**Update:** Revised reference to Fages and Rizk time series paper — see below.

The slides for this lecture review Linear Temporal Logic (LTL), with several examples of LTL formulae for expressing properties of Petri Net behaviour.

Boardwork then gave the expansion from linear time to branching time, the syntax for CTL formulae, and some small discussion of their interpretation.

The final slide gives some further reading, reproduced below. All articles are linked to web pages where you should be able to download copies: you may be asked to log in with the EASE authentication service along the way.

- M. Heiner, D. Gilbert, and R. Donaldson. Petri Nets for systems and synthetic biology. In
*Formal Methods for Computational Systems Biology*, Lecture Notes in Computer Science 5016, pages 215–264. Springer-Verlag, 2008. -
Following the detailed material on Petri Net properties, with their associated transition systems and reachability graphs, there is some discussion of how to apply temporal logics to these: Section 4.2(6)

*Model Checking of Special Behavioural Properties.* - P. T. Monteiro, D. Ropers, R. Mateescu, A. T. Freitas, and H. de Jong. Temporal logic patterns for querying dynamic models of cellular interaction networks.
*Bioinformatics*, 24(16):227–233, 2008. -
This is from the proceedings of the ECCB 2008 conference on Computational Biology, where the authors presented a talk on this work.

The authors describe a system of templates for converting readable sentences that capture “recurring biological questions” into statements of temporal logic, thereby enabling automated model-checking of these questions on specific networks of molecular reactions.

The paper uses the Genetic Network Analyzer (GNA) tool. You can find out more about this at the GNA home page, which includes more papers and a tutorial guide to using the tool.

- François Fages and Aurélien Rizk. On the Analysis of Numerical Data Time Series in Temporal Logic. In
*Computational Methods in Systems Biology: Proceedings of the International Conference CMSB 2007*, Lecture Notes in Bioinformatics 4695, pages 48–63. Springer-Verlag, 2007. -
**Note:**The slides originally referred to the full journal version of this paper,*On Temporal Logic Constraint Solving for the Analysis of Numerical Data Time series.*You can read that too, of course, but it contains much material on the algorithmics involved, rather than systems biology; in particular the efficient implementation of model-checking the kind of arithmetic constraints that are used in the time-series analysis.Fages and Rizk use Linear Temporal Logic to describe, and check, properties of time-series data taken from simulations of biochemical systems. These properties refer to real-valued variables over time, and the authors are able to compute not just whether a statement is true or false, but the range of real values which make it true.

You can find out more about the Biochemical Abstract Machine (BIOCHAM) they used at the BIOCHAM web site.