.
+It checks if f (or !f) is det-Büchi realizable via a reduction
+to deterministic-Rabin in order to tell if f belongs to the
+recurrence (or the persistance) class.
+.IP 3
+It checks if f (or !f) is det-Büchi realizable via a reduction
+to deterministic-parity in order to tell if f belongs to the
+recurrence (or the persistance) class.
 .RE
 .RE
 
diff --git a/doc/org/hierarchy.org b/doc/org/hierarchy.org
index b19a6ab68..b0c0e4d80 100644
--- a/doc/org/hierarchy.org
+++ b/doc/org/hierarchy.org
@@ -520,7 +520,7 @@ automata.
 For the subclass of /obligation/ properties, using =-D= is a sure way
 to obain a deterministic automaton (and even a minimal one), but for
 the /recurrence/ properties that are not /obligations/ the translator
-does not make any special effort to produce deterministic automata,
+does not make /too much/ effort to produce deterministic automata,
 even with =-D= (this might change in the future).
 
 All properties that are not in the /persistence/ class (this includes
@@ -585,106 +585,40 @@ $txt
 [[file:hier-recurrence-3.svg]]
 
 One way to obtain a deterministic Büchi automaton (it has to exist, since this is
-a /recurrence/ property), is to chain a few algorithms implemented in Spot:
+a /recurrence/ property), is to request a deterministic automaton with parity
+acceptance using =-P=.  The number of color output with =-P= is always reduced
+to the minimal number possible, so for a /recurrence/ property the output
+automaton can only have one of three possible acceptance: =Inf(0)=, =t=, or =f=.
 
- 1. Determinize the non-deterministic automaton to obtain a
-    deterministic automaton with parity acceptance: this is done by
-    using =ltl2tgba -P -D=, with option =-P= indicating that parity
-    acceptance is desired.
-
-    #+NAME: hier-recurrence-4
-    #+BEGIN_SRC sh :exports code
-    ltl2tgba -P -D 'G(Gb | Fa)' -d
-    #+END_SRC
-    #+BEGIN_SRC dot :file hier-recurrence-4.svg :var txt=hier-recurrence-4 :exports results
-    $txt
-    #+END_SRC
-    #+RESULTS:
-    [[file:hier-recurrence-4.svg]]
-
- 2. Transform the parity acceptance into Rabin acceptance: this is
-    done with =autfilt --generalized-rabin=.  Because of the type of
-    parity acceptance used, the result will actually be Rabin and not
-    generalized Rabin.
-
-    #+NAME: hier-recurrence-5
-    #+BEGIN_SRC sh :exports code
-    ltl2tgba -P -D 'G(Gb | Fa)' |
-      autfilt --generalized-rabin -d
-    #+END_SRC
-    #+BEGIN_SRC dot :file hier-recurrence-5.svg :var txt=hier-recurrence-5 :exports results
-    $txt
-    #+END_SRC
-
-    #+RESULTS:
-    [[file:hier-recurrence-5.svg]]
-
-    (The only change here is in the acceptance condition.)
-
- 3. In step 4 we are going to convert the automaton to state-based
-    Büchi, and this sometimes works better if the input Rabin automaton
-    also uses state-based acceptance.  So let us add =-S= to the
-    previous command:
-
-    #+NAME: hier-recurrence-6
-    #+BEGIN_SRC sh :results verbatim :exports code
-    ltl2tgba -P -D 'G(Gb | Fa)' |
-      autfilt -S --generalized-rabin -d
-    #+END_SRC
-
-    #+BEGIN_SRC dot :file hier-recurrence-6.svg :var txt=hier-recurrence-6 :exports results
-    $txt
-    #+END_SRC
-
-    #+RESULTS:
-    [[file:hier-recurrence-6.svg]]
-
- 4. Finally, convert the resulting automaton to BA, using =autfilt
-    -B=.  Spot can convert automata with any acceptance condition to
-    BA, but when the input is a deterministic Rabin automaton, it uses
-    a dedicated algorithm that preserves determinism whenever possible
-    (and we know it is possible, because we are working on a
-    recurrence formula).  Adding =-D= here to suggest that we are
-    trying to obtain a deterministic automaton does not hurt, as it
-    will enable simplifications as a side-effect (without =-D= we
-    simply get a larger deterministic automaton).
-
-    #+NAME: hier-recurrence-7
-    #+BEGIN_SRC sh :exports code
-    ltl2tgba -P -D 'G(Gb | Fa)' |
-      autfilt -S --generalized-rabin |
-      autfilt -B -D -d
-    #+END_SRC
-
-    #+BEGIN_SRC dot :file hier-recurrence-7.svg :var txt=hier-recurrence-7 :exports results
-    $txt
-    #+END_SRC
-
-    #+RESULTS:
-    [[file:hier-recurrence-7.svg]]
-
-Here we are lucky that the deterministic Büchi automaton is even
-smaller than the original non-deterministic version.  As said earlier,
-passing =-S= to the first =autfilt= was optional, but in this case it
-helps producing a smaller automaton.  Here is what we get without it:
-
-#+NAME: hier-recurrence-8
+#+NAME: hier-recurrence-4
 #+BEGIN_SRC sh :exports code
-ltl2tgba -P -D 'G(Gb | Fa)' |
-  autfilt --generalized-rabin |
-  autfilt -B -D -d
+ltl2tgba -P -D 'G(Gb | Fa)' -d
 #+END_SRC
-
-#+BEGIN_SRC dot :file hier-recurrence-8.svg :var txt=hier-recurrence-8 :exports results
+#+BEGIN_SRC dot :file hier-recurrence-4.svg :var txt=hier-recurrence-4 :exports results
 $txt
 #+END_SRC
-
 #+RESULTS:
-[[file:hier-recurrence-8.svg]]
+[[file:hier-recurrence-4.svg]]
 
