## A Really Temporal Logic

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Citations: | 259 - 27 self |

### BibTeX

@MISC{Alur_areally,

author = {Rajeev Alur and Thomas A. Henzinger},

title = {A Really Temporal Logic },

year = {}

}

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### Abstract

We introduce a temporal logic for the specification of real-time systems. Our logic, TPTL, employs a novel quantifier construct for referencing time: the freeze quantifier binds a variable to the time of the local temporal context. TPTL is both a natural language for specification and a suitable formalism for verification. We present a tableau-based decision procedure and a model checking algorithm for TPTL. Several generalizations of TPTL are shown to be highly undecidable.

### Citations

4159 |
Introduction to Automata Theory, Languages, and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...inistic version of the tableau-based decision procedure [Wol83]; the hardness part is patterned after a proof that the universality problem of regular expressions with exponentiation is EXPSPACE-hard =-=[HU79]-=-. [EXPSPACE] It suffices to show that the complementary problem of checking the satisfiability of a TPTL-formula is in nondeterministic EXPSPACE and, hence, by Savitch's theorem, in (deterministic) EX... |

1618 | The Temporal Logic of Reactive and Concurrent Systems - Manna, Pnueli - 1991 |

1314 | The temporal logic of programs - Pnueli - 1977 |

891 |
Theory of Recursive Functions and Effective Computability
- Rogers
- 1967
(Show Context)
Citation Context ...1 1 -hard problem of 2-counter machines to the respective satisfiability problems. It follows that they cannot even be (recursively) axiomatized (for an exposition of the analytical hierarchy consult =-=[Rog67]-=-). 4.1 A \Sigma 1 1 -complete problem A nondeterministic 2-counter machine M consists of two counters C and D, and a sequence of n instructions, each of which may increment or decrement one of the cou... |

384 | Model checking for real-timed systems - Alur, Courcoubetis, et al. - 1990 |

348 |
The complexity of propositional linear temporal logics
- Sistla, Clarke
- 1985
(Show Context)
Citation Context ...OE-paths) can be omitted. 3.2 Complexity of TPTL The following theorem establishes TPTL as being exponentially harder to decide than its untimed base PTL, which has a PSPACE-complete decision problem =-=[SC85]-=-. The extra exponential is caused by the succinct representation of time constants in TPTL and is typical for many real-time specification languages [AH90]. Theorem 2 (Complexity of TPTL) The validity... |

302 |
First-Order Logic
- Smullyan
- 1995
(Show Context)
Citation Context ...s subsection, we are given a formula OE of TPTL and wish to determine if OE is satisfiable. The tableau method searches systematically for a model of OE. It originated with the propositional calculus =-=[Smu68]-=- and was first applied to obtain a decision procedure for a modal logic of computation in the case of dynamic logic [Pra80]. We follow the standard presentation of the tableau-based decision procedure... |

242 |
On the Temporal Analysis of Fairness
- Gabbay, Pnueli, et al.
- 1980
(Show Context)
Citation Context ... of a formula will be important for locating the computational complexity of problems whose input includes formulas of TPTL. The temporal operators. TPTL is based on the two temporal operators of PTL =-=[GPSS80]-=-. The next formula flp asserts about a timed state sequence that the second state in the sequence satisfies the proposition p. The until formula p U q asserts about a timed state sequence that there i... |

230 |
Safety analysis of timing properties in real-time systems
- Jahanian, Mok
- 1986
(Show Context)
Citation Context ...plete with the addition of a single unary predicate. The proof technique we used to show Presburger TPTL undecidable can be applied to many real-time specification languages, including the logics RTL =-=[JM86]-=-, GCTL [Har88], RTTL [Ost90], and MTL [Koy90]. All of these formalisms admit addition over time as a primitive, which renders them undecidable (see [AH90]). 4.4 Dense TPTL An alternative way of extend... |

219 | T.A.: Real-Time Logics: Complexity and Expressiveness
- Alur, Henzinger
- 1990
(Show Context)
Citation Context ...ms for verification. While the tableau method for TPTL will be developed in Section 3, the validity problem for TPTL with unconstrained classical quantification was recently shown to be nonelementary =-=[AH90]. TPTL ver-=-sus bounded temporal operators Several researchers have proposed to add an infinite supply of real-time modalities such as 3ffi ("eventually within ffi time units") to PTL [PH88, Koy90] or b... |

