CAESAR.BDD manual page

Table of Contents## Name

caesar.bdd - structural and behavioural analysis of Nested-Unit Petri
Nets
## Synopsis

**caesar.bdd** *option* [ *filename***.nupn** ]
## Description

Taking as input
*filename***.nupn** (or by default the standard input), which contains a Nested-Unit
Petri Net (NUPN) encoded in the **nupn**
format, **caesar.bdd** performs
various structural or behavioural analyses, depending on the *option* specified
on the command line, and writes the corresponding results to the standard
output. Error and warning messages, if any, are displayed on the standard
error. ## Options

## Interrupts

**caesar.bdd** handles certain POSIX interrupt signals (namely, SIGINT,
SIGQUIT, SIGALRM, and SIGTERM) and adapt its behaviour in consequence.
Upon reception of such a signal:
## User-specified Timeouts

The
environment variable *N*, the exploration of the marking graph will stop after
*N* seconds. If this variable is set to zero, the exploration of the marking
graph is disabled, meaning only the initial marking is visited. If this
variable is not set (or set to an invalid value, such as "" or -1), the
exploration of the marking graph is performed without timeout. Upon expiration
of a timeout:
## User-specified
Iteration Bound

The environment variable *N*, the exploration
of the marking graph will stop after the *N*-th iteration has been performed.
If this variable is set to zero, the exploration of the marking graph is
disabled, meaning only the initial marking is visited. If this variable
is not set (or set to an invalid value, such as "" or -1), the exploration
of the marking graph is performed without upper bound on the number of
iterations.
## Compression Algorithm

The output of the four options **-concurrent-places**,
**-concurrent-units**, **-dead-places**, and **-dead-transitions** is compressed using a
simple algorithm based on run-length encoding: when a character is followed
by an integer *n* greater than 3 enclosed between parentheses, it means that
this character must be repeated *n* times, including its initial occurrence.
For instance, the following sequence of characters: ## Decompression Algorithm

The compressed output of the
four options **-concurrent-places**, **-concurrent-units**, **-dead-places**, and **-dead-transitions**
is decompressed using a simple algorithm, implemented as follows in the
C language: ## Exit Status

The exit status of **caesar.bdd** may take the following
values:

## Authors

Damien Bergamini
and Hubert Garavel (INRIA Rhone-Alpes)
## Files

## Credits

To perform its behavioural analyses, **caesar.bdd** uses
the Binary Decision Diagram package CUDD (Release 3.0) developed by Fabio
Somenzi at the University of Colorado (Boulder, CO, USA).
## Bibliography

[Gar19]
Hubert Garavel. "Nested-Unit Petri Nets". Journal of Logical and Algebraic
Methods in Programming, vol. 104, pages 60-85, April 2019. Available from
http://cadp.inria.fr/publications/Garavel-19.html
## See Also

**caesar**
,
**nupn**
, **nupn_info**
## Bugs

Please
report new bugs to Hubert.Garavel@inria.fr

See the **nupn**
manual page for a detailed definition of the
NUPN file format.

For performance reasons, most options of **caesar.bdd** assume
that the contents of *filename***.nupn** are correct. When dealing with an unknown
Nested-Unit Petri Net, it is thus advisable to first analyze its contents
with **-check** before using any other option. Indeed, these other options often
perform less stringent verifications than **-check**, or even no verification
at all. In particular, options **-dead-places**, **-dead-transitions**, **-concurrent-places**,
and **-concurrent-units** rely on an optimized encoding of markings, which is
only correct if markings are safe or unit-safe, and may may produce invalid
results if this is not the case.

Also, most options of **caesar.bdd** do not
take into account the various pragmas that may be present in *filename***.nupn**.
As a general rule, the pragmas "`!multiple_arcs`

", "`!multiple_initial_tokens`

",
and "`!unit_safe`

" (see the section on pragmas in the **nupn**
format
definition) are ignored, unless explicitly stated otherwise below.

