* doc/org/ltlmix.org: Fix several typos.

doc/org/ltlmix.org

@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-#+TITLE: =ltlgrind=
+#+TITLE: =ltlmix=
 #+DESCRIPTION: Spot command-line tool for combining LTL formulas randomly
 #+INCLUDE: setup.org
 #+HTML_LINK_UP: tools.html
@@ -14,7 +14,7 @@ some handwritten, meaningful formulas, and combine those formulas to
 build larger sets that are possibly more challenging.
 
 Here is a very simple example that builds five formulas that are
-Boolean combination of formulas from taken in the set
+Boolean combination of formulas taken from the set
 $\{\mathsf{GF}a,\mathsf{FG}b,\mathsf{X}c\}$:
 
 #+BEGIN_SRC sh :exports both
@@ -34,15 +34,15 @@ ltlmix -f GFa -f FGb -f Xc -n 5
 
 For each formula that it generates, =ltlmix= constructs a random
 syntax-tree of a certain size (5 by default) in which internal nodes
-represent operators selected randomly from a list of operator, and
+represent operators selected randomly from a list of operators.
-leaves are subformulas selected randomly from the set of input
+Leaves of that tree are subformulas selected randomly from the set of
-formulas.  As an example, the syntax tree of =!φ₁ xor !φ₂= has size 5,
+input formulas.  As an example, the syntax tree of =!φ₁ xor !φ₂= has
-and its leaves =φ₁= and =φ₂= will be taken randomly from the set of
+size 5, and its leaves =φ₁= and =φ₂= will be taken randomly from the
-input formulas.
+set of input formulas.
 
-The algorithm is actually the same as for =randltl=, except
+The algorithm is actually the same as for =randltl=, except that
-that =randltl= use random atomic propositions as leaves when =ltlmix=
+=randltl= uses random atomic propositions as leaves when =ltlmix= uses
-uses random formulas.
+random formulas.
 
 The same input formula can be picked several times to be used on
 multiple leaves of the tree.  Note that because Spot implements some
@@ -104,10 +104,10 @@ and	1
 or	1
 #+end_example
 
-In the above list, =false= and =true= represent the Boolean constants
+In the above list, =false= and =true= represent the Boolean constants:
-(which are usually undesirable when building random Boolean formulas),
+those are usually undesirable when building random Boolean formulas,
-and =sub= represent a random formula drawn from the list of input
+especially with Spot's trivial rewritings.  =sub= represents a random
-formulas.
+formula drawn from the list of input formulas.
 
 The above command shows that each operator has a weight, called
 /priority/.  When the priority is 0, the operator is never used.  When
@@ -146,7 +146,7 @@ Ge <-> (!Fc <-> !Xa)
 ** Boolean or LTL syntax tree
 
 By default, the syntax tree generated on top of the randomly selected
-input formula uses only Boolean operators.
+input formulas uses only Boolean operators.
 
 Using option =-L= will use LTL operators instead.
 
@@ -195,21 +195,24 @@ and	1
 or	1
 #+end_example
 
-Note that in the LTL case, =false= and =true= can be generated by default.
+Note that in the LTL case, =false= and =true= can be generated by
+default: when building leave, =alse= and =true= have the same
+probability to be selected as any input formula.
+example).  He
 
 * Randomizing atomic propositions with =-A= or =-P=
 
 Options =-A= or =-P= can be used to change the atomic propositions
-used in the input formulas.  This works as follows: if =-A N= was
+used in the input formulas.  This works as follows: if option =-A N=
-given, every time an input formula φ is selected, its atomic
+was given, every time an input formula φ is selected, its atomic
 propositions are replaced by atomic propositions randomly selected in
 a set of size $N$.  If φ uses $i$ atomic propositions and $i\ge N$,
 then those $i$ atomic proposition will be remapped to $i$ distinct
-atomic propositions chosen randomly in that set.  if $i>N$, some of
+atomic propositions chosen randomly in that set.  If $i>N$, some of
 the new atomic propositions may replace several of the original atomic
 propositions.
 
-Option =-P N= is similar to =-A N= except that the selected atomic
+Option =-P N= is similar to =-A N=, except that the selected atomic
 propositions can possibly be negated.
 
 
@@ -218,17 +221,18 @@ These options solve two problems:
 - They lessen the issue that a formula selected several times can lead
   to syntax tree such as =φ | φ | φ= that reduces to =φ=.  Now, each
   occurrence of =φ= as a chance to use different atomic propositions.
-  (The larger =N= is, the more likely it is that these copies of φ
+  The larger =N= is, the more likely it is that these copies of φ
-  will be different).
+  will be different.
 
 - They allow combining formulas that had completely different sets of
-  atomic propositions, in such a way that they are now interdependent
+  atomic propositions, in such a way that they are now interdependent.
-  (the smaller N is the more likely it is that subformulas will share
+  The smaller N is the more likely it is that subformulas will share
-  atomic propositions).
+  atomic propositions.
 
 
-Here is an example with a single formula, =GFa=, whose atomic proposition
+Here is that same example with a single formula, =GFa=, whose atomic
-will be randomly replaced by one of $\{p_0,p_1,p_2,p_3,p_4\}$.
+proposition will be randomly replaced by one of
+$\{p_0,p_1,p_2,p_3,p_4\}$.
 
 #+BEGIN_SRC sh :exports both
   ltlmix -fGFa -A5 --tree-size=8 -n10
@@ -274,8 +278,8 @@ GF!p1 xor (!GF!p2 | (GF!p1 <-> GFp0))
 ** Mixing the DAC patterns
 
 The command [[file:genltl.org][=genltl --dac-pattern=]] will print a list of 55 LTL
-formulas representing various specification patterns listed by Dwyer
+formulas representing various specification patterns listed by [[https://doi.org/10.1145/302405.30267][Dwyer
-et al. (FMSP'98).  Using =--stat=%x= to count the atomic propositions
+et al. (FMSP'98)]].  Using =--stat=%x= to count the atomic propositions
 in each formula, and some standard unix tools, we can compute that
 they use at most 6 atomic propositions.
 
@@ -324,9 +328,9 @@ so is uses atomic propositions $\{p_0,p_1,...\}$ starting at 0 and without gap.
 
 ** Random conjunctions
 
-Some benchmarks (e.g., [[https://www.cs.rice.edu/~vardi/papers/time13.pdf][for LTL satisfiability]]) are built by
+Some benchmarks (e.g., [[https://www.cs.rice.edu/~vardi/papers/time13.pdf][for LTL satisfiability]]) are built as
-conjunction of $L$ random formulas picked from a set of basic
+conjunctions of $L$ random formulas picked from a set of basic
-formulas.  Each picked formula has its atomic proposition mapped to
+formulas.  Each picked formula has its atomic propositions mapped to
 random literals built from a subset of $m$ atomic variables.
 
 Given a value for $m$, option =-P m= will achieve the second part of
@@ -335,7 +339,7 @@ need to ask for a tree of size $2L-1$ in which only the =and= operator
 is allowed.
 
 Here is an example with $L=10$ (hence =--tree-size=19=) and $m=50$.
-The example use a small set of three basic formulas
+The example uses a small set of three basic formulas
 $\{\mathsf{G}a,\mathsf{F}a,\mathsf{X}a\}$ for illustration, but in
 practice you should replace these =-f= options by =-F FILENAME=
 pointing to a file containing all the input formulas to select from.
@@ -359,9 +363,9 @@ Xp27 & Xp5 & Fp28 & Xp18 & G!p13 & Gp35 & Gp38 & G!p45 & G!p48 & Gp12
 Xp7 & G!p48 & Xp14 & Fp24 & Xp43 & Fp47 & Fp14 & Gp30 & Xp23 & G!p31
 #+end_example
 
-In fact building random conjunctions is common enough to have its own
+Random conjunctions is common enough to have its own flag.  Using =-C
-flag.  Using =-C N= will see the tree size to $2N-1$ and disable all
+N= will see the tree size to $2N-1$ and disable all operators but
-operators but =and=.  The above command can therefore be reduced to
+=and=.  The above command can therefore be reduced to
 
 #+BEGIN_SRC sh :exports both
   ltlmix -fGa -fFa -fXa -n10 -P50 -C10
@@ -381,7 +385,6 @@ Xp27 & Xp5 & Fp28 & Xp18 & G!p13 & Gp35 & Gp38 & G!p45 & G!p48 & Gp12
 Xp7 & G!p48 & Xp14 & Fp24 & Xp43 & Fp47 & Fp14 & Gp30 & Xp23 & G!p31
 #+end_example
 
