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Manfred Kufleitner

Publications and source records attributed to Manfred Kufleitner.

At least 19 recordsLinked to original sources
arXiv2024

The Word Problem for $(\omega - 1)$-Terms over $\mathrm{DAb}$

We give a ranker-based description using finite-index congruences for the variety $\boldsymbol{\mathrm{DAb}}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for general pseudowords over $\boldsymbol{\mathrm{DAb}}$. For $(\omega - 1)$-terms, this normal form is computable, which yields an algorithm for the word problem for $(\omega - 1)$-terms of $\boldsymbol{\mathrm{DAb}}$.

cs.FL↗
arXiv2022

Yet another proof of Parikh's Theorem

Parikh's Theorem says that the Parikh image of a context-free language is semilinear. We give a short proof of Parikh's Theorem using the formulation of Verma, Seidl, and Schwentick in terms of Presburger arithmetic. The proof relies on an Eulerian property of derivation trees of context-free languages and was inspired by Hierholzer's algorithm; it does not use the Chomsky normal form.

cs.FL↗
arXiv2022

Reachability Games and Parity Games

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. The results in this paper are not meant to be original. The positional determinacy result is shown for possibly infinite parity games using the ideas of Zielonka which he published in 1998. In order to show quasi-polynomial time, we follow Lehtinen's register games, which she introduced in 2018. Although the time complexity of Lehtinen's algorithm is not optimal, register games are conceptually simple and interesting in their own right. Various of our proofs are either new or simplifications of the original proofs. The topics in this paper include the definition and the computation of optimal attractors for reachability games, too.

cs.GT↗
arXiv2021

Conelikes and Ranker Comparisons

For every fixed class of regular languages, there is a natural hierarchy of increasingly more general problems: Firstly, the membership problem asks whether a given language belongs to the fixed class of languages. Secondly, the separation problem asks for two given languages whether they can be separated by a language from the fixed class. And thirdly, the covering problem is a generalization of separation problem to more than two given languages. Most instances of such problems were solved by the connection of regular languages and finite monoids. Both the membership problem and the separation problem were also extended to ordered monoids. The computation of pointlikes can be interpreted as the algebraic counterpart of the covering problem. In this paper, we consider the extension the computation of pointlikes to ordered monoids. This leads to the notion of conelikes for the corresponding algebraic framework. We apply this framework to the Trotter-Weil hierarchy and both the full and the half levels of the $\text{FO}^2$ quantifier alternation hierarchy. As a consequence, we solve the covering problem for the resulting subvarieties of $\mathbf{DA}$. An important combinatorial tool are uniform ranker characterizations for all subvarieties under consideration; these characterizations stem from order comparisons of ranker positions.

cs.FL↗
arXiv2021

Deciding FO2 Alternation for Automata over Finite and Infinite Words

We consider two-variable first-order logic $\text{FO}^2$ and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword-patterns. For certain types of subword-patterns there exists a non-deterministic logspace algorithm to decide their presence or absence in a given automaton. In particular, this leads to $\mathbf{NL}$ algorithms for deciding the levels of the $\text{FO}^2$ quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are $\mathbf{NL}$-hard and, hence, $\mathbf{NL}$-complete.

cs.FL↗
arXiv2021

The Inclusion Problem for Forest Languages under Substitutions

We consider algorithms and lower bounds for various problems over forest languages; as input models we allow forest algebras, deterministic forest automata and nondeterministic forest automata. For the equivalence problem, we give an almost-linear-time algorithm for both forest algebras and deterministic forest automata; this is complemented by a polynomial time hardness result. The emptiness problem is complete for polynomial time over each of the three models. Additionally, we consider the emptiness of intersection problem for forest algebras and deterministic forest automata; this problem turns out to be complete for exponential time. It is well-known that the corresponding problems for word languages are complete for nondeterministic logarithmic space and for polynomial space, respectively. Equipped with this toolbox of algorithms and lower bounds, we consider various inclusion problems for regular forest languages under substitutions. The substitutions in this paper replace leaf variables by forest languages. Depending on the direction of the inclusion, the problem for a given substitution is either complete for polynomial time or for exponential time; in particular, the equivalence problem under substitutions is complete for exponential time and, hence, more difficult than the equivalence problem for forest languages without substitutions. If we ask whether there exists a substitution such that a given inclusion holds, then this problem is either complete for NP or exponential time, depending on whether we consider inclusion or equivalence; moreover, the problem is undecidable if the substitution is applied on both sides.

cs.FL↗
arXiv2020

Nesting negations in FO2 over infinite words

We consider two-variable first-order logic FO2 over infinite words. Restricting the number of nested negations defines an infinite hierarchy; its levels are often called the half-levels of the FO2 quantifier alternation hierarchy. For every level of this hierarchy, we give an effective characterization. For the lower levels, this characterization is a combination of an algebraic and a topological property. For the higher levels, algebraic properties turn out to be sufficient. Within two-variable first-order logic, each algebraic property is a single ordered identity of omega-terms. The topological properties are the same as for the lower half-levels of the quantifier alternation hierarchy without the two-variable restriction (i.e., the Cantor topology and the alphabetic topology). Our result generalizes the corresponding result for finite words. The proof uses novel techniques and is based on a refinement of Mal'cev products for ordered monoids.

