Formal verification has emerged as a promising method to ensure the safety and reliability of neural networks. However, many relevant properties, such as fairness or global robustness, pertain to the entire input space. If one applies verification techniques naively, the neural network is checked even on inputs that do not occur in the real world and have no meaning. To tackle this shortcoming, we propose the VeriFlow architecture as a flow-based density model tailored to allow any verification approach to restrict its search to some data distribution of interest. We argue that our architecture is particularly well suited for this purpose because of two major properties. First, we show that the transformation that is defined by our model is piecewise affine. Therefore, the model allows the usage of verifiers based on constraint solving with linear arithmetic. Second, upper density level sets (UDL) of the data distribution are definable via linear constraints in the latent space. As a consequence, representations of UDLs specified by a given probability are effectively computable in the latent space. This property allows for effective verification with a fine-grained, probabilistically interpretable control of how (a-)typical the inputs subject to verification are.

Automatic structures are structures that admit a finite presentation via automata. Their most prominent feature is that their theories are decidable. In the literature, one finds automatic structures with non-elementary theory (e.g., the complete binary tree with equal-level predicate) and automatic structures whose theories are at most 3-fold exponential (e.g., Presburger arithmetic or infinite automatic graphs of bounded degree). This observation led Durand-Gasselin to the question whether there are automatic structures of arbitrary high elementary complexity. We give a positive answer to this question. Namely, we show that for every h >=0 the forest of (infinitely many copies of) all finite trees of height at most h+2 is automatic and it's theory is complete for STA(*, exp_h(n, poly(n)), poly(n)), an alternating complexity class between h-fold exponential time and space. This exact determination of the complexity of the theory of these forests might be of independent interest.

We investigate the recently introduced concept of uniformly tree-automatic classes in the realm of parameterized complexity theory. Roughly speaking, a class of finite structures is uniformly tree-automatic if it can be presented by a set of finite trees and a tuple of automata. A tree t encodes a structure and an element of this structure is encoded by a labeling of t. The automata are used to present the relations of the structure. We use this formalism to obtain algorithmic meta-theorems for first-order logic and in some cases also monadic second-order logic on classes of finite Boolean algebras, finite groups, and graphs of bounded tree-depth. Our main concern is the efficiency of this approach with respect to the hidden parameter dependence (size of the formula). We develop a method to analyze the complexity of uniformly tree-automatic presentations, which allows us to give upper bounds for the runtime of the automata-based model checking algorithm on the presented class. It turns out that the parameter dependence is elementary for all the above mentioned classes. Additionally we show that one can lift the FPT results, which are obtained by our method, from a class C to the closure of C under direct products with only a singly exponential blow-up in the parameter dependence.

We investigate the decidability of logical aspects of graphs that arise as Cayley-graphs of the so-called queue monoids. These monoids model the behavior of the classical (reliable) fifo-queues. We answer a question raised by Huschenbett, Kuske, and Zetzsche and prove the decidability of the first-order theory of these graphs with the help of an - at least for the authors - new combination of the well-known method from Ferrante and Rackoff and an automata-based approach. On the other hand, we prove that the monadic second-order of the queue monoid's Cayley-graph is undecidable.

We study structures that are automatic with advice. These are structures that admit a presentation by finite automata (over finite or infinite words or trees) with access to an additional input,called an advice. Over finite words, a standard example of a structure that is automatic with advice, but not automatic in the classical sense, is the additive group of rational numbers (Q,+). By using a set of advices rather than a single advice, this leads to the new concept of a parameterised automatic presentation as a means to uniformly represent a whole class of structures. The decidability of the first-order theory of such a uniformly automatic class reduces to the decidability of the monadic second-order theory of the set of advices that are used in the presentation. Such decidability results also hold for extensions of first-order logic by regularity preserving quantifiers, such as cardinality quantifiers and Ramsey quantifiers. To investigate the power of this concept, we present examples of structures and classes of structures that are automatic with advice but not without advice, and we prove classification theorems for the structures with an advice automatic presentation for several algebraic domains. In particular, we prove that the class of all torsion-free Abelian groups of rank one is uniformly omega-automatic and that there is a uniform omega-tree-automatic presentation of the class of all Abelian groups up to elementary equivalence and of the class of all countable divisible Abelian groups. On the other hand we show that every uniformly omega-automatic class of Abelian groups must have bounded rank. While for certain domains, such as trees and Abelian groups, it turns out that automatic presentations with advice are capable of presenting significantly more complex structures than ordinary automatic presentations, there are other domains, such as Boolean algebras, where this is provably not the case. Further, advice seems to not be of much help for representing some particularly relevant examples of structures with decidable theories, most notably the field of reals. Finally we study closure properties for several kinds of uniformly automatic classes, and decision problems concerning the number of non-isomorphic models in uniformly automatic classes with the unique representation property.

We study the descriptive complexity of summation problems in Abelian groups and semigroups. In general, an input to the summation problem consists of an Abelian semigroup G, explicitly represented by its multiplication table, and a subset X of G. The task is to determine the sum over all elements of X. Algorithmically this is a very simple problem. If the elements of X come in some order, then we can process these elements along that order and calculate the sum in a trivial way. However, what makes this fundamental problem so interesting for us is that from the viewpoint of logical definability its tractability is much more delicate. If we consider the semigroup G as an abstract structure and X as an abstract set, without a linear order and hence without a canonical way to process the elements one by one, then it is unclear how to define the sum in any logic that does not have the power to quantify over a linear order. Indeed the trivial summation algorithm cannot be expressed in any polynomial-time logic or, in fact, in any computational model which works on abstract mathematical structures in an isomorphism-invariant way without violating polynomial resource bounds. The surprising difficulty, in terms of logical definability, of this basic mathematical problem is the reason why Ben Rossman asked, more than ten years ago, whether it can be expressed in the logic Choiceless Polynomial Time with counting (CPT). Note that, to date, CPT is one of the most powerful known candidates for a logic that might be capable of defining every polynomial-time property of finite structures. In this paper we clarify the status of the definability for the summation problem for Abelian groups and semigroups in important polynomial-time logics. In our first main result we show that the problem can be defined in fixed-point logic with counting (FPC). Since FPC is contained in CPT this settles Rossman's question. Our proof is based on a dynamic programming approach and heavily uses the counting mechanism of FPC. In our second main result we give a matching lower bound and show that the use of counting operators cannot be avoided: the summation problem, even over Abelian groups, cannot be defined in pure fixed-point logic without counting. Our proof is based on a probabilistic argument.

We investigate structural properties of omega-automatic presentations of infinite structures in order to sharpen our methods to determine whether a given structure is omega-automatic. We apply these methods to show that no field of characteristic 0 admits an injective omega-automatic presentation, and that uncountable fields with a definable linear order cannot be omega-automatic.