Table of Contents
bcg_min - minimization of normal, probabilistic, or stochastic labeled
transitions systems (LTS) encoded in the BCG format
bcg_min [
bcg_options]
[
-strong |
-branching] [
-normal |
-prob |
-rate [
-self]] [
-epsilon eps] [
-format format_string]
[
-class]
input.bcg [
output.bcg]
where bcg_options is defined below (see GENERAL
OPTIONS).
bcg_min takes as input the BCG graph input.bcg, minimizes this
graph according to some bisimulation relation, and writes the resulting
reduced graph to output.bcg, replacing input.bcg if output.bcg is omitted.
bcg_min implements various algorithms to perform minimization
of graphs encoded in the BCG format according to strong or branching bisimulation
[GH02]. A graph input or output by
bcg_min can be:
- either a "normal" LTS,
whose transitions are either labelled with "normal" labels or with the
"internal" label (usually noted "tau" in the scientific literature and
displayed as the character string "
i" by the various BCG tools), - or a
"probabilistic LTS": these are LTS with "normal" labelled transitions,
as well as "special" transitions, whose labels are either of the form
"
prob %p" or "label; prob %p", where %p denotes a floating-point number in
the range ]0..1] and label denotes a character string that does not contain
the ";" character. For each state, the sum of "%p" values on special transitions
leaving the state must be less or equal than 1 (see ANNEX 1 below for a
discussion on how the probabilistic LTS model of bcg_min generalizes other
theoretical models published in the literature), - or a "stochastic LTS":
these are LTS with "normal" labelled transitions, as well as "special"
transitions, whose labels are either of the form "
rate %f" or "label; rate %f",
where %f denotes a strictly positive floating-point number and label denotes
a character string that does not contain the ";" character (see ANNEX 2
below for a discussion on how the stochastic LTS model of bcg_min generalizes
other theoretical models published in the literature).
The
following
bcg_options are currently supported:
-version,
-create,
-update,
-remove,
-cc,
-tmp,
-uncompress,
-compress,
-register,
-short,
-medium, and
-size.
See the
bcg
manual page for a description of these options.
The following options are also supported:
- -strong
- Perform LTS reduction
according to strong bisimulation. On (Discrete or Continuous Time) Markov
Chains and on Markov Reward Models, this reduction agrees with lumpability
of [KS76] (see ANNEX 1, ANNEX 2, and BIBLIOGRAPHY below). To compute strong
bisimulation on a normal LTS, bcg_min uses the algorithm of [KS90]. To
compute strong bisimulation on probabilistic and stochastic LTS bcg_min
uses the algorithm of [HS99]. Default option.
- -branching
- Perform LTS reduction
according to branching bisimulation. To compute branching bisimulation on
a normal LTS, bcg_min uses the algorithm of [GV90]. To compute branching
bisimulation on a stochastic (resp. probabilistic) LTS, bcg_min uses a
variant of the algorithm of [HS99]: the branching bisimulation condition
is applied only to the "normal" fragment of the transition relation, while
the stochastic (resp. probabilistic) fragments are mimimized with respect
to strong bisimulation. It is worth noticing that the notion of branching
bisimulation is rather meaningless for probabilistic systems. Not a default
option.
- -normal
- Consider input.bcg as a normal LTS. With this option, labels
of the form "
rate %f" or "label; rate %f" or "prob %p" or "label; prob %p" are
processed as ordinary labels, without special meaning attached. Default
option.
- -prob
- Consider input.bcg as a probabilistic LTS. With this option,
each label of the form "
prob %p" or "label; prob %p" is recognized as denoting
a probabilistic transition with probability %p. bcg_min will stop with an
error message if, for some probabilistic transition, %p is out of ]0..1],
or if the probabilistic transitions going out of the same state have a
cumulated sum strictly greater than 1. With this option, labels of the
form "rate %f" or "label; rate %f" are processed as ordinary labels. Not a
default option.
- -rate [ -self ]
- Consider input.bcg as a stochastic LTS. With
this option, each label of the form "
rate %f" or "label; rate %f" is recognized
as denoting a stochastic transition with rate %f. bcg_min will stop with
an error message if, for some stochastic transition, %f is less or equal
to 0. If the -branching option is selected, and some state has both an outgoing
stochastic transition and an outgoing internal (i.e., "tau") transition,
bcg_min will print a warning and remove the stochastic transition in order
to preserve the notion of maximal progress. With this option, labels of
the form "prob %p" or "label; prob %p" are processed as ordinary labels.
Not a default option.
If -self sub-option is given, all self loops (i.e., transitions that remain
within the same equivalence class) having labels of the form "rate %f"
are removed. This implements the weak Markovian equivalences described in
[Bra02] and [BHKW05]. Not a default sub-option.
- -epsilon eps
- Set the precision
of floating-point comparisons to eps, where eps is a real value. When eps
is out of [0..1[, bcg_min reports an error. Default value for eps is 1E-6.
