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DOI: 10.15514/ISPRAS-2020-32(6)-12
Automated Analysis of DP-systems Using Timed-Arc Petri Nets via TAPAAL Tool
1 A.A. Izmaylov, ORCID: 0000-0003-1721-3347 <zinoviy23@gmail.com> 1 L.W. Dworzanski, ORCID: 0000-0002-0074- 7660 <leo@mathtech.ru> 1 National Research University Higher School of Economics, Myasnitskaya st., 20, Moscow, 101000, Russia
Abstract. Timed-Arcs Petri nets (TaPN-nets) are a time extension of Petri nets that allows assigning clocks to tokens. System of dynamic points on a metric graph (DP-systems) is another dynamical model that is studied in discrete geometry dynamics and motivated by study of localized Gaussian wave packets scattering on thin structures; as well, DP-systems could be utilized to overapproximate the dynamics of messages scattering in distributed systems. In the latter case, time-temporal properties of DP-systems become a matter of interest. However, there are no tools that enable us to analyse them. In this work, we provide a new approach to automated analysis of DP-systems using the translation of a DP-system into a TaPN-net which is implemented as a TAPAAL plugin. The translation let us use the comprehensive tool support for TaPN-nets (TAPAAL/UPPAAL) to analyze DP-systems dynamical characteristics expressed in TCTL language. We demonstrated how to express some of them and verify time-temporal properties of a DP-system using the suggested approach, and performed experiments to obtain empirical estimates of the tool performance.
Keywords: metric graphs; timed-arc Petri nets; time temporal properties
For citation: Izmaylov A.A., Dworzanski L.W. Analysis of DP-systems Using Timed-Arc Petri Nets via TAPAAL Tool. Trudy ISP RAN/Proc. ISP RAS, vol. 32, issue 6, 2020, pp. 155-166. DOI: 10.15514/ISPRAS-2020-32(6)-12
Acknowledgements. This work was supported by the RFBR grant 20-07-01103 a.
Автоматический анализ дискретных динамических систем на метрических графах с помощью сетей Петри с временными дугами и инструмента TAPAAL
1 А.А. Измайлов, ORCID: 0000-0003-1721-3347 <zinoviy23@gmail.com> 1 Л.В. Дворянский, ORCID: 0000-0002-0074- 7660 <leo@mathtech.ru> 1 Национальный исследовательский университет «Высшая школа экономики», Россия, 101000, г. Москва, ул. Мясницкая, д. 20
Аннотация. Сети Петри с временными дугами - это временное расширение сетей Петри (TaPN-сети), которое позволяет присваивать таймеры фишкам. Система динамических точек на метрическом графе (DP-система) это другая динамическая модель, которая рассматривается в теории геометрических дискретных динамических систем и, исторически, ее изучение мотивировано изучением распространения локализованных гауссовых волновых пакетов по тонким структурам; кроме того, DP-системы могут использоваться для приближенного представления динамики распространения сообщений в распределенных системах. В этой работе, мы описываем новый подход для автоматического анализа DP-систем используя трансляцию в TaPN сеть, которая реализована как расширение инструмента TAPAAL. Подход позволяет использовать мощные инструменты верификации (TAPAAL/UPPAAL) для анализа динамических характеристик DP-систем,
представленных на языке TCTL. В работе продемонстрировано, как можно кодировать временно-темпоральные свойства DP-систем в рамках предложенного подхода, и приведены результаты экспериментальных тестов.
Ключевые слова: метрические графы; сети Петри с временными дугами; темпорально-временные динамические свойства
Для цитирования: Измайлов А.А., Дворянский Л.В. Анализ дискретных динамических систем на метрических графах с помощью сетей Петри с временными дугами и инструмента TAPAAL. Труды ИСП РАН, том 32, вып. 6, 2020 г., стр. 155-166 (на английском языке). DOI: 10.15514/ISPRAS-2020-32(6)-12
Благодарности: Работа выполнена при поддержке гранта РФФИ 20-07-01103 a.
1. Introduction
The notion of a Petri nets evolved from a chemistry process model to a model of indefinitely expandable computing system consistent with laws of physics in C.A. Petri works [1]. Nowadays, Petri nets are widely-used to model the behaviour of distributed concurrent computer systems and concurrent processes in biology, chemistry, physics, and other fields [1]. Timed-arc Petri nets (TaPN-nets) are an extension of Petri nets with time semantics: tokens are assigned clocks [2]; an inscription on an incoming arc of a transition define tokens of which age can be consumed by the firing of the transition.
Metric graphs are graphs with lengths assigned to edges. A dynamical system consisting of a metric graph and dynamic points moving along the graph edges (DP-system) is a geometrical discrete dynamical system originally motivated by the problem of evolution of wave packets in thin structures. It could be considered as a simplified discrete model of a quantum graph - a metric graph equipped with functions on its edges, a differential operator acting on such functions, and matching conditions on its vertices [3], [4] - with narrow localized wave packets [5], [6]. Quantum graphs occurred as a model or tool in a number of problems in chemistry, physics, engineering, and mathematics since 1930s [7]. It was shown that there exists a correspondence between the statistics of localized solutions on a quantum graph and the dynamics of a DP-system [6]. Points in a DP-system may represent supports of Gaussian wave packets in a quantum graph and/or the projection of wave propagation on medium geodesics. Both models, Petri nets and DP-systems, embrace realtime dynamics of discrete entities moving within a topological structure defined by a graph. Some results towards the characteristics of the dynamics of DP-systems were recently obtained in [8-11]. The growth of the number of points moving along edges and its asymptotic are studied for metric trees in [10] and, for some special cases, in [6]. In [11], polynomial approximation for the growth of the number of dynamic points moving along metric graphs is studied, and explicit formulae for the first two terms of the polynomial approximation are given. In [6], stabilization of the number of points in a DP-system is studied and explicit formulae for graphs with edges of the same length and star graphs are given, exploiting a connection with analytic number theory problems. While the mentioned quantitative dynamic statistics of DP-systems are studied, finer temporal properties of DP-systems are not; like, for example, which node vi or V2 will receive a point first, or whether ei will have received ten or more points by the time e2 received its first point. In this work, we provide a translation of a DP-system into a TaPN-net [12], which allows reducing the analysis of temporal properties of DP-system to analysis of TaPN-net and conduct it using the effective widely-used TAPAAL tool [13], which is under active development (two golds, a silver, and a bronze medal in MCC'20 [14]).
Temporal properties of Petri nets, such as liveness, boundedness, and reversibility, and their complexities have been extensively studied (see comprehensive review [15]); and, many of them are in PSPACE or, worse, in EXPSPACE complexity classes. The study and development of Petri nets analysis methods is still active; for example, the long-standing question on whether reachability and coverability problems have the same complexity has been recently resolved in [16] showing that
reachability is not even elementary, thus, impacting complexity of many problems in other fields, such as formal languages, logic, linear algebra, etc. However, for some restricted Petri net subclasses, many important properties, which are undecidable in general case, are decidable or even have polynomial time complexity.
There is a lot of ways to introduce time constraints in Petri nets [17-18]. Unfortunately, it was shown that almost any of semantical extensions makes Petri nets Turing-complete and, by Rice-Uspensky theorem, many of general behavioural problems immediately become undecidable. The widely known time extensions of Petri nets - Time Petri nets [19] and Timed (Duration) Petri nets [20] -are Turing-complete as they admit urgency and allow to model unbounded counters. Time extensions of Petri nets, as well as other real-time models, are under active study as, for many real-world software/hardware systems, time related aspects like performance, time-outs, delays, and latency are crucial for correct functioning [21-23]. A time semantics with restricted urgency was recently suggested for TaPN-nets in [24]; the suggested semantics allows urgent transitions to consume tokens only from the bounded places of a Petri net, and this restriction makes some behavioural problems decidable for TaPN-nets. In [25], author suggested an approach to use timing specifications to improve the behavioural properties of an untimed P/T-net with an example of making live and unbounded untimed P/T-net live and bounded by cutting off token generators using time-based constraints.
TaPN-nets attract our attention as they feature dynamics that makes it possible to model DP-systems. In [26], timed-arc Petri nets were consistently combined with 'nets-within-nets' hierarchical structure and sound time semantics was provided; also, the decidability of behavioural coverability-related properties using the notion of well-structured transition systems was established. For recent review of results on TAPN-nets and their verification, we refer to [27]. Moreover, TaPN-nets have comprehensive tool support via TAPAAL and UPPAAL software. Tool UPPAAL makes it possible to model and verify networks of timed automata. TAPAAL is a tool for modelling, simulation, and verification of TaPN-nets; it can utilize UPPAAL as a verification engine. While a lot of results on temporal properties of timed and untimed versions of Petri nets were obtained, quantitative statistics of the behaviour of Petri nets and their extensions are studied to much lesser extent. The translation of TaPN-nets into DP-systems can be used to overapproximate the number of different age-values in the TAPN-net. This translation is a part of the tool implemented within the frame of this works - it extracts a metric graph in GraphML format from a TaPN-net allowing one to use existing software for metric graphs to overapproximate the stabilization time or growth rate of the number of clocks in the TaPN-net.
In Section 2, basic notions and notations are given. Section 3 covers the translation between DP-systems and TaPN-nets and implementation details. In Section 4, we demonstrate how to verify time-temporal properties of a DP-system using the suggested approach; results of performance experiments are provided. Section 5 concludes the paper with some discussion.
2. Preliminaries
By N, Q>0, R, we denote the sets of natural, non-negative rational, and real numbers, respectively. The set of open and closed intervals over Q>0 и (то) is denoted by I (Q>0). For a set S, a bag (multiset) m over S is a mapping m : 5 ^ N. The set of all bags over S is denoted by N5. We denote addition and subtraction of two bags by + and -, the number of all elements in m taking into account the multiplicity by |m|, and comparisons of bags by =, <, >, <, >. We start by giving the definition of a metric graph [3].
Definition 1 (Metric graph). A graph Г is said to be a metric graph, if
• each arc a is assigned a positive length 1(a) E (0, то). An arc with 1(a) = то has only one incident vertex and is called a lead;
• the lengths of the arcs that are reversals of each other are assumed to be equal;
• a coordinate x(a) E [0,1(a)] increasing in the direction of the arc is assigned on each arc;
• the relation x(a') = 1(a) — x(a) holds between the coordinates on mutually reversed arcs.
• for arc a, the length 1(e) of its support edge e is equal to 1(a).
A state s of a DP-system D on a metric graph F is a set of points located on the arcs of F. When time starts to flow, each point p moves along its arc a direction from 0 to 1(a) with the same velocity; its position is denoted by x(p). If p reaches a vertex v, new points are issued to all the outgoing arcs of v (intuitively, this corresponds to wave packet scattering). New points generation may result in more than one point on some arcs. Each produced point p' starts moving along corresponding e. When more than one points reach a vertex simultaneously at t, on each outgoing arc, only one point is produced, as if only one point has reached the vertex at t; i.e., points met on a vertex fuse, and each coordinate of an arc can carry only one dynamic point. However, points do not collide anywhere on edges except vertices, i.e., if two points met on an edge, they both continue their movement towards their own directions. In fig. 1, the initial set of points consists of two points in vertices v1 and v3. The point in v1 produces a new point on edge [v1, v2], The point in v3 produces points on edges leading to vertices v2, v4, v5, v6, v7. After a time unit, there are no points in v1 and v3 (coloured gray), but there are points (coloured black) moving from v2 and v3 to their adjacent vertices.
Fig.1. System of dynamic points D on metric graph r
The stabilization time ts(r) of a metric graph F with commensurable edge lengths is the value of the period of time from the starting point to the point in time when the number of points Nr(t) on the graph has been stabilized [6], i.e., Vt > ts(r) ■ Nr(t) = Nr(ts(r)). Henceforth, when stabilization time is discussed, discrete time instants when points collapse at vertices are excluded from consideration as, strictly, at these instants the number of points can decrease. A place/transition net (PT-net) is a Petri net with indistinguishable tokens. Definition 2 (Place/transition nets). A PT-net is a tuple < P, T, F, y >, where
• P and T are disjoint finite sets of places, respectively, transitions;
• F Q (P X T) U (T X P) is a set of arcs (flow relation);
• y ■ F — N is a weight function;
For an element x £ P U T, an arc < y,x > is called an input arc, and arc < x,y > - an output arc. A preset • x and a postset x • are subsets of P U T such that • x = {y | < y,x > £ F} and x • = {yl < x,y > £ F}. A marking of N is a function m: P ^ N. A pair < N,m> of a PT-net and a marking is called a marked net.
Let N = < P,T,F,y > be a PT-net. A transition t £ T is enabled in a marking m iff Vp £ • t ^ m(p) > y <p,t>. An enabled transition t can fire yielding a new marking m'(p) = m(p) — y < p,t > + y < t,p > for each p £ P (denoted m —■^ m' ).
Now, we provide the definition of Timed-Arc Petri nets with token-based time semantics and urgency [2], [12], [24].
Definition 3 (Timed-Arc Petri net with Urgency). A TaPN-net is a tuple TaPN = < N.y1 ,U >, where
• N = < P,T,F,y > is a PT-net called the skeleton of TaPN and denoted by S(TaPN);
• yt'- P * T ^ I(Q > 0) is a set of token-age constraints on arcs;
• U : T ^ Q>0 is a set of urgency constraints on transitions.
The marking m = < ms, mt, mu > of a TaPN-net TaPN consists of a marking ms of S(TaPN), a time marking mt : Tok(ms) ^ Q>0 that assigns clocks to tokens, and an urgency marking mu : T(ms) ^ Q>0 that assigns clocks to transitions, where T(ms) comprises all transitions and Tok(ms) comprises all tokens of the marked PT-net < S(TaPN),ms >. The urgency constraint U(t) means that t must fire if t has been enabled for U(t) units of time. The token-age constraint Yt('P, t) means that t may fire only by consuming a token z in p with mt(z) E yt('P, 0. The urgency U of a transition is depicted as a number near the transition. The time constraints yl of an arc are depicted as an interval on the arc.
The operational semantics of TaPN-nets is defined by incorporating time constraints into the firing rules of PT-nets. A transition t is enabled in the marking m = < ms,mt,mu >, if t is enabled in < S(TaPN), ms > and time constraints of t are satisfied, i.e., each token z from a place p involved in the firing of t satisfies mt(z) E Y*(p, t).
A time elapsing step corresponds to the elapsing 5 time units in each clock of the marking m. We assume that all token clocks and transition clocks run at the same pace. We denote by m + S the marking with all its clocks increased by 5, i.e., for each token z E ms : (mt + S)(z) = mt(z) + S, and for each transition t : (mu + S)(t) = mu(t) + S. Under urgency restrictions, a time elapsing step S is allowed if there are no S' E [0, S) such that the m + S' marking has urgent transitions.
3. Automated translation of DP-systems into TaPN-nets
In the context of communication networks and computer systems, we may consider a distributed system of communicating reactive sensor-nodes (or other computational devices). One of the nodes is a server that initiates communication by sending control messages (signals) while other sensor-nodes react on such signals. On the assumption that the processing speed of nodes is much higher than the speed of signal propagation, when a node receives one or more signals from its neighbour nodes, it responds instantly by sending signals to some of its neighbours. In an extreme case, each node sends signals to all of its neighbours upon each message arrival. This extreme behaviour corresponds to the dynamics of a DP-system enabling us to use metric graphs to overapproximate messages scattering in networks.
In the latter case, time-temporal properties of DP-systems become a matter of interest [28]. However, up to the knowledge of authors, there are no tools that enable us to conduct analysis of such natural time-temporal properties as:
• Is it possible that more than N signals come to node X in K seconds;
• Is it possible that in the next K minutes two messages come with the difference of their time moments less that e milliseconds;
• For the system initial phase of duration N, if a point comes to node X at time T, then no points comes to node Y within K seconds from T;
In this section, we provide an algorithm for the translation of DP-systems in TaPN-nets. Tool TAPAAL supports verification of specifications in timed computation tree logic (TCTL) language, which allows expressing time and temporal properties of process dynamics [29] [30]. In the next section, we demonstrate how to express these properties in TCTL-properties of converted TaPN-nets.
To simulate DP-systems with models of Petri nets with real-time semantics, we need to represent points moving along edges using clocks in a Petri net. The advance of a point along an edge in a metric graph is modelled with the progress of the corresponding clock in a Petri net. The number of points on a metric graph is not fixed and may grow (infinitely when edge lengths are incommensurable); therefore, it is not possible to model DP-system using clocks in time or timed Petri nets [19], [20] as these models have finite structurally-determined number of clocks. In TaPN-nets, clocks are assigned to tokens [12], and the number of clocks can grow along with the number of tokens in the net.
We start with the description of DP-system to TaPN-net translation procedure, and, later, cover some design and technical details on the implementation of the translation.
We denote the set of points on the edge e by P(e). For a lead < v1,v2 >, one of v1 and v2 is marked with infinity. Note that, as movement of a point along an edge corresponds to a continuous process, we model edges of DP-system with TaPN-net places; and, as arrival of a point at a vertex corresponds to a discrete event, we model vertices of DP-system with TaPN-net transitions.
3.1 Algorithm for translation from a DP-system to a TAPN-net procedure
Step 1. For each edge e connecting vertices a and b in D, we create places eab and eba in NTA; Step 2. For each lead < a, b > in D with b marked with infinity, an arc from vertex a to a distinct infinity-vertex is introduced in NTA. We need to handle points moving away from vertex a separately.
(a) create place eab, transition tab with urgency 0, and an arc from eab to tab with time interval [0,0];
(b) for each point on the arc < a, b >
i. create a token with 0 time at place eab;
ii. create transition tba with urgency 0;
(c) for each point pt on arc < b, a >
i. create place elba with a token;
ii. create an arc from elba to tba with time interval [x(Pi),x(Pi)];
(d) for each arc from a to c, where c ^ b, create an arc from tba to eac;
Step 3. For each already created place eab in NTA such that vertex a is not marked with infinity in D
(a) for each point p on arc < a, b >
i. create token at eab with timer set to x(p);
(b) create transition tab with 0 urgency if b is not marked with infinity;
(c) create an arc with time interval [1(e), 1(e)] from eab to tab;
(d) for each arc < b,c > in D, create an arc from tab to ebc in NTA;
(e) create transition t'ab, an arc from eab to t'ab with multiplicity 2, urgency 0, and zero time interval, and arc from t'ab back to eab.
An example of such conversion is illustrated in fig. 2. As each undirected edge in D corresponds to two opposite arcs, the number of places in NTA on the right is twice as the number of vertices in D on the left. Transitions corresponds to the event of a point reaching a vertex; auxiliary transitions are introduced to clean redundant tokens from edge-places in NTA to handle events when multiple point came to a vertex simultaneously.
3.2 Algorithm for translation from timed-arc Petri nets to metric graphs
To use the results on the asympotics and estimates on growth and stabilization time of the number of dynamic points in DP-system [6] to overapproximate TaPN-net the number of different token age-values, we implement the translation of a restricted class of TaPN-nets into DP-systems.
л * * * ■ "*■ - + i ■ * ^ \
Fig.2. Metric graph D on the left and the resulting TAPN Nta on the right Let us consider a class of TaPN-nets under the following restrictions:
1) The skeleton of TaPN-net is a marked graph, i.e., every place has exactly one incoming arc, and exactly one outgoing arc [1];
2) All urgency time specifications are 0;
3) All arcs are labelled with point-intervals only, i.e., intervals of form [t, t].
Let NTA be an input TaPN-net and D be the resulting DP-system. The next procedure converts the TaPN-net to a DP-system. The DP-system D generates point each time NTA produces a new token with timer.
1) For each transition t^ in NTA, create vertex Vi in D;
2) For each transition ti, place p, and transition tj, such that there is an arc from ti to p and an arc from p to tj
a) create an edge connecting vertices vt and Vj (which correspond to transitions tt and tj, respectively) with a length equal to the time interval on the arc from p to tt; this procedure may introduce multiple edges or self-loops, thus, an intermediate graph may be not a metric graph; we provide a resolving strategy after the procedure;
b) for each token in p, put a point on beginning of edge from vi to v, i. e., at vertex v,;
3) For each place p, which has only inbound arcs or only outbound arcs
a) for each adjacent transition t, create a lead from corresponding vertex v;
b) if p has only outbound arcs, then for each token in p, for each adjacent transition th put a point on the lead incident to corresponding vertex v, moving towards v, and located on the distance equal to the point-interval on arc <p,tt>.
To resolve multiple edges and self-loops introduced during the translation, we break one of edges e into two edges by introduce a new vertex; lengths of these new edges are equal to the half 1(e). For self loops, we need to introduce two vertices, and they breaks edge e into three parts with lengths equal to a third of 1(e). This step, firstly, breaks the loop with one edge, and, secondly, breaks multiple edges; the result of such step for a loop transition is illustrated in fig. 3.
3.3 Implementation details
The translation is implemented in tool mg-to-tapn as a TAPAAL extension; the tool uses TAPAAL internal representation for storing and analysis of TaPN-nets. A JSON-based file format for processing DP-systems was developed and its support was implemented. The file format is an extension of JSON Graph Format [31]. To allow visualization of DP-systems, a support for GraphML-based format compatible with yEd Graph Editor [32] was implemented. As mg-to-tapn is an extension of TAPAAL, the following functions can be utilized both through command line interface or graphical user interface of TAPAAL.
• Translate DP-system in JSON format to TaPN-net in TAPAAL representation;
• Translate TaPN-net in TAPAAL representation to DP-system in JSON format;
• Translate TaPN-net in TAPAAL representation to DP-system in GraphML-based yEd-compatible format.
4. Checking TCTL properties of DP-system and experimental results
In this section, we demonstrate how to check DP-system time-temporal properties of a metric graph using the implemented translation and provide results of performance experiments.
4.1 Checking time-temporal properties of a metric graph
Tool TAPAAL does not directly support the whole class of TCTL properties; therefore, for timetemporal properties, we need to combine supported verification engines with additional auxiliary net fragments to capture such properties. We demonstrate how to encode the following propertiesusing this technique.
i. Is it possible that more than N signals come to node X in K seconds. Firstly, we need to capture the property that more than N signals come to node X. Arrival of a signal to X from Y along arc <Y,X > corresponds to a firing of transition tYX in TaPN (see naming strategy in section 3). We add a place p'x, which will be used as a counter for the number of signals-tokens that came to X. Then, we connect each tYX to p'x with an arc; each firing of tYX will produce a token in p'x. We call such a place counter for X. In addition, we need to determine a time point when K seconds has passed. We introduce a new place pstart and put a token in pstart. We add urgent transition ttime and connect pstart and ttimewith an arc specified by time interval [K, K]. So, place pstart becomes empty when K seconds has passed. We call such a net fragment consisting of place pstart with a token, transition ttime, and arc < pstart, ttime > specified by [K, K] as timer for K.
Finally, we can express the property as whether there is a reachable state when p'x has K or more tokens, and pstart has exactly one token. This property is purely temporal while time features are captured using time specifications in timer for K. In CTL, the property is expressed as EF((m(p'x) > N) & (m(pstart) = 1)).
ii. Is it possible that in the next K minutes two messages come with the difference of their time moments less that £ milliseconds. To check the property, we add timer for K construction, counter for X, and transition t'. We add arc < p'x, t' > labelled with [e,e]. To make , t' urgent, we use age invariant Inv: < e on p'x. If there are two or more tokens in p'x, , then those tokens come to node X with time difference less than e. The corresponding CTL property is EF ((p'x >2) & (pstart = 1))-
iii. For the system initial phase of duration N, if a point comes to node X at time T, then no points comes to node Y within К seconds from T. We add counter for X, counter for Y,
and timer for N. We add transition t'x, arc < p'x, t'x >labelled with [К, K], age invariant Inv: < K' on p'x making t'x urgent, urgent transition t'Y, and arc < p'Y , t'Y >. If a token is in p'x for К seconds, then t'x fires consuming it; if a token came to p'Y, it leaves the system via t'Y immediately.
The property become violated if p'x and p'Y hold tokens simultaneously, as it corresponds to a state where a token, which timer value is less than K, is in X, and a token came to Y . The resulting CTL property is AG-((p'x > 1) & (p'Y > 1) & (pstart = 1)).
Table 1. Performance results for verifying the first property
Graph type/ size (in nodes) 3 5 10 20 30
Complete 0.05s 0.11s 1.36s (11mb) 41.29s (26mb) -
Key 0.03s 0.07s 0.09s 0.4s (6mb) 11.91s (86mb)
Cycle 0.02s 0.03s 0.03s 0.02s 0.03s
Star 0.05s 0.09s 1.10s (13mb) 51.53s (353mb) > 300s (> 1.8gb)
Table 2. Performance results for verifying the third property
Graph type/ size (in nodes) 3 5 10 20 30
Complete 0.01s 0.01s 0.03s 0.65s (11mb) -
Key 0.01s 0.01s 0.08s 4.21s (45mb) > 300s (> 1.8gb)
Cycle 0.01s 0.01s 0.01s 0.01s 0.06s / 2mb
Star 0.01s 0.01s > 300s (> 1.6gb) 14.16s (120mb) > 300s (> 1.1gb)
4.2 Experimental results
We conducted experiments on metric graphs of different topology: star graph, complete graph, cycle graph, and «key» graph (linear graph joined with a cycle of 3 elements on the end). The metric graphs were converted to TaPN-nets in TAPAAL format. The first and third property, encoded as suggested in the previous section, were checked for the converted metric graphs; verification time and memory consumed were measured.
The results for the first property are given in Table 1. For this experiment, we checked the first property for the initial fragment of 1000 time units. Time and consumed memory were measured; if the amount of consumed memory was negligible, the value is not provided. The results for metric graph K30 were not obtained as more than a thousand nodes in the resulting TaPN-net exceeded some internal restrictions of TAPAAL. The low verification time for a cycle metric graph is stipulated by the low node degrees in the graph thus leading to a smaller number of transitions in the converted TaPN-net. The results for the third property are given in Table 2. For this experiment, we checked the second property with K = 20 and N = 1000. We see almost the same picture with the exception of that for key metric graphs the memory restrictions come in play earlier. The experiments were performed on a personal computer with 2 GHz Quad-Core Intel Core i5 processor and 16 GB of memory (3733 MHz LPDDR4X) running under macOS 11.0.1 operating system.
5. Conclusion
In this paper, we developed an approach that enable us to analyze TCTL-properties of DP-systems based on the implementation of the translation between DP-systems and TaPN-nets. We demonstrated how to conduct analysis of TCTL properties of DP-systems by adding an auxiliary net
fragment. For several time-temporal properties of metric graphs, we showed how to express the properties in TAPAAL using an auxiliary subnet and TAPAAL TCTL queries. The automated translation allows using the well-known efficient tool TAPAAL to carry out simulation and verification (analysis of TCTL properties) of DP-system dynamics and interpreting results for DP-systems on a given restricted subclass of TaPN-nets.
The experiments showed that for scarce mid-sized graphs analysis could be done quite effectively. As these experiments were conducted for the first time, to get more straightforward results, we neither used any optimizations nor tried to encode properties in a more compressed manner. Therefore, we expect that optimizations could considerably improve performance on the benchmark tests as complete and star graphs possess much symmetry.
To improve the precision of the overapproximation of a TaPN-net with a DP-system, the already known counting functions and asymptotics for undirected metric graphs shall be extended to directed metric graphs. This direction is being under current research.
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Information about authors / Информация об авторах
Леонид Владимирович ДВОРЯНСКИИ- доктор естественных наук (Doctor rerum naturalium), независимый исследователь. Сфера научных интересов: методы анализа динамических свойств дискретных динамических систем, моделей параллельных и распределенных систем, моделей систем реального времени, вполне структурированных систем переходов.
Leonid Wladimirovich DWORZANSKI - doctor of natural sciences (Doctor rerum naturalium), independent researcher. Research interests: analysis methods for discrete event dynamical systems, models for parallel and distributed computation, models for real-time systems, well-structured transition systems.
Александр Александрович ИЗМАЙЛОВ проходит обучение в департаменте программной инженерии факультет компьютерных наук НИУ ВШЭ.
Aleksandr Aleksandrovich IZMAYLOV studies at School of Software Engineering Faculty of Computer Science National Research University Higher School of Economics.
