The paper shows that there is a deep structure on certain sets of bisimilar Probabilistic Automata (PA). The key prerequisite for these structures is a notion of compactness of PA. It is shown that compact bisimilar PA form lattices. These results are then used in order to establish normal forms not only for finite automata, but also for infinite automata, as long as they are compact.
{"title":"Lattice structures for bisimilar Probabilistic Automata","authors":"Johann Schuster, M. Siegle","doi":"10.4204/EPTCS.140.1","DOIUrl":"https://doi.org/10.4204/EPTCS.140.1","url":null,"abstract":"The paper shows that there is a deep structure on certain sets of bisimilar Probabilistic Automata (PA). The key prerequisite for these structures is a notion of compactness of PA. It is shown that compact bisimilar PA form lattices. These results are then used in order to establish normal forms not only for finite automata, but also for infinite automata, as long as they are compact.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"130 1","pages":"1-15"},"PeriodicalIF":0.0,"publicationDate":"2014-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76365079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.
{"title":"Algorithmic Verification of Continuous and Hybrid Systems","authors":"O. Maler","doi":"10.4204/EPTCS.140.4","DOIUrl":"https://doi.org/10.4204/EPTCS.140.4","url":null,"abstract":"We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"94 1","pages":"48-69"},"PeriodicalIF":0.0,"publicationDate":"2014-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89754884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-09-01DOI: 10.22460/INFINITY.V2I2.P156-168
H. Sugilar
Penelitian ini merupakan penelitian kuasi eksperimen dengan disain kelompok kontrol tidak ekivalen karena tidak adanya pengacakan dalam menentukan subjek penelitian. Peneliti tidak membentuk kelas baru berdasarkan pemilihan sampel secara acak. Subjek sampel diambil dua kelas dari kelas VII siswa MTs Negeri Cikembar Kabupaten Sukabumi, satu kelas sebagai kelas eksperimen dengan pembelajaran generatif dan satu kelas sebagai kelas kontrol dengan pembelajaran konvensional. Instrumen yang digunakan adalah tes dan non tes. Hasil studi penelitian ini adalah: 1) peningkatan kemampuan berpikir kreatif siswa yang mengikuti pembelajaran generatif lebih baik daripada siswa yang mengikuti pembelajaran matematika secara konvensional ditinjau dari pencapaian hasil belajar dan peningkatan kemampuan berpikir kreatif. Kemampuan berpikir kreatif kelas eksperimen termasuk pada kategori sedang sedangkan kelas kontrol termasuk kategori rendah.2) terdapat perbedaan peningkatan kemampuan berpikir kreatif matematik antara siswa kemampuan tinggi, sedang, dan rendah yang mendapat pembelajaran generatif, 3) disposisi matematik siswa yang mengikuti pembelajaran matematika melalui pembelajaran generatif lebih baik daripada siswa yang mengikuti pembelajaran matematika secara konvensional, disposisi matematik siswa pada kelas eksperimen termasuk pada kategori sedang, sedangkan pada kelas kontrol disposisi matematik termasuk pada kategori rendah. 4) terdapat interaksi antara model pembelajaran dan tingkat kemampuan awal siswa dalam menghasilkan kemampuan berpikir kreatif. 