217 | T.: The benefits of relaxing punctuality - Alur, Feder, et al. - 1996 |

216 | An old-fashioned recipe for real time - Abadi, Lamport - 1994 |

211 |
Specifying real-time properties with metric temporal logic
- Koymans
- 1990
(Show Context)
Citation Context ...edicate. The proof technique we used to show Presburger TPTL undecidable can be applied to many real-time specification languages, including the logics RTL [JM86], GCTL [Har88], RTTL [Ost90], and MTL =-=[Koy90]-=-. All of these formalisms admit addition over time as a primitive, which renders them undecidable (see [AH90]). 4.4 Dense TPTL An alternative way of extending the expressive power of TPTL is to relax ... |

211 |
Temporal logic can be more expressive
- Wolper
- 1983
(Show Context)
Citation Context ...OE-path. This can be achieved by a straightforward modification of the standard techniques for marking all vertices of a graph that lie on an infinite path along which all eventualities are satisfied =-=[Wol83]-=-. The remaining state graph is called the final tableau for OE. It follows that a TPTL-formula OE has a \Delta-bounded model iff its final tableau is not empty. The procedure for finding the final tab... |

194 | Logics and models of real time: a survey
- Alur, Henzinger
- 1992
(Show Context)
Citation Context ... this paper was published [AH89], many new results concerning realtime temporal logics have been obtained [ACD90, AH90, AFH91, WME92]. We point to [Hen90] for a complete axiomatization of TPTL and to =-=[AH92]-=- for a survey of recent results. 2 Timed Temporal Logic We define Timed Propositional Temporal Logic, TPTL, and demonstrate its adequacy as a real-time specification language. 2.1 Timed state sequence... |

175 | The Temporal Logic of Branching Time - Ben-Ari, Manna, et al. - 1983 |

172 | Synthesis of Communicating Processes from Temporal-Logic Specifications - Wolper - 1982 |

148 | Proving Liveness Properties of Concurrent Programs - Owicki, Lamport - 1982 |

126 | Quantitative temporal reasoning
- Emerson, Mok, et al.
- 1990
(Show Context)
Citation Context ...cidability of finite-state formalisms with explicit time. Alternative approaches to the automatic verification of real-time systems using temporal logic include work on the branching-time logic RTCTL =-=[EMSS89]-=- and the explicit-clock logic XCTL [HLP90]. RTCTL makes the simplifying assumption of modeling synchronous real-time systems, all of whose events occur with the ticks of a global clock. In the case of... |

120 | What good are digital clocks - Henzinger, Manna, et al. - 1992 |

108 |
Temporal Logic for Real-Time Systems
- Ostroff
- 1989
(Show Context)
Citation Context ...a single unary predicate. The proof technique we used to show Presburger TPTL undecidable can be applied to many real-time specification languages, including the logics RTL [JM86], GCTL [Har88], RTTL =-=[Ost90]-=-, and MTL [Koy90]. All of these formalisms admit addition over time as a primitive, which renders them undecidable (see [AH90]). 4.4 Dense TPTL An alternative way of extending the expressive power of ... |

64 |
Explicit clock temporal logic
- Harel, Lichtenstein, et al.
- 1990
(Show Context)
Citation Context ...explicit time. Alternative approaches to the automatic verification of real-time systems using temporal logic include work on the branching-time logic RTCTL [EMSS89] and the explicit-clock logic XCTL =-=[HLP90]-=-. RTCTL makes the simplifying assumption of modeling synchronous real-time systems, all of whose events occur with the ticks of a global clock. In the case of XCTL, all formulas are quantifier-free an... |

49 |
A near-optimal method for reasoning about action
- Pratt
- 1980
(Show Context)
Citation Context ...tematically for a model of OE. It originated with the propositional calculus [Smu68] and was first applied to obtain a decision procedure for a modal logic of computation in the case of dynamic logic =-=[Pra80]-=-. We follow the standard presentation of the tableau-based decision procedure for PTL [BMP81, Wol83] and begin by constructing the initial tableau for OE. Checking the satisfiability of OE can then be... |

