A Proof of the Factorization Forest Theorem

We show that for every homomorphism $\Gamma^+ \to S$ where $S$ is a finite semigroup there exists a factorization forest of height \leq 3 \abs{S}. The proof is based on Green's relations.Comment: 4 page

On Bijective Variants of the Burrows-Wheeler Transform

The sort transform (ST) is a modification of the Burrows-Wheeler transform (BWT). Both transformations map an arbitrary word of length n to a pair consisting of a word of length n and an index between 1 and n. The BWT sorts all rotation conjugates of the input word, whereas the ST of order k only uses the first k letters for sorting all such conjugates. If two conjugates start with the same prefix of length k, then the indices of the rotations are used for tie-breaking. Both transforms output the sequence of the last letters of the sorted list and the index of the input within the sorted list. In this paper, we discuss a bijective variant of the BWT (due to Scott), proving its correctness and relations to other results due to Gessel and Reutenauer (1993) and Crochemore, Desarmenien, and Perrin (2005). Further, we present a novel bijective variant of the ST.Comment: 15 pages, presented at the Prague Stringology Conference 2009 (PSC 2009

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.Comment: Full version of a paper accepted to FSTTCS 201

Fragments of first-order logic over infinite words

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language classes of certain (unambiguous) polynomials. An immediate consequence is the decidability of the membership problem of these classes, but this was shown before by Wilke and Bojanczyk and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on Theoretical Aspects of Computer Science, STACS 200

Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

One Quantifier Alternation in First-Order Logic with Modular Predicates

Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison $x<y$ and predicates for $x \equiv i \mod n$ has been investigated by Barrington, Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that definability in the two-variable fragment FO2[<,MOD] is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in Sigma2[<,MOD]. The fragment Sigma2 consists of first-order formulas in prenex normal form with two blocks of quantifiers starting with an existential block. In addition we show that Delta2[<,MOD], the largest subclass of Sigma2[<,MOD] which is closed under negation, has the same expressive power as two-variable logic FO2[<,MOD]. This generalizes the result FO2[<] = Delta2[<] of Therien and Wilke to modular predicates. As a byproduct, we obtain another decidable characterization of FO2[<,MOD]

The FO^2 alternation hierarchy is decidable

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Th\'erien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and Vollmer

Ehrenfeucht-Fraisse Games on Omega-Terms

Fragments of first-order logic over words can often be characterized in terms of finite monoids or finite semigroups. Usually these algebraic descriptions yield decidability of the question whether a given regular language is definable in a particular fragment. An effective algebraic characterization can be obtained from identities of so-called omega-terms. In order to show that a given fragment satisfies some identity of omega-terms, one can use Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting proofs often require a significant amount of book-keeping with respect to the constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on omega-terms. To this end we assign a labeled linear order to every omega-term. Our main theorem shows that a given fragment satisfies some identity of omega-terms if and only if Duplicator has a winning strategy for the game on the resulting linear orders. This allows to avoid the book-keeping. As an application of our main result, we show that one can decide in exponential time whether all aperiodic monoids satisfy some given identity of omega-terms, thereby improving a result of McCammond (Int. J. Algebra Comput., 2001)

Star-free languages and local divisors

A celebrated result of Schützenberger says that a language is star-free if and only if it is is recognized by a finite aperiodic monoid. We give a new proof for this theorem using local divisors. © 2014 Springer International Publishing

Solutions of Word Equations over Partially Commutative Structures

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 first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n log n). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups