## An Expressively Complete Linear Time Temporal Logic for Mazurkiewicz Traces (1997)

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Citations: | 45 - 5 self |

### BibTeX

@INPROCEEDINGS{Thiagarajan97anexpressively,

author = {P. S. Thiagarajan and I. Walukiewicz},

title = {An Expressively Complete Linear Time Temporal Logic for Mazurkiewicz Traces},

booktitle = {},

year = {1997},

pages = {183--194},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

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### Abstract

A basic result concerning LTL, the propositional temporal logic of linear time, is that it is expressively complete; it is equal in expressive power to the first order theory of sequences. We present here a smooth extension of this result to the class of partial orders known as Mazurkiewicz traces. These partial orders arise in a variety of contexts in concurrency theory and they provide the conceptual basis for many of the partial order reduction methods that have been developed in connection with LTL-specifications. We show that LTrL, our linear time temporal logic, is equal in expressive power to the first order theory of traces when interpreted over (finite and) infinite traces. This result fills a prominent gap in the existing logical theory of infinite traces. LTrL also constitutes a characterisation of the so called trace consistent (robust) LTL-specifications. These are specifications expressed as LTL formulas that do not distinguish between different linearisations of the same trace and hence are amenable to partial order reduction methods.

### Citations

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Citation Context ...the first order theory of traces. There is also a more pragmatic motivation for studying temporal logics interpreted over traces. To bring this out, we first recall that, as first suggested by Pnueli =-=[23]-=-, LTL is often interpreted over the runs of a distributed system. It is known that these runs can be grouped together into equivalence classes; two runs are equated in case they differ only in the ord... |

514 |
Finite Model Theory
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Citation Context ...ve: A⊕ B ϕ iff there exists i ∈ {1, 2, . . . , k} with A ψi and B ψ ′ i. Proof The proof is a standard application of Ehrenfeucht-Fräıssé games. For a description of the games see for example =-=[4]-=-. We denote the n-move game on structures A and B by Gn(A,B). Let us denote by qd(θ) the quantifier depth of the sentence θ. We define an n-theory of a structure C as the set of sentences Thn(C) = {θ ... |

506 |
Automata on infinite objects
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Citation Context ...ed [3]. This theory may be viewed as a smooth generalization of the classical theory of sequences. It turns out that most of the algebraic, automata-theoretic and logical results concerning sequences =-=[27]-=- have a natural extension to the setting of traces. There has been however one prominent gap to date in the logical theory of traces. Our main result concerning LTrL fills this gap. To bring this out,... |

318 | Partial-Order Methods for the Verification of Concurrent Systems - An Approach to the State-Explosion Problem
- Godefroid
- 1996
(Show Context)
Citation Context ...e computational resources used during the verification of trace consistent requirements can be substantial. This is the insight that underlies many of the so called partial order reduction techniques =-=[11, 20, 31]-=-. There is an alternative way to exploit the non-sequential nature of the behaviour of distributed systems and the consequent partial order based reduction techniques. It consists of developing tempor... |

273 |
A Stubborn Attack on State Explosion
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- 1991
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Citation Context ...e computational resources used during the verification of trace consistent requirements can be substantial. This is the insight that underlies many of the so called partial order reduction techniques =-=[11, 20, 31]-=-. There is an alternative way to exploit the non-sequential nature of the behaviour of distributed systems and the consequent partial order based reduction techniques. It consists of developing tempor... |

273 | A classification of models for concurrency
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Citation Context ...rticular, traces constitute the natural tool for capturing the non-interleaved linear time behaviour of models such as Petri nets, event structures and distributed transition systems of various kinds =-=[30]-=-. The theory of traces is well-developed [3]. This theory may be viewed as a smooth generalization of the classical theory of sequences. It turns out that most of the algebraic, automata-theoretic and... |