**-check**- Perform syntactic, structural and behavioural analysis of the Nested-Unit
Petri Net to verify whether the stated constraints of the NUPN format are
satisfied. The
**-check**option performs additional checks and warns about redundant or void units. In addition to these static checks, the**-check**option also explores the marking graph using BDDs (Binary Decision Diagrams), prints the number of states reached at each iteration, and checks whether all reachable markings are one safe and unit safe; such checks are inconclusive if the marking graph is too large for being generated. The warning and error messages emitted by this option may differ depending on whether the "`!unit_safe`

" pragma is present or absent. **-arcs**- Display the number of arcs in the Nested-Unit Petri Net.
**-arcs-pt**- Display the number of place-transition arcs in the Nested-Unit Petri Net.
**-arcs-tp**- Display the number of transition-place arcs in the Nested-Unit Petri Net.
**-bits**- Display the number of bits needed to encode (using the best encoding) the markings of the Nested-Unit Petri Net.
**-concurrent-places**- Explore
the set of reachable markings to determine the pairs of "concurrent" places,
i.e., pairs of places Pi and Pj such that Pi = Pj or there exists a reachable
marking containing both Pi and Pj. The exploration of the set of reachable
markings may be incomplete if a timeout (see below) is specified. The output
is defined as follows: Let P be the number of places in the Nested-Unit
Petri Net;
**caesar.bdd**writes to the standard output a list of P*(P+1)/2 character values corresponding to the lower half of a matrix indexed according to increasing place numbers; this list of characters is displayed as a sequence of lines of increasing length; the upper half of the matrix is not displayed because it enjoys a symmetry property (see below).

In a first approximation, the character written at the intersection of row i and column j is either "`1`

" if places Pi and pj are concurrent, "`0`

" if these places are not concurrent, or "`.`

" if the answer is unknown.

Actually, beyond these three values, there are six more possible values: "`=`

", "`<`

", and "`>`

", which are equivalent to "0" (i.e., non-concurrent), and "`~`

", "`[`

", and "`]`

", which are equivalent to "." (i.e., unknown).

The meaning of these nine values is defined as follows. The NUPN is said to be "presumably unit safe" if either the exploration was complete and the NUPN was found to be unit safe, or the exploration was incomplete and the "`!unit_safe`

" pragma is present. Line i corresponds to place Pi (belonging to unit Ui) and row j corresponds to place Pj (belonging to unit Uj). The character written at the intersection of row i and column j (with i != j, i.e., for each non-diagonal element) is equal to:

- "`=`

": if the NUPN is presumably unit safe and both places belong to the same unit, i.e., Ui = Uj;

- "`<`

": if the NUPN is presumably unit safe and the unit of Pi is contained in the unit of Pj, i.e., Sub* (Ui, Uj), where the Sub* predicate is specified in the**nupn**format definition;

- "`>`

": if the NUPN is presumably unit safe and the unit of Pi contains the unit of Pj, i.e., Sub* (Uj, Ui);

- "`1`

": if none of the above applies, and a marking containing both Pi and Pj was reached;

- "`~`

": if the NUPN is not presumably unit safe and both places are different and belong to the same unit, i.e., (Pi != Pj) and (Ui = Uj);

- "`[`

": if the NUPN is not presumably unit safe and the unit of Pi is contained in the unit of Pj, i.e., Sub* (Ui, Uj);

- "`]`

": if the NUPN is not presumably unit safe and the unit of Pi contains the unit of Pj, i.e., Sub* (Uj, Ui);

- "`0`

": if none of the above applies, and the exploration was complete and no marking containing both Pi and Pj was reached;

- "`.`

": if none of the above applies, and the exploration was incomplete and no marking containing both Pi and Pj was reached.

Concerning diagonal elements, the character written at the intersection of row i and column i is equal to:

- "`1`

": if a marking containing Pi was reached (i.e., Pi is not a dead place);

- "`0`

": if the exploration was complete and no marking containing Pi was reached (i.e., Pi is a dead place);

- "`.`

": if the exploration was incomplete and no marking containing Pi was reached.

The lower half and upper half of the matrix are symmetric modulo permutations of "`<`

" and "`>`

", and of "`[`

" and "`]`

". Each line of the lower half of the matrix is compressed using the algorithm described below in the section entitled "COMPRESSION ALGORITHM". **-concurrent-units**- Explore the set of reachable markings
and determine the pairs of "concurrent" units, i.e., pairs of units Ui and
Uj that satisfy the predicate Disjoint (Ui, Uj) specified in the
**nupn**format definition, and such that there exists at least one reachable marking that contains a place transitively belonging to Ui and a place transitively belonging to Uj. The output is defined as follows: Let U be the number of units in the Nested-Unit Petri Net;**caesar.bdd**writes to the standard output a list of U*(U+1)/2 character values corresponding to the lower half of a matrix indexed according to increasing unit numbers; this list of characters is displayed as a sequence of lines of increasing length; the upper half of the matrix is not displayed because it is symmetric; the character written at the intersection of row i and column j is either "`1`

