## A Propositional Modal Logic of Time Intervals (1996)

Venue: | Journal of the ACM |

Citations: | 120 - 2 self |

### BibTeX

@ARTICLE{Halpern96apropositional,

author = {Joseph Y. Halpern and Yoav Shoham},

title = {A Propositional Modal Logic of Time Intervals},

journal = {Journal of the ACM},

year = {1996},

volume = {38},

pages = {279--292}

}

### Years of Citing Articles

### OpenURL

### Abstract

: In certain areas of artificial intelligence there is need to represent continuous change and to make statements that are interpreted with respect to time intervals rather than time points. To this end we develop a modal temporal logic based on time intervals, a logic which can be viewed as a generalization of pointbased modal temporal logic. We discuss related logics, give an intuitive presentation of the new logic, and define its formal syntax and semantics. We make no assumption about the underlying nature of time, allowing it to be discrete (such as the natural numbers) or continuous (such as the rationals or the reals), linear or branching, complete (such as the reals) or not (such as the rationals). We show, however, that there are formulas in the logic that allow us to distinguish all these situations. We also give a translation of our logic into first-order logic, which allows us to apply some results on first-order logic to our modal one. Finally, we consider the difficulty o...

### Citations

2293 | Maintaining Knowledge about Temporal Intervals
- Allen
- 1983
(Show Context)
Citation Context ...since in addition to "after" we have "immediately after", "during", "beginning", "ending" "overlapping", etc. It turns out that we can expre=-=ss all twelve relations between two distinct intervals (see [1]) by six modal operators: hBi, hEi, and -=-hAi (for "begin", "end" and "after"), and their "transposes" hBi, hEi, and hAi. (In fact, as pointed out by Venema [33], we can express these twelve relations u... |

1215 |
The temporal logic of programs
- Pnueli
- 1977
(Show Context)
Citation Context ...ery developed in theoretical computer science so far is inadequate for our purposes. Modal temporal logic, developed in formal philosophy (e.g., [24]) and first applied to reasoning about programs in =-=[21]-=-, interprets formulas over time points. Also, in most formulations, time is assumed to be discrete (the one exception of which we are aware is [4]). A similar comment applies to dynamic logic [22]: fo... |

837 |
Theory of Recursive Functions and Effective Computability
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(Show Context)
Citation Context ...lem We now turn our attention to the complexity of the validity problem for the logic. We begin with a brief review of the notions of \Pi 1 1 and its dual \Sigma 1 1 . Further details can be found in =-=[25]-=- or any other standard textbook of recursive function theory. Formulas of second-order arithmetic with set variables consist of formulas of first-order arithmetic (that is, in the language with consta... |

828 |
Towards a general theory of action and time
- Allen
- 1984
(Show Context)
Citation Context ...jects? In philosophy one finds logic of both kinds. In computer science almost all interval-based logics construct intervals out of points (the 2 only exception of which we are aware is Allen's logic =-=[2]-=-). We will join the majority, and construct intervals out of points. 2. Commitment to a particular underlying temporal structure. With no exception, all intervalbased temporal logics in computer scien... |

618 |
A logic-based calculus of events
- Kowalski, Sergot
- 1986
(Show Context)
Citation Context ...th modal interval logics, which are the topic of this comparison section. We will therefore not discuss recent work by Allen and Hayes [3], Ladkin [14], Ladkin and Maddux [15], or Kowalski and Sergot =-=[13]-=-. An concise overview of this and other literature is offered in van Benthem's [32]. Although most of the work on modal temporal logic has been point-based, recent years have seen a growing interest i... |

601 |
A Mathematical Introduction to Logic
- Enderton
- 1972
(Show Context)
Citation Context ... any two dense, linear and unbounded structures are elementarily equivalent : they cannot be distinguished by formulas in the first-order logic whose only relation symbols are `=' and `!' (see, e.g., =-=[7]-=-). In particular, it follows that Q and R are elementarily equivalent. However, as we are about to show, Q and R are distinguishable in our logic (although, as we shall show in section 7, it is the ca... |

