Volker Diekert

Universität Stuttgart · Institut für Formale Methoden der Informatik

Parity games are positionally determined. This is a fundamental and classical result. In 2010, Calude et al. showed a breakthrough result for finite parity games: the winning regions and their positional winning strategies can be computed in quasi-polynomial time.In the present paper we give a self-contained and detailed proofs for both results. Th...

Parity games are positionally determined. This is a fundamental and classical result. In 2010, Calude et al. showed a breakthrough result for finite parity games: the winning regions and their positional winning strategies can be computed in quasi-polynomial time. In the present paper we give a self-contained and detailed proofs for both results. T...

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property Φ. What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying Φ in the family? We approach this question by using...

We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $\sigma$ is a substitution such that $\sigma(x)$ is a set of trees in $T(\Sigma\cup...

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property \(\varPhi \). What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there exists one graph satisfying \(\varPhi \)? We approach this que...

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $\Phi$.What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question if is there exists one graph satisfying$\Phi$? We approach this question by using...

Regular matching problems have a long history. In 1964, Ginsburg and Hibbard showed that on input regular languages $L$ and $R$ it is decidable whether there is a generalized sequential machine which maps $L$ onto $R$, and noticed that the results cannot be extended to context-free languages. We investigate regular matching problems with respect to...

It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper, we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in [Formula...

In this work we extend previously known decidability results for $2\times 2$ matrices over $\mathbb{Q}$. Namely, we introduce a notion of flat rational sets: if $M$ is a monoid and $N\leq M$ is its submonoid, then flat rational sets of $M$ relative to $N$ are finite unions of the form $L_0g_1L_1 \cdots g_t L_t$ where all $L_i$s are rational subsets...

The word problem of a finitely generated group is the set of words over the generators that are equal to the identity in the group. The word problem is therefore a formal language. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In the seminal paper Mulle...

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. Moreover the problem becomes hard for DEXPTIME, as soon as the quotient group of the free...

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which fo...

We prove that the set of all solutions for twisted word equations with regular constraints is an EDT0L language and can be computed in PSPACE. It follows that the set of solutions to equations with rational constraints in a context-free group (= finitely generated virtually free group) in reduced normal forms is EDT0L. We can also decide (in PSPACE...

This volume presents the lecture notes from the authors’ three summer courses offered during the program “Automorphisms of Free Groups: Geometry, Topology, and Dynamics” held at the Centre de Recerca Matemàtica (CRM) in Bellaterra, Spain. The first two chapters present the basic tools needed, from formal language theory (regular and context-free l...

The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag-Solitar gr...

In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. Thus, the question arises “how many” such elements there are. This question can be formalized using the notion of strongly generic sets and lower bounds can...

This paper presents a PSPACE algorithm which yields a finite graph of exponential size that describes the set of all solutions of equations in free groups as well as the set of all solutions of equations with rational constraints in free monoids. This became possible due to the recent recompression technique. While this technique was successfully a...

QuickHeapsort is a combination of Quicksort and Heapsort. We show that the expected number of comparisons for QuickHeapsort is always better than for Quicksort if a usual median-of-constant strategy is used for choosing pivot elements. In order to obtain the result we present a new analysis for QuickHeapsort splitting it into the analysis of the pa...

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existenc...

The idea behind this book is to provide the mathematical foundations for assessing modern developments in the Information Age. It deepens and complements the basic concepts, but it also considers instructive and more advanced topics. The treatise starts with a general chapter on algebraic structures; this part provides all the necessary knowledge f...

We give NSPACE(n log n) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the firs...

In this paper we continue a classical work of Sch\"utzenberger on codes with bounded synchronization delay. He was interested to characterize those regular languages where the groups in the syntactic monoid belong to a variety $H$. He allowed operations on the language side which are union, intersection, concatenation and modified Kleene-star invol...

When a property needs to be checked against an unknown or very complex system, classical exploration techniques like model-checking are not applicable anymore. Sometimes a monitor can be used, that checks a given property on the underlying system at runtime. A monitor for a property L is a deterministic finite automaton \(\mathcal {M}_L\) that afte...

Geometry and Diophantine equations have been ever-present in mathematics. According to the existing literature the work of Diophantus of Alexandria was mentioned before 364 AD, but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a D...

When a property needs to be checked against an unknown or very complex system, classical exploration techniques like model-checking are not applicable anymore. Sometimes a~monitor can be used, that checks a given property on the underlying system at runtime. A monitor for a property $L$ is a deterministic finite automaton $M_L$ that after each fini...

We show that, given a word equation over a finitely generated free group, the set of all solutions in reduced words forms an EDT0L language. In particular, it is an indexed language in the sense of Aho. The question whether a formal language description of solution sets in reduced words as indexed language is possible has been been open for some ye...

In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs: In this work we exami...

In a series of papers, Brzozowski together with Tamm, Davies, and Szykula studied the quotient complexities of atoms of regular languages [6, 7, 3, 4]. The authors obtained precise bounds in terms of binomial sums for the most complex situations in the following five cases: (G): general, (R.): right ideals, (L): left ideals, (T): two-sided ideals a...

