Marco B. Caminati

Marco B. Caminati

In a recent paper, new theorems linking apparently unrelated mathematical objects (event structures from concurrency theory and full graphs arising in computational biology) were discovered by cross-site data mining on huge databases, and building on existing Isabelle-verified event structures enumeration algorithms. Given the origin and newness of such theorems, their formal verification is particularly desirable. This paper presents such a verification via Isabelle/HOL definitions and theorems, and exposes the technical challenges found in the process. The introduced formalisation completes the verification of Isabelle-verified event structure enumeration algorithms into a fully verified framework to link event structures to full graphs.

Marco Caminati

A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case. As a supplement, the same conjecture is shown, through computing a counterexample, to be false when unimodularity hypothesis is dropped.

Marco B. Caminati

Classifying formal languages according to the expressiveness of grammars able to generate them is a fundamental problem in computational linguistics and, therefore, in the theory of computation. Furthermore, such kind of analysis can give insight into the classification of abstract algebraic structure such as groups, for example through the correspondence given by the word problem. While many such classification problems remain open, others have been settled. Recently, it was proved that $n$-balanced languages (i.e., whose strings contain the same occurrences of letters $a_i$ and $A_i$ with $1\leq i \leq n$) can be generated by multiple context-free grammars (MCFGs), which are one of the several slight extensions of context free grammars added to the classical Chomsky hierarchy to make the mentioned classification more precise. This paper analyses the existing proofs from the computational and the proof-theoretical point of views, systematically studying whether each proof can lead to a verified (i.e., checked by a proof assistant) algorithm parsing balanced languages via MCFGs. We conclude that none of the existing proofs is realistically suitable against this practical goal, and proceed to provide a radically new, elementary, extremely short proof for the crucial case $n \leq 2$. A comparative analysis with respect to the existing proofs is finally performed to justify why the proposed proof is a substantial step towards concretely obtaining a verified parsing algorithm for $O_2$.

Juliana Bowles, Marco B. Caminati

Complex systems are usually modelled through a combination of structural and behavioural models, where separate behavioural models make it easier to design and understand partial behaviour. When partial models are combined, we need to guarantee that they are consistent, and several automated techniques have been developed to check this. We argue that in some cases it is impossible to guarantee total consistency, and instead we want to find execution paths across such models with minimal conflicts with respect to a certain metric of interest. We present an efficient and scalable solution to find optimal paths through a combination of the theorem prover Isabelle with the constraint solver Z3. Our approach has been inspired by a healthcare problem, namely how to detect conflicts between medications taken by patients with multiple chronic conditions, and how to find preferable alternatives automatically.

Marco B. Caminati, Manfred Kerber, Colin Rowat

We present an original theorem in auction theory: it specifies general conditions under which the sum of the payments of all bidders is necessarily not identically zero, and more generally not constant. Moreover, it explicitly supplies a construction for a finite minimal set of possible bids on which such a sum is not constant. In particular, this theorem applies to the important case of a second-price Vickrey auction, where it reduces to a basic result of which a novel proof is given. To enhance the confidence in this new theorem, it has been formalized in Isabelle/HOL: the main results and definitions of the formal proof are re- produced here in common mathematical language, and are accompanied by an informal discussion about the underlying ideas.