## Local LTL with past constants is expressively complete for Mazurkiewicz traces (2003)

Venue: | In Proc. of MFCS’03, number 2747 in LNCS |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Gastin03localltl,

author = {Paul Gastin and Madhavan Mukund and K. Narayan Kumar},

title = {Local LTL with past constants is expressively complete for Mazurkiewicz traces},

booktitle = {In Proc. of MFCS’03, number 2747 in LNCS},

year = {2003},

pages = {429--438},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

Abstract. To obtain an expressively complete linear-time temporal logic (LTL) over Mazurkiewicz traces that is computationally tractable, we need to intepret formulas locally, at individual events in a trace, rather than globally, at configurations. Such local logics necessarily require past modalities, in contrast to the classical setting of LTL over sequences. Earlier attempts at defining expressively complete local logics have used very general past modalities as well as filters (side-conditions) that “look sideways ” and talk of concurrent events. In this paper, we show that it is possible to use unfiltered future modalities in conjunction with past constants and still obtain a logic that is expressively complete over traces.

### Citations

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Citation Context ...tion with past constants and still obtain a logic that is expressively complete over traces. Keywords Temporal logics, Mazurkiewicz traces, concurrency 1 Introduction Linear-time temporal logic (LTL) =-=[17]-=- has established itself as a useful formalism for specifying the interleaved behaviour of reactive systems. To combat the combinatorial blow-up involved in describing computations of concurrent system... |

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Citation Context ...m for describing the behaviour of static networks of communicating finite-state agents [24]. LTL over Σ-labelled sequences is equivalent to FOΣ(<), the first-order logic over Σ-labelled linear orders =-=[12]-=- and thus defines the class of aperiodic languages over Σ. Though FOΣ(<) permits assertions about both the past and the future, future modalities suffice for establishing the expressive completeness o... |

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Citation Context ...f aperiodic languages over Σ. Though FOΣ(<) permits assertions about both the past and the future, future modalities suffice for establishing the expressive completeness of LTL with respect to FOΣ(<) =-=[8]-=-. From a practical point of view, a finite-state program may be checked against an LTL specification relatively efficiently. ⋆ Partial support of CEFIPRA-IFCPAR Project 2102-1 (ACSMV) is gratefully ac... |

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Citation Context ...osition which is a consequence of several results on the equivalence between aperiodic word languages, star-free word languages, first order word languages and word languages expressible in LTLT (XU) =-=[18, 12, 14, 20, 8, 15, 16, 2]-=-. Proposition 10. Every aperiodic word language K ∈ T ∞ is expressible in LTLT (XU). We fix T = h(bΓ ) and we define the mapping σ : (bΓ ) ∞ → T ∞ by σ(t) = h(bt1)h(bt2) · · · if t = bt1bt2 · · · with... |

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Citation Context ...leteness. Many proofs have had to be omitted in this extended abstract. A full version of the paper is available in [11]. 2 Preliminaries We briefly recall some notions about Mazurkiewicz traces (see =-=[5]-=- for background). A dependence alphabet is a pair (Σ, D) where the alphabet Σ is a finite set of actions and the dependence relation D ⊆ Σ ×Σ is reflexive and symmetric. The independence relation I is... |

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Citation Context ...osition which is a consequence of several results on the equivalence between aperiodic word languages, star-free word languages, first order word languages and word languages expressible in LTLT (XU) =-=[18, 12, 14, 20, 8, 15, 16, 2]-=-. Proposition 10. Every aperiodic word language K ∈ T ∞ is expressible in LTLT (XU). We fix T = h(bΓ ) and we define the mapping σ : (bΓ ) ∞ → T ∞ by σ(t) = h(bt1)h(bt2) · · · if t = bt1bt2 · · · with... |

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Citation Context ... b) /∈ D, a and b are deemed to be independent actions that may occur concurrently. Traces are a natural formalism for describing the behaviour of static networks of communicating finite-state agents =-=[24]-=-. LTL over Σ-labelled sequences is equivalent to FOΣ(<), the first-order logic over Σ-labelled linear orders [12] and thus defines the class of aperiodic languages over Σ. Though FOΣ(<) permits assert... |

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Citation Context ...ly. ⋆ Partial support of CEFIPRA-IFCPAR Project 2102-1 (ACSMV) is gratefully acknowledged.s430 The first expressively complete temporal logic over traces was described in [6] for finite traces and in =-=[19]-=- for infinite traces. The result was refined in [4] to show expressive completeness without past modalities, using an extension of the proof technique developed for LTL in [23]. Formulas in both these... |

