Unlikely intersection in higher-dimensional formal groups and pp-adic dynamical systems

In this work, we study arithmetic dynamical questions for formal groups and $p$-adic dynamical systems in higher dimensions. As a generalization of Berger's result in 1-dimensional case, in the paper, we prove that for any two higher-dimensional formal groups over the ring of $p$-adic integers, if they have infinitely many torsion points in common, then they coincide. Finally, we prove if $\mathcal{D}_1$ and $\mathcal{D}_2$ are $d$-dimensional stable $p$-adic dynamical systems with infinitely many preperiodic points in common then $\mathcal{D}_1$ and $\mathcal{D}_2$ coincide.Comment: 16 pages, In this work we have studied arithmetic dynamical questions in formal groups and non-Archimedean dynamical systems. Comments are welcome

Threshold functions and Poisson convergence for systems of equations in random sets

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "$\mathcal{A}$ contains a non-trivial solution of $M\cdot\textbf{x}=\textbf{0}$", where $\mathcal{A}$ is a random set and each of its elements is chosen independently with the same probability from the interval of integers $\{1,\dots,n\}$. Our study contains a formal definition of trivial solutions for any combinatorial structure, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the behaviour of the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page

Collaborative Verification-Driven Engineering of Hybrid Systems

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Often, hybrid systems are rather complex in that they require expertise from many domains (e.g., robotics, control systems, computer science, software engineering, and mechanical engineering). Moreover, despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires nontrivial human guidance, since hybrid systems verification tools solve undecidable problems. It is, thus, not uncommon for development and verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (i) graphical (UML) and textual modeling of hybrid systems, (ii) exchanging and comparing models and proofs, and (iii) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks

Computational reverse mathematics and foundational analysis

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only $\omega$-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a $\Pi^1_1$ complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than $\Pi^1_1\text{-}\mathsf{CA}_0$.Comment: Submitted. 41 page

A Vision of Collaborative Verification-Driven Engineering of Hybrid Systems

Abstract. Hybrid systems with both discrete and continuous dynamics are an important model for real-world physical systems. The key challenge is how to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires significant human guidance, since hybrid systems verification tools solve undecidable problems. It is thus not uncommon for verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (i) modeling hybrid systems, (ii) exchanging and comparing models and proofs, and (iii) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks.