# Petra Wolf

PhD Student in Theoretical Computer Science

I am currently working at the University of Trier as a researcher in the Theoretical Computer Science group led by Prof. Dr. Henning Fernau.

I am interested in Quantum Computing, Formal Languages, Complexity Theory, Computability, and Automata Theory.

## List of Publications

[PV] = Link to Publisher's Version

[Pre] = Link to Preprint Version

Accepted at ICTCS 2020, arXiv [Pre]

### From Decidability to Undecidability by Considering Regular Sets of Instances

(Petra Wolf)

We are lifting classical problems from single instances to regular sets of instances. The task of finding a positive instance of the combinatorial problem P in a potentially infinite given regular set is equivalent to the so called intreg-problem of P, which asks for a given DFA A, whether the intersection of P with L(A) is non-empty. The intreg-problem generalizes the idea of considering multiple instances at once and connects classical combinatorial problems with the field of automata theory. While the question of the decidability of the intreg-problem has been answered positively for several NP- and even PSPACE-complete problems, we are presenting natural problems even from L with an undecidable intreg-problem. We also discuss alphabet sizes and different encoding-schemes elaborating the boundary between problem-variants with a decidable respectively undecidable intreg-problem.

Accepted at MFCS 2020, arXive [Pre]

### Synchronizing Deterministic Push-Down Automata Can Be Really Hard

(Henning Fernau, Petra Wolf, Tomoyuki Yamakami)

The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata.

There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.

arXiv [Pre]

### Synchronization of Deterministic Visibly Push-Down Automata

(Henning Fernau, Petra Wolf)

We generalize the concept of synchronizing words for finite automata, which map all states of the automata to the same state, to deterministic visibly push-down automata. Here, a synchronizing word w does not only map all states to the same state but also fulfills some conditions on the stack content of each run after reading w. We consider three types of these stack constraints: after reading w, the stack (1) is empty in each run, (2) contains the same sequence of stack symbols in each run, or (3) contains an arbitrary sequence which is independent of the other runs. We show that in contrast to general deterministic push-down automata, it is decidable for deterministic visibly push-down automata whether there exists a synchronizing word with each of these stack constraints, i.e., the problems are in EXPTIME. Under the constraint (1) the problem is even in P. For the sub-classes of deterministic very visibly push-down automata the problem is in P for all three types of constraints. We further study variants of the synchronization problem where the number of turns in the stack height behavior caused by a synchronizing word is restricted, as well as the problem of synchronizing a variant of a sequential transducer, which shows some visibly behavior, by a word that synchronizes the states and produces the same output on all runs.

arXiv [Pre]

### Regular Intersection Emptiness of Graph Problems: Finding a Needle in a Haystack of Graphs with the Help of Automata

(Heinning Fernau, Petra Wolf)

The Int_reg-problem of a combinatorial problem P asks, given a nondeterministic automaton M as input, whether the language L(M) accepted by M contains any positive instance of the problem P. We consider the Int_reg-problem for a number of different graph problems and give general criteria that give decision procedures for these Int_reg-problems. To achieve this goal, we consider a natural graph encoding so that the language of all graph encodings is regular. Then, we draw the connection between classical pumping- and interchange-arguments from the field of formal language theory with the graph operations induced on the encoded graph. Our techniques apply among others to the Int_reg-problem of well-known graph problems like Vertex Cover and Independent Set, as well as to subgraph problems, graph-edit problems and graph-partitioning problems, including coloring problems.

arXiv [Pre]

### Synchronization under Dynamic Constraints

(Petra Wolf)

Imagine an assembly line where a box with a lid and liquid in it enters in some unknown orientation. The box should leave the line with the open lid facing upwards with the liquid still in it. To save costs there are no complex sensors or image recognition software available on the assembly line, so a reset sequence needs to be computed. But how can the dependencies of the deforming impact of a transformation of the box, such as 'do not tilt the box over when the lid is open' or 'open the lid again each time it gets closed' be modeled? We present three attempts to model constraints of these kinds on the order in which the states of an automaton are transitioned by a synchronizing word. The first two concepts relate the last visits of states and form constraints on which states still need to be reached, whereas the third concept concerns the first visits of states and forms constraints on which states might still be reached. We examine the computational complexity of different variants of the problem, whether an automaton can be synchronized with a word that respects the constraints defined in the respective concept, and obtain nearly a full classification. While most of the problems are PSPACE-complete we also observe NP-complete variants and variants solvable in polynomial time.

One of them is the careful synchronization problem for partial weakly acyclic automata (which are partial automata, the states of which can be ordered such that no transition leads to a smaller state), which is shown to be solvable in time O(k^2 n^2 log(n)) where n is the size of the state set and k is the size of the alphabet. The algorithm even computes a synchronizing words as a witness. This is quite surprising as the careful synchronization problem uses to be a hard problem for most classes of automata.

We will also observe a drop of the complexity if we track the orders of states on several paths simultaneously instead of tracking the set of active states. Further, we give upper bounds on the length of a synchronizing word depending on the size of the input relation and show that the Cerny conjecture holds for partial weakly acyclic automata.

MFCS 2019 [PV]

### Computational Complexity of Synchronization under Regular Constraints

(Henning Fernau, Vladimir V. Gusev, Stefan Hoffmann, Markus Holzer, Mikhail V. Volkov, Petra Wolf)

Many variations of synchronization of finite automata have been studied in the previous decades. Here, we suggest studying the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We show that this synchronization problem becomes PSPACE-complete even for some constraint automata with two states and a ternary alphabet. In addition, we characterize constraint automata with arbitrarily many states for which the constrained synchronization problem is polynomial-time solvable. We classify the complexity of the constrained synchronization problem for constraint automata with two states and two or three letters completely and lift those results to larger classes of finite automata.

DCFS 2019 [PV]

### On the Decidability of Finding a Positive ILP-Instance in a Regular Set of ILP-Instances

(Petra Wolf)

The regular intersection emptiness problem for a decision problem P (intreg(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since intreg(P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the intreg -problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the intreg-problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of intreg(ILP).

FUN 2018 [PV]

We study the game Greedy Spiders, a two-player strategic defense game, on planar graphs and show PSPACE-completeness for the problem of deciding whether one player has a winning strategy for a given instance of the game. We also generalize our results in metatheorems, which consider a large set of strategic defense games. We achieve more detailed complexity results by restricting the possible strategies of one of the players, which leads us to Sigma^p_2- and Pi^p_2-hardness results.

LATA 2018 [PV]

### Deciding Regular Intersection Emptiness of Complete Problems for PSPACE and the Polynomial Hierarchy

(Demen Güler, Andreas Krebs, Klaus-Jörn Lange, Petra Wolf)

For a regular set R of quantified Boolean formulae we decide whether R contains a true formula. We conclude that there is a PSPACE-complete problem for which emptiness of intersection with a regular set is decidable. Furthermore, by restricting depth and order of quantification we obtain complete problems for each level of the polynomial hierarchy with this decidability as well.

## Bachelor and Master Thesis

Bachelor Thesis (12.2016)

### Computational Complexity of Computer Games - Gaming under Time Pressure

Identification of combination of game elements which give a game, in which they

occur, a certain computational complexity. Abstracted game elements have been studied instead of

concrete games. The work focused on the stopwatch as a game element.

Master Thesis (10.2018)

### Decidability of the Regular Intersection

Emptiness Problem

We determine the decidability of the intreg-problem for several languages. The intreg-problem of a fixed language L asks whether the intersection of L with a given regular language R is empty or not. We prove undecidability of the intreg-problem for variations of the Machine Language, Bounded Tiling, Corridor Tiling, Bounded PCP, Equivalence of Regular Expressions in a shuffled encoding, and String Equivalence Modulo Padding in a shuffled encoding. We prove decidability of the intreg-problem for String Equivalence Modulo Padding in a sequential encoding as well as over a unary alphabet in both encodings, SAT, k-TQBF, True Quantified Boolean Formula, Integer Linear Programming, Vertex Cover, Independent Set, Knapsack, and Integer Knapsack. In the most cases we show the decidability of the intreg-problem by constructing a condensed automaton condensed(A) from a given DFA A. In L(condensed(A)) only finitely many words have to be tested for membership in the intersection (defined by intreg) under which there is one element in the intersection if and only there is one in the original regular language described by A. We develop the three techniques merging, separating, and replacing of constructing the automaton condensed(A) and demonstrate them on the problems Vertex Cover, Independent Set, and Knapsack.

## Vita

### Eberhard Karls Universität Tübingen

Bachelor of Science in Computer Science 2016

Master of Science in Computer Science 2018

### Universität Trier

In progress: PhD supervised by Prof. Dr. Henning Fernau

### Scholarships and Awards

Master degree with distinction (Master Abschluss mit Auszeichnung)

Deutschlandstipendium 2017 - 2019

JSPS Summer Program (2020)

### Participation in Workshops and Schools

Tübingen's Frontiers of Theory: Lectures and Research (02.2016, 09.2016, 02.2017, 06,2017)

Theorietage (2017, 2018, 2019)

Dagstuhl Seminar: Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative (2018)

Workshop on Kernelization (2019)

## Teaching

Exercises/Course Administration

Universität Trier:

2019, 2020 Kleines Studienprojekt LaTeX-Kurs

Eberhard Karls Universität Tübingen:

2015/2016

Theoretische Informatik

2016

Komplexitätstheorie

2016/2017

Theoretische Informatik

Mathematischer Vorkurs

2017

Berechenbarkeit

Formale Sprachen

Proseminar Theoretische Informatik - Reguläre Sprachen

2017/2018

Theoretische Informatik

Proseminar Theoretische Informatik - Große Informatiker der Geschichte

2018

Proseminar Komplexität von Computerspielen

## Contact

Petra Wolf

Theoretische Informatik

Campus II

Universität Trier

D-54286 Trier

wolfp[at]uni-trier.de

„Computer Science is no more about computers than astronomy is about telescopes.“