# -*- coding: utf-8 -*- #+TITLE: =dstar2tgba= #+SETUPFILE: setup.org #+HTML_LINK_UP: tools.html This tool converts deterministic Rabin and Streett automata, presented in [[http://www.ltl2dstar.de/docs/ltl2dstar.html][the format output by =ltl2dstar=]], into Büchi automata. It's usage is almost similar to [[file:ltl2tgba.org][=ltl2tgba=]] except that instead of supplying a formula to translate, you should specify a filename containing the deterministic Rabin or Streett automaton to convert. * Two quick examples Here are some brief examples before we discuss the behavior of =dstar2tgba= in more detail. ** From Rabin to Büchi The following command instructs =ltl2dstar= to: 1. run =ltl2tgba -sD= to build a Büchi automaton for =Fa & GFb=, and then 2. convert that Büchi automaton into a deterministic Rabin automaton (DRA) stored in =fagfb=. Additionally we use =ltlfilt= to convert our formula to the prefix format used by =ltl2dstar=. #+BEGIN_SRC sh :results verbatim :exports node ltlfilt -f 'Fa & GFb' -l | ltl2dstar --ltl2nba=spin:ltl2tgba@-sD - fagfb #+END_SRC By looking at the file =fagfb= you can see the =ltl2dsar= actually produced a 3-state DRA: #+BEGIN_SRC sh :results verbatim :exports both cat fagfb #+END_SRC #+RESULTS: #+begin_example DRA v2 explicit Comment: "Safra[NBA=3]" States: 3 Acceptance-Pairs: 1 Start: 1 AP: 2 "a" "b" --- State: 0 Acc-Sig: +0 2 2 0 0 State: 1 Acc-Sig: 1 0 1 0 State: 2 Acc-Sig: 2 2 0 0 #+end_example =dstar2tgba= can now be used to convert this DRA into a TGBA, a BA, or a Monitor, using the same options as [[file:ltl2tgba.org][=ltl2tgba=]]. For instance here is the conversion to a Büchi automaton (=-B=) in [[http://www.graphviz.org/][GraphViz]]'s format: #+BEGIN_SRC sh :results verbatim :exports code dstar2tgba -B fagfb #+END_SRC #+BEGIN_SRC sh :results verbatim :exports results SPOT_DOTEXTRA= dstar2tgba -B fagfb --dot= #+END_SRC #+RESULTS: #+begin_example digraph G { rankdir=LR I [label="", style=invis, width=0] I -> 1 0 [label="0", peripheries=2] 0 -> 0 [label="b"] 0 -> 2 [label="!b"] 1 [label="1"] 1 -> 0 [label="a"] 1 -> 1 [label="!a"] 2 [label="2"] 2 -> 0 [label="b"] 2 -> 2 [label="!b"] } #+end_example Which can be rendered as (note that in this documentation we use some [[file:oaut.org][environment variables]] to produce a more colorful output by default): #+NAME: fagfb2ba #+BEGIN_SRC sh :results verbatim :exports code dstar2tgba -B fagfb #+END_SRC #+RESULTS: fagfb2ba #+begin_example digraph G { rankdir=LR fontname="Lato" node [fontname="Lato"] edge [fontname="Lato"] node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7] I [label="", style=invis, width=0] I -> 1 0 [label=<0
⓿>] 0 -> 0 [label=] 0 -> 2 [label=] 1 [label=<1>] 1 -> 0 [label=] 1 -> 1 [label=] 2 [label=<2>] 2 -> 0 [label=] 2 -> 2 [label=] } #+end_example #+BEGIN_SRC dot :file fagfb2ba.png :cmdline -Tpng :var txt=fagfb2ba :exports results $txt #+END_SRC #+RESULTS: [[file:fagfb2ba.png]] But we could as well require the output to be output as a never claim for Spin (option =-s=): #+BEGIN_SRC sh :results verbatim :exports both dstar2tgba -s fagfb #+END_SRC #+RESULTS: #+begin_example never { T0_init: if :: ((!(a))) -> goto T0_init :: ((a)) -> goto accept_S2 fi; accept_S2: if :: ((b)) -> goto accept_S2 :: ((!(b))) -> goto T0_S3 fi; T0_S3: if :: ((b)) -> goto accept_S2 :: ((!(b))) -> goto T0_S3 fi; } #+end_example ** Streett to TGBA :PROPERTIES: :CUSTOM_ID: streett_to_tgba_example :END: Here is the translation of =GFa & GFb= to a 4-state Streett automaton: #+BEGIN_SRC sh :results verbatim :exports both ltlfilt -f 'GFa & GFb' -l | ltl2dstar --automata=streett --ltl2nba=spin:ltl2tgba@-Ds - gfagfb cat gfagfb #+END_SRC #+RESULTS: #+begin_example DSA v2 explicit Comment: "Streett{Union{Safra[NBA=2],Safra[NBA=2]}}" States: 4 Acceptance-Pairs: 2 Start: 0 AP: 2 "a" "b" --- State: 0 Acc-Sig: -0 -1 3 2 1 0 State: 1 Acc-Sig: +0 -1 3 2 1 0 State: 2 Acc-Sig: -0 +1 3 2 1 0 State: 3 Acc-Sig: +0 +1 3 2 1 0 #+end_example And now its conversion by =dstar2tgba= to a 2-state Büchi automaton. We don't pass any option to =dstar2tgba= because converting to TGBA in GraphViz's format is the default: #+NAME: gfagfb2ba #+BEGIN_SRC sh :results verbatim :exports code dstar2tgba gfagfb #+END_SRC #+RESULTS: gfagfb2ba #+begin_example digraph G { rankdir=LR fontname="Lato" node [fontname="Lato"] edge [fontname="Lato"] node[style=filled, fillcolor="#ffffa0"] edge[arrowhead=vee, arrowsize=.7] I [label="", style=invis, width=0] I -> 0 0 [label="0"] 0 -> 1 [label=<1>] 1 [label="1"] 1 -> 1 [label=] 1 -> 1 [label=⓿>] 1 -> 1 [label=❶>] 1 -> 1 [label=⓿❶>] } #+end_example #+BEGIN_SRC dot :file gfagfb2ba.png :cmdline -Tpng :var txt=gfagfb2ba :exports results $txt #+END_SRC #+RESULTS: [[file:gfagfb2ba.png]] (Obviously the resulting automaton could be simplified further, by starting with the second state right away.) * Details ** General behavior The =dstar2tgba= tool implement a 4-step process: 1. read the DRA/DSA 2. convert it into TGBA 3. postprocess the resulting TGBA (simplifying the automaton, a degeneralizing it into a BA or Monitor if requested) 4. output the resulting automaton ** Controlling output The last two steps are shared with =ltl2tgba= and use the same options. The type of automaton to produce can be selected using the =-B= or =-M= switches: #+BEGIN_SRC sh :results verbatim :exports results dstar2tgba --help | sed -n '/Output automaton type:/,/^$/p' | sed '1d;$d' #+END_SRC #+RESULTS: : -B, --ba Büchi Automaton : -M, --monitor Monitor (accepts all finite prefixes of the given : formula) : --tgba Transition-based Generalized Büchi Automaton : (default) And these may be refined by a translation intent, should the post-processor routine had a choice to make: #+BEGIN_SRC sh :results verbatim :exports results dstar2tgba --help | sed -n '/Translation intent:/,/^$/p' | sed '1d;$d' #+END_SRC #+RESULTS: : -a, --any no preference : -C, --complete output a complete automaton (combine with other : intents) : -D, --deterministic prefer deterministic automata : --small prefer small automata (default) The effort put into post-processing can be limited with the =--low= or =--medium= options: #+BEGIN_SRC sh :results verbatim :exports results dstar2tgba --help | sed -n '/Optimization level:/,/^$/p' | sed '1d;$d' #+END_SRC #+RESULTS: : --high all available optimizations (slow, default) : --low minimal optimizations (fast) : --medium moderate optimizations For instance using =-a --low= will skip any optional post-processing, should you find =dstar2tgba= too slow. Finally, the output format can be changed with the following [[file:oaut.org][common ouput options]]: #+BEGIN_SRC sh :results verbatim :exports results dstar2tgba --help | sed -n '/Output format:/,/^$/p' | sed '1d;$d' #+END_SRC #+RESULTS: #+begin_example -8, --utf8 enable UTF-8 characters in output (ignored with --lbtt or --spin) --dot[=c|h|n|N|s|t|v] GraphViz's format (default). Add letters to chose (c) circular nodes, (h) horizontal layout, (v) vertical layout, (n) with name, (N) without name, (s) with SCCs, (t) always transition-based acceptance. -H, --hoaf[=s|t|m|l] Output the automaton in HOA format. Add letters to select (s) state-based acceptance, (t) transition-based acceptance, (m) mixed acceptance, (l) single-line output --lbtt[=t] LBTT's format (add =t to force transition-based acceptance even on Büchi automata) --name=FORMAT set the name of the output automaton -s, --spin[=6|c] Spin neverclaim (implies --ba). Add letters to select (6) Spin's 6.2.4 style, (c) comments on states --stats=FORMAT output statistics about the automaton #+end_example The =--stats= options can output statistics about the input and the output automaton, so it can be useful to search for particular pattern. For instance here is a complex command that will 1. generate an infinite stream of random