599 research outputs found
Formal Reasoning About Finite-State Discrete-Time Markov Chains in HOL
Markov chains are extensively used in modeling different aspects of engineering and scientific systems, such as performance of algorithms and reliability of systems. Different techniques have been developed for analyzing Markovian models, for example, Markov Chain Monte Carlo based simulation, Markov Analyzer, and more recently probabilistic model-checking. However, these techniques either do not guarantee accurate analysis or are not scalable. Higher-order-logic theorem proving is a formal method that has the ability to overcome the above mentioned limitations. However, it is not mature enough to handle all sorts of Markovian models. In this paper, we propose a formalization of Discrete-Time Markov Chain (DTMC) that facilitates formal reasoning about time-homogeneous finite-state discrete-time Markov chain. In particular, we provide a formal verification on some of its important properties, such as joint probabilities, Chapman-Kolmogorov equation, reversibility property, using higher-order logic. To demonstrate the usefulness of our work, we analyze two applications: a simplified binary communication channel and the Automatic Mail Quality Measurement protocol
Formal analysis techniques for gossiping protocols
We give a survey of formal verification techniques that can be used to corroborate existing experimental results for gossiping protocols in a rigorous manner. We present properties of interest for gossiping protocols and discuss how various formal evaluation techniques can be employed to predict them
Formal Availability Analysis using Theorem Proving
Availability analysis is used to assess the possible failures and their
restoration process for a given system. This analysis involves the calculation
of instantaneous and steady-state availabilities of the individual system
components and the usage of this information along with the commonly used
availability modeling techniques, such as Availability Block Diagrams (ABD) and
Fault Trees (FTs) to determine the system-level availability. Traditionally,
availability analyses are conducted using paper-and-pencil methods and
simulation tools but they cannot ascertain absolute correctness due to their
inaccuracy limitations. As a complementary approach, we propose to use the
higher-order-logic theorem prover HOL4 to conduct the availability analysis of
safety-critical systems. For this purpose, we present a higher-order-logic
formalization of instantaneous and steady-state availability, ABD
configurations and generic unavailability FT gates. For illustration purposes,
these formalizations are utilized to conduct formal availability analysis of a
satellite solar array, which is used as the main source of power for the Dong
Fang Hong-3 (DFH-3) satellite.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1505.0264
Formalization of Discrete-time Markov Chains in HOL
Markov chains are extensively used in the modeling and analysis of engineering and scientific problems which can be expressed as random processes with the memoryless property. Usually, paper-and-pencil proofs, simulation or computer algebra software are used to analyze Markovian models. However, these techniques either are not scalable or do not guarantee accurate results, which are vital in safety-critical systems. To improve the accuracy of the analysis, probabilistic model checking has been recently proposed to formally analyze Markovian systems. However, model checking suffers from the inherent state-explosion problem and thus has a very limited scope in terms of analyzing Markovian models.\newline
\indent In order to overcome the above mentioned limitations, this thesis advocates the usage of higher-order-logic theorem proving for conducting the analysis of Markov chains. We present the higher-order-logic formalization of Discrete-time Markov Chains with finite number of discrete states. We also verify some of their most widely used properties using a theorem prover. These foundations allow us to formally express and reason about Markov chains within the sound core of a theorem prover and thus attain precise results. Moreover, by building upon these foundational results, this thesis also presents the formalization of classified discrete-time Markov chains and hidden Markov chains in higher-order logic. These are widely used concepts in the analysis of Markovian models and thus allow us to tackle the formal analysis of a wide range of engineering and scientific systems. For illustration purposes, the thesis also presents some applications including a binary communication channel, the automatic mail quality measurement (AMQM) protocol, a DNA sequence, a least recently used (LRU) stack model and the birth-death process
Formalization of Continuous Time Markov Chains with Applications in Queueing Theory
The performance analysis of engineering systems have become very critical due to their usage in safety and mission critical domains such as military and biomedical devices. Such an analysis is often carried out based on the Markovian (or Markov Chains based) models of underlying software and hardware components. Furthermore, some important properties can only be captured by queueing theory which involves Markov Chains with continuous time behavior. Classically, the analysis of such models has been performed using paper-and-pencil based proofs and computer simulation, both of which cannot provide perfectly accurate results due to the error-prone nature of manual proofs and the non-exhaustive nature of simulation. Recently, model checking based formal methods have also been used to analyze Markovian and queuing systems. However, such an approach is only applicable for small systems and cannot certify generic properties due to the sate-space explosion problem.
In this thesis, we propose to use higher-order-logic theorem proving as a complementary approach to conduct the formal analysis of queueing systems. To this aim, we present the higher-order-logic formalization of the Poisson process which is the foremost step to model queueing systems. We also verify some of its classical properties such as exponentially distributed inter-arrival time, memoryless property and
independent and stationary increments. Moreover, we used the formalization of the Poisson process to model and verify the error probability of a generic optical communication system. Then we present the formalization of Continuous-Time Markov Chains along with the Birth-Death process. Lastly, we demonstrate the utilization of our developed infrastructure by presenting the formalization of an M/M/1 queue which is widely used to model telecommunication systems. We also formally verified the generic result about the average waiting time for any given queue
Formal verification of higher-order probabilistic programs
Probabilistic programming provides a convenient lingua franca for writing
succinct and rigorous descriptions of probabilistic models and inference tasks.
Several probabilistic programming languages, including Anglican, Church or
Hakaru, derive their expressiveness from a powerful combination of continuous
distributions, conditioning, and higher-order functions. Although very
important for practical applications, these combined features raise fundamental
challenges for program semantics and verification. Several recent works offer
promising answers to these challenges, but their primary focus is on semantical
issues.
In this paper, we take a step further and we develop a set of program logics,
named PPV, for proving properties of programs written in an expressive
probabilistic higher-order language with continuous distributions and operators
for conditioning distributions by real-valued functions. Pleasingly, our
program logics retain the comfortable reasoning style of informal proofs thanks
to carefully selected axiomatizations of key results from probability theory.
The versatility of our logics is illustrated through the formal verification of
several intricate examples from statistics, probabilistic inference, and
machine learning. We further show the expressiveness of our logics by giving
sound embeddings of existing logics. In particular, we do this in a parametric
way by showing how the semantics idea of (unary and relational) TT-lifting can
be internalized in our logics. The soundness of PPV follows by interpreting
programs and assertions in quasi-Borel spaces (QBS), a recently proposed
variant of Borel spaces with a good structure for interpreting higher order
probabilistic programs
Foundational (co)datatypes and (co)recursion for higher-order logic
We describe a line of work that started in 2011 towards enriching Isabelle/HOL's language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)induction. In contrast with other systems offering codatatypes, no additional axioms or logic extensions are necessary with our approach
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