cs.FL↗
arXiv2018

Testing Simon's congruence

Piecewise testable languages are a subclass of the regular languages. There are many equivalent ways of defining them; Simon's congruence $\sim_k$ is one of the most classical approaches. Two words are $\sim_k$-equivalent if they have the same set of (scattered) subwords of length at most k. A language L is piecewise testable if there exists some k such that L is a union of $\sim_k$-classes. For each equivalence class of $\sim_k$, one can define a canonical representative in shortlex normal form, that is, the minimal word with respect to the lexicographic order among the shortest words in $\sim_k$. We present an algorithm for computing the canonical representative of the $\sim_k$-class of a given word $w \in A^*$ of length n. The running time of our algorithm is in O(|A|n) even if $k \le n$ is part of the input. This is surprising since the number of possible subwords grows exponentially in k. The case $k > n$ is not interesting since then, the equivalence class of w is a singleton. If the alphabet is fixed, the running time of our algorithm is linear in the size of the input word. Moreover, for fixed alphabet, we show that the computation of shortlex normal forms for $\sim_k$ is possible in deterministic logarithmic space. One of the consequences of our algorithm is that one can check with the same complexity whether two words are $\sim_k$-equivalent (with k being part of the input).

cs.FL↗
arXiv2018

The Intersection Problem for Finite Monoids

We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Therien.

cs.FL↗
arXiv2017

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

For two given $ω$-terms $α$ and $β$, the word problem for $ω$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $α=β$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for $ω$-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by $ω$-terms (this property is also known as $ω$-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.

cs.FL↗
arXiv2017

Green's Relations in Finite Transformation Semigroups

We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.

cs.FL↗
arXiv2016

Operations on Weakly Recognizing Morphisms

Weakly recognizing morphisms from free semigroups onto finite semigroups are a classical way for defining the class of omega-regular languages, i.e., a set of infinite words is weakly recognizable by such a morphism if and only if it is accepted by some Büchi automaton. We consider the descriptional complexity of various constructions for weakly recognizing morphisms. This includes the conversion from and to Büchi automata, the conversion into strongly recognizing morphisms, and complementation. For some problems, we are able to give more precise bounds in the case of binary alphabets or simple semigroups.

cs.FL↗
arXiv2016

Solutions of Word Equations over Partially Commutative Structures

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ its quotient group (for example, a right-angled Artin or Coxeter group). We show that for any system of word equations over $M(A,I)$ with recognizable constraints, the solution set - in $M(A,I)$ or in $G(A,I)$ - is an EDT0L language. It is given by an NFA $\mathcal{A}$ recognizing endomorphisms over some extended monoid. Furthermore, if the input size is $n$, then the automaton $\mathcal{A}$ can be constructed effectively by an NSPACE$(n\log n)$-transducer. As a consequence, both Satisfiability (whether the system admits a solution) and Finiteness (whether the solution set is infinite) are decidable in NSPACE$(n \log n)$. For a natural subclass of constraints, we conjecture that these problems are NP-complete.

cs.FL↗
arXiv2015

Efficient Algorithms for Morphisms over Omega-Regular Languages

Morphisms to finite semigroups can be used for recognizing omega-regular languages. The so-called strongly recognizing morphisms can be seen as a deterministic computation model which provides minimal objects (known as the syntactic morphism) and a trivial complementation procedure. We give a quadratic-time algorithm for computing the syntactic morphism from any given strongly recognizing morphism, thereby showing that minimization is easy as well. In addition, we give algorithms for efficiently solving various decision problems for weakly recognizing morphisms. Weakly recognizing morphism are often smaller than their strongly recognizing counterparts. Finally, we describe the language operations needed for converting formulas in monadic second-order logic (MSO) into strongly recognizing morphisms, and we give some experimental results.

cs.FL↗
arXiv2015

Level Two of the Quantifier Alternation Hierarchy over Infinite Words

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment $Σ_2$ over infinite words. Here, $Σ_2$ consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is $\mathbb{B}Σ_2$. Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an $ω$-regular language $L$, it is decidable whether $L$ is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun's recent decidability result for $\mathbb{B}Σ_2$ from finite to infinite words.

cs.FL↗
arXiv2014

Two-Variable Ehrenfeucht-Fraisse Games over Omega-Terms

Fragments of first-order logic over words can often be characterized in terms of finite monoids, and identities of omega-terms are an effective mechanism for specifying classes of monoids. Huschenbett and the first author have shown how to use infinite Ehrenfeucht-Fraisse games on linear orders for showing that some given fragment satisfies an identity of omega-terms (STACS 2014). After revisiting this result, we show that for two-variable logic one can use simpler linear orders.

cs.LO↗
arXiv2014

A Survey on the Local Divisor Technique

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 Factorization Forest Theorem. We also introduce the notion of localizable language class as a new abstract concept which unifies some of the proofs for the results above. The current arXiv-version includes some additional material about codes of bounded synchronization delay as well as some updates concerning related literature.

cs.FL↗