Journal of Logical and Algebraic Methods in Programming最新文献

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Reversible debugging of concurrent Erlang programs: Supporting imperative primitives 并发 Erlang 程序的可逆调试支持命令式基元
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2024-01-17 DOI: 10.1016/j.jlamp.2024.100944
Pietro Lami , Ivan Lanese , Jean-Bernard Stefani , Claudio Sacerdoti Coen , Giovanni Fabbretti
{"title":"Reversible debugging of concurrent Erlang programs: Supporting imperative primitives","authors":"Pietro Lami ,&nbsp;Ivan Lanese ,&nbsp;Jean-Bernard Stefani ,&nbsp;Claudio Sacerdoti Coen ,&nbsp;Giovanni Fabbretti","doi":"10.1016/j.jlamp.2024.100944","DOIUrl":"10.1016/j.jlamp.2024.100944","url":null,"abstract":"<div><p>Reversible computing is a programming paradigm allowing one to execute programs both in the standard, forward direction as well as backwards, recovering past states. A relevant application of reversible computing is causal-consistent reversible debugging, which allows one to explore concurrent computations backwards and forwards to find a bug. The basic idea is that any action can be undone, provided that its consequences are undone beforehand. This approach has been put into practice in CauDEr, a Causal-consistent reversible Debugger<span> for the Erlang programming language. CauDEr provides the ability to explore a concurrent computation back and forward in a step-by-step way as well as to undo an action far in the past including all and only its consequences (rollback), and to replay an action from a log, together with its causes. CauDEr supports the functional, concurrent and distributed fragment of Erlang. However, Erlang also includes imperative primitives to manage a map (shared among all the processes of a same node) associating process identifiers to names. Here we extend CauDEr and the related theory, including rollback and replay, to support such imperative primitives. From a theoretical point of view, the imperative primitives create different causal structures to those derived from the concurrent Erlang fragment previously handled in CauDEr, yet we show that the main results proved for previous versions of CauDEr are still valid. From a practical point of view, this allows one to debug a larger subset of Erlang programs, as shown with a small case study of a server providing mathematical functionalities.</span></p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"138 ","pages":"Article 100944"},"PeriodicalIF":0.9,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139498019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Trace preservation in B and Event-B refinements B 和 Event-B 精炼中的微量保留
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2024-01-05 DOI: 10.1016/j.jlamp.2024.100943
Sebastian Stock , Atif Mashkoor , Michael Leuschel , Alexander Egyed
{"title":"Trace preservation in B and Event-B refinements","authors":"Sebastian Stock ,&nbsp;Atif Mashkoor ,&nbsp;Michael Leuschel ,&nbsp;Alexander Egyed","doi":"10.1016/j.jlamp.2024.100943","DOIUrl":"10.1016/j.jlamp.2024.100943","url":null,"abstract":"<div><p>Refinement guarantees that the concrete version of a model does not violate the constraints introduced at the abstract level. The peculiarity of refinement, however, is that we have no guarantee about the preservation of the behavior of the model. For example, a trace (a set of desirable states and transitions) created on the abstract model may not replay on the concrete model. Its manual recreation, usually via animation, is necessary to run the trace, as the model may have changed significantly during refinement. However, this is a labor-intensive and error-prone task. To this end, this article presents an automatic trace refining technique and tool called <em>BERT</em> (<u>B</u> and <u>E</u>vent-B Trace <u>R</u>efinement <u>T</u>echnique) that allows modelers to ensure the behavioral integrity of high-level traces at the concrete level. The cost- and time-effectiveness of BERT are shown in industrial-strength case studies from the automotive and aviation domains.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100943"},"PeriodicalIF":0.9,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2352220824000014/pdfft?md5=014ec80e27c6d35f0f8d4a7e25c9564e&pid=1-s2.0-S2352220824000014-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139375928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Protocol choice and iteration for the free cornering 自由转角的协议选择和迭代
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-12-29 DOI: 10.1016/j.jlamp.2023.100942
Chad Nester , Niels Voorneveld
{"title":"Protocol choice and iteration for the free cornering","authors":"Chad Nester ,&nbsp;Niels Voorneveld","doi":"10.1016/j.jlamp.2023.100942","DOIUrl":"10.1016/j.jlamp.2023.100942","url":null,"abstract":"<div><p>We extend the free cornering of a symmetric monoidal category, a double categorical model of concurrent interaction, to support branching communication protocols and iterated communication protocols. We validate our constructions by showing that they inherit significant categorical structure from the free cornering, including that they form monoidal double categories. We also establish some elementary properties of the novel structure they contain. Further, we give a model of the free cornering in terms of strong functors and strong natural transformations, inspired by the literature on computational effects.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100942"},"PeriodicalIF":0.9,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139068328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bridging formal methods and machine learning with model checking and global optimisation 用模型检查和全局优化连接形式化方法和机器学习
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-12-28 DOI: 10.1016/j.jlamp.2023.100941
Saddek Bensalem , Xiaowei Huang , Wenjie Ruan , Qiyi Tang , Changshun Wu , Xingyu Zhao
{"title":"Bridging formal methods and machine learning with model checking and global optimisation","authors":"Saddek Bensalem ,&nbsp;Xiaowei Huang ,&nbsp;Wenjie Ruan ,&nbsp;Qiyi Tang ,&nbsp;Changshun Wu ,&nbsp;Xingyu Zhao","doi":"10.1016/j.jlamp.2023.100941","DOIUrl":"10.1016/j.jlamp.2023.100941","url":null,"abstract":"<div><p>Formal methods and machine learning are two research fields with drastically different foundations and philosophies. Formal methods utilise mathematically rigorous techniques for software and hardware systems' specification, development and verification. Machine learning focuses on pragmatic approaches to gradually improve a parameterised model by observing a training data set. While historically, the two fields lack communication, this trend has changed in the past few years with an outburst of research interest in the robustness verification of neural networks. This paper will briefly review these works, and focus on the urgent need for broader and more in-depth communication between the two fields, with the ultimate goal of developing learning-enabled systems with excellent performance and acceptable safety and security. We present a specification language, MLS<sup>2</sup>, and show that it can express a set of known safety and security properties, including generalisation, uncertainty, robustness, data poisoning, backdoor, model stealing, membership inference, model inversion, interpretability, and fairness. To verify MLS<sup>2</sup> properties, we promote the global optimisation-based methods, which have provable guarantees on the convergence to the optimal solution. Many of them have theoretical bounds on the gap between current solutions and the optimal solution.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100941"},"PeriodicalIF":0.9,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2352220823000950/pdfft?md5=524bb8cc97eab39538606c56c0fd3849&pid=1-s2.0-S2352220823000950-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139068888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Formally verified animation for RoboChart using interaction trees 使用交互树为 RoboChart 制作经过正式验证的动画
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-12-28 DOI: 10.1016/j.jlamp.2023.100940
Kangfeng Ye, Simon Foster, Jim Woodcock
{"title":"Formally verified animation for RoboChart using interaction trees","authors":"Kangfeng Ye,&nbsp;Simon Foster,&nbsp;Jim Woodcock","doi":"10.1016/j.jlamp.2023.100940","DOIUrl":"10.1016/j.jlamp.2023.100940","url":null,"abstract":"<div><p>RoboChart is a core notation in the RoboStar framework. It is a timed and probabilistic domain-specific and state machine-based language for robotics. RoboChart supports shared variables and communication across entities in its component model. It has formal denotational semantics given in CSP. The semantic technique of Interaction Trees (ITrees) represents behaviours of reactive and concurrent programs interacting with their environments. Recent mechanisation of ITrees, ITree-based CSP semantics and a Z mathematical toolkit in Isabelle/HOL bring new applications of verification and animation for state-rich process languages, such as RoboChart. In this paper, we use ITrees to give RoboChart novel operational semantics, implement it in Isabelle, and use Isabelle's code generator to generate verified and executable animations. We illustrate our approach using an autonomous chemical detector and patrol robot models, exhibiting nondeterminism and using shared variables. With animation, we show two concrete scenarios for the chemical detector when the robot encounters different environmental inputs and three for the patrol robot when its calibrated position is in other corridor sections. We also verify that the animated scenarios are trace refinements of the CSP denotational semantics of the RoboChart models using FDR, a refinement model checker for CSP. This ensures that our approach to resolve nondeterminism using CSP operators with priority is sound and correct.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100940"},"PeriodicalIF":0.9,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2352220823000949/pdfft?md5=5312a452edca31ae541827e9147a0d84&pid=1-s2.0-S2352220823000949-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139067982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On proving that an unsafe controller is not proven safe 关于证明不安全的控制器不能证明是安全的
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-12-19 DOI: 10.1016/j.jlamp.2023.100939
Yuvaraj Selvaraj , Jonas Krook , Wolfgang Ahrendt , Martin Fabian
{"title":"On proving that an unsafe controller is not proven safe","authors":"Yuvaraj Selvaraj ,&nbsp;Jonas Krook ,&nbsp;Wolfgang Ahrendt ,&nbsp;Martin Fabian","doi":"10.1016/j.jlamp.2023.100939","DOIUrl":"10.1016/j.jlamp.2023.100939","url":null,"abstract":"<div><p>Cyber-physical systems are often safety-critical and their correctness is crucial, such as in the case of automated driving. Using formal mathematical methods is one way to guarantee correctness and improve safety. Although these methods have shown their usefulness, care must be taken because modelling errors might result in proving a faulty controller safe, which is potentially catastrophic in practice. This paper deals with two such modelling errors in <em>differential dynamic logic</em>, a formal specification and verification language for <em>hybrid systems</em>, which are mathematical models of cyber-physical systems. The main contributions are to provide conditions under which these two modelling errors cannot cause a faulty controller to be proven safe, and to show how these conditions can be proven with help of the interactive theorem prover KeYmaera X. The problems are illustrated with a real world example of a safety controller for automated driving, and it is shown that the formulated conditions have the intended effect both for a faulty and a correct controller. It is also shown how the formulated conditions aid in finding a <em>loop invariant</em> candidate to prove properties of hybrid systems with feedback loops. Furthermore, the relation between such a loop invariant and the characterisation of the <em>maximal control invariant set</em> is discussed.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100939"},"PeriodicalIF":0.9,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2352220823000937/pdfft?md5=a0255d67f0ba6855c0bca13111b7ef6d&pid=1-s2.0-S2352220823000937-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138816599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bounded model checking for interval probabilistic timed graph transformation systems against properties of probabilistic metric temporal graph logic 基于概率度量时间图逻辑的区间概率时间图变换系统的有界模型检验
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-12-01 DOI: 10.1016/j.jlamp.2023.100938
Sven Schneider, Maria Maximova, Holger Giese
{"title":"Bounded model checking for interval probabilistic timed graph transformation systems against properties of probabilistic metric temporal graph logic","authors":"Sven Schneider,&nbsp;Maria Maximova,&nbsp;Holger Giese","doi":"10.1016/j.jlamp.2023.100938","DOIUrl":"10.1016/j.jlamp.2023.100938","url":null,"abstract":"<div><p>Cyber-physical systems often encompass complex concurrent behavior with timing constraints and probabilistic failures on demand. The analysis whether such systems with probabilistic timed behavior adhere to a given specification is essential. The formalism of Interval Probabilistic Timed Graph Transformation Systems (IPTGTSs) is often a suitable choice to model cyber-physical systems because <em>(a)</em> its rule-based approach to graph transformation can capture a wide range of system's structure dynamics when the states of the system can be represented by graphs while <em>(b)</em><span> it employs interval specifications for probabilistic behavior as well as lower and upper bounds on delays of steps to support systems where precise probabilities and delays are not known or may change during the runtime of the system. Probabilistic Metric Temporal Graph Logic (PMTGL) has been introduced as a powerful specification language to express worst-case/best-case probabilistic timed requirements such as actor-based soft deadlines using </span><em>(a)</em> path properties relying on its Metric Temporal Graph Logic fragment to track individual graph elements and <em>(b)</em><span> an operator inherited from Probabilistic Timed Computation Tree Logic to express worst-case/best-case probabilistic requirements identifying worst-case/best-case resolutions of non-determinism. Bounded Model Checking (BMC) support for Probabilistic Timed Graph Transformation Systems (PTGTSs) w.r.t. properties specified using PMTGL has been already presented. However, for IPTGTSs no analysis support w.r.t. PMTGL properties has been developed for stating metric temporal properties on identified subgraphs and their structural changes over time.</span></p><p>In this paper, we adapt the BMC approach developed for PTGTSs to the case of IPTGTSs extending modeling and analysis support to the usage of probability intervals more appropriately covering cyber-physical systems where probabilistic effects cannot be specified precisely and need to be approximated instead. In our evaluation, we apply an implementation of our BMC approach in <span>AutoGraph</span> to a novel running example demonstrating the effect of using probability intervals instead of precise probability values.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100938"},"PeriodicalIF":0.9,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138512609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Back to the format: A survey on SOS for probabilistic processes 回到格式:对概率过程的SOS的调查
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-11-13 DOI: 10.1016/j.jlamp.2023.100929
Valentina Castiglioni , Ruggero Lanotte , Simone Tini
{"title":"Back to the format: A survey on SOS for probabilistic processes","authors":"Valentina Castiglioni ,&nbsp;Ruggero Lanotte ,&nbsp;Simone Tini","doi":"10.1016/j.jlamp.2023.100929","DOIUrl":"10.1016/j.jlamp.2023.100929","url":null,"abstract":"<div><p>In probabilistic process algebras the classic qualitative description of process behaviour is enriched with quantitative information on it, usually modelled in terms of probabilistic weights and/or distributions over the qualitative behaviour. In this setting, we use behavioural equivalences to check whether two processes show exactly the same behaviour, and, if this is not the case, we can use behavioural metrics to measure the distance between them. Compositional reasoning requires that equivalence, or closeness, of behaviour of two processes are not destroyed when language operators are applied on top of them in order to build larger processes. Formally, the equivalence must be a congruence, and the metric must be uniformly continuous, with respect to language operators. Instead of verifying these compositional properties by hand, operator-by-operator, it is much more convenient to prove them for a class of operators once for all, and to check that the operators one is dealing with are in that class. This is achieved by means of SOS specification formats: they consist in a set of syntactical constraints characterising a class of operators on the patterns of SOS rules, that define the operational semantics of languages. With this survey, we aim to collect and describe the specification formats that have been proposed in the literature to guarantee the compositional properties of (variants of) bisimulation equivalences and bisimulation metrics in the probabilistic setting.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"137 ","pages":"Article 100929"},"PeriodicalIF":0.9,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2352220823000834/pdfft?md5=ce48feb3ed771dee50dbe8f80129e90b&pid=1-s2.0-S2352220823000834-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135715196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Business processes resource management using rewriting logic and deep-learning-based predictive monitoring 业务流程资源管理使用重写逻辑和基于深度学习的预测监控
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-11-07 DOI: 10.1016/j.jlamp.2023.100928
Francisco Durán , Nicolás Pozas , Camilo Rocha
{"title":"Business processes resource management using rewriting logic and deep-learning-based predictive monitoring","authors":"Francisco Durán ,&nbsp;Nicolás Pozas ,&nbsp;Camilo Rocha","doi":"10.1016/j.jlamp.2023.100928","DOIUrl":"https://doi.org/10.1016/j.jlamp.2023.100928","url":null,"abstract":"<div><p>A significant task in business process optimization is concerned with streamlining the allocation and sharing of resources. This paper presents an approach for analyzing business process provisioning under a resource prediction strategy based on deep learning. A timed and probabilistic rewrite theory specification formalizes the semantics of business processes. It is integrated with an external oracle in the form of a long short-term memory neural network that can be queried to predict how traces of the process may advance within a time frame. Comparison of execution time and resource occupancy under different parameters is included for several case studies, as well as details on the construction of the deep learning model and its integration with Maude.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"136 ","pages":"Article 100928"},"PeriodicalIF":0.9,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91987265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Time distance-based computation of the DBM over-approximation of preemptive real-time systems 基于时间距离的抢占式实时系统DBM过逼近计算
IF 0.9 4区 数学
Journal of Logical and Algebraic Methods in Programming Pub Date : 2023-11-04 DOI: 10.1016/j.jlamp.2023.100927
Abdelkrim Abdelli
{"title":"Time distance-based computation of the DBM over-approximation of preemptive real-time systems","authors":"Abdelkrim Abdelli","doi":"10.1016/j.jlamp.2023.100927","DOIUrl":"https://doi.org/10.1016/j.jlamp.2023.100927","url":null,"abstract":"<div><p>The verification of preemptive real-time systems is a crucial aspect in ensuring their correctness and reliability to meet strict time constraints. Generally, the analysis of the behaviors of such systems requires the computation of the reachability graphs encoding their state space. However, the construction of the latter is computationally expensive and resource-consuming as it involves, for each graph node, managing and solving polyhedral constraints whose complexity is exponential.</p><p>In this paper, we explore a novel approach that builds an over-approximation of the state space of preemptive real-time systems. Our graph construction extends the expression of a node to the time-distance system that encodes the quantitative properties of past-fired subsequences. This makes it possible to restore relevant time information that is used to compute in a polynomial time a tighter difference bound matrix over-approximation of the polyhedral constraints. We show that the obtained graph is more appropriate to restore the quantitative properties of the model. The simulation results show that our graphs are almost of the same size as the exact graphs, while improving by far the times needed for their computation.</p></div>","PeriodicalId":48797,"journal":{"name":"Journal of Logical and Algebraic Methods in Programming","volume":"136 ","pages":"Article 100927"},"PeriodicalIF":0.9,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91987267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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