-It is likely that =ltl2tgba -B -D= will implement all this processing
-chain in the future, but so originally =-D= was only expressing a
-preference not a requirement.
+Note that if the acceptance is =t=, the property is a monitor, and if
+its =f=, the property is =false=.  In any way, if you would like to
+obtain a DBA for any recurrent property, a sure way to avoid these
+difference is to pipe the result through =autfilt -B=
+
+#+NAME: hier-recurrence-5
+#+BEGIN_SRC sh :exports code
+ltl2tgba -P -D 'G(Gb | Fa)' | autfilt -B -d
+#+END_SRC
+#+BEGIN_SRC dot :file hier-recurrence-5.svg :var txt=hier-recurrence-5 :exports results
+$txt
+#+END_SRC
+#+RESULTS:
+[[file:hier-recurrence-5.svg]]
+
+
+It is likely that =ltl2tgba -B -D= will implement these steps in the
+future, but so originally =-D= was only expressing a preference not a
+requirement.
 
 ** Persistence
 
@@ -725,8 +659,9 @@ $txt
 
 Note that in this example, we know that =GFa= is trivial enough that
 =ltl2tgba -D GFa= will generate a deterministic automaton.  In the
-general case we might have to determinize the automaton as we did in
-the previous section (we will do it again below).
+general case we might have to determinize the automaton using =-P -D= as
+we did in the previous section.  For persistence properties, =-P -D= should
+return an automaton whose acceptance is one of =Fin(0)=, =t=, or =f=.
 
 /Persistence/ properties can be represented by weak Büchi automata.
 The translator is aware of that, so when it detects that the input
@@ -739,10 +674,8 @@ optimization is simply not applied.)
 If the input is a weak property that is not syntactically weak, the
 output will not necessarily be weak.  One costly way to obtain a weak
 automaton for a formula $\varphi$ would be to first compute a
-deterministic Büchi automaton of the recurrence $\lnot\varphi$ then
-complement the acceptance of the resulting automaton, yielding a
-deterministic co-Büchi automaton, and then transform that into a Büchi
-automaton.
+deterministic co-Büchi automaton $\varphi$ then transform that into a
+Büchi automaton.
 
 Let's do that on the persistence formula =F(G!a | G(b U a))=, just for
 the fun of it.
@@ -766,15 +699,11 @@ $txt
 #+RESULTS:
 [[file:hier-persistence-3.svg]]
 
-Furthermore it appears that =ltl2tgba -D= does generate a deterministic
-Büchi automaton for the complement, instead we get a non-deterministic
-generalized Büchi automaton:
+So let's determinize using parity acceptance:
 
 #+NAME: hier-persistence-4
 #+BEGIN_SRC sh :exports code
-ltlfilt --negate -f 'F(G!a | G(b U a))' |
-  ltl2tgba -D |
-  autfilt --highlight-nondet=5 -d
+ltl2tgba -P -D 'F(G!a | G(b U a))' -d
 #+END_SRC
 #+BEGIN_SRC dot :file hier-persistence-4.svg :var txt=hier-persistence-4 :exports results
 $txt
@@ -783,58 +712,16 @@ $txt
 #+RESULTS:
 [[file:hier-persistence-4.svg]]
 
-So let us use the same tricks as in the previous section,
-determinizing this automaton into a Rabin automaton, and then back to
-deterministic Büchi:
-
-#+NAME: hier-persistence-5
-#+BEGIN_SRC sh :exports code
-ltlfilt --negate -f 'F(G!a | G(b U a))' |
-  ltl2tgba -P -D |
-  autfilt --generalized-rabin |
-  autfilt --tgba -D -d
-#+END_SRC
-#+BEGIN_SRC dot :file hier-persistence-5.svg :var txt=hier-persistence-5 :exports results
-$txt
-#+END_SRC
-
-#+RESULTS:
-[[file:hier-persistence-5.svg]]
-
-This is a deterministic Büchi automaton for the negation of our formula.
-Now we can complement it to obtain a deterministic co-Büchi automaton for =F(G!a | G(b U a))=:
-
-#+NAME: hier-persistence-6
-#+BEGIN_SRC sh :exports code
-ltlfilt --negate -f 'F(G!a | G(b U a))' |
-  ltl2tgba -P -D |
-  autfilt --generalized-rabin |
-  autfilt --tgba -D |
-  autfilt --complement -d
-#+END_SRC
-
-#+BEGIN_SRC dot :file hier-persistence-6.svg :var txt=hier-persistence-6 :exports results
-$txt
-#+END_SRC
-
-#+RESULTS:
-[[file:hier-persistence-6.svg]]
-
-And finally we convert the result back to Büchi:
+And finally we convert the result back to Büchi with =autfilt -B=.
 
 #+NAME: hier-persistence-7
 #+BEGIN_SRC sh :exports code
-ltlfilt --negate -f 'F(G!a | G(b U a))' |
-  ltl2tgba -P -D |
-  autfilt --generalized-rabin |
-  autfilt --tgba -D |
-  autfilt --complement -B -d
+ltl2tgba -P -D 'F(G!a | G(b U a))' | autfilt -B --highlight-nondet --small -d
 #+END_SRC
-
 #+BEGIN_SRC dot :file hier-persistence-7.svg :var txt=hier-persistence-7 :exports results
 $txt
 #+END_SRC
 #+RESULTS:
 [[file:hier-persistence-7.svg]]
 
-That is indeed, a weak automaton.
+That is indeed, a weak non-deterministic automaton.
diff --git a/spot/tl/hierarchy.cc b/spot/tl/hierarchy.cc
index 870a46ae6..d069fd65b 100644
--- a/spot/tl/hierarchy.cc
+++ b/spot/tl/hierarchy.cc
@@ -1,5 +1,5 @@
 // -*- coding: utf-8 -*-
-// Copyright (C) 2017, 2018 Laboratoire de Recherche et Développement de
+// Copyright (C) 2017-2019 Laboratoire de Recherche et Développement de
 // l'Epita (LRDE)
 //
 // This file is part of Spot, a model checking library.
@@ -65,79 +65,91 @@ namespace spot
       return !cobuchi->intersects(not_aut);
     }
 
+    [[noreturn]] static void invalid_spot_pr_check()
+    {
+      throw std::runtime_error("invalid value for SPOT_PR_CHECK "
+                               "(should be 1, 2, or 3)");
+    }
+
+    static prcheck
+    algo_to_perform(bool is_persistence, bool aut_given)
+    {
+      static unsigned value = [&]()
+        {
+          int val;
+          try
+            {
+              auto s = getenv("SPOT_PR_CHECK");
+              val = s ? std::stoi(s) : 0;
+            }
+          catch (const std::exception& e)
+            {
+              invalid_spot_pr_check();
+            }
+          if (val < 0 || val > 3)
+            invalid_spot_pr_check();
+          return val;
+        }();
+      switch (value)
+        {
+        case 0:
+          if (aut_given && !is_persistence)
+            return prcheck::via_Parity;
+          else
+            return prcheck::via_CoBuchi;
+        case 1:
+          return prcheck::via_CoBuchi;
+        case 2:
+          return prcheck::via_Rabin;
+        case 3:
+          return prcheck::via_Parity;
+        default:
+          SPOT_UNREACHABLE();
+        }
+      SPOT_UNREACHABLE();
+      return prcheck::via_Parity;
+    }
+
     static bool
     detbuchi_realizable(const twa_graph_ptr& aut)
     {
       if (is_universal(aut))
         return true;
 
-      // if aut is a non-deterministic TGBA, we do
-      // TGBA->DPA->DRA->(D?)BA.  The conversion from DRA to
-      // BA will preserve determinism if possible.
+      bool want_old = algo_to_perform(false, true) == prcheck::via_Rabin;
+      if (want_old)
+        {
+          // if aut is a non-deterministic TGBA, we do
+          // TGBA->DPA->DRA->(D?)BA.  The conversion from DRA to
+          // BA will preserve determinism if possible.
+          spot::postprocessor p;
+          p.set_type(spot::postprocessor::Generic);
+          p.set_pref(spot::postprocessor::Deterministic);
+          p.set_level(spot::postprocessor::Low);
+          auto dpa = p.run(aut);
+          if (dpa->acc().is_generalized_buchi())
+            {
+              assert(is_deterministic(dpa));
+              return true;
+            }
+          else
+            {
+              auto dra = to_generalized_rabin(dpa);
+              return rabin_is_buchi_realizable(dra);
+            }
+        }
+      // Converting reduce_parity() will produce a Büchi automaton (or
+      // an automaton with "t" or "f" acceptance) if the parity
+      // automaton is DBA-realizable.
       spot::postprocessor p;
-      p.set_type(spot::postprocessor::Generic);
+      p.set_type(spot::postprocessor::Parity);
       p.set_pref(spot::postprocessor::Deterministic);
       p.set_level(spot::postprocessor::Low);
       auto dpa = p.run(aut);
-      if (dpa->acc().is_generalized_buchi())
-        {
-          assert(is_deterministic(dpa));
-          return true;
-        }
-      else
-        {
-          auto dra = to_generalized_rabin(dpa);
-          return rabin_is_buchi_realizable(dra);
-        }
+      return dpa->acc().is_f() || dpa->acc().is_generalized_buchi();
     }
   }
 
-  [[noreturn]] static void invalid_spot_pr_check()
-  {
-    throw std::runtime_error("invalid value for SPOT_PR_CHECK "
-                             "(should be 1 or 2)");
-  }
-
-  static prcheck
-  algo_to_perform(bool is_persistence, bool aut_given)
-  {
-    static prcheck env_algo = [&]()
-      {
-        int val;
-        try
-          {
-            auto s = getenv("SPOT_PR_CHECK");
-            val = s ? std::stoi(s) : 0;
-          }
-        catch (const std::exception& e)
-          {
-            invalid_spot_pr_check();
-          }
-        if (val == 0)
-          {
-            if (aut_given && !is_persistence)
-              return prcheck::via_Rabin;
-            else if (is_persistence || !aut_given)
-              return prcheck::via_CoBuchi;
-            else
-              SPOT_UNREACHABLE();
-          }
-        else if (val == 1)
-          {
-            return prcheck::via_CoBuchi;
-          }
-        else if (val == 2)
-          {
-            return prcheck::via_Rabin;
-          }
-        else
-          {
-            invalid_spot_pr_check();
-          }
-      }();
-    return env_algo;
-  }
-
   bool
   is_persistence(formula f, twa_graph_ptr aut, prcheck algo)
   {
@@ -161,6 +173,7 @@ namespace spot
                                   ltl_to_tgba_fm(f, make_bdd_dict(), true));
 
       case prcheck::via_Rabin:
+      case prcheck::via_Parity:
         return detbuchi_realizable(ltl_to_tgba_fm(formula::Not(f),
                                                   make_bdd_dict(), true));
 
