# Advances in Verification of Time Petri Nets and Timed by Doc.dr.hab. Wojciech Penczek, Dr. Agata Pólrola (auth.)

By Doc.dr.hab. Wojciech Penczek, Dr. Agata Pólrola (auth.)

This monograph provides a accomplished advent to timed automata (TA) and

time Petri nets (TPNs) which belong to the main typical types of real-time

systems. the various latest equipment of translating time Petri nets to timed

automata are provided, with a spotlight at the translations that correspond to the

semantics of time Petri nets, associating clocks with numerous parts of the

nets. "Advances in Verification of Time Petri Nets and Timed Automata – A Temporal

Logic technique" introduces timed and untimed temporal specification languages

and provides version abstraction equipment in keeping with country classification techniques for TPNs

and on partition refinement for TA. additionally, the monograph provides a contemporary growth

in the improvement of 2 version checking tools, in accordance with both exploiting

abstract nation areas or on program of SAT-based symbolic recommendations.

The ebook addresses learn scientists in addition to graduate and PhD scholars

in laptop technological know-how, logics, and engineering of actual time systems.

**Read or Download Advances in Verification of Time Petri Nets and Timed Automata: A Temporal Logic Approach PDF**

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**Extra resources for Advances in Verification of Time Petri Nets and Timed Automata: A Temporal Logic Approach**

**Example text**

The following abbreviations are used in the deﬁnitions of both TA and TPNs. Let IR denote the set of real numbers, Q – the set of rational numbers, ZZ – the set of integers, and IN – the set of naturals (including 0). For each S ∈ {IR, Q, ZZ} by S0+ (S+ ) we denote a subset of S consisting of all its nonnegative (respectively positive) elements. Moreover, by IN+ we mean the set of positive natural numbers. When we deal with elements of IR0+ ∪ {∞}, by the notations “≤ b” and “[a, b]” we mean “< b” and “[a, b)” if a ∈ IR0+ and b = ∞.

PnP } is a ﬁnite set of places, T = {t1 , . . , tnT } is a ﬁnite set of transitions, where P ∩ T = ∅, F : (P × T ) ∪ (T × P ) −→ IN is the ﬂow function, and m0 : P −→ IN is the initial marking of P. Intuitively, Petri nets are directed weighted graphs of two types of nodes: places (representing conditions) and transitions (representing events), whose arcs correspond to these elements in the domain of the ﬂow function, for which the value of this function is positive. The arcs are assigned positive weights according to the values of F .

A concrete state σ N of N is deﬁned as an ordered pair (m, clock N ), where • m is a marking, and • clock N : I −→ IR0+ is a function which for each index i ∈ I gives the time elapsed since the marked place p ∈ Pi of the process Ni of N became marked most recently. For δ ∈ IR0+ , by clock N + δ we denote the function given by (clock N + δ)(i) = clock N (i)+δ for all i ∈ I. Moreover, let (m, clock N )+δ denote (m, clock N +δ). The (dense) concrete state space of N is now a transition system CcN (N ) = (Σ N , (σ N )0 , →N c ), where • Σ N is the set of all the concrete states of N , • (σ N )0 = (m0 , clock0N ) with clock0N (i) = 0 for each i ∈ I is the initial state, and • a timed consecution relation →N c ⊆ Σ N × (T ∪ IR0+ ) × Σ N is deﬁned by action- and time successors as follows: δ – for δ ∈ IR0+ , (m, clock N ) →N c (m, clock N + δ) iﬀ · for each t ∈ en(m) there exists i ∈ I with •t ∩ Pi = ∅ such that (clock N + δ)(i) ≤ Lf t(t) (time successor), t – for t ∈ T , (m, clock N ) →N c (m1 , clock1N ) iﬀ · t ∈ en(m), · for each i ∈ I with •t ∩ Pi = ∅ we have clock N (i) ≥ Ef t(t), · there is i ∈ I with •t ∩ Pi = ∅ such that clock N (i) ≤ Lf t(t), · m1 = m[t , and · for all i ∈ I we have clock1N (i) = 0 if •t ∩ Pi = ∅ and clock1N (i) = clock N (i) otherwise (action successor).