Deriving Petri nets from finite transition systems

This paper presents a novel method to derive a Petri net from any specification model that can be mapped into a state-based representation with arcs labeled with symbols from an alphabet of events (a Transition System, TS). The method is based on the theory of regions for Elementary Transition Systems (ETS). Previous work has shown that, for any ETS, there exists a Petri Net with minimum transition count (one transition for each label) with a reachability graph isomorphic to the original Transition System. Our method extends and implements that theory by using the following three mechanisms that provide a framework for synthesis of safe Petri nets from arbitrary TSs. First, the requirement of isomorphism is relaxed to bisimulation of TSs, thus extending the class of synthesizable TSs to a new class called Excitation-Closed Transition Systems (ECTS). Second, for the first time, we propose a method of PN synthesis for an arbitrary TS based on mapping a TS event into a set of transition labels in a PN. Third, the notion of irredundant region set is exploited, to minimize the number of places in the net without affecting its behavior. The synthesis method can derive different classes of place-irredundant Petri Nets (e.g., pure, free choice, unique choice) from the same TS, depending on the constraints imposed on the synthesis algorithm. This method has been implemented and applied in different frameworks. The results obtained from the experiments have demonstrated the wide applicability of the method.Peer ReviewedPostprint (published version

Permutation Games for the Weakly Aconjunctive μ\mu-Calculus

We introduce a natural notion of limit-deterministic parity automata and present a method that uses such automata to construct satisfiability games for the weakly aconjunctive fragment of the $\mu$-calculus. To this end we devise a method that determinizes limit-deterministic parity automata of size $n$ with $k$ priorities through limit-deterministic B\"uchi automata to deterministic parity automata of size $\mathcal{O}((nk)!)$ and with $\mathcal{O}(nk)$ priorities. The construction relies on limit-determinism to avoid the full complexity of the Safra/Piterman-construction by using partial permutations of states in place of Safra-Trees. By showing that limit-deterministic parity automata can be used to recognize unsuccessful branches in pre-tableaux for the weakly aconjunctive $\mu$-calculus, we obtain satisfiability games of size $\mathcal{O}((nk)!)$ with $\mathcal{O}(nk)$ priorities for weakly aconjunctive input formulas of size $n$ and alternation-depth $k$. A prototypical implementation that employs a tableau-based global caching algorithm to solve these games on-the-fly shows promising initial results

New region-based algorithms for deriving bounded Petri nets

The theory of regions was introduced in the early nineties as a method to bridge state and event-based models. This paper tackles the problem of deriving a Petri net from a state-based model, using the theory of regions. Some of the restrictions required in the traditional approach are dropped in this paper, together with significant extensions that make the approach applicable in new scenarios. One of these scenarios is Process Mining, where accepting (discovering) additional behavior in the synthesized Petri net is sometimes valued. The algorithmic emphasis used in this paper contributes to the demystification of the theory of regions as been only a good theoretical exercise, opening the door for its application in the industrial domain.Peer ReviewedPostprint (published version

Reverse-engineering of polynomial dynamical systems

Multivariate polynomial dynamical systems over finite fields have been studied in several contexts, including engineering and mathematical biology. An important problem is to construct models of such systems from a partial specification of dynamic properties, e.g., from a collection of state transition measurements. Here, we consider static models, which are directed graphs that represent the causal relationships between system variables, so-called wiring diagrams. This paper contains an algorithm which computes all possible minimal wiring diagrams for a given set of state transition measurements. The paper also contains several statistical measures for model selection. The algorithm uses primary decomposition of monomial ideals as the principal tool. An application to the reverse-engineering of a gene regulatory network is included. The algorithm and the statistical measures are implemented in Macaulay2 and are available from the authors

Automatic Generation of Minimal Cut Sets

A cut set is a collection of component failure modes that could lead to a system failure. Cut Set Analysis (CSA) is applied to critical systems to identify and rank system vulnerabilities at design time. Model checking tools have been used to automate the generation of minimal cut sets but are generally based on checking reachability of system failure states. This paper describes a new approach to CSA using a Linear Temporal Logic (LTL) model checker called BT Analyser that supports the generation of multiple counterexamples. The approach enables a broader class of system failures to be analysed, by generalising from failure state formulae to failure behaviours expressed in LTL. The traditional approach to CSA using model checking requires the model or system failure to be modified, usually by hand, to eliminate already-discovered cut sets, and the model checker to be rerun, at each step. By contrast, the new approach works incrementally and fully automatically, thereby removing the tedious and error-prone manual process and resulting in significantly reduced computation time. This in turn enables larger models to be checked. Two different strategies for using BT Analyser for CSA are presented. There is generally no single best strategy for model checking: their relative efficiency depends on the model and property being analysed. Comparative results are given for the A320 hydraulics case study in the Behavior Tree modelling language.Comment: In Proceedings ESSS 2015, arXiv:1506.0325

Region-based algorithms for process mining and synthesis of Petri nets

The theory of regions was introduced in the early nineties as a bridge between state-based and event-based specifications. Since then, much attention has been paid to theoretical extensions of this theory, but less advances have appeared in the application domain. This paper provides contributions in both dimensions: the theory of bisimulation-based synthesis from Cortadella {em et al.} is generalized and adapted to the area of Process Mining. On the application domain, efficient methods and data structures to support the synthesis problem are developed, together with a practical implementation. Experiments reported witness the practicality of the approach described in this paper.Postprint (published version

Inadmissible Class of Boolean Functions under Stuck-at Faults

Many underlying structural and functional factors that determine the fault behavior of a combinational network, are not yet fully understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in irredundant two-level circuits even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response from an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. We report some bounds and enumerate the total number of root functions up to 6 variables. Finally, we point out several open problems and possible applications of root functions in logic design and testing