## A robust class of context-sensitive languages (2007)

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Venue: | In LICS |

Citations: | 28 - 6 self |

### BibTeX

@INPROCEEDINGS{Torre07arobust,

author = {Salvatore La Torre and P. Madhusudan and Gennaro Parlato},

title = {A robust class of context-sensitive languages},

booktitle = {In LICS},

year = {2007},

pages = {161--170},

publisher = {IEEE Computer Society}

}

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### Abstract

We define a new class of languages defined by multi-stack automata that forms a robust subclass of context-sensitive languages, with decidable emptiness and closure under boolean operations. This class, called multi-stack visibly pushdown languages (MVPLs), is defined using multi-stack pushdown automata with two restrictions: (a) the pushdown automaton is visible, i.e. the input letter determines the operation on the stacks, and (b) any computation of the machine can be split into�stages, where in each stage, there is at most one stack that is popped. MVPLs are an extension of visibly pushdown languages that captures noncontext free behaviors, and has applications in analyzing abstractions of multithreaded recursive programs, significantly enlarging the search space that can be explored for them. We show that MVPLs are closed under boolean operations, and problems such as emptiness and inclusion are decidable. We characterize MVPLs using monadic second-order logic over appropriate structures, and exhibit a Parikh theorem for them. 1.

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Citation Context ... Regular word languages have played an important role in this regard, and have found applications in searching documents, defining logics for behaviors, and algorithms for model-checking systems (see =-=[8, 11, 19, 20, 22]-=-). In certain linear models, there is an implicit relation on the positions in the model that can be inferred from its labels. A standard example of these are XML/SGML docuThe author was partially sup... |

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Citation Context ...nder boolean operations and projection), we get (see [20]): Theorem 11 A language is a�-MVPL over� � � ���� �� iff there is an MSO�sentence over� � � � with . The Parikh mapping, introduced by Parikh =-=[15]-=-, associates a word with the vector of natural numbers that reflect ���������� � the number of occurrences of the symbols in the word. Formally, the Parikh image of a word , over the alphabet���������... |

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Citation Context ...with interpreted relations for the left-child and right-child, and boolean operators [20]. The following theorem relates the class of regular trees and the class of trees definable by MSO. Theorem 1 (=-=[7, 18]-=-) A tree language is regular iff it is MSO definable. 3. The Emptiness Problem Given a�-MVPA�over�, the emptiness problem for �is to decide whether �is empty. In this section, we show that the emptine... |

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Citation Context ...with interpreted relations for the left-child and right-child, and boolean operators [20]. The following theorem relates the class of regular trees and the class of trees definable by MSO. Theorem 1 (=-=[7, 18]-=-) A tree language is regular iff it is MSO definable. 3. The Emptiness Problem Given a�-MVPA�over�, the emptiness problem for �is to decide whether �is empty. In this section, we show that the emptine... |

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Citation Context ...urally present, which relates calls to procedures with their corresponding returns. Viewing the nesting structure as explicit edges in the document led to the recently studied theory of nested words (=-=[1, 2]-=-). The notion of regularity for nested words captures a different class of languages than regular word languages (i.e. regular nested word languages can be context-free when the nesting edge relation ... |

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Citation Context ... that many errors are already found after a few number of context-switches (switches between stacks), and in fact for any�, checking if an error state is reached within� context switches is decidable =-=[16]-=-. Context-bounded checking of concurrent recursive programs has shown to discover several errors in programs (see [5, 16, 17]). MVPLs generalize this result by showing that checking whether an error s... |

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Citation Context ...any�, checking if an error state is reached within� context switches is decidable [16]. Context-bounded checking of concurrent recursive programs has shown to discover several errors in programs (see =-=[5, 16, 17]-=-). MVPLs generalize this result by showing that checking whether an error state is reached within�phases, where in each phase all processes can evolve, but only one of them is allowed to return from p... |

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Citation Context ...urally present, which relates calls to procedures with their corresponding returns. Viewing the nesting structure as explicit edges in the document led to the recently studied theory of nested words (=-=[1, 2]-=-). The notion of regularity for nested words captures a different class of languages than regular word languages (i.e. regular nested word languages can be context-free when the nesting edge relation ... |

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Citation Context ...ations reached in�-phases is not regular. Other results on deciding reachability, including bounded-context switching and nested locks, crucially rely on the fact that the reachable sets are regular (=-=[3, 4, 12, 16]-=-). (In [4] the forward reachable sets are not regular, but backward reachable sets are.) We believe that analyzing�-phase executions of a program model explore a much larger space of configurations th... |

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Citation Context ...a non-context-free set of configurations. Other extensions of context-free languages include the class of languages accepted by stack automata [9] and those accepted by higher-order pushdown automata =-=[13]-=-. While these classes have a decidable emptiness problem, they are not robust (for example, they are not closed under complement). We do not know whether�-MVPLs are contained in these classes, but we ... |

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Citation Context ...ations reached in�-phases is not regular. Other results on deciding reachability, including bounded-context switching and nested locks, crucially rely on the fact that the reachable sets are regular (=-=[3, 4, 12, 16]-=-). (In [4] the forward reachable sets are not regular, but backward reachable sets are.) We believe that analyzing�-phase executions of a program model explore a much larger space of configurations th... |

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Citation Context ...any�, checking if an error state is reached within� context switches is decidable [16]. Context-bounded checking of concurrent recursive programs has shown to discover several errors in programs (see =-=[5, 16, 17]-=-). MVPLs generalize this result by showing that checking whether an error state is reached within�phases, where in each phase all processes can evolve, but only one of them is allowed to return from p... |

30 | Reachability analysis of multithreaded software with asynchronous communication
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Citation Context ...ations reached in�-phases is not regular. Other results on deciding reachability, including bounded-context switching and nested locks, crucially rely on the fact that the reachable sets are regular (=-=[3, 4, 12, 16]-=-). (In [4] the forward reachable sets are not regular, but backward reachable sets are.) We believe that analyzing�-phase executions of a program model explore a much larger space of configurations th... |

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Citation Context ...he first stack with the other two stacks empty, which is a non-context-free set of configurations. Other extensions of context-free languages include the class of languages accepted by stack automata =-=[9]-=- and those accepted by higher-order pushdown automata [13]. While these classes have a decidable emptiness problem, they are not robust (for example, they are not closed under complement). We do not k... |

10 |
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Citation Context ... be decidable when regular expressions in the logic are replaced with particular classes of non-regular languages. The non-regular but context-free language extensions that were known to be decidable =-=[10]-=- were recently generalized in [14], where it was shown that PDL with visiblypushdown language modalities is decidable. Whether one can obtain a generalization of the small number of decidable extensio... |

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Citation Context ...ions in the logic are replaced with particular classes of non-regular languages. The non-regular but context-free language extensions that were known to be decidable [10] were recently generalized in =-=[14]-=-, where it was shown that PDL with visiblypushdown language modalities is decidable. Whether one can obtain a generalization of the small number of decidable extensions of PDL with non-context-free la... |