#### DMCA

## ABSTRACT Formalization of Discrete-time Markov Chains in HOL (2013)

### Citations

5891 | A tutorial on hidden Markov models and selected applications in speech recognition,”
- Rabiner
- 1989
(Show Context)
Citation Context ...otal probability theorem given in Equation (2.3c) and Theorem 5.1, as well as Definition 5.1. This theorem is frequently used in estimating the parameters of a HMM using the maximum likelihood method =-=[96]-=-, in which the computations of the parameters mainly depend on the joint probability of a given observable sequence. Furthermore, it plays an important role in the Baum-Welch algorithm [96]. 5.2.3 Bes... |

1935 | A new approach to the economic analysis of nonstationary time series and the business cycle.
- Hamilton
- 1989
(Show Context)
Citation Context ... [97]). Moreover, the enzyme activity and the growth of copolymers can be modeled as a Markov chain in chemistry [104]. Furthermore, a variety of economics and finance phenomena, such as asset prices =-=[35]-=- and market crashes [93], are described as Markov chains. Some music composition algorithms [54] based on the Markov chain theory are employed in software. In addition, the Markov chain theory has bee... |

1344 | Profile hiddenMarkovmodels,”
- Eddy
- 1998
(Show Context)
Citation Context ...f each base. In this DNA model, the first state is called exon (E), the second one 5’ splice site (5), and the third one intron (I) [10]. This model is described and studied very naturally using HMMs =-=[21]-=-: a DTMC over the states E, 5, and I is used in order to know in which state the proteins are, then another random process is defined which characterizes the emission of A, G, T or C according to the ... |

1076 | A theory of type polymorphism in programming.
- Milner
- 1978
(Show Context)
Citation Context ... depends, [80] is built upon theorem prover HOL4, hence, we select the theorem prover HOL4 as our proof assistant. Based on Robin Milner’s proof-checking program, Logic for Computable Functions (LCF) =-=[82]-=-, HOL4 is developed for conducting proofs in higher-order logic by using the strongly-typed functional Meta-Language (ML) [91]. As a system of deduction with a precise semantics, HOL4 is capable of ve... |

860 |
Bayesian Networks and Decision Graphs.
- Jensen, Nielsen
- 2007
(Show Context)
Citation Context ...The third one is the Total Probability Theorem and the fourth one is a lemma of the Total Probability Theorem. The last theorem is the Additivity Theorem. Mathematically, the conditional independence =-=[53]-=- is an important concept, which is the foundation of graphical models and mainly used in Bayesian Network. The mathematical definition of conditional independence is: Definition 2.2. (Conditional Inde... |

622 |
der Vorst. Templates for the Solution of Linear Systems
- Barret, Berry, et al.
- 1993
(Show Context)
Citation Context ...eal numbers obtained for the corresponding results are mainly calculated by applying some numerical methods, such as the Jacobi Over-Relaxation, Gauss-Seidel and Successive Over-Relaxation algorithms =-=[6]-=-, for the affordable computation reason. In this case, the results include approximations. 1.1.3 Probabilistic Model Checking Recently, probabilistic model checking [100] has been proposed for analyzi... |

565 |
Markov chains,
- Norris
- 1998
(Show Context)
Citation Context ...v Process is usually called the Discrete-Time Markov Chain (DTMC) [8]. A DTMC model consists of a list of the possible states of the system along with the possible transition paths among these states =-=[88]-=-. This kind of a simple structure of a DTMC model makes it handy to study various behaviors such as transient behavior and limiting behavior, of a stochastic process with discrete spaces by solving th... |

555 |
Markov chains: Gibbs fields, Monte Carlo simulation, and queues, volume 31.
- Bremaud
- 1999
(Show Context)
Citation Context ...ce [48], physics and statistical genetics [73]. On the other hand, our formalization of DTMC is general enough to describe time-inhomogeneous DTMCs, which are part of the notations in MCMC algorithms =-=[11]-=- and are also applied in many fields, like the adiabatic theorem [60] in quantum computations, by means of instantiating the transition probability (in Definition 3.3), which is a function taking the ... |

497 |
Queueing systems volume I: theory
- Kleinrock
- 1975
(Show Context)
Citation Context ...en the birth-death coefficients are bi = λ and di = µ (λ and µ are constants) for all i in the state space, then the model described in Definition 72 4.13 represents a classical M/M/1 queueing system =-=[57]-=- (in this case, the average inter arrival time becomes 1 λ and the average service time is 1 µ ). Thereafter, the verified theorems can be directly applied to analyzing the ergodicity of M/M/1 queuein... |

429 |
Probability Statistics with Reliability, Queuing and Computer Science Applications
- Trivedi
- 2002
(Show Context)
Citation Context ....1. For instance, in biologic science, a typical Markov chain namely, the BirthDTMC Sports Biology Economics &sFinance MusicChemistry Physics Figure 1.1: Markov Chain Application Fields Death process =-=[109]-=-, is applied in modeling biological populations. Also, the Markov chain theory has been applied in physics (such as thermodynamics [65] and statistical 2 mechanics [97]). Moreover, the enzyme activity... |

