## Lecture 11: Markov Chains and Stochastic Logics

18 February 2010

Small Petri net example in the form of a signal cascade. Extracting from this in turn: a transition system; a DTMC with transition probabilities; and a CTMC with rates. Corresponding refinements of temporal logic: LTL/CTL; PCTL with probabilities; CSL with probabilities and duration. How this extends expressiveness, and how replacing discrete “always” with probabilistic “almost surely” gives results more closely matching intuition.

Probabilistic symbolic model checking and the PRISM tool. Examples from:

### Homework

Read the remainder of the Kwiatkowska et al. tutorial, §§4–6.

Also, recall this short report:

In the meantime, however, Fisher and her fellow executable-biology enthusiasts have a lot of convincing to do, says Stephen Oliver, a biologist at the University of Cambridge, UK. “Modelling in general is regarded sceptically by many biologists,” he points out.

Do you think this is true? Can you find any evidence for or against?

## Lecture 9: Continuous time

8 February 2010

Note: The slides have been revised since the lecture, removing some untruths, adding more about BDDs and exponential distribution, and providing further references.

Symbolic Model Checking: How advanced algorithms test assertions on systems with states than you can imagine possible.

Road Map Review: Shifting between high-level intuitive models and low-level mathematical ones, with multiple choices and analyses.

Continuous Time: Negative exponential distributions and why we use it; starting with Stochastic Petri Nets.

### Homework

Read the following short articles for Thursday.

## Lecture 8: Mix, match and use temporal logics

6 February 2010

Revisit example valuation of CTL formula, now corrected. Further possible logics: Hennessy-Milner, modal μ-calculus, and monadic second-order logic. Presentation of syntax, semantics, and small examples. Relative expressiveness. Mixing and matching modalities and operators for convenience, distinct from expressiveness: CTL- as an example.

Application to specific biological models, as discussed in these articles: Fages and Rizk evaluating LTL formulae over time series data; de Jong et al. on mapping textual descriptions into CTL.

## Lecture 6: Branching Time and CTL

28 January 2010

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.

## Lecture 5: Linear Temporal Logic

25 January 2010

The slides for this lecture review the notions of labelled transition systems, runs and traces, and give examples of the expansion of Petri net behaviour into labelled transition systems.

Boardwork then gave definitions of LTL formula, their meaning, and some small examples of application to LTS runs and Petri net behaviours.

Heiner et al. discuss how even finite reachability graphs for Petri nets may blow up very fast (§4(5)). However, they then jump straight to branching-time with CTL.

Huth and Ryan present the syntax and semantics of LTL (§3(2)) as well as some motivation on model checking.

The SPIN tool is based on LTL model checking.