LTL formulas with [[file:randltl.org][=randltl=]], 2. use [[file:ltlfilt.org][=ltlfilt=]] to rewrite the W and M operators away (=--remove-wm=), simplify the formulas (=-r=), remove duplicates (=u=) as well as formulas that have a size less then 3 (=--size-min=3=), 3. use =head= to keep only 10 of such formula 4. loop to process each of these formula: - print it - then convert the formula into =ltl2dstar='s input format, process it with =ltl2dstar= (using =ltl2tgba= as the actual LTL->BA transltor), and process the result with =dstar2tgba= to build a Büchi automaton (=-B=), favoring determinism if we can (=-D=), and finally displaying some statistics about this conversion. The statistics displayed in this case are: =%S=, the number of states of the input (Rabin) automaton, =%s=, the number of states of the output (Büchi) automaton, =%d=, whether the output automaton is deterministic, and =%p= whether the automaton is complete. #+BEGIN_SRC sh :results verbatim :exports both randltl -n -1 --tree-size=10..14 a b c | ltlfilt --remove-wm -r -u --size-min=3 | head -n 10 | while read f; do echo "$f" ltlfilt -l -f "$f" | ltl2dstar --ltl2nba=spin:ltl2tgba@-Ds - - | dstar2tgba -B --stats=' DRA: %Sst.; BA: %sst.; det.? %d; complete? %p' done #+END_SRC #+RESULTS: #+begin_example (b | Fa) R Fc DRA: 9st.; BA: 6st.; det.? 1; complete? 1 Ga U (Gc R (!a | Gc)) DRA: 7st.; BA: 7st.; det.? 0; complete? 0 GFc DRA: 3st.; BA: 3st.; det.? 1; complete? 1 !a | (a R b) DRA: 3st.; BA: 2st.; det.? 1; complete? 0 Xc R (G!c R (b | G!c)) DRA: 4st.; BA: 2st.; det.? 1; complete? 0 c & G(b | F(a & c)) DRA: 5st.; BA: 3st.; det.? 1; complete? 0 XXFc DRA: 4st.; BA: 4st.; det.? 1; complete? 1 XFc | Gb DRA: 5st.; BA: 4st.; det.? 1; complete? 1 G(((!a & Gc) | (a & F!c)) U (!a | Ga)) DRA: 6st.; BA: 5st.; det.? 1; complete? 1 a & !b DRA: 3st.; BA: 2st.; det.? 1; complete? 0 #+end_example An important point you should be aware of when comparing these numbers of states is that the deterministic automata produced by =ltl2dstar= are complete, while the automata produced by =dstar2tgba= (deterministic or not) are not complete by default. This can explain a difference of one state (the so called "sink" state). You can instruct =dstar2tgba= to output a complete automaton using the =--complete= option (or =-C= for short). ** Conversion from Rabin and Streett to TGBA The algorithms used to convert Rabin and Streett into TGBA/BA are different. *** Rabin to BA The conversion implemented is a variation of Krishnan et al.'s "Deterministic ω-Automata vis-a-vis Deterministic Büchi Automata" (ISAAC'94) paper. They explain how to convert a deterministic Rabin automaton (DRA) into a deterministic Büchi automaton (DBA) when such an automaton exist. The surprising result is that when a DRA is DBA-realizable, a DBA can be obtained from the DRA without changing its transition structure. Spot implements a slight refinement to the above technique: any DRA will be converted into a BA, and the determinism will be conserved only in strongly connected components where determinism can be conserved. *** Streett to TGBA Streett automata are converted into non-deterministic TGBA. When a Streett automaton uses multiple acceptance pairs, we use generalized acceptance conditions in the TGBA to limit the combinatorial explosion. A straightforward translation from Streett to BA, as described for instance by [[http://www.automata.rwth-aachen.de/~loeding/diploma_loeding.pdf][Löding's diploma thesis]], will create a BA with $|Q|\cdot(4^n-3^n+2)$ states if the input Streett automaton has $|Q|$ states and $n$ acceptance pairs. Our translation to TGBA limits this to $|Q|\cdot(2^n+1)$ states. Sometimes, as in the [[#streett_to_tgba_example][example for =GFa & GFb=]] the output of this conversion happens to be deterministic. This is pure luck: Spot does not implement any algorithm to preserve the determinism of Streett automata.