420 |
Principles of model checking.
- Baier, Katoen
- 2008
(Show Context)
Citation Context ... models automatically and provide counter-examples if the verification of a system model failed. Although they offer exact solutions, they are drastically limited by the state-space explosion problem =-=[5]-=- and the limited expressiveness of the property specification language. Furthermore, the time for analyzing the properties of a system is largely dependent on the convergence speed of the underlying a... |

381 |
Markov Chains and Mixing Times.
- Levin, Peres, et al.
- 2008
(Show Context)
Citation Context ...ization of DTMC and the verification of the most important properties, in which the concepts of reversibility and stationary properties accommodate the formal reasoning about Markov chain mixing time =-=[67]-=- and the formalizations of stationary process and stationary distribution grant the formal verification of the stationary properties of numerous other stochastic 12 processes [Bio-Jr-1, Bio-Cf-1]. We ... |

273 |
Markov Chains with Stationary Transition Probabilities,
- CHUNG
- 1967
(Show Context)
Citation Context ...e process consists of all the possible state sequences based on a given state space S. Now, based on these definitions, a Markov process can be 23 defined as a stochastic process with Markov property =-=[13]-=-. If a Markov process has finite or countably infinite state space Ω, then it is called a Markov chain and satisfies the following Markov property: For 0 ≤ t0 ≤ · · · ≤ tn and f0, · · · , fn+1 in the ... |

259 |
Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse Der Mathematik; translated as Foundations of Probability,
- Kolmogorov
- 1933
(Show Context)
Citation Context ...ned on open sets and this constrains the verification of some applications. More recently, Mhamdi [80] improved the development based on the axiomatic definition of probability proposed by Kolmogorov =-=[58]-=-. Mhamdi’s theory provides a mathematical consistent for assigning and deducing probabilities of events. Hölzl [46] has also formalized three chapters of measure theory in Isabelle/HOL. Affeldt [2] s... |

221 |
Hidden Markov and other models for discrete-valued time series.
- IL, Zucchini
- 1997
(Show Context)
Citation Context ...g DTMC etc., where these classified Markov chains [88] are widely used to simplify the analysis of long-term behaviors for most applications. As an evolved Markov chain, the Hidden Markov Model (HMM) =-=[72]-=- is a stochastic process involving an underlying Markov chain which changes the output of the random functions associated with each state in the Markov chain. The observer can visualize the output of ... |

197 | SPNP: Stochastic Petri Net Package,”
- Ciardo, Muppala, et al.
- 1989
(Show Context)
Citation Context ...er the capability of analyzing large and complex models. The Markov chain of an SPN is modeled by means of a reachability graph [61]. The prevailing software 5 tools of stochastic petri nets are SPNP =-=[14]-=- and GreatSPN [30]. These tools can model, validate, and evaluate distributed systems and analyze the dynamic events of the models by means of embedded Markov chain theory. For example, the quantitati... |

144 |
Stochastic Processes with applications,
- Bhattacharya, Waymire
- 1990
(Show Context)
Citation Context ...me dependant and so is the case with chemical reactions. Thus, various probabilistic models are employed to describe the behaviors of systems. Diverse random processes exhibit the memoryless property =-=[8]-=-, which means that the future state depends only on the current state and is independent of any past state. In science and engineering domains, numerous applications desire to predict the future state... |

139 | Finite Markov Chains and Algorithmic Applications. Cambridge:
- Haggstrom
- 2002
(Show Context)
Citation Context ..., i.e., their limit probability distributions are independent of the initial distributions. For this reason, we now focus on the verification of some key properties of aperiodic and irreducible DTMCs =-=[34]-=-. 4.3.1 Positive Transition Probability Based on the the aperiodicity and irreducibility characteristics of certain DTMCs, we can prove the most important properties of them. One of the results is tha... |

119 | Theorem Proving with the Real Numbers,
- Harrison
- 1998
(Show Context)
Citation Context ...e ML functions that simplify goals into subgoals. Over the past few decades, the formalization of many foundational mathematical theories has led to tremendous progress in HOL4. For example, Harrison =-=[36]-=- formalized real numbers, topology, limits, sequences and series as well as differentiation and integration. His work is part of the current distribution of HOL. Hurd [49] developed a probability theo... |

96 |
A Modern Course
- Reichl
- 2004
(Show Context)
Citation Context ...plication Fields Death process [109], is applied in modeling biological populations. Also, the Markov chain theory has been applied in physics (such as thermodynamics [65] and statistical 2 mechanics =-=[97]-=-). Moreover, the enzyme activity and the growth of copolymers can be modeled as a Markov chain in chemistry [104]. Furthermore, a variety of economics and finance phenomena, such as asset prices [35] ... |

76 |
Matrix theory.
- Franklin
- 1968
(Show Context)
Citation Context ...ented in Section 3.3. The absorbing DTMCs are frequently applied in modeling social-psychological problems [64]. Combining the formalization of absorbing DTMC 73 in Section 4.2 with the matrix theory =-=[24]-=-, the properties of absorbing DTMC can also be formally verified using higher-order logic. The proof scripts contain about 8000 lines of HOL4 code and are available at [68]. The major challenge of the... |

75 |
Task Analysis Methods for Instructional Design.
- Jonassen, Tessmer, et al.
- 1999
(Show Context)
Citation Context ...he reliability, maintainability or safety of systems using Markov methods, which offer simplistic modeling approaches and are more flexible compared to traditional approaches, for example, Fault Tree =-=[18]-=-. Some prevalent tool examples include Möbius [84] and SHARPE [105]. These tools mainly provide the services on analyzing the failure or repair of a model, which may occur in the lifetime of any prod... |

74 |
σ-fields and probability.
- Nędzusiak
- 1990
(Show Context)
Citation Context ...i). A random process denotes a collection of random variables Xt (t ∈ T ). If the indices (t) of random variables Xt are discrete, then this random process is a discretetime random process. Nedzusiak =-=[86]-=- and Bialas [9] were among the first to propose to formalize some measure and probability theories in higher-order-logic. Later, Hurd [49] implemented their work and formalized a certain measure space... |

72 | Implementation of Symbolic Model Checking for Probabilitics Systems.
- Parker
- 2002
(Show Context)
Citation Context ...operty specification language. Furthermore, the time for analyzing the properties of a system is largely dependent on the convergence speed of the underlying algorithms. For example, the Power method =-=[90]-=-, which is a well-known iterative method, is applied to compute the steady-state probabilities (or limiting probabilities) of Markov chains in PRISM. Such algorithms are mainly based on numerical meth... |

66 |
On dominance relations and the structure of animal societies, III: The condition for secure structure.
- Landau
- 1953
(Show Context)
Citation Context ...Cs can be verified by applying their definitions, given in Section 4.2, and the theorems presented in Section 3.3. The absorbing DTMCs are frequently applied in modeling social-psychological problems =-=[64]-=-. Combining the formalization of absorbing DTMC 73 in Section 4.2 with the matrix theory [24], the properties of absorbing DTMC can also be formally verified using higher-order logic. The proof script... |

65 |
Mathematical Techniques for Analyzing Concurrent and Probabilisitc Systems.
- Rutten, Kwaiatkowska, et al.
- 2004
(Show Context)
Citation Context ...sive Over-Relaxation algorithms [6], for the affordable computation reason. In this case, the results include approximations. 1.1.3 Probabilistic Model Checking Recently, probabilistic model checking =-=[100]-=- has been proposed for analyzing Markovian systems by modeling stochastic behaviors using probabilistic state machines and exhaustively reasoning about the probabilistic properties of numerous Markovi... |

53 | Formal verification of probabilistic algorithms,
- Hurd
- 2002
(Show Context)
Citation Context ...developed a formalization of time-homogeneous DTMC with 11 finite state space [69]. However, the definition of DTMC is not general enough due to the inherit disadvantage of the probability theorem in =-=[49]-=- and the verified theorems are not rich enough to deal with various DTMC models. Our work, in this thesis, mainly relies on the most recent and general formalization of probability theory [81] and all... |

52 | Extended Markovian process algebra
- Bernardo, Gorrieri
- 1996
(Show Context)
Citation Context ... which may occur in the lifetime of any product. Some other software tools used for evaluating performance, e.g., MACOM [102] and HYDRA [85], take the advantage of a popular Markovian process algebra =-=[7]-=-, i.e., PEPA [92], to model systems and efficiently compute passage time densities and quantities in large-scale Markov chains. Another technique, Stochastic Petri Nets (SPN ) [32], has been found as ... |

51 |
Stochastic Petri Nets: Modelling, Stability, Simulation.
- Haas
- 2002
(Show Context)
Citation Context ...vian process algebra [7], i.e., PEPA [92], to model systems and efficiently compute passage time densities and quantities in large-scale Markov chains. Another technique, Stochastic Petri Nets (SPN ) =-=[32]-=-, has been found as a powerful method for modeling large-scale systems because it allows local state modeling instead of global modeling. SPN are utilized to model the stochastic systems and offer the... |

49 |
Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers.
- Yates, Goodman
- 2005
(Show Context)
Citation Context ...ices. A queue system is usually described as a Birth-Death process in which the population consists of customers. This potential project may start by formalizing the count process and poisson process =-=[112]-=-, as well as the continuous-time Markov chain. Various queues can be found in the open literature, such as M/G/1, M/G/∞, GI/GI/1 [101] and they can be used to model diverse interesting applications. •... |

46 | A formal analysis of Bluetooth device discovery
- DUFLOT, KWIATKOWSKA, et al.
- 2006
(Show Context)
Citation Context ...ify the properties of this DTMC model with around 600 lines of code in HOL. In the telecommunications domain, various systems, such as the time elapsing in a synchronous fashion of a Bluetooth device =-=[20]-=-, wireless LAN [63] and broadcast protocols [22], are described using state diagrams, which can be modeled as the DTMCs. The existing library of the formalized DTMC presented in this chapter can be ap... |

