## Temporal and modal logic (1995)

Venue: | HANDBOOK OF THEORETICAL COMPUTER SCIENCE |

Citations: | 1136 - 17 self |

### BibTeX

@INPROCEEDINGS{Emerson95temporaland,

author = {E. Allen Emerson},

title = {Temporal and modal logic},

booktitle = {HANDBOOK OF THEORETICAL COMPUTER SCIENCE},

year = {1995},

pages = {995--1072},

publisher = {Elsevier}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a comprehensive and unifying survey of the theoretical aspects of Temporal and modal logic.

### Citations

1255 | The Temporal Logic of Programs, in - Pnueli - 1977 |

825 |
Design and Synthesis of Synchronization Skeletons using Branching Time Temporal Logic
- Clarke, Emerson
- 1981
(Show Context)
Citation Context ...corresponds to what one might naturally rst think of as a branching time logic. CTL is closely related to branching time logics proposed in [La80], [EC80], [QS81], [BPM81], and was itself proposed in =-=[CE81]-=-. However, as we shall see, its syntactic restrictions signi cantly limit its expressive power. We therefore also consider the much richer language CTL*, which is sometimes referred to informally as f... |

615 | A Mathematical Introduction to Logic - Enderton - 1972 |

607 |
An automata-theoretic approach to automatic program verification
- Vardi, Wolper
- 1986
(Show Context)
Citation Context ...oach. First, it does provide the only known elementary time decision procedures for some logics. Secondly, automata can provide a general, uniform framework encompassing temporal reasoning (cf. [VW5] =-=[VW86]-=- [V87]). Automata themselves have been proposed as a potentially useful speci cation language. Automata, moreover,bearanobvious relation to temporal structures, abstract concurrent programs, etc. This... |

505 | Knowledge and common knowledge in a distributed environment
- Halpern, Moses
- 1990
(Show Context)
Citation Context ...ted systems, it is a natural metaphor to refer to what a process knows. Logics of knowledge represent an e ort to provide a formal basis for such reasoning. Anumber of systems have been proposed (cf. =-=[HM84]-=-, [Le84], [LR86], [DM86]). Typical modalities include K ip which means that \process i knows p" and Cp which means that \p is common knowledge" in the sense that \all processes know p, all processes k... |

434 | Introduction to VLSI Systems - Mead, Conway - 1980 |

363 |
Propositional dynamic logic of regular programs
- Fischer, Ladner
- 1979
(Show Context)
Citation Context ...tates are identi ed to collapse a possibly in nite model to a small nite one. An example of a quotient construction is its application to yield a decision procedure for Propositional Dynamic Logic of =-=[FL79]-=-, discussed in [KT89]. There the equivalence relation is de ned so that, in essence, two states are equivalent when they agree (i.e., have the same truth value) on all subformulae of the formula p0 be... |

256 |
sometimes” and ”not never” revisited: On branching versus linear time temporal logic
- Emerson, Halpern
- 1986
(Show Context)
Citation Context ...by allowing quanti cation over possible futures. Both approaches have been applied to program reasoning, and it is a matter of debate as to whether branching or linear time is preferable (cf. [La80], =-=[EH86]-=-, [Pn85]) 2.4 Points versus Intervals Most temporal logic formalisms developed for program reasoning have been based on temporal operators that are evaluated as true or false of points in time. Some f... |

244 |
Tense Logic and the Theory of Linear Order
- Kamp
- 1968
(Show Context)
Citation Context ...il P U > q X(p U q) is of particular interest. Note that false U > q X(false U q) Xq. The single modality strict, strong until is enough to de ne all the other linear time operators (as shown by Kamp =-=[Ka68]-=-.) Remark: One other common variation is simply notational. Some authors use 2p for Gp, 3p for Fp, and p for Xp. Another minor variation is to change the underlying structure to be any initial segment... |