5) terdapat asosiasi antara kemampuan berpikir kreatif matematik dengan disposisi matematik, kategori asosiasinya tinggi. Kata Kunci : Pembelajaran Generatif, Berpikir Kreatif, Disposisi Matematik This research quasi-experiment with design controls not equivalent as there is not a beating in determining the subject research. Researchers does not set up new class based on the election samples randomly. The subject samples taken two classes of class VII students MTs Cikembar Sukabumi, a class as class experiment with learning generative and one class as control classes with conventional teaching. Instruments that used is testing and non-test. Result of the study this research are: 1) increase the capacity and capability creative thinking students attending generative learning better than students who follow in mathematics teaching conventional learning achievement in terms of and increase the capacity and capability creative thinking. Ability to think creative class experiments, including in category is while control classes include category rendah.2 increase the capacity and capability) there are differences between the students think creatively mathematical ability, is low, and who got a lesson generative, 3) Mathematical Disposition students attending mathematics lessons by learning generative better than students who follow in mathematics teaching conventional, this mathematical disposition students in the cl
该研究是一种实验准研究，该实验对象的设计因其在确定主题时缺乏预设而没有受到保护。研究人员不会根据随机选择样本来创建一个新的类。样本对象从苏加梅县cigenebumi的七年级学生MTs上取下了两个班，一个是模拟学习的实验班，一个是传统学习控制班。使用的仪器是测试和非测试。本研究的结果是:1)在传统的数学学习中，通过学习成绩和创造性思维能力的提高学生的创造性思维能力。创造性思维能力包括在类别而实验控制包括年级低。2)类别中增加数学创造性思维能力区别能力高、中等和低的学生得到学习生成语法,3)生成语法学习数学的学生参加学习数学通过性格比传统跟随学习数学的学生,实验课上的学生在数学上的学习规程属于中等类别，而在课堂上的数学规程则属于较低的类别。4)学习模式与学生产生创造性思维能力的早期能力水平之间存在互动。5)在数学思维能力和数学成绩差之间存在一种联想，这是一个高度社会化的类别。关键词:泛化学习、创造性思考、设计控制实验的数学缺陷审查人员并没有在随机的元素样本上建立新的类别。样本包括两个班的学生MTs cits Sukabumi的课程，这是一个班的学习生成实验，其中一个班是控制传统教学的学生。使用的工具是测试和不测试。这项研究的结果是:1)增加不断发展的知识分子和不断发展的知识分子，比那些在数学教学中不断学习的学生更好，也比那些在数学教学中不断发展、不断发展创造力思考的学生更好。有能力思考创作类实验，包括控制细分，同时控制细分。2增加《capacity和-能力)之间有分歧学生觉得creatively mathematical不在乎,是低,与世卫组织有一个generative课,3)mathematical Disposition学生attending数学学习generative偏课比世卫组织学生跟着在数学conventional教书,这mathematical Disposition experiments班中的学生,在类别在内的控制住,而课堂这个mathematical Disposition在内的类别是低。4)模式之间存在着教学和早期技能水平的创作能力之间的关系。5)有一种关系，即有能力思考受虐受虐，有纪律，有教养会。关键字:学习生成性，创造性思维，数学推理
{"title":"Meningkatkan Kemampuan Berpikir Kreatif Dan Disposisi Matematik Siswa Madrasah Tsanawiyah Melalui Pembelajaran Generatif","authors":"H. Sugilar","doi":"10.22460/INFINITY.V2I2.P156-168","DOIUrl":"https://doi.org/10.22460/INFINITY.V2I2.P156-168","url":null,"abstract":"Penelitian ini merupakan penelitian kuasi eksperimen dengan disain kelompok kontrol tidak ekivalen karena tidak adanya pengacakan dalam menentukan subjek penelitian. Peneliti tidak membentuk kelas baru berdasarkan pemilihan sampel secara acak. Subjek sampel diambil dua kelas dari kelas VII siswa MTs Negeri Cikembar Kabupaten Sukabumi, satu kelas sebagai kelas eksperimen dengan pembelajaran generatif dan satu kelas sebagai kelas kontrol dengan pembelajaran konvensional. Instrumen yang digunakan adalah tes dan non tes. Hasil studi penelitian ini adalah: 1) peningkatan kemampuan berpikir kreatif siswa yang mengikuti pembelajaran generatif lebih baik daripada siswa yang mengikuti pembelajaran matematika secara konvensional ditinjau dari pencapaian hasil belajar dan peningkatan kemampuan berpikir kreatif. Kemampuan berpikir kreatif kelas eksperimen termasuk pada kategori sedang sedangkan kelas kontrol termasuk kategori rendah.2) terdapat perbedaan peningkatan kemampuan berpikir kreatif matematik antara siswa kemampuan tinggi, sedang, dan rendah yang mendapat pembelajaran generatif, 3) disposisi matematik siswa yang mengikuti pembelajaran matematika melalui pembelajaran generatif lebih baik daripada siswa yang mengikuti pembelajaran matematika secara konvensional, disposisi matematik siswa pada kelas eksperimen termasuk pada kategori sedang, sedangkan pada kelas kontrol disposisi matematik termasuk pada kategori rendah. 4) terdapat interaksi antara model pembelajaran dan tingkat kemampuan awal siswa dalam menghasilkan kemampuan berpikir kreatif. 