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22
A classification of symbolic transition systems
- ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2005
"... We define five increasingly comprehensive classes of infinite-state systems, called STS1--STS5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.STS1 These are the systems with finite bisimilarity quotients. They can be analyzed symbolica ..."
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Cited by 54 (6 self)
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We define five increasingly comprehensive classes of infinite-state systems, called STS1--STS5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.STS1 These are the systems with finite bisimilarity quotients. They can be analyzed symbolically by iteratively applying predecessor and Boolean operations on state sets, starting from a finite number of observable state sets. Any such iteration is guaranteed to terminate in that only a finite number of state sets can be generated. This enables model checking of the μ-calculus.STS2 These are the systems with finite similarity quotients. They can be analyzed symbolically by iterating the predecessor and positive Boolean operations. This enables model checking of the existential and universal fragments of the μ-calculus.STS3 These are the systems with finite trace-equivalence quotients. They can be analyzed symbolically by iterating the predecessor operation and a restricted form of positive Boolean operations (intersection is restricted to intersection with observables). This enables model checking of all ω-regular properties, including linear temporal logic.STS4 These are the systems with finite distance-equivalence quotients (two states are equivalent if for every distance d, the same observables can be reached in d transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new state sets are generated. This enables model checking of the existential conjunction-free and universal disjunction-free fragments of the μ-calculus.STS5 These are the systems with finite bounded-reachability quotients (two states are equivalent if for every distance d, the same observables can be reached in d or fewer transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new states are encountered (this is a weaker termination condition than above). This enables model checking of reachability properties.
Software Model Checking
"... Software model checking is the algorithmic analysis of programs to prove properties of their executions. It traces its roots to logic and theorem proving, both to provide the conceptual framework in which to formalize the fundamental questions and to provide algorithmic procedures for the analysis o ..."
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Cited by 50 (0 self)
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Software model checking is the algorithmic analysis of programs to prove properties of their executions. It traces its roots to logic and theorem proving, both to provide the conceptual framework in which to formalize the fundamental questions and to provide algorithmic procedures for the analysis of logical questions. The undecidability theorem [Turing 1936] ruled out the possibility of a sound and complete algorithmic solution for any sufficiently powerful programming model, and even under restrictions (such as finite state spaces), the correctness problem remained computationally intractable. However, just because a problem is hard does not mean it never appears in practice. Also, just because the general problem is undecidable does not imply that specific instances of the problem will also be hard. As the complexity of software systems grew, so did the need for some reasoning mechanism about correct behavior. (While we focus here on analyzing the behavior of a program relative to given correctness specifications, the development of specification mechanisms happened in parallel, and merits a different survey.) Initially, the focus of program verification research was on manual reasoning, and
Fair Simulation Relations, Parity Games, and State Space Reduction for Büchi Automata
"... We give efficient algorithms, beating or matching optimal known bounds, for computing a variety of simulation relations on the state space of a Buchi automaton. Our algorithms are derived via a unified and simple paritygame framework. This framework incorporates previously studied notions like fair ..."
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Cited by 37 (2 self)
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We give efficient algorithms, beating or matching optimal known bounds, for computing a variety of simulation relations on the state space of a Buchi automaton. Our algorithms are derived via a unified and simple paritygame framework. This framework incorporates previously studied notions like fair and direct simulation, but our main motivation is state space reduction, and for this purpose we introduce a new natural notion of simulation, called delayed simulation. We show that, unlike fair simulation, delayed simulation preserves the automaton language upon quotienting, and that it allows substantially better state reduction than direct simulation. We use the parity-game approach, based on a recent algorithm by Jurdzinski, to efficiently compute all the above simulation relations. In particular, we obtain an O(mn 3 )-time and O(mn)-space algorithm for computing both the delayed and fair simulation relations. The best prior algorithm for fair simulation requires time O(n 6 ) ([HKR97]). Our framework also allows one to compute bisimulations efficiently: we compute the fair bisimulation relation in O(mn 3 ) time and O(mn) space, whereas the best prior algorithm for fair bisimulation requires time O(n 10 ) ([HR00]). 1
Adding Regular Expressions to Graph Reachability and Pattern Queries
- Frontiers of Computer Science
, 2012
"... Abstract—It is increasingly common to find graphs in which edges bear different types, indicating a variety of relationships. For such graphs we propose a class of reachability queries and a class of graph patterns, in which an edge is specified with a regular expression of a certain form, expressin ..."
