## Branching Time and Partial Order in Temporal Logics (1995)

Venue: | Time and Logic: A Computational Approach |

Citations: | 13 - 2 self |

### BibTeX

@INPROCEEDINGS{Penczek95branchingtime,

author = {Wojciech Penczek},

title = {Branching Time and Partial Order in Temporal Logics},

booktitle = {Time and Logic: A Computational Approach},

year = {1995},

pages = {179--228},

publisher = {UCL Press}

}

### OpenURL

### Abstract

The aim of this paper is to present existing propositional temporal logics with branching and partially ordered time. These logics are used for specifying and proving properties of programs and systems. The branching time approach is useful e.g. for non-deterministic programs and can be applied also for concurrent programs. The partial order approach is especially useful for concurrent programs and allows one to study more subtle properties than those based on branching time. A survey of branching time logics, computation tree logics, partial order temporal logics and logics based on event structures is given. The following issues are concerned in this paper: the completeness of proof systems, the finite model property, decidability, model checking and expressiveness of the logics. 1 Introduction The aim of this paper is to present existing formal languages of propositional temporal logic with frames based on branching time structures or, more general, partial orders. Bran...

### Citations

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Design and synthesis of synchronization skeletons using branching time temporal logic
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Citation Context ...e year later Ben-Ari, Manna, and Pnueli defined the Unified Branching Time System (UB) [3]. In the same year, Clarke and Emerson extended expressiveness of UB by defining Computation Tree Logic (CTL) =-=[4]-=-; their results are fully described in [5]. In 1983, Emerson and Halpern gave the definition of CTL* [12]. The first logic based on partial orders (POTL) was defined by Pinter and Wolper in 1984 [34] ... |

375 |
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Citation Context ... length of the tested formula. 2 It turns out that there is no better algorithm for testing satisfiability since the complexity of the algorithm matches a lower bound for satisfiability. Theorem 2.4 (=-=[11, 17]-=-) There is a deterministic exponential time lower bound for UB satisfiability. 2 The proof sketches of the above theorems are given for CTL - an extension of UB. 2.3 Computation Tree Logic (CTL) Compu... |

267 | Event structures - Winskel - 1987 |

174 |
The temporal logic of branching time
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Citation Context ... sequential agents. The first ideas about branching time logics appeared in the papers of Abrahamson [1], [2]. One year later Ben-Ari, Manna, and Pnueli defined the Unified Branching Time System (UB) =-=[3]-=-. In the same year, Clarke and Emerson extended expressiveness of UB by defining Computation Tree Logic (CTL) [4]; their results are fully described in [5]. In 1983, Emerson and Halpern gave the defin... |

147 | Decision procedures and expressiveness in the temporal logic of branching time
- Emerson, Halpern
- 1985
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Citation Context ... length of the tested formula. 2 It turns out that there is no better algorithm for testing satisfiability since the complexity of the algorithm matches a lower bound for satisfiability. Theorem 2.4 (=-=[11, 17]-=-) There is a deterministic exponential time lower bound for UB satisfiability. 2 The proof sketches of the above theorems are given for CTL - an extension of UB. 2.3 Computation Tree Logic (CTL) Compu... |

92 |
sometimes" and "not never" revisited: on branching versus linear time temporal logic
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- 1983
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Citation Context ...year, Clarke and Emerson extended expressiveness of UB by defining Computation Tree Logic (CTL) [4]; their results are fully described in [5]. In 1983, Emerson and Halpern gave the definition of CTL* =-=[12]-=-. The first logic based on partial orders (POTL) was defined by Pinter and Wolper in 1984 [34] and then extended by Kornatzky and Pinter [18]. Three years later Katz and Peled presented Interleaving S... |

85 |
Modalities for model checking: Branching time logic strikes back
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- 1987
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Citation Context ...y of checking satisfiability and performing model checking. There are, however, many logics "between" CTL and CTL* (which were not mentioned here) like CTL + [11], ECTL and ECTL + [10], [7],=-= and FCTL [14]-=-, which extend expressiveness of CTL, but have still less complicated algorithms of testing satisfiablity and of model checking than CTL*. It should also be mentioned that there are branching time log... |

61 |
Events in Computation
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Citation Context ... was put forward by Lodaya and Thiagarajan [25]. The next section introduces event structures and related notions. 4.1 Event Structures The notion of an event structure has been introduced by Winskel =-=[37]-=-. An event structure represents the behaviour of a distributed system by means of a set of event occurrences, a causality relation that partially orders the event occurrences and a conflict relation w... |