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Citation Context ...osition which is a consequence of several results on the equivalence between aperiodic word languages, star-free word languages, first order word languages and word languages expressible in LTLT (XU) =-=[18, 12, 14, 20, 8, 15, 16, 2]-=-. Proposition 10. Every aperiodic word language K ∈ T ∞ is expressible in LTLT (XU). We fix T = h(bΓ ) and we define the mapping σ : (bΓ ) ∞ → T ∞ by σ(t) = h(bt1)h(bt2) · · · if t = bt1bt2 · · · with... |

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Citation Context ...or finite traces and in [19] for infinite traces. The result was refined in [4] to show expressive completeness without past modalities, using an extension of the proof technique developed for LTL in =-=[23]-=-. Formulas in both these logics are defined at global configurations (maximal antichains). Unfortunately, reasoning at the level of global configurations makes the complexity of deciding satisfiabilit... |

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Citation Context ...r all s ∈ S. A trace language L ⊆ R is aperiodic if it is recognized by some morphism to a finite and aperiodic monoid. First-order definability coincides with aperiodicity for traces.s432 Theorem 1 (=-=[6, 7]-=-). A language L ⊆ R(Σ, D) is expressible in FOΣ(<) if and only if it is aperiodic. 3 Local temporal logic We denote by LocTL i Σ the set of (internal) formulas over the alphabet Σ. They are given by t... |

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Citation Context ...h these logics are defined at global configurations (maximal antichains). Unfortunately, reasoning at the level of global configurations makes the complexity of deciding satisfiability non-elementary =-=[21]-=-. Computational tractability seems to require interpreting formulas at local states—effectively at individual events. Recently, in [10], a local temporal logic has been defined over traces and shown t... |

19 | LTL is expressively complete for Mazurkiewicz traces
- Diekert, Gastin
(Show Context)
Citation Context ...-1 (ACSMV) is gratefully acknowledged.s430 The first expressively complete temporal logic over traces was described in [6] for finite traces and in [19] for infinite traces. The result was refined in =-=[4]-=- to show expressive completeness without past modalities, using an extension of the proof technique developed for LTL in [23]. Formulas in both these logics are defined at global configurations (maxim... |

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Satisfiability and model checking for MSO-definable temporal logics are in PSPACE
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Citation Context ...st formulas.) As in [3, 4, 10], we show expressive completeness using an extension to traces of the proof technique introduced in [23] for LTL over sequences. From the recent general result proved in =-=[9]-=-, it follows that the satisfiability problem for this new logic is also in Pspace. The paper is organized as follows. We begin with some preliminaries about traces. In Section 3 we define our new temp... |

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Citation Context ...fication relatively efficiently. ⋆ Partial support of CEFIPRA-IFCPAR Project 2102-1 (ACSMV) is gratefully acknowledged.s430 The first expressively complete temporal logic over traces was described in =-=[6]-=- for finite traces and in [19] for infinite traces. The result was refined in [4] to show expressive completeness without past modalities, using an extension of the proof technique developed for LTL i... |

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An elementary expressively complete temporal logic for Mazurkiewicz traces
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(Show Context)
Citation Context ...s makes the complexity of deciding satisfiability non-elementary [21]. Computational tractability seems to require interpreting formulas at local states—effectively at individual events. Recently, in =-=[10]-=-, a local temporal logic has been defined over traces and shown to be expressively complete and tractable (the satisfiability problem is in Pspace). This logic uses both future and past modalities (si... |

9 | Local Logics of Traces
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(Show Context)
Citation Context ...ny past operators is expressively complete over words but this cannot be the case for traces: there exist two first-order inequivalent traces that cannot be distinguished using only future modalities =-=[22]-=-. In this paper, we show that a very limited ability to talk about the past is sufficient to obtain expressive completeness over traces. Our logic uses unfiltered future modalities and a finite number... |

8 | Local temporal logic is expressively complete for cograph dependence alphabets
- Diekert, Gastin
- 2001
(Show Context)
Citation Context ...red future modalities and a finite number of past constants. (In particular, there is no nesting of past operators and for that matter even future formulas cannot be nested into past formulas.) As in =-=[3, 4, 10]-=-, we show expressive completeness using an extension to traces of the proof technique introduced in [23] for LTL over sequences. From the recent general result proved in [9], it follows that the satis... |

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Recent results on automata and infinite words
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Complete and tractable local linear time temporal logics over traces
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Citation Context ...s (side-conditions). It was also shown that for finite traces, a restricted form of past modalities suffices, but only in conjunction with filtered future modalities. Another proposal is presented in =-=[1]-=- and this logic also uses the since operator. LTL without any past operators is expressively complete over words but this cannot be the case for traces: there exist two first-order inequivalent traces... |