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 offers a complete creation to timed automata (TA) and
time Petri nets (TPNs) which belong to the main commonly used types of real-time
systems. a few of the present tools 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 procedure" introduces timed and untimed temporal specification languages
and supplies version abstraction equipment in line with kingdom category methods for TPNs
and on partition refinement for TA. additionally, the monograph offers a up to date development
in the improvement of 2 version checking equipment, in accordance with both exploiting
abstract nation areas or on program of SAT-based symbolic options.

The publication addresses learn scientists in addition to graduate and PhD scholars
in desktop technological know-how, logics, and engineering of actual time systems.

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1): • the constraints of the form xi ∼ c with xi ∈ X , ∼ ∈ {<, ≤, ≥, >} and c ∈ IN are expressed by xi − x0 ∼ c, • these of the form xi = c or xi − xj = c – by the conjunctions xi − x0 ≤ c ∧ x0 − xi ≤ −c and xi − xj ≤ c ∧ xj − xi ≤ −c, respectively, • true is expressed by x0 − x0 < ∞, and • all the constraints of the form xi − xj > c with xi , xj ∈ X + and c ∈ IN are replaced by xj − xi < −c, whereas these of the form xi − xj ≥ c – by xj − xi ≤ −c. The constraints of the form xi − xj < −∞ are introduced to express constraints which are never satisfied.

For simplicity of the description below, let ((m(p1 ), . . , m(p8 )), (clock T (t1 ), . . , clockT (t6 ))) denote the concrete state (m, clock T ) ∈ Σ T . 5, 0)) →T c . . A state σ R ∈ Σ R is reachable if there exists a (σ R )0 -run ρR and i ∈ IN such that σ R = σiR + δ for some 0 ≤ δ ≤ δi , where σiR + δi is an element of ρR . The set of all the reachable states of N is denoted by ReachR N . A marking m . The set of all the reachable is reachable if there is a state (m, ·) ∈ ReachR N markings of N is denoted by RMN (it is easy to see that this set does not depend on the way the concrete states of the net are defined).

By a partition of a set B we mean a family of its disjoint subsets B such that B = B. B ∈B 32 • • • • 2 Timed Automata X Z := {v ∈ IRn0+ | (∃v ∈ Z) v ≤ v}, Z ⇑ Z = {v ∈ Z | (∃v ∈ Z ) v ≤ v ∧ (∀v ≤ v ≤ v ) v ∈ Z ∪ Z }, Z[X := 0] = {v[X := 0] | v ∈ Z}, X | v[X := 0] ∈ Z}. , Z[X := 0] and [X := 0]Z) and the standard intersection preserve zones [8, 159]. A description of the implementation of Z\Z , following [8], is given also in Sect. 3. Some examples of the operations are presented in Fig. 2. x2 x2 5 x2 Z \ Z = {Z1 , Z2 } 7 6 Z \ Z = {Z1 , Z2 , Z3 } 6 6 Z3 Z Z Z1 3 3 1 1 3 4 6 x1 x2 Z1 3 Z2 1 3 4 5 6 x1 x2 6 Z2 3 4 5 6 x1 6 x1 x2 6 6 Z Z ∩Z 3 Z 3 1 x1 3 4 x1 3 2 x2 x2 x2 6 6 6 Z⇑Z 3 3 Z ⇑Z Z[x1 := 0] 1 3 4 x1 x2 4 x1 x1 x2 x2 7 Z [x1 := 0] 5 [x1 := 0]Z = ∅ [x1 := 0]Z 3 3 x1 x1 Fig.

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