## Saturation Unbound (2003)

Venue: | Proc. TACAS |

Citations: | 41 - 21 self |

### BibTeX

@INPROCEEDINGS{Ciardo03saturationunbound,

author = {Gianfranco Ciardo and Robert Marmorstein and Radu Siminiceanu},

title = {Saturation Unbound},

booktitle = {Proc. TACAS},

year = {2003},

pages = {379--393},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

In previous work, we proposed a "saturation" algorithm for symbolic state-space generation characterized by the use of multi-valued decision diagrams, boolean Kronecker operators, event locality, and a special iteration strategy. This approach outperforms traditional BDDbased techniques by several orders of magnitude in both space and time but, like them, assumes a priori knowledge of each submodel's state space. We introduce a new algorithm that merges explicit local statespace discovery with symbolic global state-space generation. This relieves the modeler from worrying about the behavior of submodels in isolation.

### Citations

2920 | Graph-Based Algorithms for Boolean Function Manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...odeler from worrying about the behavior of submodels in isolation. 1 Introduction Since their introduction, implicit methods for symbolic model checking, such as decision diagrams, in particular BDDs =-=[5, 6, 8]-=-, have been enormously successful. However, the systems targeted have been mainly synchronous VLSI designs and protocols, where the possible values of each state variable can be easily determined a pr... |

873 | Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams
- Bryant
- 1992
(Show Context)
Citation Context ...odeler from worrying about the behavior of submodels in isolation. 1 Introduction Since their introduction, implicit methods for symbolic model checking, such as decision diagrams, in particular BDDs =-=[5, 6, 8]-=-, have been enormously successful. However, the systems targeted have been mainly synchronous VLSI designs and protocols, where the possible values of each state variable can be easily determined a pr... |

342 | Modeling with General Stochastic Petri Nets
- Marsan
- 1995
(Show Context)
Citation Context ...ate can reach asnite number of states in a single step. A subtle problem remains, however, if an innite number of states can be reached in one step. For example, in Generalized Stochastic Petri Nets [=-=1]-=-, immediate transitions, such as v in Fig. 6, are processed not by themselves, but as events that can take place instantaneously after thesring of timed transitions, such as t and u (somewhat analogou... |

323 |
Symbolic model checking: 10 states and beyond
- Burch, Clarke, et al.
- 1990
(Show Context)
Citation Context ...odeler from worrying about the behavior of submodels in isolation. 1 Introduction Since their introduction, implicit methods for symbolic model checking, such as decision diagrams, in particular BDDs =-=[5, 6, 8]-=-, have been enormously successful. However, the systems targeted have been mainly synchronous VLSI designs and protocols, where the possible values of each state variable can be easily determined a pr... |

226 | CUDD: CU decision diagram package - release 2.4.2. http://vlsi.colorado.edu/˜fabio/CUDD/ cuddIntro.html
- Somenzi
- 2009
(Show Context)
Citation Context ...0.92 | 5 Results We compare the space and runtime required by our new algorithm with those of its pregeneration predecessor [12] and of NuSMV [14], a symbolic verier built on top of the CUDD library [=-=27]-=-. We use a 2.4 Ghz Pentium IV with 1GB of memory. Our examples include four models from [12], parametrized by an integer N : dining philosophers, slotted ring, round robin mutual exclusion, andsexible... |

151 | Symbolic model checking with partitioned transition relations
- Burch, Clarke, et al.
- 1991
(Show Context)
Citation Context ...n our saturationsalgorithm [12], which uses an MDD to store the global states and boolean Kronecker matrices to encode the transition relation. By using a disjunctivelypartitioned transition relation =-=[20]-=-, exploiting event locality [23], performing inplace updates [11] of MDD nodes, and using an innovative iteration strategy, ? Work supported in part by the National Aeronautics and Space Administratio... |

137 |
On the stochastic structure of parallelism and synchronisation models for distributed algorithms
- Plateau
- 1985
(Show Context)
Citation Context ...uires an ecient encoding of the next-state function. Unlike BDD approaches, where N , or each N e , is encoded in a 2K-level BDD, we adopt a Kronecker representation inspired by work on Markov chains =-=[2, 7, 2-=-5]. As in [11, 12, 23], we use a consistent model partition, where each N e is decomposed into K local next-state functions N e;k , for Kk1, which satisfy 8(i K ; : : : ; i 1 ) 2 b S ; N e (i K ; : : ... |

124 | Abstract regular model checking
- Bouajjani, Habermehl, et al.
(Show Context)
Citation Context ...c second-order logic, can be used to verify parametric systems without relying on a proof by induction. These types of approach can be generally classied under the umbrella of regular model checking [=-=4-=-]. Our goal in this paper is more modest, since we only target models with asnite state space, but it is also very dierent. The saturation approach we introduced in [12] has been shown to be vastly su... |

121 | MONA: Monadic second-order logic in practice
- Henriksen, Jensen, et al.
- 1995
(Show Context)
Citation Context ...ue BDDs of [18] allow one to model systems with asnite number of boolean variables plus one or more unbounded queues, as long as the contents of the queue can be represented by a DFA. The MONA system =-=[19-=-], implementing monadic second-order logic, can be used to verify parametric systems without relying on a proof by induction. These types of approach can be generally classied under the umbrella of re... |

82 |
Multi-valued decision diagrams: theory and applications
- Kam, Villa, et al.
- 1998
(Show Context)
Citation Context ...ct local behavior without aecting the global behavior, a dicult and error-prone endeavor. In this paper, we address this problem with an algorithm that produces a multi-valued decision diagram (MDD) [=-=21] repr-=-esentation of thesnal state-space and a separately stored representation of the \minimal" local state spaces. The algorithm interleaves explicit local exploration of each submodel with symbolic e... |

62 | Efficient reachability set generation and storage using decision diagrams
- Miner, Ciardo
- 1999
(Show Context)
Citation Context ..., the burden of this task has been placed on the user. In NuSMV [14], for example, the domain of multi-valued variables must be explicitly specied as a set or integer range. In our own previous work [=-=11, 12, 23], the-=- input (a Petri net) must be partitioned so that the state space of each \local subnet" can be generated in isolation. This practice requires careful addition of inhibitor arcs or other construct... |

59 |
A simple and fast algorithm to obtain all invariants of a generalized petri net
- Martinez, Silva
- 1982
(Show Context)
Citation Context ... bound our local state spaces and proceed using pregeneration, but our on-the- y approach is superior because it has a small overhead, while invariant analysis is very expensive in pathological cases =-=[22]-=-. More importantly, the invariant approach is limited: a net might not be fully covered by invariants yet be bounded because of inhibitor arcs or other constructs; invariant analysis alone might sugge... |

39 | Efficient symbolic state-space construction for asynchronous systems
- Ciardo, Lüttgen, et al.
(Show Context)
Citation Context ..., the burden of this task has been placed on the user. In NuSMV [14], for example, the domain of multi-valued variables must be explicitly specied as a set or integer range. In our own previous work [=-=11, 12, 23], the-=- input (a Petri net) must be partitioned so that the state space of each \local subnet" can be generated in isolation. This practice requires careful addition of inhibitor arcs or other construct... |

31 | Symbolic guided search for CTL model checking
- Bloem, Ravi, et al.
- 2000
(Show Context)
Citation Context ... breadth-rst approach of traditional symbolic state-space generation, with the goal of reducing the peak number of nodes, but they often still have some vestiges of breadth-rst search. For example, [3=-=, 26] im-=-prove eciency by exploring only a portion of the newly-found states, those encoded by the \densest " nodes in the decision diagram. The closest to our saturation approach is the \modied breadth-r... |

26 |
Variable ordering algorithms for ordered binary decision diagrams and their evaluation
- Fujita, Fujisawa, et al.
- 1993
(Show Context)
Citation Context ...plan to investigate reordering, splitting, and merging of MDD levels. These are more general than the reordering of BDD variables, but we hope to extend some heuristics already known for that problem =-=[16]-=-. Also, our algorithms are currently implemented in SMART [10], a simulation and modeling tool for logic and stochastic analysis. Eventually, we intend to make them available as C++ libraries. Referen... |

21 | Using edge-valued decision diagrams for symbolic generation of shortest paths
- Ciardo, Siminiceanu
- 2002
(Show Context)
Citation Context ...very dierent. The saturation approach we introduced in [12] has been shown to be vastly superior to traditional breadthrst approaches for globally-asynchronous locally-synchronous systems (see also [1=-=3]-=- for its application to edge-valued decision diagrams). However, it was limited to models for which bounds on the state variables are known a priori. Here, we extended it to bounded models with unknow... |

20 |
Petri nets with marking-dependent arc multiplicity: properties and analysis
- Ciardo
- 1994
(Show Context)
Citation Context ...ing. Indeed, invariant analysis does not take into account the presence of inhibitor arcs, and it can deal only with a limited class of marking-dependent arc cardinalities used in self-modifying nets =-=[9-=-], yet both constructs are very useful in practice to dene compact and realistic models. Our solution is fast, general, user-friendly, and terminates in at least all cases where previous algorithms te... |