@@ -195,6 +208,7 @@ namespace spot
                                                  make_bdd_dict(), true));
 
       case prcheck::via_Rabin:
+      case prcheck::via_Parity:
         return detbuchi_realizable(aut ? aut :
                                    ltl_to_tgba_fm(f, make_bdd_dict(), true));
 
diff --git a/spot/tl/hierarchy.hh b/spot/tl/hierarchy.hh
index 49b66aa6a..347d175f8 100644
--- a/spot/tl/hierarchy.hh
+++ b/spot/tl/hierarchy.hh
@@ -1,5 +1,5 @@
 // -*- coding: utf-8 -*-
-// Copyright (C) 2017, 2018 Laboratoire de Recherche et Développement
+// Copyright (C) 2017, 2018, 2019 Laboratoire de Recherche et Développement
 // de l'Epita (LRDE)
 //
 // This file is part of Spot, a model checking library.
@@ -37,6 +37,9 @@ namespace spot
   ///
   /// If PR_via_CoBuchi, they will check if the formula is cobuchi_realizable.
   ///
+  /// If PR_via_Parity, they will check if the formula is
+  /// detbuchi/cobuchi-realizable by calling reduce_parity() on a DPA.
+  ///
   /// Note that is_persistence() and is_recurrence() will work on a formula f
   /// or its negation because of the duality of both classes
   /// (see https://spot.lrde.epita.fr/hierarchy.html for details).
@@ -47,6 +50,7 @@ namespace spot
     Auto,
     via_CoBuchi,
     via_Rabin,
+    via_Parity,
   };
 
   /// \ingroup tl_hier
diff --git a/tests/core/hierarchy.test b/tests/core/hierarchy.test
index a7794c5c9..0c962d211 100755
--- a/tests/core/hierarchy.test
+++ b/tests/core/hierarchy.test
@@ -1,7 +1,7 @@
 #!/bin/sh
 # -*- coding: utf-8 -*-
-# Copyright (C) 2017 Laboratoire de Recherche et Développement de
-# l'Epita (LRDE).
+# Copyright (C) 2017, 2019 Laboratoire de Recherche et Développement
+# de l'Epita (LRDE).
 #
 # This file is part of Spot, a model checking library.
 #
@@ -70,31 +70,33 @@ ltlfilt -q --safety --guarantee -f '(Gb R (b xor Gb)) W (a W Xa)'
 # make sure SPOT_PR_CHECK is read.
 randltl -n -1 a b c | ltlfilt -v --syntactic-safety --syntactic-guarantee \
   --syntactic-obligation --syntactic-recurrence --syntactic-persistence \
-  -n 100 > res
-cat res | SPOT_PR_CHECK=x ltlfilt --recurrence 2>err && exit 1
-cat res | SPOT_PR_CHECK=9 ltlfilt --recurrence 2>err && exit 1
+  -n 500 > res
+SPOT_PR_CHECK=x ltlfilt res --recurrence 2>err && exit 1
+SPOT_PR_CHECK=9 ltlfilt res --recurrence 2>err && exit 1
 grep SPOT_PR_CHECK err
 
 # Testing Recurrence.
-cat res | ltlfilt --recurrence > r0
-for i in 1 2; do
-  cat res | SPOT_PR_CHECK=$i ltlfilt --recurrence > r$i
+ltlfilt res --recurrence > r0
+for i in 1 2 3; do
+    SPOT_PR_CHECK=$i ltlfilt res --recurrence > r$i
 done
 diff r0 r2
 diff r1 r2
-cat res | ltlfilt -o %h.ltl
+diff r2 r3
+ltlfilt res -o %h.ltl
 cat R.ltl O.ltl G.ltl S.ltl B.ltl | sort > rogsb.ltl
 sort r2 > r3
 diff r3 rogsb.ltl
 
 # Testing Persistence.
-cat res | ltlfilt --persistence > p0
-for i in 1 2; do
-  cat res | SPOT_PR_CHECK=$i ltlfilt --persistence > p$i
+ltlfilt res --persistence > p0
+for i in 1 2 3; do
+  SPOT_PR_CHECK=$i ltlfilt res --persistence > p$i
 done
 diff p0 p2
 diff p1 p2
-cat res | ltlfilt -o %h.ltl
+diff p2 p3
+ltlfilt res -o %h.ltl
 cat P.ltl O.ltl G.ltl S.ltl B.ltl | sort > pogsb.ltl
 sort p2 > p3
 diff p3 pogsb.ltl