" if units Ui and Uj are concurrent, "`0`

" if these units are not concurrent, or "`.`

" if the answer is unknown because the exploration of reachable markings has been interrupted. Each line of the lower half of the matrix is compressed using the algorithm described below in the section entitled "COMPRESSION ALGORITHM". **-creator**- Display the name of the creator tool specified by the
`!creator`

pragma if it is present in the Nested-Unit Petri Net, or the empty string if this pragma is absent. This pragma is specified in the "PRAGMA CREATOR" section of the**nupn**format definition. **-dead-places**- Explore (parts
of) the set of reachable markings to determine the set of "dead" places,
i.e., places that do not belong to any reachable marking. The output is defined
as follows: Let P be the number of places in the Nested-Unit Petri Net;
**caesar.bdd**writes to the standard output a line of P character values corresponding to each place and ordered according to increasing place numbers; the character written for each place is either "`1`

" if the place is dead, "`0`

" if the place is not dead, or "`.`

" if the answer is unknown because the exploration of reachable markings has been interrupted. The`!unit_safe`

pragma, if present, is used to speed up calculations. The output line is compressed using the algorithm described below in the section entitled "COMPRESSION ALGORITHM". **-dead-transitions**- Explore (parts of) the set of reachable markings to determine
the set of "dead" transitions, i.e., transitions that are not enabled in
any reachable marking. The output is defined as follows: Let T be the number
of transitions in the Nested-Unit Petri Net;
**caesar.bdd**writes to the standard output a line of T character values corresponding to each transition and ordered according to increasing transition numbers; the character written for each transition is either "`1`

" if the transition is dead, "`0`

" if the transition is not dead (i.e., quasi-live), or "`.`

" if the answer is unknown because the exploration of reachable markings has been interrupted. The`!unit_safe`

pragma, if present, is used to speed up calculations. The output line is compressed using the algorithm described below in the section entitled "COMPRESSION ALGORITHM". **-density**- Display the density of the incidence matrice of the the Nested-Unit Petri Net, i.e., the number of arcs divided by twice the product of the number of places by the number of transitions. The density is equal to zero if there are no transitions (hence, no arcs).
**-encodings**- Display statistics about the number of bits required to represent the markings of the Nested-Unit Petri Net using various possible encodings, namely those encodings described in Section 6 of [Gar19].
**-height**- Display the height of the unit tree of the Nested-Unit Petri Net. All leaf units have height one, and the root unit only increases the height if this unit is not void (i.e., has at least one local place).
**-hwb**- Display the HWB code
of the Nested-Unit Petri Net. This code has the form
*height*-*width*-*bits*, where the three fields are those numbers computed by options**-height**,**-width**, and**-bits**, respectively. Fields*height*and*width*are omitted if*width*is equal to the number of places, i.e., if the NUPN is trivial. **-idle-units**- Display the list of idle units (i.e., non-void units that have no local place in the initial marking, and such that all transitions having an output place in such a unit also have an input place in this unit) or the empty string if there are no such units in the Nested-Unit Petri Net. Idle units are thus a subset of dead units (i.e., non-void units having no local place in any reachable marking).
**-initial-places**- Display the list of places present in the initial marking or the empty string if there are no such places in the Nested-Unit Petri Net.
**-initial-tokens**- Display the number of tokens present in the initial
marking. If present, the
`!multiple_initial_tokens`

pragma is taken into account. **-initial-units**- Display the list of initial units (i.e., units that contain at least one place present in the initial marking) or the empty string if there are no initial places. The same unit can be displayed multiple times if it contains several places in the initial marking, meaning that the Nested-Unit Petri Net is not unit-safe.
**-input-places**- Display the number of input places of the transition with the most input places; if there is no transition, this number is equal to zero.
**-input-transitions**- Display the number of input transitions of the place with the most input transitions; if there is no transition, this number is equal to zero.
**-leaf-units**- Display list of leaf units (i.e., units that have no sub-unit) in the Nested-Unit Petri Net.
**-max-concurrency**- Display (an over-approximation of) the upper bound for
the maximal number of tokens in any reachable marking. This option is expected
to quickly compute a result, even if the Nested-Unit Petri Net is large;
consequently, no exhaustive exploration of reachable markings is done. This
option assumes that the Nested-Unit Petri Net is safe, otherwise the result
displayed is undefined. The
`!unit_safe`

pragma, if present, is used to speed up calculations. **-max-place**- Display the highest place number in the Nested-Unit Petri Net.
**-max-transition**- Display the highest transition number in the Nested-Unit Petri Net; if there is no transition, this number is equal to zero.
**-max-unit**- Display the highest unit number in the Nested-Unit Petri Net.
**-mcc**- Undocumented
option used to prepare or complete the model forms of the Model Checking
Contest. This option computes various structural and behavioural properties
of the Nested-Unit Petri Net. The
`!unit_safe`

