# Revisiting separation: algorithms and complexity

## Abstract

Abstract Linear temporal logic (LTL) with Since and Until modalities is expressively equivalent, over the class of complete linear orders, to a fragment of first-order logic known as FOMLO (first-order monadic logic of order). It turns out that LTL, under some basic assumptions, is expressively complete if and only if it has the property, called separation, that every formula is equivalent to a Boolean combination of formulas that each refer only to the past, present or future. Herein we present simple algorithms and their implementations to perform separation of the LTL with Since and Until, over discrete and complete linear orders, and translation from FOMLO formulas into equivalent temporal logic formulas. We additionally show that the separation of a certain fragment of LTL results in at most a double exponential size growth.

- Authors:

- Departamento de Matemática, Instituto Superior Técnico, U Lisboa, Portugal
- Departamento de Matemática, Instituto Superior Técnico, U Lisboa, Portugal and Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, U Lisboa, Portugal

- Publication Date:

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE), Fuel Cycle Technologies (NE-5)

- OSTI Identifier:
- 1600564

- Grant/Contract Number:
- UIDB/04561/2020

- Resource Type:
- Published Article

- Journal Name:
- Logic Journal of the IGPL

- Additional Journal Information:
- Journal Name: Logic Journal of the IGPL; Journal ID: ISSN 1367-0751

- Publisher:
- Oxford University Press

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

### Citation Formats

```
Oliveira, Daniel, and Rasga, João. Revisiting separation: algorithms and complexity. Country unknown/Code not available: N. p., 2020.
Web. doi:10.1093/jigpal/jzz081.
```

```
Oliveira, Daniel, & Rasga, João. Revisiting separation: algorithms and complexity. Country unknown/Code not available. doi:10.1093/jigpal/jzz081.
```

```
Oliveira, Daniel, and Rasga, João. Tue .
"Revisiting separation: algorithms and complexity". Country unknown/Code not available. doi:10.1093/jigpal/jzz081.
```

```
@article{osti_1600564,
```

title = {Revisiting separation: algorithms and complexity},

author = {Oliveira, Daniel and Rasga, João},

abstractNote = {Abstract Linear temporal logic (LTL) with Since and Until modalities is expressively equivalent, over the class of complete linear orders, to a fragment of first-order logic known as FOMLO (first-order monadic logic of order). It turns out that LTL, under some basic assumptions, is expressively complete if and only if it has the property, called separation, that every formula is equivalent to a Boolean combination of formulas that each refer only to the past, present or future. Herein we present simple algorithms and their implementations to perform separation of the LTL with Since and Until, over discrete and complete linear orders, and translation from FOMLO formulas into equivalent temporal logic formulas. We additionally show that the separation of a certain fragment of LTL results in at most a double exponential size growth.},

doi = {10.1093/jigpal/jzz081},

journal = {Logic Journal of the IGPL},

number = ,

volume = ,

place = {Country unknown/Code not available},

year = {2020},

month = {2}

}

DOI: 10.1093/jigpal/jzz081

Works referenced in this record:

##
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journal, July 2000

- Etessami, Kousha; Wilke, Thomas
- Information and Computation, Vol. 160, Issue 1-2

##
The complexity of propositional linear temporal logics

journal, July 1985

- Sistla, A. P.; Clarke, E. M.
- Journal of the ACM (JACM), Vol. 32, Issue 3

##
A Proof of Kamp's theorem

journal, January 2014

- Rabinovich, Alexander
- Logical Methods in Computer Science, Vol. 10, Issue 1