37 |
Improved upper and lower bounds for modal logics of programs: Preliminary report
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Citation Context ... theorem consists in showing that the satisfiability problem can be reduced to testing non-emptiness of tree automata. A lower bound of deterministic double exponential time has also been established =-=[42]-=-. In [6], a deterministic double exponential algorithm for CTL* interpreted over fair structures has been defined. 2.4.2 Model checking for CTL* Model checking problem for CTL is solvable in determini... |

35 |
Reasoning with time and chance
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Citation Context ...e branching time logics with syntax like CTL or CTL*, the formulas of which are interpreted over fair structures [6], Abrahamson structures (suffix and fusion closed) [2] and probabilistic structures =-=[24]-=-. These logics have also been shown to be decidable and to have the finite model property. 3 Temporal Logics on Partial Orders The aim of this section is to present the existing formal languages of te... |

32 |
Interleaving set temporal logic
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- 1987
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Citation Context ...based on partial orders (POTL) was defined by Pinter and Wolper in 1984 [34] and then extended by Kornatzky and Pinter [18]. Three years later Katz and Peled presented Interleaving Set Temporal Logic =-=[21]-=- based on configuration structures. The history of event structure logics is much shorter. The first logic dealing with n-sequential event Introduction 205 structures was defined by Lodaya and Thiagar... |

32 |
A temporal logic for reasoning about partially ordered computations
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Citation Context ...) [4]; their results are fully described in [5]. In 1983, Emerson and Halpern gave the definition of CTL* [12]. The first logic based on partial orders (POTL) was defined by Pinter and Wolper in 1984 =-=[34]-=- and then extended by Kornatzky and Pinter [18]. Three years later Katz and Peled presented Interleaving Set Temporal Logic [21] based on configuration structures. The history of event structure logic... |

28 |
Using branching time logic to synthesize synchronization skeletons
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(Show Context)
Citation Context ...s the high complexity of checking satisfiability and performing model checking. There are, however, many logics "between" CTL and CTL* (which were not mentioned here) like CTL + [11], ECTL a=-=nd ECTL + [10]-=-, [7], and FCTL [14], which extend expressiveness of CTL, but have still less complicated algorithms of testing satisfiablity and of model checking than CTL*. It should also be mentioned that there ar... |

23 | Alternative semantics for temporal logics - Emerson - 1983 |

20 |
Decidability and expressiveness of logics of processes
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- 1980
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Citation Context ... states of an entire system whereas events in event structure models represent local states of sequential agents. The first ideas about branching time logics appeared in the papers of Abrahamson [1], =-=[2]-=-. One year later Ben-Ari, Manna, and Pnueli defined the Unified Branching Time System (UB) [3]. In the same year, Clarke and Emerson extended expressiveness of UB by defining Computation Tree Logic (C... |

17 | Automata, tableaux, and temporal logics - Emerson - 1985 |

17 |
An efficient verification method for parallel and distributed programs
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(Show Context)
Citation Context ... logics UB and CTL by past modalities. However, their semantic structures can be linked with partial orderings representing runs of concurrent systems. 3.3 Interleaving Set Temporal Logic (ISTL) [21],=-=[22]-=- So far, temporal logics have been interpreted over pre-order structures. Now, Interleaving Set Temporal Logic, interpreted over partial order structures of global states, is introduced. The main aim ... |

14 |
A temporal logic for event structures
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- 1988
(Show Context)
Citation Context ...s. The history of event structure logics is much shorter. The first logic dealing with n-sequential event Introduction 205 structures was defined by Lodaya and Thiagarajan [25] in 1987. Then, Penczek =-=[29]-=- presented a temporal logic based on unrestricted event structures. This logic has been extended and refined consequently in [30], [27], and [28]. In this paper the logics mentioned above are presente... |

11 |
Reasoning about fair concurrent programs
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Citation Context ...consists in showing that the satisfiability problem can be reduced to testing non-emptiness of tree automata. A lower bound of deterministic double exponential time has also been established [42]. In =-=[6]-=-, a deterministic double exponential algorithm for CTL* interpreted over fair structures has been defined. 2.4.2 Model checking for CTL* Model checking problem for CTL is solvable in deterministic lin... |