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Cited by 29 (5 self)
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Abstract—It is increasingly common to find graphs in which edges bear different types, indicating a variety of relationships. For such graphs we propose a class of reachability queries and a class of graph patterns, in which an edge is specified with a regular expression of a certain form, expressing the connectivity in a data graph via edges of various types. In addition, we define graph pattern matching based on a revised notion of graph simulation. On graphs in emerging applications such as social networks, we show that these queries are capable of finding more sensible information than their traditional counterparts. Better still, their increased expressive power does not come with extra complexity. Indeed, (1) we investigate their containment and minimization problems, and show that these fundamental problems are in quadratic time for reachability queries and are in cubic time for pattern queries. (2) We develop an algorithm for answering reachability queries, in quadratic time as for their traditional counterpart. (3) We provide two cubictime algorithms for evaluating graph pattern queries based on extended graph simulation, as opposed to the NP-completeness of graph pattern matching via subgraph isomorphism. (4) The effectiveness, efficiency and scalability of these algorithms are experimentally verified using real-life data and synthetic data. I.
Strong preservation as completeness in abstract interpretation
- In Proc. European Symp. Programming, LNCS 2986
, 2004
"... Abstract. Many algorithms have been proposed to minimally refine abstract transition systems in order to get strong preservation relatively to a given temporal specification language. These algorithms compute a state equivalence, namely they work on abstractions which are partitions interpretation-b ..."
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Cited by 24 (6 self)
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Abstract. Many algorithms have been proposed to minimally refine abstract transition systems in order to get strong preservation relatively to a given temporal specification language. These algorithms compute a state equivalence, namely they work on abstractions which are partitions interpretation-based view, state partitions are just one particular type of abstraction, and therefore it could well happen that the refined partition constructed by the algorithm is not the optimal generic abstraction. On the other hand, it has been already noted that the well-known concept of complete abstract interpretation is related to strong preservation of abstract model checking. This paper establishes a precise correspondence between complete abstract interpretation and strongly preserving abstract model checking, by showing that the problem of minimally refining an abstract model checking in order to get strong preservation can be formulated as a complete domain refinement in abstract interpretation, which always admits a fixpoint solution. As a consequence of these results, we show that some well-known behavioural equivalences used in process algebra like simulation and bisimulation can be elegantly characterized in pure abstract interpretation as completeness properties. 1
Generalized strong preservation by abstract interpretation
- J. Logic and Computation
, 2007
"... Standard abstract model checking relies on abstract Kripke structures which approximate concrete models by gluing together indistinguishable states, namely by a partition of the concrete state space. models that are more general than abstract Kripke structures. Accordingly, strong preservation is ge ..."
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Cited by 14 (9 self)
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Standard abstract model checking relies on abstract Kripke structures which approximate concrete models by gluing together indistinguishable states, namely by a partition of the concrete state space. models that are more general than abstract Kripke structures. Accordingly, strong preservation is generalized to abstract interpretation-based models and precisely related to the concept of completeness in abstract interpretation. The problem of minimally refining an abstract model in order to make it strongly preserving for some language L can be formulated as a minimal domain refinement in abstract interpretation in order to get completeness w.r.t. the logical/temporal operators of L. It turns out that this refined strongly preserving abstract model always exists and can be characterized as a greatest fixed point. As a consequence, some well-known behavioural equivalences, like bisimulation, simulation and stuttering, and their corresponding partition refinement algorithms can be elegantly characterized in abstract interpretation as completeness properties and refinements.
Capturing Topology in Graph Pattern Matching
"... Graph pattern matching is often defined in terms of subgraph isomorphism, an np-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be conducted in cubic-time. However, they fall short of capturing ..."
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Cited by 13 (6 self)
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Graph pattern matching is often defined in terms of subgraph isomorphism, an np-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be conducted in cubic-time. However, they fall short of capturing the topology of data graphs, i.e., graphs may have a structure drastically different from pattern graphs they match, and the matches found are often too large to understand and analyze. To rectify these problems, this paper proposes a notion of strong simulation, a revision of graph simulation, for graph pattern matching. (1) We identify a set of criteria for preserving the topology of graphs matched. We show that strong simulation preserves the topology of data graphs and finds a bounded number of matches. (2) We show that strong simulation retains the same complexity as earlier extensions of simulation, by providing a cubic-time algorithm for computing strong simulation. (3) We present the locality property of strong simulation, which allows us to effectively conduct pattern matching on distributed graphs. (4) We experimentally verify the effectiveness and efficiency of these algorithms, using real-life data and synthetic data. 1.