-
 Selecting 10 random conjuncts out of 3×50×2=300 possibilities has a
 13.7% chance that at least 2 conjuncts will be identical (see [[https://en.wikipedia.org/wiki/Birthday_problem][Birthday
 paradox]]), so because of Spot's trivial rewritings, some of the above
@@ -398,10 +401,10 @@ a setup similar to the above, except that when atomic proposition are
 randomized, we must make sure not to change their input or output
 nature.
 
-They use small examples from the [[http://www.ist.tugraz.at/staff/jobstmann/lily/][Lily]] distribution has basic formulas
+They use small examples from the [[http://www.ist.tugraz.at/staff/jobstmann/lily/][Lily]] distribution as basic formulas
-to combine.  Spot can print those using =genltl --lily=.  There are 23
+to combine.  Spot can print those formulas using =genltl --lily=.
-of them, we will limit ourselves to four of them for illustrative
+There are 23 of them, but we will limit ourselves to four of them for
-purpose.
+illustrative purpose.
 
 #+BEGIN_SRC sh :exports both
   genltl --lily=8..11
@@ -413,12 +416,13 @@ purpose.
 : (GFi1 | Fi0) -> (GFo1 | Go0)
 : !(G(i1 -> Fo0) & G(i0 -> Fo1))
 
-Notice that atomic proposition either start with =i= (for input) or
+Notice how these atomic propositions either start with =i= (for input)
-=o= for output.  This allows Spot to infer their nature.  For instance,
+or =o= for output.  This allows Spot to infer their nature.  For
-we could feed those directly to [[file:ltlsynt.org][=ltlsynt=]]:
+instance, we could feed those directly to [[file:ltlsynt.org][=ltlsynt=]] to decide if they
+are realizable:
 
-#+BEGIN_SRC sh :exports both :prologue true
+#+BEGIN_SRC sh :exports both :epilogue true
-  genltl --lily=8..11 | ltlsynt -q
+  genltl --lily=8..11 | ltlsynt --realizability
 #+END_SRC
 
 #+RESULTS:
@@ -427,7 +431,6 @@ we could feed those directly to [[file:ltlsynt.org][=ltlsynt=]]:
 : REALIZABLE
 : UNREALIZABLE
 
-
 When randomizing the atomic propositions in these formulas before
 combining them, we want to replace each input (resp. output)
 proposition by a random input (resp. output) proposition.  This is
@@ -453,8 +456,8 @@ set of 5 (resp. 4).
 : !(G(i3 -> Fo1) & G(i1 -> Fo3)) & !(G(i3 -> Fo0) & G(i0 -> Fo2)) & ((GFi0 | Fi3) -> (GFo0 | Go1))
 : ((Fi1 | GFi4) -> (Go0 | GFo2)) & !(G(i0 -> Fo2) & G(i4 -> Fo1)) & !(G(i3 -> Fo2) & G(i1 -> Fo1))
 
-#+BEGIN_SRC sh :exports both :prologue true
+#+BEGIN_SRC sh :exports both :epilogue true
-  genltl --lily=8..11 | ltlmix -A5,4 -C3 -n6 | ltlsynt -q
+  genltl --lily=8..11 | ltlmix -A5,4 -C3 -n6 | ltlsynt --realizability
 #+END_SRC
 
 #+RESULTS:
@@ -468,11 +471,11 @@ set of 5 (resp. 4).
 Note that because one of the original pattern is unrealizable, any
 conjunction involving it will be unrealizable.  Even if we had only
 realizable specifications to combine, the smaller the atomic
-propositions set are, the more likely the random conjuncts will be in
+proposition sets are, the more likely the random conjuncts will be in
 conflict.  Therefore, increasing the number of atomic propositions to
 chose from may help to get more realizable formulas.
 
-#+BEGIN_SRC sh :exports both :prologue true
+#+BEGIN_SRC sh :exports both :epilogue true
   genltl --lily=8..11 | ltlmix -A50,50 -C3 -n6 | ltlsynt -q
 #+END_SRC