43 |
Checking that finite-state concurrent programs satisfy their linear specifications
- Lichtenstein, Pnueli
- 1985
(Show Context)
Citation Context ...oral logic. The verification problem asks if all possible runs of the system S satisfy the specification OE. In the case of PTL, the tableau construction can be used to solve the verification problem =-=[LP84]-=-. The initial tableau T (:OE) for the negated specification :OE captures precisely the models of the formula :OE. Hence the system S meets the specification OE iff there is no infinite path that is co... |

39 | Applications of Temporal Logic to the Specification of Real Time Systems - Pnueli, Harel - 1988 |

37 | Theory of Recursive Functions and E ective Computability - Rogers - 1967 |

35 |
The temporal specification and verification of real-time systems
- Henzinger
- 1991
(Show Context)
Citation Context ...esent finite-state real-time systems by abstract state graphs with timing information. Typically, it is not difficult to compile a given concrete syntax into timed state graphs (consult, for example, =-=[Hen91]-=- for the translation of timed transition systems into timed state graphs). Model checking is an algorithmic verification technique that compares the temporal-logic specification of a system against a ... |

30 | Half-order modal logic: How to prove real-time properties
- Henzinger
- 1990
(Show Context)
Citation Context ...restrictions unnecessary. Since an earlier version of this paper was published [AH89], many new results concerning realtime temporal logics have been obtained [ACD90, AH90, AFH91, WME92]. We point to =-=[Hen90]-=- for a complete axiomatization of TPTL and to [AH92] for a survey of recent results. 2 Timed Temporal Logic We define Timed Propositional Temporal Logic, TPTL, and demonstrate its adequacy as a real-t... |

25 | Proving Real-time Properties of Programs with Temporal Logic - Bernstein, Harter - 1981 |

20 | The bene ts of relaxing punctuality - Alur, Feder, et al. - 1991 |

18 | Presburger Arithmetic With Unary Predicates is Π 1 -Complete
- Halpern
- 1991
(Show Context)
Citation Context ...As a corollary we infer that the first-order theory of the natural numbers with , multiplication by 2, and monadic predicates is \Pi 1 1 -complete. A similar result has been obtained independently in =-=[Hal91]-=-, where it is shown that Presburger arithmetic becomes \Pi 1 1 -complete with the addition of a single unary predicate. The proof technique we used to show Presburger TPTL undecidable can be applied t... |

15 | Temporal logic of real-time systems - Ostro |

8 | Checking that nitestate concurrent programs satisfy their linear apecication - Lichtenstein, Pnueli - 1985 |

8 | Applications of temporal logic to the speci cation of real-time systems - Pnueli, Harel - 1988 |

7 |
Temporal analysis of real-time systems
- Harel
- 1988
(Show Context)
Citation Context ...e addition of a single unary predicate. The proof technique we used to show Presburger TPTL undecidable can be applied to many real-time specification languages, including the logics RTL [JM86], GCTL =-=[Har88]-=-, RTTL [Ost90], and MTL [Koy90]. All of these formalisms admit addition over time as a primitive, which renders them undecidable (see [AH90]). 4.4 Dense TPTL An alternative way of extending the expres... |

6 |
Propositional dynamic logic of regular programs
- Harel, Pnueli, et al.
- 1983
(Show Context)
Citation Context ...ng 0. The problem of deciding if a nondeterministic Turing machine has, over the empty tape, a computation in which the starting state is visited infinitely often, has been shown \Sigma 1 1 -complete =-=[HPS83]-=-. Along the same lines we obtain the following result. Lemma 8 (Complexity of 2-counter machines) The problem of deciding if a given nondeterministics2-counter machine has a recurring computation, is ... |

6 | The Temporal Speci cation and Veri cation of Real-time Systems - Henzinger - 1991 |

4 | Half-order modal logic: how toprove real-time properties - Henzinger - 1990 |

2 | Asynchronous propositional temporal logic - Wang, Mok, et al. - 1992 |