Distributed Graph Pattern Matching
"... Graph simulation has been adopted for pattern matching to reduce the complexity and capture the need of novel applications. With the rapid development of the Web and social networks, data is typically distributed over multiple machines. Hence a natural question raised is how to evaluate graph simula ..."
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Cited by 8 (1 self)
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Graph simulation has been adopted for pattern matching to reduce the complexity and capture the need of novel applications. With the rapid development of the Web and social networks, data is typically distributed over multiple machines. Hence a natural question raised is how to evaluate graph simulation on distributed data. To our knowledge, no such distributed algorithms are in place yet. This paper settles this question by providing evaluation algorithms and optimizations for graph simulation in a distributed setting. (1) We study the impacts of components and data locality on the evaluation of graph simulation. (2) We give an analysis of a large class of distributed algorithms, captured by a message-passing model, for graph simulation. We also identify three complexity measures: visit times, makespan and data shipment, for analyzing the distributed algorithms, and show that these measures are essentially controversial with each other. (3) We propose distributed algorithms and optimization techniques that exploit the properties of graph simulation and the analyses of distributed algorithms. (4) We experimentally verify the effectiveness and efficiency of these algorithms, using both real-life and synthetic data. Categories and Subject Descriptors H.2.8 [Database Management]: Database applications— graph data, data mining
An Efficient Simulation Algorithm based on Abstract Interpretation
, 709
"... A number of algorithms for computing the simulation preorder are available. Let Σ denote the state space, → the transition relation and Psim the partition of Σ induced by simulation equivalence. The algorithms by Henzinger, Henzinger, Kopke and by Bloom and Paige run in O(|Σ||→|)-time and, as far a ..."
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Cited by 8 (4 self)
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A number of algorithms for computing the simulation preorder are available. Let Σ denote the state space, → the transition relation and Psim the partition of Σ induced by simulation equivalence. The algorithms by Henzinger, Henzinger, Kopke and by Bloom and Paige run in O(|Σ||→|)-time and, as far as time-complexity is concerned, they are the best available algorithms. However, these algorithms have the drawback of a space complexity that is more than quadratic in the size of the state space. The algorithm by Gentilini, Piazza, Policriti — subsequently corrected by van Glabbeek and Ploeger — appears to provide the best compromise between time and space complexity. Gentilini et al.’s algorithm runs in O(|Psim | 2 |→|)-time while the space complexity is in O(|Psim | 2 + |Σ | log |Psim|). We present here a new efficient simulation algorithm that is obtained as a modification of Henzinger et al.’s algorithm and whose correctness is based on some techniques used in applications of abstract interpretation to model checking. Our algorithm runs in O(|Psim||→|)-time and O(|Psim||Σ | log |Σ|)-space. Thus, this algorithm improves the best known time bound while retaining an acceptable space complexity that is in general less than quadratic in the size of the state space. An experimental evaluation showed good comparative results with respect to Henzinger, Henzinger and Kopke’s algorithm. 1
Saving Space in a Time Efficient Simulation Algorithm
"... A number of algorithms are available for computing the simulation relation on Kripke structures and on labelled transition systems representing concurrent systems. Among them, the algorithm by Ranzato and Tapparo [2007] has the best time complexity, while the algorithm by Gentilini et al. [2003] – ..."
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Cited by 6 (1 self)
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A number of algorithms are available for computing the simulation relation on Kripke structures and on labelled transition systems representing concurrent systems. Among them, the algorithm by Ranzato and Tapparo [2007] has the best time complexity, while the algorithm by Gentilini et al. [2003] – successively corrected by van Glabbeek and Ploeger [2008] – has the best space complexity. Both space and time complexities are critical issues in a simulation algorithm, in particular memory requirements are crucial in the context of model checking when dealing with large state spaces. We propose here a new simulation algorithm that is obtained as a space saving modification of the time efficient algorithm by Ranzato and Tapparo: a symbolic representation of sets is embedded in this algorithm so that any set of states manipulated by the algorithm can be efficiently stored as a set of blocks of a suitable state partition. It turns out that this new simulation algorithm retains a space complexity comparable with Gentilini et al.’s algorithm while improving on Gentilini et al.’s time bound. 1.