11 | On udecidability of propositional temporal logics on trace systems
- Penczek
- 1992
(Show Context)
Citation Context ...n of ISTL The price to be paid for extra expressiveness of ISTL is its undecidability. It was shown that if some more conditions are required about ISTL models, then ISTL is undecidable. Theorem 3.31 =-=[31]-=- ISTL interpreted over full executions of trace systems is undecidable. 2 The satisfiability problem for ISTL is reduced to the recurring tiling problem, which is known to be undecidable and \Pi 1 1 -... |

7 |
A Temporal Logic for the Local Specification of Concurrent Systems
- Penczek
- 1989
(Show Context)
Citation Context ...model property, but it is decidable. The next extension of Event Structure Logic (ESL) has been motivated by a great interest in specifying discrete systems. 4.6 Discrete Event Structure Logic (DESL) =-=[30]-=- The class of frames is restricted to discrete event structures. A new modality reflecting a successor relation is introduced to the formal language of ESL. 4.6.1 Syntax of DESL The set of DESL formul... |

5 |
A 1981 Branching Time Temporal Logics and the Design of Correct Concurrent Programs
- Emerson
(Show Context)
Citation Context ...high complexity of checking satisfiability and performing model checking. There are, however, many logics "between" CTL and CTL* (which were not mentioned here) like CTL + [11], ECTL and ECT=-=L + [10], [7]-=-, and FCTL [14], which extend expressiveness of CTL, but have still less complicated algorithms of testing satisfiablity and of model checking than CTL*. It should also be mentioned that there are bra... |

5 | Towards a temporal logic of causality and choice - Reisig - 1989 |

4 |
A Model Checker for Partial Order Temporal Logic, EE Pub n
- Kornatzky, Pinter
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Citation Context ... of POTL), then the proof system has to be extended by formulas of the form 9 fl T, 9 fl T, and all the results are still valid with small changes in the proofs. 3.2.5 Model checking for POTL[U ; U ] =-=[19]-=- Model checking for POTL[U ; U ] is more complicated than for CTL. There are two reasons for this. Firstly, formulas contain backward modalities. And secondly, formulas are interpreted over models cor... |

4 | Temporal logics for trace systems
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- 1993
(Show Context)
Citation Context ...roblem for ISTL is reduced to the recurring tiling problem, which is known to be undecidable and \Pi 1 1 -complete. As far no complete proof system for the whole language of ISTL has been defined. In =-=[32]-=-, a complete infinitary axiomatization for ISTL without formulas of the form 9 2 p was presented. There has also been defined the logic ISTL* [21], which uses the language of CTL* with the same models... |

2 |
Modal logic of concurrent programs
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(Show Context)
Citation Context ...esent states of an entire system whereas events in event structure models represent local states of sequential agents. The first ideas about branching time logics appeared in the papers of Abrahamson =-=[1]-=-, [2]. One year later Ben-Ari, Manna, and Pnueli defined the Unified Branching Time System (UB) [3]. In the same year, Clarke and Emerson extended expressiveness of UB by defining Computation Tree Log... |

2 |
Sistla (1983): Automatic Verification of Finite State Concurrent Systems Using Temporal Logic Spesifications: A Practical Approach
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Citation Context ...fined the Unified Branching Time System (UB) [3]. In the same year, Clarke and Emerson extended expressiveness of UB by defining Computation Tree Logic (CTL) [4]; their results are fully described in =-=[5]-=-. In 1983, Emerson and Halpern gave the definition of CTL* [12]. The first logic based on partial orders (POTL) was defined by Pinter and Wolper in 1984 [34] and then extended by Kornatzky and Pinter ... |

2 |
An extension to partial order temporal logic (potl
- Kornatzky, Pinter
- 1987
(Show Context)
Citation Context .... In 1983, Emerson and Halpern gave the definition of CTL* [12]. The first logic based on partial orders (POTL) was defined by Pinter and Wolper in 1984 [34] and then extended by Kornatzky and Pinter =-=[18]-=-. Three years later Katz and Peled presented Interleaving Set Temporal Logic [21] based on configuration structures. The history of event structure logics is much shorter. The first logic dealing with... |

2 |
Thiagarajan P S 1989 An axiomatization of event structures, LNCS
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Citation Context ...as defined by Lodaya and Thiagarajan [25] in 1987. Then, Penczek [29] presented a temporal logic based on unrestricted event structures. This logic has been extended and refined consequently in [30], =-=[27]-=-, and [28]. In this paper the logics mentioned above are presented. The same pattern of presentation is followed for each logic. Firstly, the syntax and the semantics of a logic is given. Then, it is ... |

2 |
The Complexity of
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Citation Context ...or CTL*. Unfortunately, the method of assigning the subformulas of a tested formula to the states of the model cannot be applied. One has to use more powerful automata theoretic methods. Theorem 2.15 =-=[36]-=- Model checking for CTL* is PSPACE-complete. 2 The main reason that CTL* is not broadly applied is the high complexity of checking satisfiability and performing model checking. There are, however, man... |

1 |
Jutla (1988): The complexity of free automata and logics of programs
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Citation Context ...y of CTL* is still open. Checking whether a CTL* formula is satisfiable is much more difficult than in the case of CTL. But this is still decidable. Temporal Logics on Partial Orders 220 Theorem 2.14 =-=[13]-=- There is an algorithm for deciding whether a CTL* formula is satisfiable of deterministic double exponential complexity in the length of the formula. 2 The proof of this theorem consists in showing t... |

1 | Sistla (1984): Deciding Full Branching Time Logic - Emerson, P |

1 | Branching Time Temporal Logic, LNCS - Emerson, Srinivasan - 1988 |

1 |
Hyper Finite State Systems: A Hypergraph Model for Distributed Finite State Systems
- Kornatzky, Pinter
- 1986
(Show Context)
Citation Context ...d by the fact that the initial or the terminal state of some process may have no predecessors or successors, respectively. This kind of frames is required, for example, for reasoning about Petri Nets =-=[20]-=-. The semantics of POTL[U ; U ] formulas (except for the new ones) is like for POTL, involving the following new definition of a forward (backward) fullpath. A forward (backward) fullpath is a maximal... |

1 |
Thiagarajan (1993): Decidability of a Partial Order Based Temporal Logic. LNCS 700
- Lodaya, Ramanujam, et al.
(Show Context)
Citation Context ...the completeness theorem is infinite because the logic lacks the finite model property. The proof of this fact is similar to the corresponding proof for ESL, which will be given further. Theorem 4.39 =-=[23]-=- SESL is decidable. 2 Logics on Event Structures 239 In fact in [23] the decidability of a slightly different logic is shown, but with a few changes, the same proof can be applied for SESL. There are ... |

1 |
Thiagarajan (1987): A Modal logic for a Subclass of Event Structures
- Lodaya, S
(Show Context)
Citation Context ...d on configuration structures. The history of event structure logics is much shorter. The first logic dealing with n-sequential event Introduction 205 structures was defined by Lodaya and Thiagarajan =-=[25]-=- in 1987. Then, Penczek [29] presented a temporal logic based on unrestricted event structures. This logic has been extended and refined consequently in [30], [27], and [28]. In this paper the logics ... |

1 |
S.Thiagarajan (1992): An Axiomatization of Well Branching Prime Event
- Mukund, P
(Show Context)
Citation Context ... by Lodaya and Thiagarajan [25] in 1987. Then, Penczek [29] presented a temporal logic based on unrestricted event structures. This logic has been extended and refined consequently in [30], [27], and =-=[28]-=-. In this paper the logics mentioned above are presented. The same pattern of presentation is followed for each logic. Firstly, the syntax and the semantics of a logic is given. Then, it is shown how ... |

1 |
Event Structure Semantics for CCS and Realated Languages
- Winskel
- 1982
(Show Context)
Citation Context ...subclass of event structures, i.e., for n-agent event structures, is presented. For example, the semantics of asynchronous comunicating sequential processes can be defined by n-agent event structures =-=[38]. 4.2-=-.1 Syntax of SESL Let AP be a countable set of atomic propositions. The set T = ft 1 ; :::; t ng consisting of n atomic type propositions is fixed. It is assumed that AP "T = ; and \Phi = AP [T .... |

1 | Event Structures. LNCS 255 - Winskel - 1986 |

1 |
S4 + S5 together - S4+5
- Vakarelow
- 1987
(Show Context)
Citation Context ...atisfies all the properties of pseudo-models. The next step in the proof consists in unwinding the canonical pseudo-model M 0 into a modally equivalent model of ESL. The method, called "copying&q=-=uot; (see [41]-=-), is used. The structure M 1 = ((E 1 ;s1 , ] 1 ); V 1 ) is defined as follows: ffl E 1 = E 0 \Theta I, where I is a set of integers; e j abbreviates (e; j) for e 2 E 0 and j 2 I. ffls1 = f(e i ; e 0 ... |