Michael Ludwig’s research while affiliated with University of Tübingen and other places

In the last two decades visibly pushdown languages (VPLs) have found many applications in diverse areas such as formal verification and processing of XML documents. Recently, there has been a significant interest in studying quantitative versions of finite-state systems as well as visibly pushdown systems. In this work, we take forward this study for visibly pushdown systems by considering a functional version of visibly pushdown automata. Our version is formally a generalization of cost register automata (CRA) defined by [Alur et al., 2013]. We observe that our model continues to have all the good properties of the CRAs in spite of being a generalization. Apart from studying the functional properties of the model, we also study the complexity theoretic aspects. Recently such a study was conducted by [Allender and Mertz, 2014] with respect to CRAs. Here we show that CRAs when appended with a visible stack (i.e. in the model defined here), continue to have the same complexity theoretic upper bounds as are known for CRAs. Moreover, we observe that one of the upper bounds shown by Allender et al. which was not tight for CRAs becomes tight for our model. Hence, it answers one of the questions raised in their work.

We extend the familiar program of understanding circuit complexity in terms of regular languages to visibly counter languages. Like the regular languages, the visibly counter languages are $\mathrm {NC}^{1}$- complete. We investigate what the visibly counter languages in certain constant depth circuit complexity classes are. We have initiated this in a previous work for $\mathrm {AC}^{0}$. We present characterizations and decidability results for various logics and circuit classes. In particular, our approach yields a way to understand $\mathrm {TC}^{0}$, where the regular approach fails.

Low circuit complexity classes and regular languages exhibit very tight interactions that shade light on their respective expressiveness. We propose to study these interactions at a functional level, by investigating the deterministic rational transductions computable by constant-depth, polysize circuits. To this end, a circuit framework of independent interest that allows variable output length is introduced. Relying on it, there is a general characterization of the set of transductions realizable by circuits. It is then decidable whether a transduction is definable in $\mathrm{AC}^0$ and, assuming a well-established conjecture, the same for $\mathrm{ACC}^0$.

We obtain results within the area of dense completeness, which describes a close relation between families of formal languages and complexity classes. Previously we were able show that this relation exists between counter languages and $\mathbf {NL}$ but not between the regular languages and $\mathbf {NC^1}$. We narrow the gap between the regular languages and the counter languages by considering visibly counter languages. It turns out that they are not densely complete for $\mathbf {NC^1}$. At the same time we found a restricted counter automaton model which is densely complete for $\mathbf {NL}$. Besides counter automata we show more positive examples in terms of L-systems.

We examine visibly counter languages, which are languages recognized by visibly counter automata (a.k.a. input driven counter automata). We are able to effectively characterize the visibly counter languages in AC0 and show that they are contained in FO[+].

We extend the # operator in a natural way and derive a new type of counting complexity. While #$\mathcal C$ classes (where $\mathcal C$ is some circuit-based class like NC 1) only count proofs for acceptance of some input in circuits, one can also count proofs for rejection. The here proposed Zap-$\mathcal C$ complexity classes implement this idea. We show that Zap-$\mathcal C$ lies between #$\mathcal C$ and Gap-$\mathcal C$. In particular we consider Zap-NC 1 and polynomial size branching programs of bounded and unbounded width. We find connections to planar branching programs since the duality of positive and negative proofs can be found again in the duality of graphs and their co-graphs. This links to possible applications of our contribution, like closure properties of complexity classes.