240 | Checking that finite state concurrent programs satisfy their linear specification
- Lichtenstein, Pnueli
- 1985
(Show Context)
Citation Context ...essible within FairCTL, although they are describable in CTL* or even PLTL. The PSPACEcompleteness of these latter logics, on rst hearing, would seem to be a serious drawback. Lichtenstein and Pnueli =-=[LP85]-=- noted, however, that model checking is a problem with two input parameters, the structure and the speci cation, and then proceeded to develop a model checking algorithm for PLTL of complexity exponen... |

239 | Many-Valued Logic - Rescher - 1969 |

237 |
On the Temporal Analysis of Fairness
- Gabbay, Pnueli, et al.
- 1980
(Show Context)
Citation Context ...0) j= q]. Theorem 3.2. As measured with respect to initial equivalence, PLTLB is equivalent inexpressive power to PLTLF. This can be proved using results regarding the theory of linear orderings (cf. =-=[GPSS80]-=-): We also note the following relationship between i and g: Proposition 3.3. p g qi Gp i Gq. By convention we shall take satis able to mean initially satis able and valid to mean initially valid, unle... |

222 |
Counter-free automata
- McNaughton, Papert
- 1971
(Show Context)
Citation Context ...r expression (d) L + is de nable by a nite state automaton The equivalence of conditions (b), (c), and (d) was established using lengthy and di cult arguments in the monograph of McNaughton & Pappert =-=[MP62]-=-. The equivalence of conditions (a) and (b) in Theorem 6.4 was established in Kamp [Ka68], while for Theorem 6.5 it was established in [LPZ85]. Direct translations between PLTL and star-free regular e... |

207 |
Automata-theoretic techniques for modal logic of programs
- Vardi, Wolper
- 1986
(Show Context)
Citation Context ... by the automaton.) General automata-theoretic 48stechniques for reasoning about a number of relatively simple logics, including CTL, using Buchi tree automata have been described by Vardi and Wolper =-=[VW84]-=-. For branching time logics with richer modalities such as CTL*, the tableau construction is not directly applicable. Instead, the problem reduces to constructing a tree automaton for the branching ti... |

184 |
Using branching time temporal logic to synthesize synchronization skeletons
- Emerson, Clarke
- 1982
(Show Context)
Citation Context ...en in [Pr79], [BPM81], [BHP82], [Wo82], [Wo83], [HS84]. See also the excellent survey byWolper [Wo84]. In the sequel we describe a tableaubased decision procedure for CTL formulae, along the lines of =-=[EC82]-=- and [EH85]. The following de nitions and terminology are needed. We assume that the candidate formula p0 is in positive normal form, obtained by pushing negations inward as far as possible using de M... |

180 |
What good is temporal logic
- Lamport
- 1983
(Show Context)
Citation Context ...Approach A great deal of work has been done investigating the proof-theoretic approach toveri cation of concurrent programs using TL (cf. e.g. [Pn81], [MP81], [MP82], [MP83], [La 80], [Ha81], [OL82], =-=[La83]-=-, [SMS82]). Typically, one tries to prove, by hand, that a given program meets a certain TL speci cation using various axioms and inference rules for the system of TL. A drawback of this approach is t... |

168 |
The temporal logic of branching time
- Ben-Ari, Pnueli, et al.
- 1983
(Show Context)
Citation Context ... X (\nexttime"), or U (\until"). It corresponds to what one might naturally rst think of as a branching time logic. CTL is closely related to branching time logics proposed in [La80], [EC80], [QS81], =-=[BPM81]-=-, and was itself proposed in [CE81]. However, as we shall see, its syntactic restrictions signi cantly limit its expressive power. We therefore also consider the much richer language CTL*, which is so... |

