Decidability of Difference Logic over the Reals with Uninterpreted Unary Predicates

First-order logic fragments mixing quantifiers, arithmetic, and uninterpreted predicates are often undecidable, as is, for instance, Presburger arithmetic extended with a single uninterpreted unary predicate. In the SMT world, difference logic is a quite popular fragment of linear arithmetic which is less expressive than Presburger arithmetic. Difference logic on integers with uninterpreted unary predicates is known to be decidable, even in the presence of quantifiers. We here show that (quantified) difference logic on real numbers with a single uninterpreted unary predicate is undecidable, quite surprisingly. Moreover, we prove that difference logic on integers, together with order on reals, combined with uninterpreted unary predicates, remains decidable.Comment: This is the preprint for the submission published in CADE-29. It also includes an additional detailed proof in the appendix. The Version of Record of this contribution will be published in CADE-2

Deciding Satisfiability for Fragments with Unary Predicates and Difference Arithmetic

peer reviewedWe report our work in progress\footnote[2]{Our knowledge on this matter has significantly improved since this paper was written. An update on this work will be provided in a following article.} to build decision procedures for highly expressive languages mixing arithmetic and uninterpreted predicates. More precisely, we study a language combining difference-logic constraints and unary predicates. The decision problem is impacted by the domain of interpretation $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$. For the integer domains $\mathbb{N}$ and $\mathbb{Z}$, the problem reduces to deciding the monadic second-order theory of $\mathbb{N}$ with the successor relation (S1S), which is known to be feasible by translating formulas into automata, and then checking the emptiness of their accepted language. We also advocate the use of automata as an appropriate tool to represent the set of models for the real domain $\mathbb{R}$

Precision measurement of the structure of the CMS inner tracking system using nuclear interactions

The structure of the CMS inner tracking system has been studied using nuclear interactions of hadrons striking its material. Data from proton-proton collisions at a center-of-mass energy of 13 TeV recorded in 2015 at the LHC are used to reconstruct millions of secondary vertices from these nuclear interactions. Precise positions of the beam pipe and the inner tracking system elements, such as the pixel detector support tube, and barrel pixel detector inner shield and support rails, are determined using these vertices. These measurements are important for detector simulations, detector upgrades, and to identify any changes in the positions of inactive elements

Precision measurement of the structure of the CMS inner tracking system using nuclear interactions

The structure of the CMS inner tracking system has been studied using nuclear interactions of hadrons striking its material. Data from proton-proton collisions at a center-of-mass energy of 13 TeV recorded in 2015 at the LHC are used to reconstruct millions of secondary vertices from these nuclear interactions. Precise positions of the beam pipe and the inner tracking system elements, such as the pixel detector support tube, and barrel pixel detector inner shield and support rails, are determined using these vertices. These measurements are important for detector simulations, detector upgrades, and to identify any changes in the positions of inactive elements