46 | Probabilistic model checking of the IEEE 802.11 wireless local area network protocol.
- Kwiatkowska, Norman, et al.
- 2002
(Show Context)
Citation Context ...of this DTMC model with around 600 lines of code in HOL. In the telecommunications domain, various systems, such as the time elapsing in a synchronous fashion of a Bluetooth device [20], wireless LAN =-=[63]-=- and broadcast protocols [22], are described using state diagrams, which can be modeled as the DTMCs. The existing library of the formalized DTMC presented in this chapter can be applied to verify the... |

38 | 2004: Downscaling of daily rainfall occurrence over Northeast Brazil using a hidden Markov model
- Robertson, Kirshner, et al.
(Show Context)
Citation Context ...ls. Various simulation-based HMM analysis tools, dedicated to a particular system domain, have been reported in the literature. Some prominent examples include HMMTool [43] as part of the NHMMtoolbox =-=[98]-=- to predict daily rainfall sequence. ChIP-Seq [12], MArkov MOdeling Tool (MAMOT ) [23] and HMMER [42] are some of the popular simulation software in biological research. However, as mentioned earlier,... |

36 | VESTA: A Statistical Model-Checker and Analyzer for Probabilistic Systems. In:
- Sen, Viswanathan, et al.
- 2005
(Show Context)
Citation Context ...e machines and exhaustively reasoning about the probabilistic properties of numerous Markovian systems in a precise logic. Some of the commonly used probabilistic model checkers are PRISM [94], VESTA =-=[103]-=- and Ymer [113]. Among them, PRISM is the most widely used, where the system model is constructed as the state transition graph and the properties are specified using Probabilistic Computer Temporal L... |

33 |
L.: A skeptic’s approach to combining HOL and maple.
- Harrison, Thery
- 1998
(Show Context)
Citation Context ...he input. It is clear that “x / x = 1” holds only when x 6= 0. Moreover, CAS algorithms cannot simplify some expression. For example, the following simplification rule is not supported by Mathematica =-=[37]-=-: √ x2 = x if x ≥ 0 −x if x ≤ 0 As a result, the core of computer algebra systems cannot be considered sound enough to guarantee the correctness of the final result. Moreover, the presence of ... |

30 | Formal Verification and Simulation for Performance Analysis for Probabilistic Broadcast Protocols.
- FEHNKER, GAO
- 2006
(Show Context)
Citation Context ...d 600 lines of code in HOL. In the telecommunications domain, various systems, such as the time elapsing in a synchronous fashion of a Bluetooth device [20], wireless LAN [63] and broadcast protocols =-=[22]-=-, are described using state diagrams, which can be modeled as the DTMCs. The existing library of the formalized DTMC presented in this chapter can be applied to verify these telecommunication system m... |

30 |
Locality in page reference strings
- Shedler, Tung
- 1972
(Show Context)
Citation Context ...at this LRU stack specification does not cover the case of nonuniform page referencing behaviors of some programs. Thus, we have been able to formally verify the numerical methods result presented in =-=[106]-=-. The ability to formally verify theorems involving classified Markovian models and the proof script only consists of about 400 lines code in HOL4. The short script clearly indicates the usefulness of... |

29 | Rapture: A tool for verifying Markov Decision Processes.
- Jeannet, D’Argenio, et al.
- 2002
(Show Context)
Citation Context ...arious queues can be found in the open literature, such as M/G/1, M/G/∞, GI/GI/1 [101] and they can be used to model diverse interesting applications. • The principal of Markov Decision Process (MDP) =-=[52]-=- is a Markov chain with a reward function and a discount factor. We can develop a platform for formally analyzing MDP models using our formalization of DTMC. For instance, the formalization of discret... |

26 |
Basic Probability Theory
- Ash
- 1970
(Show Context)
Citation Context ...-homogeneous DTMC, based on the state diagram and the initial distribution and transition probability matrix. Definition 3.14. (AMQM Collection Model) ` ∀ X p a b c d. AMQM MODEL X p n = th dtmc X p (=-=[0, 3]-=-, POW [0, 3]) Li (Ltr n) ∧ (1 ≤ n) Here, the state space is represented as a pair, in which the set [0, 3] contains all the states and POW [0,3] is the sigma-algebra of the states set [0, 3], to ensur... |

25 |
HYDRA: HYpergraph-based Distributed Responsetime Analyser
- Dingle, Harrison, et al.
- 2003
(Show Context)
Citation Context ...vide the services on analyzing the failure or repair of a model, which may occur in the lifetime of any product. Some other software tools used for evaluating performance, e.g., MACOM [102] and HYDRA =-=[85]-=-, take the advantage of a popular Markovian process algebra [7], i.e., PEPA [92], to model systems and efficiently compute passage time densities and quantities in large-scale Markov chains. Another t... |

23 |
Elements of Queueing Theory, with Applications,
- Saaty
- 1961
(Show Context)
Citation Context ...ct may start by formalizing the count process and poisson process [112], as well as the continuous-time Markov chain. Various queues can be found in the open literature, such as M/G/1, M/G/∞, GI/GI/1 =-=[101]-=- and they can be used to model diverse interesting applications. • The principal of Markov Decision Process (MDP) [52] is a Markov chain with a reward function and a discount factor. We can develop a ... |

22 |
Ruin problems with compounding assets,
- Harrison
- 1977
(Show Context)
Citation Context ...ing higher-order logic by applying our formalization of DTMC interpreted in this 96 thesis. The principal of the gambler’s ruin problem offers the capabilities of modeling the risk insurance business =-=[38]-=- in higher-order logic. • Our work provides a possibility to combine both model checking and theorem proving in order to offer an integrated framework that takes advantage of both methods for formally... |

21 | Thermodynamic formalism for systems with Markov dynamics.
- Lecomte, Appert-Rolland, et al.
- 2007
(Show Context)
Citation Context ...ysics Figure 1.1: Markov Chain Application Fields Death process [109], is applied in modeling biological populations. Also, the Markov chain theory has been applied in physics (such as thermodynamics =-=[65]-=- and statistical 2 mechanics [97]). Moreover, the enzyme activity and the growth of copolymers can be modeled as a Markov chain in chemistry [104]. Furthermore, a variety of economics and finance phen... |

20 |
An introduction to Stochastic Process
- Kannan
- 1997
(Show Context)
Citation Context ...d as light blue) built upon in the first step. These pre-verified theorems include some classical theorems, such as the joint probability theorem, Chapman-Kolmogorov Equation and Absolute probability =-=[55]-=-, etc., and the stationary properties [55], which are verified in higher-order logic based on the classified discrete-time Markov chain. If the system can be described as a hidden Markov model [72], t... |

17 |
The σ-Additive Measure Theory
- Bialas
- 1990
(Show Context)
Citation Context ...cess denotes a collection of random variables Xt (t ∈ T ). If the indices (t) of random variables Xt are discrete, then this random process is a discretetime random process. Nedzusiak [86] and Bialas =-=[9]-=- were among the first to propose to formalize some measure and probability theories in higher-order-logic. Later, Hurd [49] implemented their work and formalized a certain measure space as a pair (Σ, ... |

15 | Continuous-Time Markov Decision Processes: Theory and Applications, ser. Stochastic modelling and applied probability.
- Guo, Hernández-Lerma
- 2009
(Show Context)
Citation Context ...ls using our formalization of DTMC. For instance, the formalization of discrete-time MDP is just a simple extension of Definition 3.1. It can also be further extended to formalize continuous-time MDP =-=[31]-=-, which can 95 be applied in the analysis of queueing systems, epidemic processes and population processes (including Birth-Death Processes, Birth, Death and Catastrophe Processes) [27]. • Markov Rand... |

14 | K.S.Trivedi, “Failure Correlation in Software Reliability Models
- Goseva-Popstojanova
- 2000
(Show Context)
Citation Context ...sion model in a Dynamic Spectrum Access/Cognitive Radio (DSA/CR) system [111], which is constructed for measuring spectrum occupancy, or the sdependent Bernoulli trials in software reliability models =-=[29]-=-, which is used to estimating the software failures. Also, the simple random walk with no barriers [3] can also be expressed using this DTMC model. Another interesting application presented in this ch... |

13 | Formal probabilistic analysis using theorem proving,
- Hasan
- 2008
(Show Context)
Citation Context ...as already described in the previous section, this technique is capable of conducting formal probabilistic analysis, which is the prerequisite of the Markov chain model analysis. Based on the work of =-=[39]-=- and later [79], in this thesis, we propose a comprehensive framework for the formal analysis of DTMC models by means of higher-order-logic theorem proving. A general overview of the proposed framewor... |

11 |
A.: Three Chapters of Measure Theory in Isabelle/HOL. In:
- Holzl, Heller
- 2011
(Show Context)
Citation Context ...development based on the axiomatic definition of probability proposed by Kolmogorov [58]. Mhamdi’s theory provides a mathematical consistent for assigning and deducing probabilities of events. Hölzl =-=[46]-=- has also formalized three chapters of measure theory in Isabelle/HOL. Affeldt [2] simplified the formalization of probability theory in Coq [17]. Among these works, the probability theory formalized ... |

10 |
Mechanizing Programming Logics in Higher-0rder Logic. In: Current Trends
- Gordon
- 1989
(Show Context)
Citation Context ...over, model checking suffers from state-explosion problem when analyzing larger systems and some of its underlying algorithms are not formally verified. Higher-order logic interactive theorem proving =-=[28]-=- is a formal method that provides a conceptually simple formalism with precise semantics. It allows to construct a computer based mathematical model of the system and perform mathematical reasoning to... |

8 | Markov chain-based performance analysis of multihop - Martalo, Busanelli, et al. - 2009 |

8 |
Use of the LRU stack depth distribution for simulation of paging behavior
- Turner, Strecker
- 1977
(Show Context)
Citation Context ... other, whereas this correlation property between these two sequences is not correct. Thus, most of the times we have to develop the proofs in HOL from scratch. The LRU stack model is a typical model =-=[99]-=- for simulating the paging behaviors of memory management in computer architectures. Later, this model was found to be erroneous based on simulation results [110] and the authors commented that some e... |

5 | Formal reasoning about expectation properties for continuous random variables, in: Formal Methods, in:
- Hasan, Abbasi, et al.
- 2009
(Show Context)
Citation Context ...y theory, Hasan [39] provides the formalization of random variables and the verification of statistical properties, such as expectation and variance, for both discrete and continuous random variables =-=[40]-=-. However, as mentioned above, the fact that the sample space in Hurd’s measure theory is a universal set results in a very complex definition for the arbitrary space, where the space is not the unive... |

5 | A note on adiabatic theorem for Markov chains
- Kovchegov
- 2010
(Show Context)
Citation Context ...r formalization of DTMC is general enough to describe time-inhomogeneous DTMCs, which are part of the notations in MCMC algorithms [11] and are also applied in many fields, like the adiabatic theorem =-=[60]-=- in quantum computations, by means of instantiating the transition probability (in Definition 3.3), which is a function taking the time t as its argument. The main challenge of our work is to describe... |

5 |
Discrete hidden Markov models with application to isolated user-dependent hand gesture recognition
- Mäntylä
- 2001
(Show Context)
Citation Context ... implemented in probabilistic model checker for analyzing DTMC models can be verified using theorem proving [47]. Similarly, the algorithms, such as Forward-Backward, Viterbi and Baum-Welch algorithm =-=[74]-=-, implemented in many statistical simulation tools for analyzing discrete HMMs can also be verified by using the verified HMMs theorems interpreted in Chapter 5. These algorithms can then be integrate... |

5 | On the Formalization of the Lebesgue Integration Theory in HOL.
- Mhamdi, Hasan, et al.
- 2010
(Show Context)
Citation Context ...d automatic proof methods. Due to the undecidable nature of higher-order logic, users have to verify theorems in an interactive way. The most advanced probability theories, on which our work depends, =-=[80]-=- is built upon theorem prover HOL4, hence, we select the theorem prover HOL4 as our proof assistant. Based on Robin Milner’s proof-checking program, Logic for Computable Functions (LCF) [82], HOL4 is ... |

5 |
A Markov chain occuring in enzyme kinetics
- Shapiro, Zeilberger
- 1982
(Show Context)
Citation Context ...ry has been applied in physics (such as thermodynamics [65] and statistical 2 mechanics [97]). Moreover, the enzyme activity and the growth of copolymers can be modeled as a Markov chain in chemistry =-=[104]-=-. Furthermore, a variety of economics and finance phenomena, such as asset prices [35] and market crashes [93], are described as Markov chains. Some music composition algorithms [54] based on the Mark... |

4 |
Approximate queuing models for multiprogramming computer systems.
- Avi-Itzhak, Heyman
- 1973
(Show Context)
Citation Context ...ce string is D1D2· · ·Dt· · · , which is associated with the referencing string. This distance string can be modeled as a sequence of independent and identically 63 distributed (IID) random variables =-=[4]-=-, which makes their probability mass function (PMF) as Pr(Dt = i) = ai, where i = 1, 2, . . . , n and refers to the position of the least recently used page in the stack at time t, and ∑n j=1aj = 1. T... |

4 |
What is Formal Verification?
- Bjesse
- 2005
(Show Context)
Citation Context ...ates” of the DNA, characterized by the probabilities of occurrence of each base. In this DNA model, the first state is called exon (E), the second one 5’ splice site (5), and the third one intron (I) =-=[10]-=-. This model is described and studied very naturally using HMMs [21]: a DTMC over the states E, 5, and I is used in order to know in which state the proteins are, then another random process is define... |

3 | Towards a formal verification methodology for collective robotic systems - Gjondrekaj, Loreti, et al. |

3 |
An Epidemic Process in an Open Population
- Goffman
- 1965
(Show Context)
Citation Context ...nuous-time MDP [31], which can 95 be applied in the analysis of queueing systems, epidemic processes and population processes (including Birth-Death Processes, Birth, Death and Catastrophe Processes) =-=[27]-=-. • Markov Random Field (MRF) (also called undirected graphical model)[56] is a type of stochastic process, which forms a natural generalization of Markov processes, where the time index is replaced b... |

3 |
Verifying pCTL model checking
- Hölzl, Nipkow
- 2012
(Show Context)
Citation Context ...] and all the formalizations of discrete-time Markov chain are executed in theorem prover HOL4 [28]. To the best of our knowledge, the only related work to ours is a very recent work by Hölzl et al. =-=[47]-=-, who formally defined a time-homogeneous Markov chain based on the finite state space and the transition matrix in Isabelle/HOL, where the authors they assumed no initial distribution or start state.... |

3 | Weighted markov chain model for musical composer identification
- Kaliakatsos-Papakostas, Epitropakis, et al.
- 2011
(Show Context)
Citation Context ...ain in chemistry [104]. Furthermore, a variety of economics and finance phenomena, such as asset prices [35] and market crashes [93], are described as Markov chains. Some music composition algorithms =-=[54]-=- based on the Markov chain theory are employed in software. In addition, the Markov chain theory has been applied in advanced baseball analysis [19]. A DTMC is further divided into two main categories... |

3 | Information-Theoretic Analysis using Theorem Proving.
- Mhamdi
- 2012
(Show Context)
Citation Context ...ribed in the previous section, this technique is capable of conducting formal probabilistic analysis, which is the prerequisite of the Markov chain model analysis. Based on the work of [39] and later =-=[79]-=-, in this thesis, we propose a comprehensive framework for the formal analysis of DTMC models by means of higher-order-logic theorem proving. A general overview of the proposed framework is depicted i... |

3 |
Allocation of resources on a minimized cost basis
- Prasad, Ender, et al.
- 1974
(Show Context)
Citation Context ...zyme activity and the growth of copolymers can be modeled as a Markov chain in chemistry [104]. Furthermore, a variety of economics and finance phenomena, such as asset prices [35] and market crashes =-=[93]-=-, are described as Markov chains. Some music composition algorithms [54] based on the Markov chain theory are employed in software. In addition, the Markov chain theory has been applied in advanced ba... |

2 | Formalization of Shannon’s Theorems in SSReflectCoq - Affeldt, Hagiwara - 2012 |

2 |
Random Processes for Electrical Engineering
- Probability
- 2007
(Show Context)
Citation Context ...MRF from the discrete space point of view. • The random walk is a mathematical model describing a path that consists of a succession of random steps. In fact, the random walk process is a sum process =-=[66]-=- and this type of random process has several different classes, such as the symmetric random walk and the asymmetric simple random walk [59]. Among the diverse random walk processes, the simple random... |

2 |
Formalization of finite-state discrete-time markov chains
- Liu, Hasan, et al.
- 2011
(Show Context)
Citation Context ...ges. The output certifies that the given system specification are valid for the given Markovian system. Based on [39], we developed a formalization of time-homogeneous DTMC with 11 finite state space =-=[69]-=-. However, the definition of DTMC is not general enough due to the inherit disadvantage of the probability theorem in [49] and the verified theorems are not rich enough to deal with various DTMC model... |

2 |
The Ising model in physics and statistical genetics
- Majewski, Li, et al.
(Show Context)
Citation Context ...own ferromagnetism statistical models, Ising model [67], can be formalized based on the Markov chain mixing time notations and it can be applied in neuroscience [48], physics and statistical genetics =-=[73]-=-. On the other hand, our formalization of DTMC is general enough to describe time-inhomogeneous DTMCs, which are part of the notations in MCMC algorithms [11] and are also applied in many fields, like... |

2 |
Schütte C, Generator estimation of Markov jump processes
- Metzner, Dittmer, et al.
- 2007
(Show Context)
Citation Context ... diseases (i.e. the progress of breast cancer) in medical and biological domains, or estimating the electronic system reliability. Continuous-time semi-Markov processes [51] and Markov jump processes =-=[78]-=- can also be formalized using such a CTMC formalization. • Queueing theory [11] is a mathematical study of a system which processes of the flows of customers or/and services. A queue system is usually... |

2 |
Formalization of Entropy Measures in HOL
- Mhamdi, Hasan, et al.
- 2011
(Show Context)
Citation Context ...orem in [49] and the verified theorems are not rich enough to deal with various DTMC models. Our work, in this thesis, mainly relies on the most recent and general formalization of probability theory =-=[81]-=- and all the formalizations of discrete-time Markov chain are executed in theorem prover HOL4 [28]. To the best of our knowledge, the only related work to ours is a very recent work by Hölzl et al. [... |

2 |
Isabelle: A Generic Theroem Prover, volume 828 of LNCS
- Paulson
- 1994
(Show Context)
Citation Context ...bin Milner’s proof-checking program, Logic for Computable Functions (LCF) [82], HOL4 is developed for conducting proofs in higher-order logic by using the strongly-typed functional Meta-Language (ML) =-=[91]-=-. As a system of deduction with a precise semantics, HOL4 is capable of verifying a wide variety of hardware and software as well as pure mathematics due to the high expressiveness higher-order 16 log... |

2 | MACOM - A Tool for Evaluating Communication Systems.
- Sczittnick
- 1994
(Show Context)
Citation Context ...tools mainly provide the services on analyzing the failure or repair of a model, which may occur in the lifetime of any product. Some other software tools used for evaluating performance, e.g., MACOM =-=[102]-=- and HYDRA [85], take the advantage of a popular Markovian process algebra [7], i.e., PEPA [92], to model systems and efficiently compute passage time densities and quantities in large-scale Markov ch... |

2 |
Cognitive Radio and its Application for Next Generation Cellular and Wireless Networks
- Venkataraman, Muntean
- 2012
(Show Context)
Citation Context ... usefulness of our formalization of DTMC. 50 The DTMC presented in the above example can also been interpreted as the time-dimension model in a Dynamic Spectrum Access/Cognitive Radio (DSA/CR) system =-=[111]-=-, which is constructed for measuring spectrum occupancy, or the sdependent Bernoulli trials in software reliability models [29], which is used to estimating the software failures. Also, the simple ran... |

2 |
An Introduction to Hidden Markov Models and Bayesian Networks.
- Zoubin
- 2001
(Show Context)
Citation Context ...f the random function but not the underlying states. That is why the Markov chain involved in this process is called hidden Markov chain. The observed sequence is said to be conditionally independent =-=[114]-=- on this hidden Markov chain. 3 Initially, HMMs were proposed to solve optimal linear filtering problems as the simplest dynamic Bayesian networks. However, due to their usefulness in effectively anal... |

1 |
Markov Analysis of APBA, a Baseball Simulation Game
- Davis
- 2011
(Show Context)
Citation Context ...as Markov chains. Some music composition algorithms [54] based on the Markov chain theory are employed in software. In addition, the Markov chain theory has been applied in advanced baseball analysis =-=[19]-=-. A DTMC is further divided into two main categories. It may be time homogeneous, which refers to the case where those Markov chains exhibit the constant transition probabilities between the states, o... |

1 | Distributed Response Time Analysis of GSPN Models with MapReduce
- Haggarty, Knottenbelt, et al.
- 2008
(Show Context)
Citation Context ...evaluate distributed systems and analyze the dynamic events of the models by means of embedded Markov chain theory. For example, the quantitative analysis of Generalized Stochastic Petri Nets (GSPNs) =-=[33]-=- mainly depends on a Markovian solution, in which the models are described as semi-Markov processes in order to calculate the steady state distributions of the stochastic systems. The calculations are... |

1 |
Reasoning about Conditional Probabilities in a HigherOrder-Logic Theorem Prover
- Hasan, Tahar
(Show Context)
Citation Context ...cur. Conditional probability basically reflects the dependency between the events which happen at different times in a process. The formal definition of conditional probability in HOL can be found in =-=[41]-=-, which is based on Hurd’s work [49]. In order to make use of the most advanced probability theory in our work, we improved the formalization of conditional probability as: Definition 2.1. (Conditiona... |

1 |
Neurocomputing: Foundations of Research
- Hopfield
- 1988
(Show Context)
Citation Context ...rgence Theorem [67]. One of the well-known ferromagnetism statistical models, Ising model [67], can be formalized based on the Markov chain mixing time notations and it can be applied in neuroscience =-=[48]-=-, physics and statistical genetics [73]. On the other hand, our formalization of DTMC is general enough to describe time-inhomogeneous DTMCs, which are part of the notations in MCMC algorithms [11] an... |

1 |
Introduction to Stochastic Petri Net Theroy
- Kritzinger, Bause
- 1995
(Show Context)
Citation Context ...lobal modeling. SPN are utilized to model the stochastic systems and offer the capability of analyzing large and complex models. The Markov chain of an SPN is modeled by means of a reachability graph =-=[61]-=-. The prevailing software 5 tools of stochastic petri nets are SPNP [14] and GreatSPN [30]. These tools can model, validate, and evaluate distributed systems and analyze the dynamic events of the mode... |

1 | Statistical Guarantees of Performance for RTL Designs - Kumar - 2012 |

1 |
On the Formalization of Continuous-Time Markov Chains in HOL
- Liu, Hasan, et al.
- 2013
(Show Context)
Citation Context ... (particularly, exponentially distributed) interval of time and then transit to different states. Using the formal definition of the continuous-time Markov chain and two of its verified properties in =-=[70]-=- and the limiting, derivative and integral theories, it is possible formally derive the Kolmogorov’s backward equation [109] as well as variety of its other properties in HOL4. The formalization of CT... |

1 |
Analysis of Interrogator-tag Communication Protocols
- Nokovic, Sekerinski
- 2010
(Show Context)
Citation Context ...he formalization of DTMC in higher-order logic, in this section we present two applications: a simplified binary communication channel [109] and the Automatic Mail Quality Measurement (AMQM) protocol =-=[87]-=-. 38 3.5.1 Binary Communication Model A binary communication channel [109] is a channel with binary inputs and outputs. The transmission channel is assumed to be noisy or imperfect, i.e., it is likely... |

1 | Early Quality Monitoring in the Development of Real-time Reactive Systems - Ormandjieva, Alagar, et al. |

1 |
Efficient Optimization of Large Probabilistic Models
- Struck, Gdemann, et al.
(Show Context)
Citation Context ...se approximations reduce the accuracy of the results or even execute unreliable results. Moreover, probabilistic model checking 8 tools often utilize unverified algorithms and optimization techniques =-=[108]-=-. Finally, model checking cannot be used to verify generic mathematical expressions for probabilistic analysis due to the inherent state-based nature of the approach. To the best of our knowledge, no ... |

1 |
On LRU Stack Model Suitability
- Tzelnic
- 1978
(Show Context)
Citation Context ...The LRU stack model is a typical model [99] for simulating the paging behaviors of memory management in computer architectures. Later, this model was found to be erroneous based on simulation results =-=[110]-=- and the authors commented that some experimental results had shown that the LRU stack depth distribution varies significantly among programs, even the same program running with different stimuli [110... |