166 |
Synthesis of communicating processes from temporal logic speci
- Manna, Wolper
- 1984
(Show Context)
Citation Context ...tion problems such as readers-writers and dining philophers can also be synthesized. A closely related synthesis method for CSP programs based on the use of a decision procedure for PLTL was given in =-=[MW84]-=-. In the recent [PR89] a method for synthesizing an individual component of a reactive system from a speci cation in (essentially) CTL* is described. Earlier informal e orts toward synthesis of concur... |

155 |
The temporal semantics of concurrent programs
- Pnueli
- 1981
(Show Context)
Citation Context ... 7.2 Veri cation of Concurrent Programs: Proof-Theoretic Approach A great deal of work has been done investigating the proof-theoretic approach toveri cation of concurrent programs using TL (cf. e.g. =-=[Pn81]-=-, [MP81], [MP82], [MP83], [La 80], [Ha81], [OL82], [La83], [SMS82]). Typically, one tries to prove, by hand, that a given program meets a certain TL speci cation using various axioms and inference rul... |

153 |
Myths about the mutual exclusion problem
- Peterson
- 1981
(Show Context)
Citation Context ...n be determined by inspection, considering only the potentially falsifying transitions and ignoring those which obviously cannot make false. As an example, we nowverify safety forPeterson's solution (=-=[Pe81]-=-) to the mutual exclusion problem shown in Figure 8. Each process has a noncritical section (l0, m0, resp.) in which it idles unless it needs access to its critical section (l3, m3, resp.), signalled ... |

145 |
Proving Liveness Properties of Concurrent Programs
- Owicki, Lamport
- 1982
(Show Context)
Citation Context ...rrent Programs There are a large number of correctness properties that we might wish to specify for a concurrent program. These correctness properties usually fall into two broad classes (cf. [Pn77], =-=[OL82]-=-). One class is that of \safety" properties also known as \invariance" properties. Intuitively, a safety property asserts that \nothing bad happens." The other class consists of the \liveness" propert... |

143 | Decision Procedures and Expressiveness in the Temporal Logic of Branching Time
- Emerson, Halpern
- 1985
(Show Context)
Citation Context ...], [BPM81], [BHP82], [Wo82], [Wo83], [HS84]. See also the excellent survey byWolper [Wo84]. In the sequel we describe a tableaubased decision procedure for CTL formulae, along the lines of [EC82] and =-=[EH85]-=-. The following de nitions and terminology are needed. We assume that the candidate formula p0 is in positive normal form, obtained by pushing negations inward as far as possible using de Morgan's law... |

138 |
The Glory of the Past
- Lichtenstein, Pnueli, et al.
- 1985
(Show Context)
Citation Context ...xpressive power. Recently, however, it has been advanced that use of the past tense operators might be useful simply in order to make the formulation of speci cations more natural and convenient (cf. =-=[LPZ85]-=-). Moreover, past tense operators appear to play an important role in compositional speci cation somewhat analogous to that of history variables. 3 The Technical Framework of Linear Temporal Logic 3.1... |

134 | The Complementation Problem for Büchi Automata with Applications to Temporal Logic, Theoretical Computer Science 49 - Sistla, Vardi, et al. - 1987 |

132 |
Knowledge and common knowledge in a Byzantine environment: crash failures
- Dwork, Moses
- 1990
(Show Context)
Citation Context ...ural metaphor to refer to what a process knows. Logics of knowledge represent an e ort to provide a formal basis for such reasoning. Anumber of systems have been proposed (cf. [HM84], [Le84], [LR86], =-=[DM86]-=-). Typical modalities include K ip which means that \process i knows p" and Cp which means that \p is common knowledge" in the sense that \all processes know p, all processes know that all processes k... |

126 |
Characterizing correctness properties of parallel programs using fixpoints
- Emerson, Clarke
- 1980
(Show Context)
Citation Context ... F (\sometime"), X (\nexttime"), or U (\until"). It corresponds to what one might naturally rst think of as a branching time logic. CTL is closely related to branching time logics proposed in [La80], =-=[EC80]-=-, [QS81], [BPM81], and was itself proposed in [CE81]. However, as we shall see, its syntactic restrictions signi cantly limit its expressive power. We therefore also consider the much richer language ... |