- -format format_string
- Use format_string to control the format of the floating-point numbers contained
in transition labels (these numbers correspond to the occurrences of
%f
and %p mentioned in section DESCRIPTION above). The value of format_string
should obey the same conventions as the format parameter of function sprintf(3C)
for values of type double. Default value for format_string is "%g", meaning
that floating-point numbers are printed with at most six digits after the
"." (i.e., the radix character). Other values can be used, for instance "%.9g"
to obtain nine digits instead of six, or by replacing the "%g" flag with
other flags, namely "%e", "%E", "%f", "%G", possibly combined with additional
flags (e.g., to specify precision).
- -class
- Display the equivalence classes
on the standard output. Not a default option.
Note: Options -strong and -branching
are mutually exclusive. If they occur simultaneously on the command-line,
the option occurring last is selected.
Note: Options -normal, -prob, and
-rate (with or without -self sub-option) are mutually exclusive. If they occur
simultaneously on the command-line, the option occurring last is selected.
See the
bcg
manual page for a description
of the environment variables used by all the BCG application tools.
Exit status is 0 if everything is alright, 1 otherwise.
Damien
Bergamini (INRIA/VASY), Moez Cherif (INRIA/VASY), Hubert Garavel (INRIA/VASY)
and Holger Hermanns (University of Twente).
Pepijn Crouzen added -self sub-option.
The algorithm for computing branching bisimulation was derived from a Pascal
program written by Jan Friso Groote (CWI).
- input.bcg
- BCG graph (input)
- output.bcg
- BCG graph (output)
- input@1.o
- dynamic library (input or output)
- $CADP/bin.`arch`/bcg_min
- ``bcg_min'' binary program
See the bcg
manual page for a description of the other files.
bcg
,
sprintf(3C)
Additional information is available from the CADP Web page located at http://www.inrialpes.fr/vasy/cadp
Directives for installation are given in file $CADP/INSTALLATION.
Recent
changes and improvements to this software are reported and commented in
file $CADP/HISTORY.
Please report bugs to
Hubert.Garavel@inria.fr.
The probabilistic LTS model used in
bcg_min is general
enough to support the following models, which can be considered as special
cases of probabilistic LTS:
- Discrete Time Markov Chains [Nor97]
- The graph
contains transitions of the form "
prob %p" only.
- Discrete Time Markov Reward
Models [How71]
- The graph contains transitions of the form "
prob %p" to represent
transitions not obtaining an impulse reward. State rewards are associated
to states by "normal" transitions with source and target states being
identical. The label indicates the state reward such as "reward 5". Impulse
rewards are associated to probabilistic transitions by prefixing the "prob %p"
label with a label indicating the reward obtained by taking this transition,
as in "impulse 4; prob %p".
- Alternating Probabilistic LTS [Han91]
- The graph
contains transitions of the form "
prob %p", as well as normal transitions
in such a way that there is no state possessing both normal as well as
"prob %p" transitions.
- Discrete Time Markov Decision Processes [Put94]
- The
graph contains transitions of the form "
prob %p", as well as normal transitions
in such a way that there is no state possessing both normal as well as
"prob %p" transitions. Normal and probabilistic transitions strictly alternate,
i.e. normal (resp. "prob %p") transitions are not directly followed by normal
(resp. "prob %p") transitions. Uses an encoding of Discrete Time Markov Decision
Processes into strictly Alternating Probabilistic LTS proposed in [Arg00].
- Generative Probabilistic LTS [GSS95]
- The graph contains transitions of the
type "label
; prob %p" only, and for each state the sum of "%p" values leaving
a state is equal to (or smaller than) 1.
- Reactive Probabilistic LTS [GSS95]
- The graph contains transitions of the type "label
; prob %p" only, and for
each state and each "label" the sum of "%p" values leaving a state is
equal to (or smaller than) 1.
- Stratified Probabilistic LTS [GSS95]
- The graph
contains transitions of the form "
prob %p", as well as normal transitions
in such a way that there is no state possessing both normal as well as
"prob %p" transitions. Normal transitions are not directly followed by normal
transitions.
The stochastic LTS model used in
bcg_min is general enough to support the following models, which can be
considered as special cases of stochastic LTS:
- Continuous Time Markov
Chains [Nor97]
- The graph contains transitions of the form "
rate %f" only.
- Continuous Time Markov Reward Models [How71]
- The graph contains transitions
of the form "
rate %f" to represent transitions not obtaining an impulse
reward. State rewards are assigned to states by "normal" transitions with
source and target states being identical. The label indicates the state
reward such as "reward 5". Impulse rewards are assigned to probabilistic
transitions by prefixing the "rate %f" label with a label indicating the
reward obtained, as in "impulse 4; rate %f".