pragma, if present, is taken into account. If the`!multiple_arcs`

pragma or the`!multiple_initial_tokens`

pragma is present, the**-mcc**option computes structural and behavioural properties for the original (non safe) Petri Net rather than for the NUPN itself. **-min-concurrency**- Display (an under-approximation of) the lower bound for the maximal number
of tokens in any reachable marking. This option is expected to quickly compute
a result, even if the Nested-Unit Petri Net is large; consequently, no exhaustive
exploration of reachable markings is done. This option assumes that the
Nested-Unit Petri Net is safe, otherwise the result displayed is undefined.
The
`!unit_safe`

pragma, if present, is used to speed up calculations. **-min-place**- Display the lowest place number in the Nested-Unit Petri Net.
**-min-transition**- Display the lowest transition number in the Nested-Unit Petri Net; if there is no transition, this number is equal to one.
**-min-unit**- Display the lowest unit number in the Nested-Unit Petri Net.
**-multiple-arcs**- Display the arguments
of the
`!multiple_arcs`

pragma if it is present, or the empty string if this pragma is absent. **-multiple-initial-tokens**- Display the arguments of the
`!multiple_initial_tokens`

pragma if it is present, or the empty string if this pragma is absent. **-output-places**- Display the number of output places of the transition with the most output places; if there is no transition, this number is equal to zero.
**-output-transitions**- Display the number of output transitions of the place with the most output transitions; if there is no transition, this number is equal to zero.
**-permanent-units**- Display the list of permanent units (i.e., units that have a local place in the initial marking, and such that all transitions having an input place in such a unit also have an output place in this unit) or the empty string if there are no such units in the Nested-Unit Petri Net. Permanent units are thus a subset of initial units.
**-place-order**- Display the list of place
numbers increasingly sorted according to the lexicographic order on a
76-tuple of values computed for each place; the numbers of the places equal
for this order are displayed on the same line. This option is invoked by
**nupn_info**with its**-canonical-nupn**or**-place-sort**options. **-places**- Display the number of places in the Nested-Unit Petri Net.
**-pnml**- Write to the standard
output the translation in PNML (Petri Net Markup Language) the network
given as input in the NUPN format. The PNML output will contain a "NUPN-toolspecific"
section as defined on the http://mcc.lip6.fr/nupn.php
page. If the "
`!unit_safe`

" pragma is present, it is propagated to this section. If the input network contains errors, the translation may stop prematurely, leaving an incomplete PNML file. It is therefore recommended to invoke the tool as follows:

**caesar.bdd -pnml**[*filename***.nupn**] >*filename***.pnml**|| rm -f*filename***.pnml** **-redundant-units**- Display the list of redundant units (i.e., units that have a single sub-unit) or the empty string if there are no such units in the Nested-Unit Petri Net.
**-root-unit**- Display the number of the root unit of the Nested-Unit Petri Net.
**-signature**- Display a checksum computed on the Nested-Unit Petri Net. This checksum is invariant by any permutation of places, transitions, and/or units. It does not include information about multiple arcs and/or multiple initial tokens, if any.
**-signature-multiple**- Display a checksum computed on the Nested-Unit Petri Net. This checksum is invariant by any permutation of places, transitions, and/or units. It includes information about multiple arcs and/or multiple initial tokens.
**-transition-order**- Display the list of
transition numbers increasingly sorted according to the lexicographic order
on a 11-tuple of values computed for each transition; the numbers of the
transitions equal for this order are displayed on the same line. This option
is invoked by
**nupn_info**with its**-canonical-nupn**or**-transition-sort**options. **-transitions**- Display the number of transitions in the Nested-Unit Petri Net.
**-trivial**- Display "1" if the Nested-Unit Petri Net is trivial, or "0" otherwise.
**-unit-order**- Display the list of unit numbers increasingly sorted
according to the lexicographic order on a 96-tuple of values computed on
each unit; the numbers of the units equal for this order are displayed
on the same line. This option is invoked by
**nupn_info**with its**-canonical-nupn**or**-unit-sort**options. **-units**- Display the number of units in the Nested-Unit Petri Net.
**-version**- Display the current version number of the software.
**-void-nonroot-units**- Display the list of void units (i.e., units having no local place) that are different from the root unit or the empty string if there are no such units in the Nested-Unit Petri Net. If the net is correct, the empty string should be displayed.
**-void-root-unit**- Display the number of the root unit if this unit is void (i.e., has no local place) or the empty string otherwise.
**-void-units**- Display the list of void units (i.e., units having no local place, possibly including the root unit) or the empty string if there are no such units in the Nested-Unit Petri Net.
**-width**- Display the width of the unit tree of the Nested-Unit Petri Net. This width is equal to the number of leaf units.