120 |
Expressing interesting properties of programs in propositional temporal logic (extended abstract
- Wolper
- 1986
(Show Context)
Citation Context ...essage uniqueness is essential for the correctness of the speci cation. Without it, a computation with, e.g., the same message output twice for each input message would be permitted. Recently, Wolper =-=[Wo86]-=- has provided additional insight into the power of logical formalisms for specifying message bu ers. First, he pointed out that PLTL is a priori inadequate for specifying message bu ers when the under... |

98 | Reasoning about networks with many identical finite state processes
- Browne, Clarke, et al.
- 1989
(Show Context)
Citation Context ...al successes, a potentially serious drawback totheentire model checking approach is that the size of the global state transition graph grows exponentially with the number of processes. Recent work in =-=[CG86]-=-, [SG87], [CG87] suggests that it may be possible to avoid this exponential blowup in some cases for concurrent systems with many \copies" of the same process, although this is not possible in general... |

94 | The tableau method for temporal logic: An overview. Logique et Analyse - Wolper - 1985 |

84 |
Models of program logics
- Pratt
- 1979
(Show Context)
Citation Context ... \a fullpath x in M" is understood to refer to a fullpath x in X. In the most general case X can be completely arbitrary. However, it is often helpful to impose certain requirements on X (cf. [La80], =-=[Pr79]-=-, [Ab80], [Em83]). We saythatXissu x closed provided that if computation s0s1s2... 2 X, then the su x s1s2... 2 X. Similarly, Xisfusion closed provided that whenever x1sy1, x2sy2 2 X then x1sy2 2 X. T... |

77 |
A Really Abstract Concurrent Model and its Temporal Logic
- Barringer, Kuiper, et al.
- 1986
(Show Context)
Citation Context ...nterpreted over a continuous (or dense) time structure such as the reals (or rationals) have beeninvestigated by philosophers. Their application to reasoning about concurrent programs was proposed in =-=[BKP86]-=- to facilitate the formulation of fully abstract semantics. Such continuous time logics may alsohave applications in so-called real-time programs where strict, quantitative performance requirements ar... |

66 | Using reasoning about knowledge to analyze distributed systems - Halpern - 1987 |

65 |
Modalities for model checking: Branching time strikes back
- Emerson, Lei
- 1987
(Show Context)
Citation Context ... L(s) end end end of case end end One limitation of the logic CTL is, of course, that it cannot express correctness under fair scheduling assumptions. However, the extended logic FairCTL described in =-=[EL85]-=- can express correctness under fairness (cf. [QS83]). An FCTL speci cation (p0, 0) consists of a functional assertion p0, which is a state formula, and an underlying fairness assumption 0, which is a ... |

61 | Deciding full branching time logic - Emerson, Sistla - 1984 |

51 | On the relation of programs and computations to models of temporal logic - Wolper - 1987 |

44 | Dynamic logic, Handbook of Philosophical Logic Vol - Harel - 1984 |

43 |
Reasoning about digital circuits
- Moszkowski
- 1983
(Show Context)
Citation Context ...sus Intervals Most temporal logic formalisms developed for program reasoning have been based on temporal operators that are evaluated as true or false of points in time. Some formalisms (cf. [SMV83], =-=[Mo83]-=-, [HS86]),however, have temporal operators that are evaluated over intervals of time, the claim being that use of intervals greatly simpli es the formulation of certain correctness properties. The fol... |

43 |
Fairness and related properties in transition systems – a temporal logic to deal with fairness
- Queille, Sifakis
- 1983
(Show Context)
Citation Context ...e� actually, however, there are a number of technically distinct re nements of this notion. (See, for example, the book by Francez [Fr86] as well as [Ab80], [FK84], [GPSS80], [La80], [LPS81], [Pn83], =-=[QS83]-=-, [LPZ85] and [EL85].) Some of these will be described subsequently. 22sThus to model the semantics of concurrency accurately we need fairness assumptions in addition to the computation sequences gene... |

