## Saturation: an efficient iteration strategy for symbolic state space generation (2001)

### Cached

### Download Links

Venue: | PROC. TOOLS AND ALGORITHMS FOR THE CONSTRUCTION AND ANALYSIS OF SYSTEMS (TACAS), LNCS 2031 |

Citations: | 56 - 30 self |

### BibTeX

@INPROCEEDINGS{Ciardo01saturation:an,

author = {Gianfranco Ciardo and Gerald Lüttgen and Radu Siminiceanu},

title = {Saturation: an efficient iteration strategy for symbolic state space generation},

booktitle = {PROC. TOOLS AND ALGORITHMS FOR THE CONSTRUCTION AND ANALYSIS OF SYSTEMS (TACAS), LNCS 2031},

year = {2001},

pages = {328--342},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a novel algorithm for generating state spaces of asynchronous systems using Multi–valued Decision Diagrams. In contrast to related work, we encode the next–state function of a system not as a single Boolean function, but as cross–products of integer functions. This permits the application of various iteration strategies to build a system’s state space. In particular, we introduce a new elegant strategy, called saturation, and implement it in the tool SMART. On top of usually performing several orders of magnitude faster than existing BDD–based state–space generators, our algorithm’s required peak memory is often close to the final memory needed for storing the overall state space.

### Citations

2921 | Graph-based algorithms for Boolean function manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...ring huge state spaces in the relatively small memory of a workstation. One research direction widely pursued in the literature suggests the use of decision diagrams, usually Binary Decision Diagrams =-=[7]-=- (BDDs), as a data structure for implicitly representing large sets of states in a compact fashion. This proved to be very successful for the verification of synchronous digital circuits, as it increa... |

2402 | Model checking
- Clarke, Schlingloff
- 2001
(Show Context)
Citation Context ...he final memory needed for storing the overall state space. 1 Introduction State--space generation is one of the most fundamental challenges for many formal verification tools, such as model checkers =-=[13]-=-. The high complexity of today's digital systems requires constructing and storing huge state spaces in the relatively small memory of a workstation. One research direction widely pursued in the liter... |

1425 | Petri Nets: Properties, Analysis and Applications - Murata - 1989 |

873 | Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams
- Bryant
- 1992
(Show Context)
Citation Context ...) -- A non--terminal node hk:pi has n k arcs pointing to nodes at level k \Gamma 1. If the i th arc, for i 2 S k , is to node hk \Gamma 1:qi, we write hk:pi[i] = q. Unlike in the original BDD setting =-=[7, 8]-=-, we allow for redundant nodes, having all arcs pointing to the same node. This will be convenient for our purposes, as eliminating such nodes would lead to arcs spanning multiple levels. -- A non--te... |

703 | Symbolic model checking without BDDs
- Biere, Cimatti, et al.
- 1999
(Show Context)
Citation Context ... [2], and conjunctive decompositions [21]. Finally, also note that approaches to symbolic verification have been developed, which do not rely on decision diagrams but instead on arithmetic or algebra =-=[1, 6, 26]-=-. 7 Conclusions and Future Work We presented a novel approach for constructing the state spaces of asynchronous system models using MDDs. By avoiding to encode the global next--state function as an MD... |

573 | Symbolic model checking: 10 20 states and beyond
- Burch, Clarke, et al.
- 1990
(Show Context)
Citation Context ...n of synchronous digital circuits, as it increased the manageable sizes of state spaces from about 10 6 states, with traditional explicit state–space generation techniques [14], to about 10 20 states =-=[9]-=-. Unfortunately, symbolic techniques are known not to work well for asynchronous systems, such as communication protocols, which particularly suffer from state–space explosion. The latter problem was ... |