256 |
A temporal logic for reasoning about processes and plans. Cognitive science
- McDermott
- 1982
(Show Context)
Citation Context ...rrying out outstanding tasks, managing available resources, meeting various deadlines and interacting with other agents. Here there has been some use of temporal formalisms, most notably by McDermott =-=[16]-=- and Allen [2]. Our logic is in fact related to their logics; its translation into first-order logic, either as described in a later section or as described in [29], results in a logic that is not unl... |

210 | Semantical considerations on Floyd-Hoare logic, in
- Pratt
- 1976
(Show Context)
Citation Context ... in [21], interprets formulas over time points. Also, in most formulations, time is assumed to be discrete (the one exception of which we are aware is [4]). A similar comment applies to dynamic logic =-=[22]-=-: formulas are interpreted over time points, and furthermore there is no way to state what happens during the execution of a program. There have been several extensions of these logics that deal with ... |

181 |
Present and Future
- Satellites, Past
- 1982
(Show Context)
Citation Context ...r the intuitive meaning of these statements. The machinery developed in theoretical computer science so far is inadequate for our purposes. Modal temporal logic, developed in formal philosophy (e.g., =-=[24]-=-) and first applied to reasoning about programs in [21], interprets formulas over time points. Also, in most formulations, time is assumed to be discrete (the one exception of which we are aware is [4... |

178 |
Reasoning about Change
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- 1988
(Show Context)
Citation Context ...related to their logics; its translation into first-order logic, either as described in a later section or as described in [29], results in a logic that is not unlike those of McDermott and Allen. In =-=[28]-=- we argue, however, that our logic (and its translation into first-order logic) has the advantages of very clear semantics, greater simplicity, and improved flexibility. To see how our logic lends its... |

132 |
The logic of time
- Benthem
- 1972
(Show Context)
Citation Context ...al extensions of these logics that deal with time intervals rather than just time points. Again, the initial idea of dealing with intervals goes back to the philosophers (e.g., [9], and more recently =-=[12, 26, 5, 31]). In comp-=-uter science, there has recently been work on process logic [23, 19, 11], where intervals (or "paths") represent pieces of computation, and even more recently work on interval temporal logic... |

105 |
A Common-Sense Theory of Time
- Allen, Hayes
- 1985
(Show Context)
Citation Context ...eas over the past few years. However, none of these have to do with modal interval logics, which are the topic of this comparison section. We will therefore not discuss recent work by Allen and Hayes =-=[3]-=-, Ladkin [14], Ladkin and Maddux [15], or Kowalski and Sergot [13]. An concise overview of this and other literature is offered in van Benthem's [32]. Although most of the work on modal temporal logic... |

103 |
Temporal logics in AI: Semantical and ontological considerations
- Shoham
- 1987
(Show Context)
Citation Context ...rmalisms, most notably by McDermott [16] and Allen [2]. Our logic is in fact related to their logics; its translation into first-order logic, either as described in a later section or as described in =-=[29]-=-, results in a logic that is not unlike those of McDermott and Allen. In [28] we argue, however, that our logic (and its translation into first-order logic) has the advantages of very clear semantics,... |

77 |
A Really Abstract Concurrent Model and its Temporal Logic
- Barringer, Kuiper, et al.
- 1986
(Show Context)
Citation Context ...4]) and first applied to reasoning about programs in [21], interprets formulas over time points. Also, in most formulations, time is assumed to be discrete (the one exception of which we are aware is =-=[4]-=-). A similar comment applies to dynamic logic [22]: formulas are interpreted over time points, and furthermore there is no way to state what happens during the execution of a program. There have been ... |

42 | Instants and intervals - Hamblin - 1971 |

42 |
Reasoning about digital circuits
- Moszkowski
- 1983
(Show Context)
Citation Context .... In computer science, there has recently been work on process logic [23, 19, 11], where intervals (or "paths") represent pieces of computation, and even more recently work on interval tempo=-=ral logic [17, 8]-=-. We review these logics and others in more detail in a later section. While these interval logics and process logics come closer to satisfying our goals than pointbased temporal logics do, they are s... |

40 | Process logic: expressiveness, decidability, completeness
- Harel, Kozen, et al.
- 1982
(Show Context)
Citation Context ...nts. Again, the initial idea of dealing with intervals goes back to the philosophers (e.g., [9], and more recently [12, 26, 5, 31]). In computer science, there has recently been work on process logic =-=[23, 19, 11], where in-=-tervals (or "paths") represent pieces of computation, and even more recently work on interval temporal logic [17, 8]. We review these logics and others in more detail in a later section. Whi... |

35 |
Process logic
- Pratt
- 1979
(Show Context)
Citation Context ...nts. Again, the initial idea of dealing with intervals goes back to the philosophers (e.g., [9], and more recently [12, 26, 5, 31]). In computer science, there has recently been work on process logic =-=[23, 19, 11], where in-=-tervals (or "paths") represent pieces of computation, and even more recently work on interval temporal logic [17, 8]. We review these logics and others in more detail in a later section. Whi... |

29 | The Logic of Time Representation
- Ladkin
- 1987
(Show Context)
Citation Context ... past few years. However, none of these have to do with modal interval logics, which are the topic of this comparison section. We will therefore not discuss recent work by Allen and Hayes [3], Ladkin =-=[14]-=-, Ladkin and Maddux [15], or Kowalski and Sergot [13]. An concise overview of this and other literature is offered in van Benthem's [32]. Although most of the work on modal temporal logic has been poi... |

27 | Propositional dynamic logic of nonregular programs
- Harel, Pnueli, et al.
- 1983
(Show Context)
Citation Context ..., we construct a formula that is satisfiable iff there is a computation of the TM that enters the start state infinitely often. For nondeterministic TM's, this problem is known to be \Sigma 1 1 -hard =-=[10]-=-, so this gives us that satisfiability is \Sigma 1 1 -hard, and thus that validity is \Pi 1 1 -hard. Finally, for Theorem 8.6, we construct a formula that is satisfiable iff the TM halts. We proceed a... |

27 |
Expressiveness and completeness of an interval tense logic
- Venema
- 1990
(Show Context)
Citation Context ...tions between two distinct intervals (see [1]) by six modal operators: hBi, hEi, and hAi (for "begin", "end" and "after"), and their "transposes" hBi, hEi, and =-=hAi. (In fact, as pointed out by Venema [33]-=-, we can express these twelve relations using just hBi, hEi, and their transposes.) The semantics of these operators is quite natural and simple. For example, hBi' is true of an interval hs; ti exactl... |

24 |
An interval logic for higher-level temporal reasoning
- Schwartz, Melliar-Smith, et al.
- 1983
(Show Context)
Citation Context ...veral interval-based logics. Process logic [23, 11] is a generalization of dynamic logic. Interval temporal logic [17, 8] is a generalization of point-based temporal logic. For other related work see =-=[19, 18, 27]-=-. In Section 9 we discuss these logics and their relation to ours in more detail. Here let us just point out issues that distinguish between the different logics, both in philosophy and in computer sc... |

14 |
Interval semantics for tense logic: some remarks
- Humberstone
- 1979
(Show Context)
Citation Context ...al extensions of these logics that deal with time intervals rather than just time points. Again, the initial idea of dealing with intervals goes back to the philosophers (e.g., [9], and more recently =-=[12, 26, 5, 31]). In comp-=-uter science, there has recently been work on process logic [23, 19, 11], where intervals (or "paths") represent pieces of computation, and even more recently work on interval temporal logic... |

12 |
A decidability result for second order process logic
- Parikh
- 1978
(Show Context)
Citation Context ...nts. Again, the initial idea of dealing with intervals goes back to the philosophers (e.g., [9], and more recently [12, 26, 5, 31]). In computer science, there has recently been work on process logic =-=[23, 19, 11], where in-=-tervals (or "paths") represent pieces of computation, and even more recently work on interval temporal logic [17, 8]. We review these logics and others in more detail in a later section. Whi... |

