Validation of Bitcoin transactions rely upon the successful execution of scripts written in a simple and effective, non-Turing-complete by design language, simply called SCRIPT. This makes the validation of closed scripts, i.e. those associated to actual transactions and bearing full information, straightforward. Here we address the problem of validating open scripts, i.e. we address the validation of redeeming scripts against the whole set of possible inputs, i.e. under which general conditions can Bitcoins be redeemed? Even if likely not one of the most complex languages and demanding verification problems, we advocate the merit of formal verification for the Bitcoin validation framework. We propose a symbolic verification theory for of open SCRIPT, a verifier tool-kit, and illustrate examples of use on Bitcoin transactions. Contributions include 1) a formalisation of (a fragment of) the language; 2) a novel symbolic approach to SCRIPT verification, suitable, e.g. for the verification of newly defined and non-standard payment schemas; and 3) building blocks for a larger verification theory for the developing area of Bitcoin smart contracts. The verification of smart contracts, i.e. agreements built as transaction-based protocols, is currently a difficult to formalise and computationally demanding problem.

In model-based testing (MBT) we may have to deal with a non-deterministic model, e.g. because abstraction was applied, or because the software under test itself is non-deterministic. The same test case may then trigger multiple possible execution paths, depending on some internal decisions made by the software. Consequently, performing precise test analyses, e.g. to calculate the test coverage, are not possible. This can be mitigated if developers can annotate the model with estimated probabilities for taking each transition. A probabilistic model checking algorithm can subsequently be used to do simple probabilistic coverage analysis. However, in practice developers often want to know what the achieved aggregate coverage, which unfortunately cannot be re-expressed as a standard model checking problem. This paper presents an extension to allow efficient calculation of probabilistic aggregate coverage, and moreover also in combination with k-wise coverage.

In model-based testing, we may have to deal with a non-deterministic model, e.g. because abstraction was applied, or because the software under test itself is non-deterministic. The same test case may then trigger multiple possible execution paths, depending on some internal decisions made by the software. Consequently, performing precise test analyses, e.g. to calculate the test coverage, are not possible.. This can be mitigated if developers can annotate the model with estimated probabilities for taking each transition. A probabilistic model checking algorithm can subsequently be used to do simple probabilistic coverage analysis. However, in practice developers often want to know what the achieved aggregate coverage is, which unfortunately cannot be re-expressed as a standard model checking problem. This paper presents an extension to allow efficient calculation of probabilistic aggregate coverage, and also of probabilistic aggregate coverage in combination with k-wise coverage.

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