500 |
Symbolic Model Checking: An Approach to the State Explosion Problem
- McMillan
- 1993
(Show Context)
Citation Context ... manipulation of MDDs. This is in contrast to classic BDD--based techniques which construct state spaces by iteratively applying a single, global next--state function which is itself encoded as a BDD =-=[20]-=-. Additionally, in most concurrency frameworks including Petri nets [23] and process algebras [5], next--state functions satisfy a product form allowing each component of the state vector to be update... |

323 |
Symbolic model checking: 10 states and beyond
- Burch, Clarke, et al.
- 1990
(Show Context)
Citation Context ... of synchronous digital circuits, as it increased the manageable sizes of state spaces from about 10 6 states, with traditional explicit state--space generation techniques [14], to about 10 20 states =-=[9]-=-. Unfortunately, symbolic techniques are known not to work well for asynchronous systems, such as communication protocols, which particularly suffer from state--space explosion. The latter problem was... |

202 |
The Concurrency Workbench: A semantics-based tool for the verification of finite-state systems
- Cleaveland, Parrow, et al.
- 1993
(Show Context)
Citation Context ...cessful for the verification of synchronous digital circuits, as it increased the manageable sizes of state spaces from about 10 6 states, with traditional explicit state--space generation techniques =-=[14]-=-, to about 10 20 states [9]. Unfortunately, symbolic techniques are known not to work well for asynchronous systems, such as communication protocols, which particularly suffer from state--space explos... |

168 | NuSMV: A new symbolic model verifier
- Cimatti, Clarke, et al.
- 2003
(Show Context)
Citation Context ... the significant differences of our approach to symbolic state--space generation when compared to traditional approaches reported in the literature [20], which are usually deployed for model checking =-=[12]-=-. Hence, for a fair comparison, we should extend our algorithmic implementation to that of a full model checker first. Doing this is out of the scope of the present paper and is currently work in prog... |

84 | Petri net analysis using boolean manipulation
- Pastor, Roig, et al.
- 1994
(Show Context)
Citation Context ..., including those where queens n to N still need to be placed, for any n. In this model, locality is poor, since t ij depends on levels 1 through i. -- The dining philosophers and slotted ring models =-=[10, 25]-=- are obtained by connecting N identical safe subnets in a circular fashion. The MDD has N=2 MDD levels (two subnets per level) for the former model and N levels (one subnet per level) for the latter. ... |

82 |
Multi-valued decision diagrams: theory and applications
- Kam, Villa, et al.
- 1998
(Show Context)
Citation Context ...which particularly suffer from state--space explosion. The latter problem was addressed in previous work by the authors in the context of state--space generation using Multi--valued Decision Diagrams =-=[18]-=- (MDDs), which exploited the fact that, in event--based asynchronous systems, ? This work was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1--970... |

77 | Symbolic reachability analysis based on SAT-solvers
- Abdula, Bjesse, et al.
- 2000
(Show Context)
Citation Context ...turate(1; 2) updates node h1:2i to represent the effect of firing l 1 ; the result is equal to the reserved node h1:1i. -- Snapshots (b--f ): The call Saturate(2; 2) fires event l 2 , adding arc h2:2i=-=[1] to h1:1i -=-(cf. Snapshot (c)). It also fires event e 21 which finds the "enabling pattern" (; 0; 1), with arbitrary first component, and starts building the result of the firing, through the sequence o... |

66 | Compositional and symbolic modelchecking of real-time systems
- Larsen, Pettersson, et al.
- 1995
(Show Context)
Citation Context ...pleteness, we briefly mention some other BDD--based techniques exploiting the component--based structure of many digital systems. They include partial model checking [3], compositional model checking =-=[19]-=-, partial-- order reduction [2], and conjunctive decompositions [21]. Finally, also note that approaches to symbolic verification have been developed, which do not rely on decision diagrams but instea... |