- The execution of options
**-check**,**-concurrent-places**,**-concurrent-units**,**-dead-places**,**-dead-transitions**, and**-mcc**stops the BDD-based exploration of reachable markings, and completes using pessimistic assumptions (i.e., the fact that parts of the marking graph have not been explored) delivering less precise, yet correct results. - The execution of other options terminates immediately, returning the exit code 5 (see below).

`$CAESAR_BDD_TIMEOUT`

can be used to specify a maximal
duration for marking graph exploration. If this variable is set to a strictly
positive integer - The execution of option
**-check**terminates immediately, returning the exit code 5 (see below). - The execution of options
**-concurrent-places**,**-concurrent-units**,**-dead-places**, and**-dead-transitions**stops the BDD-based exploration of reachable markings, and completes using pessimistic assumptions. - The
execution of option
**-mcc**completes by computing behavioural properties using faster algorithms that rely on pessimistic assumptions.

The functionality provided by this environment variable is not available on Windows.

`$CAESAR_BDD_ITERATIONS`

can be used
to specify a maximal number of iterations for marking graph exploration.
If this variable is set to a strictly positive integer 01110000101100000011.0000000110011111011111110.10........001111is compressed as:

01110(4)10110(6)11.0(7)11001(5)01(7)0.10.(8)001(4)

The compression algorithm is implemented as follows in the C language:

#include <stdio.h> int main () { char C, PREVIOUS = '\0'; int N, REPEAT = 0; while (1) { C = getchar (); if (C == PREVIOUS) { /* assert (C != '\0') && (C != EOF) && (C != '\n') */ ++ REPEAT; } else { /* flush the repetition buffer, if any */ if (REPEAT > 3) { printf ("(%d)", REPEAT); } else if (REPEAT > 0) { for (N = 1; N < REPEAT; ++ N) putchar (PREVIOUS); } if (C == EOF) return 0; putchar (C); if (C == '0) { PREVIOUS = '\0'; REPEAT = 0; } else { PREVIOUS = C; REPEAT = 1; } } } }

This algorithm enjoys three nice properties: (1) it can operate on the fly (e.g., using coroutines, pipes, or data streams), meaning that it is not mandatory to generate the input entirely before starting to compress it; (2) the size (in characters) of the compressed output is always less or equal to the size of the input; (3) compressing an already compressed input has no effect.

#include <stdio.h> int main () { char C, PREVIOUS = '\0'; int REPEAT = 0; while (1) { if (REPEAT > 0) { /* assert (PREVIOUS != '\0') && (PREVIOUS != '\n') */ putchar (PREVIOUS); -- REPEAT; } else { C = getchar (); if (C == EOF) return 0; if (C != '(') { putchar (C); PREVIOUS = C; } else { scanf ("%d)", &REPEAT); /* assert REPEAT > 3 */ -- REPEAT; } } } }

This algorithm can operate on the fly; for instance, one can compare two (or more) compressed files without having to decompress them entirely in advance.

- 0:
- normal termination (everything is alright)

- 1:
- memory shortage (Petri Net and/or BDDs are too large)

- 2:
- incorrect command-line arguments for
**caesar.bdd**

- 3:
*filename***.nupn**does not exist or is unreadable

- 4:
- syntax or semantic error in
*filename***.nupn**

- 5:
**caesar.bdd**terminated by the user (SIGINT or SIGQUIT) or upon timeout (SIGALRM or SIGTERM)

- 6:
- the net given in
*filename***.nupn**is not safe or not unit safe

- 7:
- system error, e.g., a pipe command that does not open or close properly

Any other value corresponds to an unexpected error.

*filename***.nupn**- Nested-Unit Petri Net (input)

[Gar15] Hubert Garavel. "Nested-Unit Petri Nets: A Structural Means to Increase Efficiency and Scalability of Verification on Elementary Nets". In R. Devillers and A. Valmari, editors, Proceedings of the 36th International Conference on Application and Theory of Petri Nets and Concurrency (PETRI NETS'15), Brussels, Belgium. Lecture Notes in Computer Science, vol. 9115, Springer, 2015. Superseded by [Gar19]. Available from http://cadp.inria.fr/publications/Garavel-15-a.html

Additional information is available from the CADP Web page located at http://cadp.inria.fr

Directives for installation
are given in files **$CADP/INSTALLATION_*.**

Recent changes and improvements
to this software are reported and commented in file **$CADP/HISTORY.**