The Interoperability Challenge: Building a Model-Driven Digital Thread Platform for CPS

With the heterogeneity of the industry 4.0 world, and more generally of the Cyberphysical Systems realm, the quest towards a platform approach to solve the interoperability problem is front and centre to any system and system-of-systems project. Traditional approaches cover individual aspects, like data exchange formats and published interfaces. They may adhere to some standard, however they hardly cover the production of the integration layer, which is implemented as bespoke glue code that is hard to produce and even harder to maintain. Therefore, the traditional integration approach often leads to poor code quality, further increasing the time and cost and reducing the agility, and a high reliance on the individual development skills. We are instead tackling the interoperability challenge by building a model driven/low-code Digital Thread platform that 1) systematizes the integration methodology, 2) provides methods and techniques for the individual integrations based on a layered Domain Specific Languages (DSL) approach, 3) through the DSLs it covers the integration space domain by domain, technology by technology, and is thus highly generalizable and reusable, 4) showcases a first collection of examples from the domains of robotics, IoT, data analytics, AI/ML and web applications, 5) brings cohesiveness to the aforementioned heterogeneous platform, and 6) is easier to understand and maintain, even by not specialized programmers. We showcase the power, versatility and the potential of the Digital Thread platform on four interoperability case studies: the generic extension to REST services, to robotics through the UR family of robots, to the integration of various external databases (for data integration) and to the provision of data analytics capabilities in R.

On the ‘Definability of Definable’ Problem of Alfred Tarski

In this paper we prove that for any m≥1 there exists a generic extension of L, the constructible universe, in which it is true that the set of all constructible reals (here subsets of ω) is equal to the set D1m of all reals definable by a parameter free type-theoretic formula with types bounded by m, and hence the Tarski ‘definability of definable’ sentence D1m∈D2m (even in the form D1m∈D21) holds for this particular m. This solves an old problem of Alfred Tarski (1948). Our methods, based on the almost-disjoint forcing of Jensen and Solovay, are significant modifications and further development of the methods presented in our two previous papers in this Journal.

Guessing models and the approachability ideal

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.

On the Δ n 1 Problem of Harvey Friedman

In this paper, we prove the following. If n≥3, then there is a generic extension of L, the constructible universe, in which it is true that the set P(ω)∩L of all constructible reals (here—subsets of ω) is equal to the set P(ω)∩Δn1 of all (lightface) Δn1 reals. The result was announced long ago by Leo Harrington, but its proof has never been published. Our methods are based on almost-disjoint forcing. To obtain a generic extension as required, we make use of a forcing notion of the form Q=Cℂ×∏νQν in L, where C adds a generic collapse surjection b from ω onto P(ω)∩L, whereas each Qν, ν<ω2L, is an almost-disjoint forcing notion in the ω1-version, that adjoins a subset Sν of ω1L. The forcing notions involved are independent in the sense that no Qν-generic object can be added by the product of C and all Qξ, ξ≠ν. This allows the definition of each constructible real by a Σn1 formula in a suitably constructed subextension of the Q-generic extension. The subextension is generated by the surjection b, sets Sω·k+j with j∈b(k), and sets Sξ with ξ≥ω·ω. A special character of the construction of forcing notions Qν is L, which depends on a given n≥3, obscures things with definability in the subextension enough for vice versa any Δn1 real to be constructible; here the method of hidden invariance is applied. A discussion of possible further applications is added in the conclusive section.

Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

INDESTRUCTIBILITY OF THE TREE PROPERTY

In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, where ${\rm{Add}}\left( {\kappa ,\lambda } \right)$ is the Cohen forcing for adding λ-many subsets of κ and $\left( {\kappa ,\lambda } \right)$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$, a generic extension of V, in which the tree property at ${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense (such as ${\kappa ^{ + + }}$-directed closed forcings or well-met forcings which are ${\kappa ^{ + + }}$-closed with the greatest lower bounds).

The special Aronszajn tree property

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal [Formula: see text], there are [Formula: see text]-Aronszajn trees, and all such trees are special.