[Paper Review] Reachability Analysis of Time Basic Petri Nets: a Time Coverage Approach

Carlo Bellettini, Lorenzo Capra|arXiv (Cornell University)|Jul 6, 2011

Petri Nets in System Modeling5 references4 citations

TL;DR

This paper proposes a symbolic reachability analysis technique for Time-Basic Petri Nets (TB nets) using a time coverage approach that overcomes the limitations of traditional time-bounded exploration. By introducing symbolic states and time-anonymous timestamps, the method constructs a finite, time-coverage-based reachability graph, enabling exact analysis of timing properties—demonstrated by successfully computing the maximum uncombusted gas concentration in a gas burner model with 20 units over 7.5 hours.

ABSTRACT

We introduce a technique for reachability analysis of Time-Basic (TB) Petri nets, a powerful formalism for real- time systems where time constraints are expressed as intervals, representing possible transition firing times, whose bounds are functions of marking's time description. The technique consists of building a symbolic reachability graph relying on a sort of time coverage, and overcomes the limitations of the only available analyzer for TB nets, based in turn on a time-bounded inspection of a (possibly infinite) reachability-tree. The graph construction algorithm has been automated by a tool-set, briefly described in the paper together with its main functionality and analysis capability. A running example is used throughout the paper to sketch the symbolic graph construction. A use case describing a small real system - that the running example is an excerpt from - has been employed to benchmark the technique and the tool-set. The main outcome of this test are also presented in the paper. Ongoing work, in the perspective of integrating with a model-checking engine, is shortly discussed.

Motivation & Objective

• To address the open problem of reachability analysis in Time-Basic Petri Nets, which are more expressive than Time Petri Nets but lack scalable analysis techniques.
• To overcome the limitations of existing tools that rely on time-bounded exploration of infinite reachability trees, restricting analysis to approximate or bounded properties.
• To develop a symbolic reachability graph construction method that supports exact verification of timing properties through time-coverage abstraction.
• To automate the analysis process with a tool-set capable of handling real-world systems, such as the gas burner case study.
• To enable the detection of critical system parameters, such as maximum uncombusted gas concentration, with precise timing constraints.

Proposed method

• The approach constructs a symbolic reachability graph using a relative time model, where timestamps are abstracted via symbolic expressions based on enabling marking time-stamps.
• It introduces the concept of 'time-anonymous' timestamps—tokens whose absolute time values do not affect system evolution, enabling state factorization and graph compression.
• A normalization procedure identifies and eliminates non-influential timestamp symbols, reducing state explosion and enabling finite time-coverage graphs.
• The method uses inclusion checks between symbolic states to determine reachability and avoid redundant state generation.
• The algorithm is implemented in a Java-based tool-set that supports input in legacy and PNML formats, and generates both binary and DOT-formatted output for visualization and property verification.
• The tool integrates with model-checking pipelines and supports property verification via condition-based marking evaluation.

Experimental results

Research questions

• RQ1 Can a symbolic reachability graph be constructed for Time-Basic Petri Nets that captures all relevant timing behaviors without time-bounded truncation?
• RQ2 How can time-anonymous timestamps be formally defined and exploited to reduce state space explosion in TB net analysis?
• RQ3 To what extent can a time-coverage abstraction replace traditional time-bounded exploration in TB net verification?
• RQ4 Can the proposed method compute exact upper bounds for safety-critical parameters, such as maximum uncombusted gas concentration, in real-time systems?
• RQ5 How does the performance of the symbolic approach scale with increasing time granularity and system complexity?

Key findings

• The tool successfully computed the maximum uncombusted gas concentration as 20 units for the gas burner model with 0.1 time granularity, a result previously unattainable with time-bounded methods.
• For the 0.1 granularity case, the analysis took approximately 7.5 hours and generated 23,635 symbolic states, of which 14,563 were reduced to a final graph of 14,563 nodes.
• The number of final states was significantly lower than the total built states, indicating effective state factorization via time-anonymous timestamp detection.
• Performance profiling showed a stable state creation rate after an initial burst, suggesting that a parallel or distributed implementation could yield substantial speedups.
• The method produced consistent results across three different time granularities (0.5, 0.25, 0.1), with maximum concentrations of 4, 8, and 20 units respectively, confirming result coherence.
• The execution time trend suggests a likely quadratic growth with respect to state space size, indicating scalability challenges for very large models.