37 |
On the complexity of omega -automata
- SAFRA
- 1988
(Show Context)
Citation Context ... on in nite objects have also been proposed to facilitate reasoning in TL's (cf. [St81], [VS85], [MP87a]). A particularly important advance in automata theory motivated by TL is Safra's construction (=-=[Sa88]-=-) for determinizing an automaton on in nite strings with only a single exponential blowup, without regard to any special structure possessed by the automaton. Not only is Safra's construction an expon... |

33 |
Probabilistic Temporal Logic for Finite and Bounded Models
- Hart, Sharir
- 1984
(Show Context)
Citation Context ... remark that the tableau construction is a rather general one, that applies to many logics. Tableau-based decision procedures for various logics are given in [Pr79], [BPM81], [BHP82], [Wo82], [Wo83], =-=[HS84]-=-. See also the excellent survey byWolper [Wo84]. In the sequel we describe a tableaubased decision procedure for CTL formulae, along the lines of [EC82] and [EH85]. The following de nitions and termin... |

32 | A temporal logic for reasoning about partially ordered computations - Pinter, Wolper - 1984 |

31 |
Deterministic Propositional Dynamic Logic: Finite Models
- Ben-Ari, Halpern, et al.
- 1982
(Show Context)
Citation Context ...nd intoagenuine model. We remark that the tableau construction is a rather general one, that applies to many logics. Tableau-based decision procedures for various logics are given in [Pr79], [BPM81], =-=[BHP82]-=-, [Wo82], [Wo83], [HS84]. See also the excellent survey byWolper [Wo84]. In the sequel we describe a tableaubased decision procedure for CTL formulae, along the lines of [EC82] and [EH85]. The followi... |

31 |
Linear and Branching Structures in the Semantics and Logics of Reactive Systems
- PNUELI
- 1985
(Show Context)
Citation Context ...ing quanti cation over possible futures. Both approaches have been applied to program reasoning, and it is a matter of debate as to whether branching or linear time is preferable (cf. [La80], [EH86], =-=[Pn85]-=-) 2.4 Points versus Intervals Most temporal logic formalisms developed for program reasoning have been based on temporal operators that are evaluated as true or false of points in time. Some formalism... |

30 | Verifying Concurrent Processes Using Temporal Logic - Hailpern - 1982 |

23 |
Yet another process logic
- Vardi, Wolper
- 1983
(Show Context)
Citation Context ...It was proposed as a unifying framework in [EH86], subsuming both CTL and PLTL, as wellasanumber of other systems. Related systems of high expressiveness are considered in [Pa79], [Ab80], [ST81], and =-=[VW83]-=-. Syntax We now give a formal de nition of the syntax of CTL*. We inductively de ne a class of state formulae (true or false of states) using rules S1-3 below and a class of path formulae (true or fal... |

20 | 1986] The logic of distributed protocols
- Ladner, Reif
(Show Context)
Citation Context ...is a natural metaphor to refer to what a process knows. Logics of knowledge represent an e ort to provide a formal basis for such reasoning. Anumber of systems have been proposed (cf. [HM84], [Le84], =-=[LR86]-=-, [DM86]). Typical modalities include K ip which means that \process i knows p" and Cp which means that \p is common knowledge" in the sense that \all processes know p, all processes know that all pro... |

19 | The taming of converse: Reasoning about two-way computations - Vardi - 1985 |

18 | Decidability and expressiveness of logics of processes - Abrahamson - 1980 |

18 | An elementary proof of the completeness of PDL. Theor - KOZEN, PARIKH - 1981 |

15 | Veri cation of Concurrent Programs: The Temporal Framework - Manna, Pnueli - 1981 |