Manfred Kufleitner

A SURVEY ON SMALL FRAGMENTS OF FIRST-ORDER LOGIC OVER FINITE WORDS

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 the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simons theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose.

The Height of Factorization Forests

We show that for every homomorphism from A+to a finite semigroup Sthere exists a factorization forest of height at most 3 i¾? Si¾? i¾? 1. Furthermore, we show that for every non-trivial group, this bound is tight. For aperiodic semigroups, we give an improved upper bound of 2 i¾? Si¾? and we show that for every ni¾? 2 there exists an aperiodic semigroup Swith nelements which reaches this bound.

Polynomials, fragments of temporal logic and the variety DA over traces

We show that some language-theoretic and logical characterizations of recognizable word languages whose syntactic monoid is in the variety DA also hold over traces. To this end we give algebraic characterizations for the language operations of generating the polynomial closure and generating the unambiguous polynomial closure over traces. We also show that there exist natural fragments of local temporal logic that describe this class of languages corresponding to DA. All characterizations are known to hold for words.

Lattices of logical fragments over words

This paper introduces an abstract notion of fragments of monadic second-order logic. This concept is based on purely syntactic closure properties. We show that over finite words, every logical fragment defines a lattice of languages with certain closure properties. Among these closure properties are residuals and inverse

AROUND DOT-DEPTH ONE

\mathcal C

On the index of Simon's congruence for piecewise testability

-morphisms. Here, depending on certain closure properties of the fragment,

The Krohn-Rhodes Theorem and Local Divisors

\mathcal C

The join levels of the trotter-weil hierarchy are decidable

is the family of arbitrary, non-erasing, length-preserving, length-multiplying, or lengthreducing morphisms. In particular, definability in a certain fragment can often be characterized in terms of the syntactic morphism. This work extends a result of Straubing in which he investigated certain restrictions of first-order formulae. As motivating examples, we present (1) a fragment which captures the stutter-invariant part of piecewise-testable languages and (2) an acyclic fragment of Σ2. As it turns out, the latter has the same expressive power as two-variable first-order logic FO2.

Languages of Dot-Depth One over Infinite Words

The dot-depth hierarchy is a classification of star-free languages. It is related to the quantifier alternation hierarchy of first-order logic over finite words. We consider subclasses of languages with dot-depth 1/2 and dot-depth 1 obtained by prohibiting the specification of prefixes or suffixes. As it turns out, these language classes are in one-to-one correspondence with fragments of alternation-free first-order logic without min- or max-predicate, respectively. For all fragments, we obtain effective algebraic characterizations. Moreover, we give new proofs for the decidability of the membership problem for dot-depth 1/2 and dot-depth 1.

On FO 2 Quantifier Alternation over Words

Simons congruence, denoted by ~ n , relates words having the same subwords of length up to n. We show that, over a k-letter alphabet, the number of words modulo ~ n is in 2 ? ( n k - 1 log ? n ) . We provide upper and lower bounds for Simons index for piecewise testability.This index depends on the length of considered subwords and the alphabet size.The previously known bounds focused on fixed length of subwords.Our bounds are relevant when alphabet size is fixed and length of subwords vary.They give a growth rate that is exponential and not doubly exponential.