5) terdapat asosiasi antara kemampuan berpikir kreatif matematik dengan disposisi matematik, kategori asosiasinya tinggi. Kata Kunci : Pembelajaran Generatif, Berpikir Kreatif, Disposisi Matematik This research quasi-experiment with design controls not equivalent as there is not a beating in determining the subject research. Researchers does not set up new class based on the election samples randomly. The subject samples taken two classes of class VII students MTs Cikembar Sukabumi, a class as class experiment with learning generative and one class as control classes with conventional teaching. Instruments that used is testing and non-test. Result of the study this research are: 1) increase the capacity and capability creative thinking students attending generative learning better than students who follow in mathematics teaching conventional learning achievement in terms of and increase the capacity and capability creative thinking. Ability to think creative class experiments, including in category is while control classes include category rendah.2 increase the capacity and capability) there are differences between the students think creatively mathematical ability, is low, and who got a lesson generative, 3) Mathematical Disposition students attending mathematics lessons by learning generative better than students who follow in mathematics teaching conventional, this mathematical disposition students in the cl","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"2 1","pages":"156-168"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68730082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Our cost model assigns token storage costs per time unit to places, and firing costs to transitions. We study the cost to reach a given control-state. In general, a cost-optimal run may not exist. However,we show that the infimum of the costs is computable.
{"title":"Petri Nets with Time and Cost","authors":"P. Abdulla, Richard Mayr","doi":"10.4204/EPTCS.107.3","DOIUrl":"https://doi.org/10.4204/EPTCS.107.3","url":null,"abstract":"We consider timed Petri nets, i.e., unbounded Petri nets where each token carries a real-valued clock. Transition arcs are labeled with time intervals, which specify constraints on the ages of tokens. Our cost model assigns token storage costs per time unit to places, and firing costs to transitions. We study the cost to reach a given control-state. In general, a cost-optimal run may not exist. However,we show that the infimum of the costs is computable.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"16 1","pages":"9-24"},"PeriodicalIF":0.0,"publicationDate":"2013-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73194964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The starting point of this work is a framework allowing to model systems with dynamic process creation, equipped with a procedure to detect symmetric executions (ie., which differ only by the identities of processes). This allows to reduce the state space, potentially to an exponentially smaller size, and, because process identifiers are never reused, this also allows to reduce to finite size some infinite state spaces. However, in this approach, the procedure to detect symmetries does not allow for computationally efficient algorithms, mainly because each newly computed state has to be compared with every already reached state. In this paper, we propose a new approach to detect symmetries in this framework that will solve this problem, thus enabling for efficient algorithms. We formalise a canonical representation of states and identify a sufficient condition on the analysed model that guarantees that every symmetry can be detected. For the models that do not fall into this category, our approach is still correct but does not guarantee a maximal reduction of state space.
{"title":"Effective Marking Equivalence Checking in Systems with Dynamic Process Creation","authors":"Lukasz Fronc","doi":"10.4204/EPTCS.107.6","DOIUrl":"https://doi.org/10.4204/EPTCS.107.6","url":null,"abstract":"The starting point of this work is a framework allowing to model systems with dynamic process creation, equipped with a procedure to detect symmetric executions (ie., which differ only by the identities of processes). This allows to reduce the state space, potentially to an exponentially smaller size, and, because process identifiers are never reused, this also allows to reduce to finite size some infinite state spaces. However, in this approach, the procedure to detect symmetries does not allow for computationally efficient algorithms, mainly because each newly computed state has to be compared with every already reached state. \u0000In this paper, we propose a new approach to detect symmetries in this framework that will solve this problem, thus enabling for efficient algorithms. We formalise a canonical representation of states and identify a sufficient condition on the analysed model that guarantees that every symmetry can be detected. For the models that do not fall into this category, our approach is still correct but does not guarantee a maximal reduction of state space.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"16 1","pages":"61-75"},"PeriodicalIF":0.0,"publicationDate":"2013-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82528541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aurel Randolph, H. Boucheneb, Abdessamad Imine, A. Quintero
The Operational Transformation (OT) approach, used in many collaborative editors, allows a group of users to concurrently update replicas of a shared object and exchange their updates in any order. The basic idea of this approach is to transform any received update operation before its execution on a replica of the object. This transformation aims to ensure the convergence of the different replicas of the object, even if the operations are executed in different orders. However, designing transformation functions for achieving convergence is a critical and challenging issue. Indeed, the transformation functions proposed in the literature are all revealed incorrect. This paper investigates the existence of transformations functions which ensure convergence. From the theoretical point of view, two properties, named TP1 and TP2, are necessary and sufficient to ensure convergence. For the basic signatures of insert and delete operations, we show, using controller synthesis technique, that there are some transformation functions, which satisfy TP1. But, there is no transformation function, which satisfies both TP1 and TP2.
{"title":"On Consistency of Operational Transformation Approach","authors":"Aurel Randolph, H. Boucheneb, Abdessamad Imine, A. Quintero","doi":"10.4204/EPTCS.107.5","DOIUrl":"https://doi.org/10.4204/EPTCS.107.5","url":null,"abstract":"The Operational Transformation (OT) approach, used in many collaborative editors, allows a group of users to concurrently update replicas of a shared object and exchange their updates in any order. The basic idea of this approach is to transform any received update operation before its execution on a replica of the object. This transformation aims to ensure the convergence of the different replicas of the object, even if the operations are executed in different orders. However, designing transformation functions for achieving convergence is a critical and challenging issue. Indeed, the transformation functions proposed in the literature are all revealed incorrect. This paper investigates the existence of transformations functions which ensure convergence. From the theoretical point of view, two properties, named TP1 and TP2, are necessary and sufficient to ensure convergence. For the basic signatures of insert and delete operations, we show, using controller synthesis technique, that there are some transformation functions, which satisfy TP1. But, there is no transformation function, which satisfies both TP1 and TP2.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"191 1","pages":"45-59"},"PeriodicalIF":0.0,"publicationDate":"2012-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74187019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Iterative imperative programs can be considered as infinite-state systems computing over possibly unbounded domains. Studying reachability in these systems is challenging as it requires to deal with an infinite number of states with standard backward or forward exploration strategies. An approach that we call Constraint-based reachability, is proposed to address reachability problems by exploring program states using a constraint model of the whole program. The keypoint of the approach is to interpret imperative constructions such as conditionals, loops, array and memory manipulations with the fundamental notion of constraint over a computational domain. By combining constraint filtering and abstraction techniques, Constraint-based reachability is able to solve reachability problems which are usually outside the scope of backward or forward exploration strategies. This paper proposes an interpretation of classical filtering consistencies used in Constraint Programming as abstract domain computations, and shows how this approach can be used to produce a constraint solver that efficiently generates solutions for reachability problems that are unsolvable by other approaches.
{"title":"Constraint-based reachability","authors":"A. Gotlieb, Tristan Denmat, Nadjib Lazaar","doi":"10.4204/EPTCS.107.4","DOIUrl":"https://doi.org/10.4204/EPTCS.107.4","url":null,"abstract":"Iterative imperative programs can be considered as infinite-state systems computing over possibly unbounded domains. Studying reachability in these systems is challenging as it requires to deal with an infinite number of states with standard backward or forward exploration strategies. An approach that we call Constraint-based reachability, is proposed to address reachability problems by exploring program states using a constraint model of the whole program. The keypoint of the approach is to interpret imperative constructions such as conditionals, loops, array and memory manipulations with the fundamental notion of constraint over a computational domain. By combining constraint filtering and abstraction techniques, Constraint-based reachability is able to solve reachability problems which are usually outside the scope of backward or forward exploration strategies. This paper proposes an interpretation of classical filtering consistencies used in Constraint Programming as abstract domain computations, and shows how this approach can be used to produce a constraint solver that efficiently generates solutions for reachability problems that are unsolvable by other approaches.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"1 1","pages":"25-43"},"PeriodicalIF":0.0,"publicationDate":"2012-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73824663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-02-01DOI: 10.22460/INFINITY.V1I1.P10-18
Rahayu Kariadinata
Learning mathematical analogy is one alternative learning that can be applied in order to cultivate the power of reason (power of reason) students. Through mathematical analogy students are required to be able to look for similarities or relationship nature of the two concepts are the same or different by comparison, then draw a conclusion from the similitude. Thus the analogy can be used as an explanation or as the basis of reasoning. Before starting the analogy of learning mathematics, teachers should examine the ability of understanding mathematical concepts of students, because of the level of understanding of students will affect the power of reason. Tasks (problems) mathematical analogy included non-routine matter, therefore the required readiness of teachers to make it. In each question contained mathematical analogy same or different concepts, so it takes quite a lot of material. Steps to make about the mathematical analogy, are: a) assemble all the concepts in mathematics student has learned; b) Similarly stacking properties / relationships contained in any concept, and c) select materials that have a nature / relationship analogous. In this paper is given two forms of matter of mathematical analogy is the analogy of mathematical models and mathematical analogy 1 models 2. Learning mathematical analogy should be carried out after a number of concepts learned. It is better to be given in classes end for many of the concepts that have been learned by the students. Reasoning power (power of reason) the student becomes an important part in the process of learning to drive them toward their future as citizens are intelligent, which will be led by the power of reason (the brain) and not by the strength (muscle) only. As noted by former US President Thomas Jefferson (in Copi, 1978: vii), which states: "In a republican nation, Whose citizens are to be led by reason and persuasion and not by force, the art of reasoning Becomes of first importance"
{"title":"Menumbuhkan Daya Nalar ( Power of Reason ) Siswa Melalui Pembelajaran Analogi Matematika","authors":"Rahayu Kariadinata","doi":"10.22460/INFINITY.V1I1.P10-18","DOIUrl":"https://doi.org/10.22460/INFINITY.V1I1.P10-18","url":null,"abstract":"Learning mathematical analogy is one alternative learning that can be applied in order to cultivate the power of reason (power of reason) students. Through mathematical analogy students are required to be able to look for similarities or relationship nature of the two concepts are the same or different by comparison, then draw a conclusion from the similitude. Thus the analogy can be used as an explanation or as the basis of reasoning. Before starting the analogy of learning mathematics, teachers should examine the ability of understanding mathematical concepts of students, because of the level of understanding of students will affect the power of reason. Tasks (problems) mathematical analogy included non-routine matter, therefore the required readiness of teachers to make it. In each question contained mathematical analogy same or different concepts, so it takes quite a lot of material. Steps to make about the mathematical analogy, are: a) assemble all the concepts in mathematics student has learned; b) Similarly stacking properties / relationships contained in any concept, and c) select materials that have a nature / relationship analogous. In this paper is given two forms of matter of mathematical analogy is the analogy of mathematical models and mathematical analogy 1 models 2. Learning mathematical analogy should be carried out after a number of concepts learned. It is better to be given in classes end for many of the concepts that have been learned by the students. Reasoning power (power of reason) the student becomes an important part in the process of learning to drive them toward their future as citizens are intelligent, which will be led by the power of reason (the brain) and not by the strength (muscle) only. As noted by former US President Thomas Jefferson (in Copi, 1978: vii), which states: \"In a republican nation, Whose citizens are to be led by reason and persuasion and not by force, the art of reasoning Becomes of first importance\"","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"48 3 1","pages":"10-18"},"PeriodicalIF":0.0,"publicationDate":"2012-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68729945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide decidability and undecidability results on the model-checking problem for infinite tree structures. These tree structures are built from sequences of elements of infinite relational structures. More precisely, we deal with the tree iteration of a relational structure M in the sense of Shelah-Stupp. In contrast to classical results where model-checking is shown decidable for MSO-logic, we show decidability of the tree model-checking problem for logics that allow only path quantifiers and chain quantifiers (where chains are subsets of paths), as they appear in branching time logics; however, at the same time the tree is enriched by the equal-level relation (which holds between vertices u, v if they are on the same tree level). We separate cleanly the tree logic from the logic used for expressing properties of the underlying structure M. We illustrate the scope of the decidability results by showing that two slight extensions of the framework lead to undecidability. In particular, this applies to the (stronger) tree iteration in the sense of Muchnik-Walukiewicz.
{"title":"Trees over Infinite Structures and Path Logics with Synchronization","authors":"Alex Spelten, W. Thomas, Sarah Winter","doi":"10.4204/EPTCS.73.5","DOIUrl":"https://doi.org/10.4204/EPTCS.73.5","url":null,"abstract":"We provide decidability and undecidability results on the model-checking problem for infinite tree structures. These tree structures are built from sequences of elements of infinite relational structures. More precisely, we deal with the tree iteration of a relational structure M in the sense of Shelah-Stupp. In contrast to classical results where model-checking is shown decidable for MSO-logic, we show decidability of the tree model-checking problem for logics that allow only path quantifiers and chain quantifiers (where chains are subsets of paths), as they appear in branching time logics; however, at the same time the tree is enriched by the equal-level relation (which holds between vertices u, v if they are on the same tree level). We separate cleanly the tree logic from the logic used for expressing properties of the underlying structure M. We illustrate the scope of the decidability results by showing that two slight extensions of the framework lead to undecidability. In particular, this applies to the (stronger) tree iteration in the sense of Muchnik-Walukiewicz.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"53 1","pages":"20-34"},"PeriodicalIF":0.0,"publicationDate":"2011-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77787978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL) are often used to describe specifications of probabilistic properties for discrete time and continuous time, respectively. In PCTL and CSL, the possibility of executions satisfying some temporal properties can be quantitatively represented by the probabilistic extension of the path quantifiers in their basic Computation Tree Logic (CTL), however, path formulae of them are expressed via the same operators in CTL. For this reason, both of them cannot represent formulae with quantitative temporal properties, such as those of the form "some properties hold to more than 80% of time points (in a certain bounded interval) on the path." In this paper, we introduce a new temporal operator which expressed the notion of frequency of events, and define probabilistic frequency temporal logic (PFTL) based on CTLstar. As a result, we can easily represent the temporal properties of behavior in probabilistic systems. However, it is difficult to develop a model checker for the full PFTL, due to rich expressiveness. Accordingly, we develop a model-checking algorithm for the CTL-like fragment of PFTL against finite-state Markov chains, and an approximate model-checking algorithm for the bounded Linear Temporal Logic (LTL) -like fragment of PFTL against countable-state Markov chains.
{"title":"A Probabilistic Temporal Logic with Frequency Operators and Its Model Checking","authors":"Takashi Tomita, Shigeki Hagihara, N. Yonezaki","doi":"10.4204/EPTCS.73.9","DOIUrl":"https://doi.org/10.4204/EPTCS.73.9","url":null,"abstract":"Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL) are often used to describe specifications of probabilistic properties for discrete time and continuous time, respectively. In PCTL and CSL, the possibility of executions satisfying some temporal properties can be quantitatively represented by the probabilistic extension of the path quantifiers in their basic Computation Tree Logic (CTL), however, path formulae of them are expressed via the same operators in CTL. For this reason, both of them cannot represent formulae with quantitative temporal properties, such as those of the form \"some properties hold to more than 80% of time points (in a certain bounded interval) on the path.\" In this paper, we introduce a new temporal operator which expressed the notion of frequency of events, and define probabilistic frequency temporal logic (PFTL) based on CTLstar. As a result, we can easily represent the temporal properties of behavior in probabilistic systems. However, it is difficult to develop a model checker for the full PFTL, due to rich expressiveness. Accordingly, we develop a model-checking algorithm for the CTL-like fragment of PFTL against finite-state Markov chains, and an approximate model-checking algorithm for the bounded Linear Temporal Logic (LTL) -like fragment of PFTL against countable-state Markov chains.","PeriodicalId":31175,"journal":{"name":"Infinity","volume":"12 1","pages":"79-93"},"PeriodicalIF":0.0,"publicationDate":"2011-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72564203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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