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. The result is proved by a direct reduction to solve language equations with one-sided conc...

Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's...

We review concepts like safety, liveness, and monitorability from a rigorous topological viewpoint. Thus, monitorability of an ω-language means that the boundary in the Cantor topology has an empty interior. We show that all ω-regular languages which are deterministic and co-deterministic are monitorable, but certain deterministic liveness properti...

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups and monoids with involution in the presence of rational constraints. This became possible due to the recently invented recompression technique of the second author. He succes...

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexi...

We consider three important and well-studied algorithmic problems in group theory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conj...

The paper is a part of an ongoing program which aims to show that the existential theory in free groups (hyperbolic groups or even toral relatively hyperbolic) is NP-complete. For that we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We re...

The word problem of a finitely generated group is the formal language of words over the generators which are equal to the identity in the group. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In a seminal paper Muller and Schupp showed the converse: A co...

In 1965 Schützenberger published his famous result that star-free languages ( $\operatorname{SF}$ ) and aperiodic languages ( $\operatorname{AP}$ ) coincide over finite words, often written as $\operatorname{SF}= \operatorname {AP}$ . Perrin generalized $\operatorname{SF} = \operatorname{AP}$ to infinite words in the mid 1980s. In 1973 Schütz...

We present a new analysis for QuickHeapsort splitting it into the analysis of the partition-phases and the analysis of the heap-phases. This enables us to consider samples of non-constant size for the pivot selection and leads to better theoretical bounds for the algorithm. Furthermore we introduce some modifications of QuickHeapsort, both in-place...

Distributed systems are notoriously difficult to understand and analyze in order to assert their correction w.r.t. given properties. They often exhibit a huge number of different behaviors, as soon as the active entities (peers, agents, processes, etc) behave in an asynchronous manner. Already the modelization of such systems is a non-trivial task,...

Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregro...

Hairpin formations arise in biochemical processes and play an important role in DNA-computing. We study language theoretical properties of hairpin formations and our new results concern the hairpin completion of two regular languages and and the iterated hairpin lengthening of any language .

Let Gamma be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structur...

This paper proves a long standing conjecture in formal language theory. It shows that all regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential, if there exists a finite confluent, and length-reducing s...

Computing normal forms in groups (or monoids) is in general harder than solving the word problem (equality testing). However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal forms for important classes of groups. We show that shortlex normal forms in graph...

The class of Church-Rosser congruential languages has been introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential (belongs to CRCL), if there is a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. To date, it is still open whether every...

We give a new proof of the Krohn-Rhodes Theorem using local divisors. The proof provides nearly as good a decomposition in terms of size as the holonomy decomposition of Eilenberg, avoids induction on the size of the state set, and works exclusively with monoids with the base case of the induction being that of a group.

The hairpin completion is an operation on formal languages that has been inspired by the hairpin formation in DNA biochemistry and by DNA computing. In this paper we investigate the hairpin completion of regular languages. It is well known that hairpin completions of regular languages are linear context-free and not necessarily regular. As regulari...

Non-Archimedean words have been introduced as a new type of infinite words which can be investigated through classical methods in combinatorics on words due to a length function. The length function, however, takes values in the additive group of polynomials ℤ[t] (and not, as traditionally, in ℕ), which yields various new properties. Non-Archimedea...

Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is in P. Before that the best known upper bound has been non-elementary. In the present paper we provide...

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at o...

The hairpin completion is an operation on formal languages which is inspired by the hairpin formation in biochemistry. Hairpin formations occur naturally within DNA-computing. It has been known that the hairpin completion of a regular language is linear context-free, but not regular, in general. However, for some time it is was open whether the reg...

In this chapter the authors consider how a dating service should optimally arrange meetings among marriage-minded ladies and gentlemen – at least from a theoretical point of view. In so doing the authors analyze the algorithm in terms of its running time, and they explain the related Marriage Theorem.

Non-Archimedean words have been introduced as a new type of infinite words which can be investigated through classical methods in combinatorics on words due to a length function. The length function, however, takes values in the additive group of polynomials ℤ[t] (and not, as traditionally, in ℕ), which yields various new properties. Non-Archimedea...

The hairpin completion is a natural operation on formal languages which has been inspired by molecular phenomena in biology and by DNA-computing. In 2009 we presented in [6] a (polynomial time) decision algorithm to decide regularity of the hairpin completion. In this paper we provide four new results: 1.) We show that the decision problem is NL-co...

C. M. Weinbaum [1] showed the following: Let w be a primitive word and a be letter in w. Then a conjugate of w can be written as uv such that a is a prefix and a suffix of u, but v neither starts nor ends with a, and u and v have a unique position in w as cyclic factors. The latter condition means that there is exactly one conjugate of w having u a...

The hairpin completion is a natural operation on formal languages which has been inspired by molecular phenomena in biology and by DNA-computing. In 2009 we presented a (polynomial time) decision algorithm to decide regularity of the hairpin completion. In this paper we provide four new results: 1.) We show that the decision problem is NL-complete....

The hairpin completion is a natural operation of formal languages which has been inspired by molecular phenomena in biology and by DNA-computing. The hairpin completion of a regular language is linear context-free and we consider the problem to decide whether the hairpin completion remains regular. This problem has been open since the first formal...

