# Decidability Results for Metric and Layered Temporal Logics

@article{Montanari1996DecidabilityRF, title={Decidability Results for Metric and Layered Temporal Logics}, author={Angelo Montanari and Alberto Policriti}, journal={Notre Dame J. Formal Log.}, year={1996}, volume={37}, pages={260-282} }

Westudy the decidability problem for metric and layered temporal logics. The logics we consider are suitable to model time granularity in various contexts, and they allow one to build granular temporal models by referring to the “natural scale” in any component of the model and by properly constraining the interactions between differently-grained components. A monadic second-order language combining operators such as temporal contextualization and projection, together with the usual…

## 38 Citations

A decidable theory of nitely-layered metric temporal structures

- Computer Science
- 1996

The theory of nitely-layered metric temporal structures is shown to be decidable and a monadic second-order language combining operators such as temporal contextualization and projection, together with the usual displacement operator of metric temporal logics, is considered.

Theories of !-layered Metric Temporal Structures: Expressiveness and Decidability

- Computer Science
- 1999

It is proved that the decidability of both the theory of metric temporal structures provided with an innnite number of arbitrarily coarse temporal layers and the theorists of metric and layered temporal logics provided with the same number of temporal layers are correct.

Decidable Theories of ω-Layered Metric Temporal Structures

- Computer Science, Philosophy
- 2000

It is proved that the decidability of both the theory of metric temporal structures provided with an infinite number of arbitrarily coarse temporal layers and the Theory of metricporal structures providedWith an infiniteNumber of arbitrarily fine temporal layers is correct.

Decidability of Interval Temporal Logics over Split-Frames via Granularity

- Computer Science, PhilosophyJELIA
- 2002

It is shown that there exists a straightforward correspondence between Split Logic and the first-order fragments of the monadic theories of time granularity proposed in the literature, which allows existing decidability results for such theories to be transferred to Split Logic.

Definability and decidability of binary predicates for time granularity

- Computer ScienceJ. Appl. Log.
- 2006

The definability and decidability of binary predicates for time granularity with respect to monadic theories over finitely and infinitely layered structures is studied.

Towards an automata-theoretic counterpart of combined temporal logics

- Computer Science
- 2002

A new class of combined automata is defined, called temporalized automata, which can be viewed as the automata-theoretic counterpart of temporalized logics, and it is successfully applied to provide the full secondorder theory of k-refinable downward unbounded layered structures with a temporal logic counterpart.

Temporalized logics and automata for time granularity

- Computer ScienceTheory and Practice of Logic Programming
- 2004

A new class of combined automata is defined, called temporalized automata, which can be proved to be the automata-theoretic counterpart of temporalized logics, and it is shown that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporalLogics, transfer from component automata to temporalized ones.

The Way to Go : Multi-Level Temporal Logics

- Computer Science
- 1999

Time granularity can be viewed not only as an important feature of a representation language, but as a formal tool to investigate expressiveness and decidability properties of temporal theories in logics, theories, and structures.

Branching within Time: An Expressively Complete and Elementarily Decidable Temporal Logic for Time Granularity

- Mathematics
- 2003

Suitable extensions of monadic second-order theories of ksuccessors have been proposed in the literature to specify in aconcise way reactive systems whose behaviour can be naturallymodeled with…

The Way to Go : Multi-Level Temporal

- Computer Science
- 1999

This paper surveys the main contributions of the research on time granularity and outlines some directions for current and future researches, and deened a uniform framework for time that combines the tense logic and the logic of program perspectives.

## References

SHOWING 1-10 OF 27 REFERENCES

A decidable theory of nitely-layered metric temporal structures

- Computer Science
- 1996

The theory of nitely-layered metric temporal structures is shown to be decidable and a monadic second-order language combining operators such as temporal contextualization and projection, together with the usual displacement operator of metric temporal logics, is considered.

Real-time logics: complexity and expressiveness

- Mathematics[1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science
- 1990

Two elementary real-time temporal logics are identified as expressively complete fragments of the theory of timed state sequences, and these two formalisms are well-suited for the specification and verification of real- time systems.

Completeness Results for Two-sorted Metric Temporal Logics

- PhilosophyAMAST
- 1995

This paper provides an axiomatization of the pure metric fragment of the logic, and proves its soundness and completeness, and shows how to obtain the metric temporal logic of linear orders by adding an ordering over displacements.

Combining Temporal Logic Systems

- PhilosophyNotre Dame J. Formal Log.
- 1996

The method of full join of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several cases, so a weaker method of restricted join is proposed and shown to transfer all three basic logical properties.

Embedding Time Granularity in a Logical Specification Language for Synchronous Real-Time Systems

- Computer ScienceSci. Comput. Program.
- 1993

Sometimes "Tomorrow" is "Sometime" - Action Refinement in a Temporal Logic of Objects

- Computer Science, PhilosophyICTL
- 1994

This work addresses the hierarchical (vertical) decomposition, or abstract implementation, of object specification in temporal logic with a tighter control of action granularity and interference, enabling the use of the “next” operator and the development of reactive systems more tractable.

Logical design for temporal databases with multiple granularities

- Computer ScienceTODS
- 1997

Temporal Boyce-Codd normal form (TBCNF) that avoids all data redundancies due to TFDs, and temporal third normal form [T3NF] that allows dependency preservation, are defined.

On the temporal analysis of fairness

- Computer SciencePOPL '80
- 1980

It is shown that with the addition of the 'until' operator -U the temporal language becomes expressively complete and two deductive systems DX and DUX are proved to be complete for the languages without and with the new operator respectively.

Automata on Infinite Objects

- Computer ScienceHandbook of Theoretical Computer Science, Volume B: Formal Models and Sematics
- 1990

Specifying Message Passing and Time-Critical Systems with Temporal Logic

- PhilosophyLecture Notes in Computer Science
- 1992

A review of modal and temporal logic with inequality finds that message passing systems and time-critical systems using modal logics with inequality are viable.