256 |
Tense Logic and the Theory of Linear Order
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Citation Context ...ng of traces. There has been however one prominent gap to date in the logical theory of traces. Our main result concerning LTrL fills this gap. To bring this out, we recall the famous theorem of Kamp =-=[13]-=- extended by Gabbay et.al. [8]. It says that LTL, the linear time temporal logic interpreted over sequences has the same expressive power as the first order theory of sequences. The surprising aspect ... |

240 |
On the temporal analysis of fairness
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- 1980
(Show Context)
Citation Context ...wever one prominent gap to date in the logical theory of traces. Our main result concerning LTrL fills this gap. To bring this out, we recall the famous theorem of Kamp [13] extended by Gabbay et.al. =-=[8]-=-. It says that LTL, the linear time temporal logic interpreted over sequences has the same expressive power as the first order theory of sequences. The surprising aspect of this result being the misma... |

240 | Modelling concurrency with partial orders - Pratt - 1986 |

193 | Fibring Logics - Gabbay - 1999 |

191 |
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- Diekert
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Citation Context ...for capturing the non-interleaved linear time behaviour of models such as Petri nets, event structures and distributed transition systems of various kinds [30]. The theory of traces is well-developed =-=[3]-=-. This theory may be viewed as a smooth generalization of the classical theory of sequences. It turns out that most of the algebraic, automata-theoretic and logical results concerning sequences [27] h... |

126 |
Concurrent program schemes and their interpretations
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- 1977
(Show Context)
Citation Context ... 2The author was partially supported by Polish KBN grant No. 8 T11C 002 11. 1 Introduction We propose a linear time temporal logic called LTrL whose models are Mazurkiewicz traces. From its inception =-=[16]-=- the class of labelled partial orders known as (Mazurkiewicz) traces has played a fundamental role in the theory of distributed systems. In particular, traces constitute the natural tool for capturing... |

85 | Automata on in objects - Thomas - 1990 |

71 |
The first order properties of products of algebraic systems
- Feferman, Vaught
- 1959
(Show Context)
Citation Context ...t and its corollaries. The major ingredients of the proof of the main result are: a decomposition result for infinite traces, which is an easy version of the Feferman-Vaught theorem for disjoint sums =-=[7]-=- and a new normal form linearisation of traces. It is the use of the Feferman-Vaught result and the new normal form that takes us past the key technical hurdles. The proof is presented in sections 4–9... |

46 |
A trace based extension of linear time temporal logic
- Thiagarajan
- 1994
(Show Context)
Citation Context ... expressively equivalent to the first order theory of traces. Thus an important corollary of our main result is that LTrL captures exactly the LTL-definable trace consistent properties. Starting with =-=[25]-=- a number of linear time temporal logics for traces have been proposed in the literature [1, 5, 18, 24]. None of these studies have been able to exhibit a logic patterned after LTL which is equivalent... |

43 | Model-checking of causality properties
- Alur, Peled, et al.
- 1995
(Show Context)
Citation Context ...our main result is that LTrL captures exactly the LTL-definable trace consistent properties. Starting with [25] a number of linear time temporal logics for traces have been proposed in the literature =-=[1, 5, 18, 24]-=-. None of these studies have been able to exhibit a logic patterned after LTL which is equivalent in expressive power to the first order theory of traces. These logics also have a semantics which has ... |

33 | Logical definability on infinite traces
- Ebinger, Muscholl
- 1996
(Show Context)
Citation Context ...ctions. One reason could be that the problem of deciding whether the property described by an LTL formula is trace consistent happens to be PSPACE–complete [22]. From the work of Ebinger and Muscholl =-=[6]-=- it is not difficult to con3 clude the following: A linear time temporal logic interpreted over traces will capture exactly the LTL-definable trace consistent properties if and only if this logic is e... |