We continue our study of stabilizers of infinite words over finite alphabets, begun in [D. Krieger, On stabilizers of infinite words, Theoret. Comput. Sci. 400 (2008), 169–181]. Let be an aperiodic infinite word over a finite alphabet, and let be its stabilizer. We show that can be partitioned into the monoid of morphisms that stabilize by finite f...

We introduce the peak normal form for elements of the Baumslag–Solitar groups BS(p,q). This normal form is very close to the length-lexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u ⁻¹ v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the...

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at o...

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for ω-languages: Σ2, FO2, FO2∩Σ2, and Δ2 (and by duality Π2 and FO2∩Π2). These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language cla...

We introduce local safety and local liveness for distributed systems whose executions are modeled by Mazurkiewicz traces. We char-acterize local safety by local closure and local liveness by local density. Restricting to first-order definable properties, we prove a decomposition theorem in the spirit of the separation theorem for linear temporal lo...

Die Dyck-Sprachen sind ein Grundbegriff aus dem Bereich der formalen Sprachen. Ausgehend von der Person des Namensgebers werden ihre Geschichte und ihre Bedeutung in der theoretischen Informatik in diesem Überblick dargestellt.

Das allgemein als Prototyp eines PSPACE-vollständigen Spiels gesehene Geographiespiel wird bezüglich seiner Komplexität genauer untersucht. Das Interesse der theoretischen Informatik an diesem Spiel wurde sehr durch die Darstellung in dem Lehrbuch von Papadimitriou [Pap94] gefördert. Allerdings bestimmt dieses Lehrbuch nicht die Komplexität des Sta...

We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of...

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and th...

Let G be a finitely generated virtually-free group. We consider the Birget–Rhodes expansion of G, which yields an inverse monoid and which is denoted by IM (G) in the following. We show that for a finite idempotent presentation P, the word problem of a quotient monoid IM (G)/P can be solved in linear time on a RAM. The uniform word problem, where G...

We give an essentially self-contained presentation of some principal results for first-order definable languages over finite and infinite words. We introduce the notion of a counter-free Büchi automaton; and we relate counter-freeness to aperiodicity and to the notion of very weak alternation. We also show that aperiodicity of a regular ∞-language...

Eine Agentur für die Vermittlung von Partnerschaften versucht, aus einer Menge von Herren und Damen eine möglichst große Zahl von Paaren zusammenzustellen. Vorausgesetzt wird dabei, dass sich die beiden Personen, die ein Paar bilden, gegenseitig sympathisch sind. Weiterhin gehen wir von der klassischen Ehe aus: Jedes Paar besteht aus genau einem He...

Mazurkiewicz traces form a model for concurrency. Temporal logic and first-order logic are important tools in order to deal with the abstract behavior of such systems. Since typical properties can be described by rather simple logical formulas one is interested in logical fragments. One focus of this paper is unary temporal logic and first-order lo...

We summarize several characterizations, inclusions, and separations on fragments of first-order logic over words and Mazurkiewicz traces. The results concerning Mazurkiewicz traces can be seen as generalizations of those for words. It turns out that over traces it is crucial, how easy concurrency can be expressed. Since there is no concurrency in w...

Mazurkiewicz traces form a model for concurrency. Temporal logic and rst-order logic are important tools in order to deal with the abstract behavior of such systems. Since typical properties can be described by rather simple logical formulas one is interested in logical fragments. One focus of this paper is unary temporal logic and rst-order logic...

We show that the existential theory of free partially commutative monoids with involution is decidable. As a consequence the existential theory of graph groups is also decidable. If the underlying alphabet of generators is fixed, we obtain a PSPACE-completeness result, otherwise (in the uniform setting) our decision procedure is in EXPSPACE. Our pr...

Confluence is an undecidable property even for finite noetherian trace-rewriting systems. Here we investigate this property for the special case of trace-rewriting systems with a single rule. For certain classes of one-rule trace-rewriting systems R we present syntactic characterizations that are necessary and sufficient for R to be confluent. Base...

The paper settles a long standing problem for Mazurkiewicz traces: the pure future local temporal logic defined with the basic modalities exists-next and until is expressively complete. This means every first-order definable language of Mazurkiewicz traces can be defined in a pure future local temporal logic. The analogous result with a global inte...

Timed automata are among the most widely studied models for real-time systems. Silent transitions, i.e., -transitions, have already been proposed in the original paper on timed automata by Alur and Dill The present paper settles this open question. Precisely, we prove that a timed automaton such that no -transition with nonempty reset set lies on a...

Finite complete replacement systems over free partially commutative monoids have a decidable word problem, and if these systems are weight-reducing, then the word problem is solvable in square time. It is unknown whether this time bound is optimal; but the restriction to semi-Thue systems or vector replacement systems yields linear time. In directi...

Free partially commutative inverse monoids are investigated. As in the case of free partially commutative monoids or groups (trace monoids or graph groups), free partially commutative inverse monoids are defined as quotients of free inverse monoids modulo a partially defined commutation relation on the generators. A quasi linear time algorithm for...