17 |
Computer-oriented formulation of transition-rate matrices via Kronecker algebra
- Amoia, Demicheli, et al.
- 1981
(Show Context)
Citation Context ...uires an ecient encoding of the next-state function. Unlike BDD approaches, where N , or each N e , is encoded in a 2K-level BDD, we adopt a Kronecker representation inspired by work on Markov chains =-=[2, 7, 2-=-5]. As in [11, 12, 23], we use a consistent model partition, where each N e is decomposed into K local next-state functions N e;k , for Kk1, which satisfy 8(i K ; : : : ; i 1 ) 2 b S ; N e (i K ; : : ... |

15 | Madre: \Symbolic Computation of the Valid States of the Sequential Machine: Algorithms and Discussion
- Coudert, C
- 1991
(Show Context)
Citation Context ...fast, general, user-friendly, and terminates in at least all cases where previous algorithms terminate. The idea of a disjunctively-partitioned transition relation is natural for asynchronous systems =-=[15, 20]-=-, and in particular for Petri nets. However, our inspiration for its (boolean) Kronecker encoding comes from theseld of Markov chains, where (real) Kronecker operators are increasingly used to encode ... |

14 |
NuSMV: a new Symbolic Model Veri
- Cimatti, Clarke, et al.
- 1999
(Show Context)
Citation Context ...d in a high-level formalism such as Petri nets or pseudocode, determining the range of the state variables is more dicult. Traditionally, the burden of this task has been placed on the user. In NuSMV =-=[14-=-], for example, the domain of multi-valued variables must be explicitly specied as a set or integer range. In our own previous work [11, 12, 23], the input (a Petri net) must be partitioned so that th... |

9 |
Saturation: an ecient iteration strategy for symbolic state space generation
- Ciardo, Luettgen, et al.
- 2001
(Show Context)
Citation Context ... of MDD nodes, memory overhead is small in practice. Expanding local state spaces. Initially, for K k 1, b S k = S k = fs k g, the k th component of the initial state s. The iteration strategy of [1=-=2]-=- saturates nodes bottom-up through an exhaustive symbolic reachability analysis: itsres globally enabled events on a node as long as new global states are found. In our new version, the MDD encodes on... |

6 |
Complexity of memorye cient Kronecker operations with applications to the solution of Markov models
- Buchholz, Ciardo, et al.
- 2000
(Show Context)
Citation Context ...uires an ecient encoding of the next-state function. Unlike BDD approaches, where N , or each N e , is encoded in a 2K-level BDD, we adopt a Kronecker representation inspired by work on Markov chains =-=[2, 7, 2-=-5]. As in [11, 12, 23], we use a consistent model partition, where each N e is decomposed into K local next-state functions N e;k , for Kk1, which satisfy 8(i K ; : : : ; i 1 ) 2 b S ; N e (i K ; : : ... |

4 |
Chaotic point iterations
- Geser, Knoop, et al.
- 1994
(Show Context)
Citation Context ...of the equation S = N (S) in which S contains fsg. Since N is composed of several functions N e , we can build S by applying each function in any order, as long as we consider each event often enough =-=[17]-=-. The systems we target can be partitioned into interacting submodels. For a model composed of K submodels, a global system state is a K-tuple (i K ; : : : ; i 1 ), where i k is the local state of sub... |

4 |
Analysis of Marked Graphs and Petri Nets by Matrix Equations
- Murata, Church
- 1975
(Show Context)
Citation Context ...st to combine symbolic generation of the global state space with exact explicit generation of the local state spaces, but the issue is related, at least for Petri nets, to the existence of invariants =-=[24]-=-. Indeed, we could use place invariants to bound our local state spaces and proceed using pregeneration, but our on-the- y approach is superior because it has a small overhead, while invariant analysi... |

4 |
Ecient computation for invariant checking
- Ravi, Somenzi
- 1999
(Show Context)
Citation Context ... breadth-rst approach of traditional symbolic state-space generation, with the goal of reducing the peak number of nodes, but they often still have some vestiges of breadth-rst search. For example, [3=-=, 26] im-=-prove eciency by exploring only a portion of the newly-found states, those encoded by the \densest " nodes in the decision diagram. The closest to our saturation approach is the \modied breadth-r... |

3 | SMART: Stochastic Model Analyzer for Reliability and Timing - Ciardo, Jones, et al. - 2001 |

2 |
1996], Symbolic protocol veri with queue BDDs
- Godefroid, Long
- 1999
(Show Context)
Citation Context ...ete-event systems has been considered before in a more general setting than ours: in most cases, the goal is the study of systems with innite but regular state spaces. For example, the Queue BDDs of [=-=18]-=- allow one to model systems with asnite number of boolean variables plus one or more unbounded queues, as long as the contents of the queue can be represented by a DFA. The MONA system [19], implement... |