64 | A tutorial on Stålmarck’s proof procedure for propositional logic
- Sheeran, Stålmarck
- 1999
(Show Context)
Citation Context ... [2], and conjunctive decompositions [21]. Finally, also note that approaches to symbolic verification have been developed, which do not rely on decision diagrams but instead on arithmetic or algebra =-=[1, 6, 26]-=-. 7 Conclusions and Future Work We presented a novel approach for constructing the state spaces of asynchronous system models using MDDs. By avoiding to encode the global next--state function as an MD... |

62 | Efficient reachability set generation and storage using decision diagrams
- Miner, Ciardo
- 1999
(Show Context)
Citation Context ...t independently of the others. Experimental results implementing these ideas of locality showed significant improvements in speed and memory consumption when compared to other state--space generators =-=[22]-=-. In this paper, we take our previous approach a significant step further by observing that the reachable state space of a system can be built by firing the system 's events in any order, as long as e... |

59 | Partial-order reduction in symbolic state space exploration
- Alur, Brayton, et al.
- 1997
(Show Context)
Citation Context ...me other BDD--based techniques exploiting the component--based structure of many digital systems. They include partial model checking [3], compositional model checking [19], partial-- order reduction =-=[2]-=-, and conjunctive decompositions [21]. Finally, also note that approaches to symbolic verification have been developed, which do not rely on decision diagrams but instead on arithmetic or algebra [1, ... |

41 | A performance study of bdd-based model checking
- Yang, Bryant, et al.
- 1998
(Show Context)
Citation Context ... systems, symbolic techniques using BDDs, which can represent state spaces in sublinear space, have been thoroughly investigated. Several implementations of BDDs are available; we refer the reader to =-=[27]-=- for a survey on BDD packages and their performance. To improve the time efficiency of BDD--based algorithms, breadth--first BDD--manipulation algorithms [4] have been explored and compared against th... |

39 | Efficient symbolic state-space construction for asynchronous systems
- Ciardo, Lüttgen, et al.
(Show Context)
Citation Context ...y Research Center, Hampton, VA 23681, USA. G. Ciardo and R. Siminiceanu were also partially supported by NASA grant No. NAG-1-2168. each event updates just a few components of a system's state vector =-=[10]-=-. Hence, firing an event only requires the application of local next--state functions and the local manipulation of MDDs. This is in contrast to classic BDD--based techniques which construct state spa... |

35 | Smart: Simulation and markovian analyzer for reliability and timing
- Ciardo, Miner
- 1996
(Show Context)
Citation Context ...ortant, the peak size of the MDD is usually kept close to its final size. Regarding future work, we plan to employ our idea of saturation for implementing an MDD--based CTL model checker within SMART =-=[11]-=-, to compare that model checker to state--of--the--art BDD--based model checkers, and to test our tool on examples that are extracted from real software. Acknowledgments. We would like to thank the an... |

32 |
Efficient breadth-first manipulation of Binary Decision Diagrams
- Ashar, Cheong
- 1994
(Show Context)
Citation Context ...DDs are available; we refer the reader to [27] for a survey on BDD packages and their performance. To improve the time efficiency of BDD--based algorithms, breadth--first BDD--manipulation algorithms =-=[4]-=- have been explored and compared against the traditional depth--first ones. However, the results show no significant speed-- ups, although breadth--first algorithms lead to more regular access pattern... |

30 | Compositional minimisation of finite state systems using interface specifications
- Graf, Steffen, et al.
- 1996
(Show Context)
Citation Context ...ither local or synchronize adjacent subnets, thus they span only two levels, except for those synchronizing subnet N with subnet 1, which span the entire MDD. -- The round--robin mutex protocol model =-=[17]-=- also has N identical safe subnets placed in a circular fashion, which represent N processes, each mapped to one MDD level. Another subnet models a resource shared by the N processes, giving raise to ... |

26 |
Variable ordering algorithms for ordered binary decision diagrams and their evaluation
- Fujita, Fujisawa, et al.
- 1993
(Show Context)
Citation Context ...lar access patterns of hash tables and caches. Regarding space efficiency, a fair amount of work has concentrated on choosing appropriate variable orderings and on dynamically re-- ordering variables =-=[15]-=-. For asynchronous software systems, symbolic techniques have been investigated less, and mostly only in the setting of Petri nets. For safe Petri nets, BDDbased algorithms for the generation of the r... |

18 |
A conjunctively decomposed boolean representation for symbolic model checking
- McMillan
- 1996
(Show Context)
Citation Context ...iting the component--based structure of many digital systems. They include partial model checking [3], compositional model checking [19], partial-- order reduction [2], and conjunctive decompositions =-=[21]-=-. Finally, also note that approaches to symbolic verification have been developed, which do not rely on decision diagrams but instead on arithmetic or algebra [1, 6, 26]. 7 Conclusions and Future Work... |

18 | Ste en. The Concurrency Workbench: A Semantics-Based Tool for the Veri cation of Concurrent Systems - Cleaveland, Parrow, et al. - 1993 |

14 | NuSMV: a new Symbolic Model Veri - Cimatti, Clarke, et al. - 1999 |

12 | Ecient reachability set generation and storage using decision diagrams - Miner, Ciardo - 1999 |

9 | Efficient encoding schemes for symbolic analysis of petri nets
- Pastor, Cortadella
- 1998
(Show Context)
Citation Context ...variable. These algorithms are capable of generating state spaces of large nets within hours. Recently, more efficient encodings of nets have been introduced, which take place invariants intosaccount =-=[24]-=-, although the underlying logic is still based on Boolean variables. In contrast, our work uses a more general version of decision diagrams, namely MDDs [18, 22], where more complex information is car... |

8 |
E cient encoding schemes for symbolic analysis of Petri nets
- Pastor, Cortadella
- 1998
(Show Context)
Citation Context ...variable. These algorithms are capable of generating state spaces of large nets within hours. Recently, more efficient encodings of nets have been introduced, which take place invariants into account =-=[24]-=-, although the underlying logic is still based on Boolean variables. In contrast, our work uses a more general version of decision diagrams, namely MDDs [18, 22], where more complex information is car... |

6 | Chaotic fixed point iterations
- Geser, Knoop, et al.
- 1994
(Show Context)
Citation Context ...approach a significant step further by observing that the reachable state space of a system can be built by firing the system 's events in any order, as long as every event is considered often enough =-=[16]-=-. We exploit this freedom by proposing a novel strategy which exhaustively fires all events affecting a given MDD node, thereby bringing it to its final saturated shape. Moreover, nodes are considered... |

5 | Partial Model Checking with ROBDDs
- Andersen, Staunstrup, et al.
- 1997
(Show Context)
Citation Context ...edded systems. For the sake of completeness, we briefly mention some other BDD--based techniques exploiting the component--based structure of many digital systems. They include partial model checking =-=[3]-=-, compositional model checking [19], partial-- order reduction [2], and conjunctive decompositions [21]. Finally, also note that approaches to symbolic verification have been developed, which do not r... |

5 | NUSMV: A New Symbolic Model Veri er - Cimatti, Clarke, et al. - 1999 |

4 | Chaotic point iterations - Geser, Knoop, et al. - 1994 |

4 | E cient breadth{ rst manipulation of binary decision diagrams - Ashar, Cheong - 1994 |

4 | alternatives for large structured state spaces - Storage - 1997 |

3 | Compositional minimisation of state systems using interface speci - Graf, Steen, et al. - 1996 |

3 | Exploiting symmetry in model checking - Clarke, Filkorn, et al. |

2 | Boolean manipulation with ordered binary-decision diagrams - Symbolic - 1992 |

1 | Ecient breadth{ manipulation of binary decision diagrams - Ashar, Cheong - 1994 |

1 | Chaotic xed point iterations - Geser, Knoop, et al. - 1994 |

1 | Compositional minimisation of nite state systems using interface speci cations - Graf, en, et al. - 1996 |