12 |
Time, logic and computation
- Benthem
- 1989
(Show Context)
Citation Context ...herefore not discuss recent work by Allen and Hayes [3], Ladkin [14], Ladkin and Maddux [15], or Kowalski and Sergot [13]. An concise overview of this and other literature is offered in van Benthem's =-=[32]-=-. Although most of the work on modal temporal logic has been point-based, recent years have seen a growing interest in interval-based modal logics. As usual, the initial idea of dealing with intervals... |

8 | The algebra of convex time intervals
- Ladkin, Maddux
(Show Context)
Citation Context ..., none of these have to do with modal interval logics, which are the topic of this comparison section. We will therefore not discuss recent work by Allen and Hayes [3], Ladkin [14], Ladkin and Maddux =-=[15]-=-, or Kowalski and Sergot [13]. An concise overview of this and other literature is offered in van Benthem's [32]. Although most of the work on modal temporal logic has been point-based, recent years h... |

8 |
Descriptively complete process logic
- Nishimura
- 1980
(Show Context)
Citation Context ...veral interval-based logics. Process logic [23, 11] is a generalization of dynamic logic. Interval temporal logic [17, 8] is a generalization of point-based temporal logic. For other related work see =-=[19, 18, 27]-=-. In Section 9 we discuss these logics and their relation to ours in more detail. Here let us just point out issues that distinguish between the different logics, both in philosophy and in computer sc... |

6 |
Intervals and Tenses
- Roper, P
- 1980
(Show Context)
Citation Context ...al extensions of these logics that deal with time intervals rather than just time points. Again, the initial idea of dealing with intervals goes back to the philosophers (e.g., [9], and more recently =-=[12, 26, 5, 31]). In comp-=-uter science, there has recently been work on process logic [23, 19, 11], where intervals (or "paths") represent pieces of computation, and even more recently work on interval temporal logic... |

4 |
A high-level semantics based on interval logic
- Halpern, Manna, et al.
- 1983
(Show Context)
Citation Context .... In computer science, there has recently been work on process logic [23, 19, 11], where intervals (or "paths") represent pieces of computation, and even more recently work on interval tempo=-=ral logic [17, 8]-=-. We review these logics and others in more detail in a later section. While these interval logics and process logics come closer to satisfying our goals than pointbased temporal logics do, they are s... |

2 |
Axioms for tense logic, ii. Time periods
- Burgess
- 1982
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Citation Context |

1 |
Special volume on qualitative reasoning and physical systems
- Bobrow, editor
- 1984
(Show Context)
Citation Context ...isions between moving objects. Although reasoning about time is clearly central to qualitative physics, the actual work that has been done makes little use of explicit temporal formalisms (see, e.g., =-=[6]). To-=- illustrate the fact that our logic lends itself nicely to this research domain, consider representing the sentence "if you open the tap then, unless someone punctures the canteen, the canteen wi... |

1 |
A Low-Level Language for Obtaining Decision
- Plaisted
- 1984
(Show Context)
Citation Context ...at the first time x equals y and ending at the first time after that when y equals 16. Intervals are assumed to consist of linearly ordered and discrete time points, and again locality is assumed. In =-=[20]-=- it is shown that satisfiability for this logic is nonelementarily decidable. Finally, Venema has recently obtained further results regarding our logic [33]. Besides the expressibility results mention... |

1 |
Global Process Logic is \Pi 1 -complete
- Streett
(Show Context)
Citation Context ...ssed, and leave open the question of decidability in the absence of this axiom of locality. Later Streett settled this question by showing that global propositional process logic is \Pi 1 1 -complete =-=[30]-=-. Interval temporal logic [8, 17] is also an extension of temporal logic in which formulas are interpreted over paths. The two modal operators considered there are fl (next) and ; (chop). The meaning ... |