32 |
Interleaving set temporal logic
- Katz, Peled
- 1987
(Show Context)
Citation Context ... associated verification task will be amenable to partial order based reduction methods. This is, in retrospect, the key feature of the branching time temporal logic ISTL introduced by Katz and Peled =-=[14]-=-. The explicit connection between ISTL and traces was later formalized and exploited by Peled and Pnueli [21] to derive proof rules for reasoning about the partially ordered runs of a concurrent progr... |

29 | Locally linear time temporal logic
- Ramanujam
- 1996
(Show Context)
Citation Context ...our main result is that LTrL captures exactly the LTL-definable trace consistent properties. Starting with [25] a number of linear time temporal logics for traces have been proposed in the literature =-=[1, 5, 18, 24]-=-. None of these studies have been able to exhibit a logic patterned after LTL which is equivalent in expressive power to the first order theory of traces. These logics also have a semantics which has ... |

29 | A logical study of distributed transition systems
- Lodaya, Ramanujam, et al.
- 1995
(Show Context)
Citation Context ...ined in the obvious way. In this case however, for all non-trivial trace alphabets (in which the independence relation is non-empty) one can easily show that the satisfiability problem is undecidable =-=[15]-=-. This is the reason why we require our atomic propositions and valuation functions to have a “local flavour”. From a pragmatic standpoint this is acceptable since atomic assertions concerning distrib... |

25 | An Algorithmic Approach for Checking Closure Properties of ω-Regular Languages
- Peled, Wilke, et al.
- 1996
(Show Context)
Citation Context ...nt properties, say, through syntactic restrictions. One reason could be that the problem of deciding whether the property described by an LTL formula is trace consistent happens to be PSPACE–complete =-=[22]-=-. From the work of Ebinger and Muscholl [6] it is not difficult to con3 clude the following: A linear time temporal logic interpreted over traces will capture exactly the LTL-definable trace consisten... |

24 | A trace consistent subset of PTL - Thiagarajan - 1995 |

24 |
On logical definability of trace languages
- Thomas
- 1990
(Show Context)
Citation Context ... σ of T . Proof In what follows, the semantic relation of satisfiability associated with the sentences of FO(Σ) will be denoted |=fo. A simple but basic observation essentially due to Wolfgang Thomas =-=[28]-=- can be stated as: Observation 6.1 For every sentence ϕ ∈ FO(Σ, I) there exists a sentence ϕ̂ ∈ FO(Σ) such that for every trace T : T |=FO ϕ iff u |=fo ϕ̂ for every u ∈ lin(T ). Recall that lin(T ) is... |

24 | Difficult configurations – on the complexity of LTrL
- Walukiewicz
- 1998
(Show Context)
Citation Context ...ll kinds. There is however a price to be paid for this transparency. The second author of this paper has recently established a non-elementary time lower bound for the satisfiability problem for LTrL =-=[29]-=-. Hence it is clear that from a practical standpoint LTrL is not the final stop in the search for the “right” linear time temporal logic for traces. However, we feel that this logic represents a vital... |

18 |
Proving partial order properties
- Peled, Pnueli
- 1994
(Show Context)
Citation Context ...ct, the key feature of the branching time temporal logic ISTL introduced by Katz and Peled [14]. The explicit connection between ISTL and traces was later formalized and exploited by Peled and Pnueli =-=[21]-=- to derive proof rules for reasoning about the partially ordered runs of a concurrent program.. At present we do not know of a characterization of the expressive power of ISTL and its variants. In a l... |

15 | Partial-Order Methods for the Veri of Concurrent Systems: An Approach to the State-Explosion Problem. LNCS 1032 - Godefroid - 1996 |

14 |
Charakterisierung von Sprachklassen unendlicher Spuren durch Logiken
- Ebinger
- 1994
(Show Context)
Citation Context ...our main result is that LTrL captures exactly the LTL-definable trace consistent properties. Starting with [25] a number of linear time temporal logics for traces have been proposed in the literature =-=[1, 5, 18, 24]-=-. None of these studies have been able to exhibit a logic patterned after LTL which is equivalent in expressive power to the first order theory of traces. These logics also have a semantics which has ... |

14 |
Past Temporal Logic
- Zuck
- 1986
(Show Context)
Citation Context ...• τ0a0 v τ1a1 v . . . v τnan. • 1 ≤ n ≤ |Σ| and ai D ai+1 for 0 ≤ i < n. All these conditions can be expressed in FO(Σ) and this easily leads to Observation 6.1. Now, by the expressiveness results of =-=[8, 32]-=-, for each sentence ϕ̂ ∈ FO(Σ) there exists α̂ ∈ LTL(Σ) such that: {u ∈ Σ∞ | u |=fo ϕ̂} = {u ∈ Σ∞ | u, ε |= α̂}. The lemma now follows at once from the definition of trace consistent formulas. Let u... |

13 |
Partial order reduction : model checking using representatives
- Peled
- 1996
(Show Context)
Citation Context ...e computational resources used during the verification of trace consistent requirements can be substantial. This is the insight that underlies many of the so called partial order reduction techniques =-=[11, 20, 31]-=-. There is an alternative way to exploit the non-sequential nature of the behaviour of distributed systems and the consequent partial order based reduction techniques. It consists of developing tempor... |

9 | A.: Logical de on in traces - Ebinger, Muscholl - 1996 |

9 |
Design and Validation of Computer Protocols. Software Series
- Holzmann
- 1991
(Show Context)
Citation Context ...eness, LTL admits a PSPACE decision procedure. As a consequence, LTL can be used as a specification logic and the model checking problem can be solved in an automated fashion using tools such as SPIN =-=[12]-=-. 31 Unfortunately, it looks unlikely at present that LTrL can be used as a specification logic. Recently, the second author has shown that the satisfiability problem for LTrL is non-elementary hard [... |

7 |
A ν-calculus with local views for sequential agents
- Niebert
- 1995
(Show Context)
Citation Context |

5 | Dicult con { on the complexity of LTrL - Walukiewicz - 1998 |

5 | A Temporal Logic for the Specification and Verification of Distributed Behaviour
- Niebert
- 1998
(Show Context)
Citation Context ...ely complete, will also admit a decision procedure with a more reasonable time complexity. The only available expressiveness results for temporal logics over traces are due to Ebinger [5] and Niebert =-=[19]-=-. Ebinger’s logic, called TLPO, has both previous state and since modalities. These past modalities are extensively used in the attempt to prove that TLPO is expressively complete when interpreted ove... |

4 | On logical de of trace languages - Thomas - 1990 |

3 | Trace Structures and other Models for Concurrency - Nielsen, Winskel - 1995 |

2 | Asynchronous cellurar automata for infinite traces - Gastin, Petit - 1992 |

2 | Asynchronous cellurar automata for in traces - Gastin, Petit - 1992 |

2 | A temporal logic for the speci and veri of distributed behaviour - Niebert - 1997 |

1 |
Linear time temporal logics over traces
- Mukund, Thiagarajan
- 1996
(Show Context)
Citation Context ...ics can be given in terms of traces. Elementary net systems, 1-safe Petri nets, trace transition systems [30] and networks of finite state automata that communicate by synchronizing on common actions =-=[17]-=- are typical examples. For all these system models, one can use LTrL formulas to specify the properties that must be satisfied by all trace runs. An example of this approach is given in Section 11. Wi... |

1 |
Private communication
- Courcelle
(Show Context)
Citation Context ...se it does not even exist for a trace alphabet consisting of just two letters which are independent. An infinite trace over such an alphabet consists of two disconnected infinite sequences. Courcelle =-=[2]-=- has shown that there is no monadic second order formula defining a linearization of this trace. Our approach to proving Lemma 3 is the following. We identify a restricted type of an infinite trace ca... |

1 | Linear time temporal llogics over traces - Mukund, Thiagarajan - 1996 |