- Continuous Time Markov Decision
Processes [Put94]
- The graph contains transitions of the form "
rate %f", as
well as normal transitions in such a way that there is no state possessing
both normal as well as "rate %f" transitions. Normal and stochastic transitions
strictly alternate, meaning that normal (resp. "rate %f") transitions are
not directly followed by normal (resp. "rate %f") transitions. Inspired by
an encoding proposed in [Arg00].
- Interactive Markov Chains [Her98]
- The graph
contains transitions of the form "
rate %f", as well as normal transitions.
- Timed Processes for Performance (TIPP) Models [HHM98]
- The graph contains
transitions of the form "label
; rate %f", as well as normal transitions.
- Performance Evaluation Process Algebra (PEPA) Models [Hil96]
- The graph contains
transitions of the form "label
; rate %f" only. Passsive transitions are represented
by abuse of the "rate" keyword. The transition label of a passive transition
is augmented by a distinguishing string to indicate that the rate value
has to be interpreted as a weight, such as in "THIS IS A PASSIVE TRANSITION
LABELLED label WITH WEIGHT; rate %f". The actual distinguishing string used
for this purpose is of no importance for bcg_min, but it must not contain
";" and must not start with the keyword "rate".
- Extended Markovian Process
Algebra (EMPA) Models [BG98]
- The graph contains transitions of the form
"label
; rate %f" only. Passive and immediate transitions are represented by
abuse of the "rate" keyword. The transition label of a passive transition
is augmented by a distinguishing string to indicate that the rate value
has to be interpreted as a weight, such as in "THIS IS A PASSIVE TRANSITION
LABELLED label WITH PRIORITY p AND WEIGHT; rate %f". The transition label
of an immediate transition is augmented in a similar way, as in "THIS IS
AN IMMEDIATE TRANSITION LABELLED label WITH PRIORITY p AND WEIGHT; rate %f".
The actual distinguishing strings used for these purposes are of no importance
for bcg_min, but they must not contain ";" and must not start with the
keyword "rate".
[Arg00] P. R. D'Argenio (2000). On the relation
among different probabilistic transition systems and probabilistic bisimulations.
CTIT Tech Report, to appear 2000.
[BG98] M. Bernardo and R. Gorrieri (1998).
A Tutorial on EMPA: A Theory of Concurrent Processes with Nondeterminism,
Priorities, Probabilities and Time. Theoretical Computer Science 202, pp.
1-54, 1998.
[BHKW05] C. Baier, H. Hermanns, J.P. Katoen, V. Wolf. Comparative branching-time
semantics for Markov chains. Information and Computation, vol. 200(2), pp.
149-214, 2005.
[Bra02] M. Bravetti. Revisiting Interactive Markov Chains. Electronic
Notes on Theoretical Computer Science, vol. 68(5), 2002.
[GH02] H. Garavel
and H. Hermanns (2002). On Combining Functional Verification and Performance
Evaluation using CADP. Proc. 11th Int. Symp. of Formal Methods Europe FME'2002
(Copenhagen, Denmark), LNCS 2391, July 2002.
[GSS95] R. J. van Glabbeek, S.
A. Smolka, and B. Steffen (1995). Reactive, generative, and stratified models
of probabilistic processes. Information and Computation 121, pp. 59-80, 1995.
[GV90] J. F. Groote and F. Vaandrager (1990). An efficient algorithm for branching
bisimulation and stuttering equivalence. Proceedings of the 17th ICALP
(Warwick), LNCS 443, pp. 626-638, 1990.
[Han91] H. A. Hansson (1991). Time and
Probability in Formal Design of Distributed Systems. PhD thesis, University
of Uppsala, 1991.
[Her98] H. Hermanns (1998). Interactive Markov Chains. Ph.D.
Thesis, University of Erlangen-Nuernberg, Germany, 1998.
[HHM98] H. Hermanns,
U. Herzog, and V. Mertsiotakis (1998). Stochastic Process Algebras - Between
LOTOS and Markov Chains. Computer Networks and ISDN Systems 30, pp. 901-924,
1998.
[Hil96] J. Hillston (1996). A Compositional Approach to Performance
Modelling. Cambridge University Press, 1996.
[How71] R. A. Howard (1971). Dynamic
Probabilistic Systems, Vol II: Semi-Markov and Decision Processes. Wiley,
1971.
[HS99] H. Hermanns and M. Siegle (1999). Bisimulation algorithms for
stochastic process algebras and their BDD-based implementation. Proceedings
of the 5th ARTS, LNCS 1601, pp. 244-265, 1999.
[KS76] J. G. Kemeny and J. L.
Snell (1976). Finite Markov Chains. Springer, 1976.
[KS90] P. C. Kanellakis
and S. A. Smolka (1990). CCS expressions, finite state processes, and three
problems of equivalence. Information and Computation 86(1), pp. 43-68, May
1990.
[Nor97] J. R. Norris (1997). Markov Chains. Cambridge University Press,
1997.
[Put94] M. L. Puterman (1994). Markov Decision Processes: Discrete Stochastic
Dynamic Programming. Wiley, New York, NY, 1994.
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