Andreas Krebs

Characterizing TC 0 in Terms of Infinite Groups

We characterize the languages in TC0 = L(Maj[<,Bit]) and L(Maj[<]) as inverse morphic images of certain groups. Necessarily these are infinite, since nonregular sets are concerned. To limit the power of these infinite algebraic objects, we equip them with a finite type set and introduce the notion of a finitely typed (infinite) monoid. Following this approach we investigate type-respecting mappings and construct a new type of block product that more adequately deals with infinite monoids. We exhibit two classes of finitely typed groups which exactly characterize TC0 and L(Maj[<]) via inverse morphisms.

An Effective Characterization of the Alternation Hierarchy in Two-Variable Logic

We give an algebraic characterization, based on the bilateral semidirect product of finite monoids, of the quantifier alternation hierarchy in two-variable first-order logic on finite words. As a consequence, we obtain a new proof that this hierarchy is strict. Moreover, by application of the theory of finite categories, we are able to make our characterization effective: that is, there is an algorithm for determining the exact quantifier alternation depth for a given language definable in two-variable logic.

Typed monoids: an Eilenberg-like theorem for non regular languages

Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid. This leads to an algebraic description of regular and non regular languages. We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid. We prove an Eilenberg-like theorem for varieties of typed monoids as well as a similar correspondence for classes of languages with weaker closure properties than varieties.

Ultrafilters on words for a fragment of logic

We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving ultrafilter equations characterising B and then projecting these to obtain profinite equations characterising B ? Reg . This suffices to establish the decidability of the membership problem for B ? Reg .

Non-definability of Languages by Generalized First-order Formulas over (N,+)

We consider first-order logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition predicate are also definable with the order predicate as the only numerical predicate. Let S be a subset of monoids. Let L be the logic closed under quantification over the monoids in S. Then we prove that L[<;,+] and L[<;] define the same neutral letter languages. Our result can be interpreted as the Crane Beach conjecture to hold for the logic L[<;,+]. As a consequence we get the result of Roy and Straubing that FO+MOD[<;,+] collapses to FO+MOD[<;]. For cyclic groups, we answer an open question of Roy and Straubing, proving that MOD[<;,+] collapses to MOD[<;]. Our result also shows that multiplication as a numerical predicate is necessary for Barringtons theorem to hold and also to simulate majority quantifiers. All these results can be viewed as separation results for highly uniform circuit classes. For example we separate FO[<;,+]-uniform CC0 from FO[<;,+]-uniform ACC0.

Streaming algorithms for recognizing nearly well-parenthesized expressions

We study the streaming complexity of the membership problem of 1-turn-Dyck2 and Dyck2 when there are a few errors in the input string. 1-turn-Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x′ ∈ 1-turn-Dyck2 such that x is obtained by flipping at most k locations of x′ using: - O(k log n) space, O(k log n) randomness, and poly(k log n) time per item and with error at most 1/nΩ(1). - O(k1+e + log n) space for every 0 ≤ e ≤ 1, O(log n) randomness, O((logO(1) n + kO(1))) time per item, with error at most 1/8. Here, we also prove that any randomized one-pass algorithm that makes error at most k/n requires at least Ω(k log(n/k)) space to accept strings which are exactly k-away from strings in 1-turn-Dyck2 and to reject strings which are exactly k + 2-away from strings in 1-turn-Dyck2. Since 1-turn-Dyck2 and the Hamming Distance problem are closely related we also obtain new upper and lower bounds for this problem. Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x′ ∈ Dyck2 such that x is obtained from x′ by changing (in some restricted manner) at most k positions using: - O(k log n + √n log n) space, O(k log n) randomness, poly(k log n) time per element and with error at most 1/nΩ(1). - O(k1+e + √n log n) space for every 0 < e ≤ 1, O(log n) randomness, O((logO(1) n + kO(1))) time per element, with error at most 1/8.

Characterizing TC 0 in terms of infinite groups

We characterize the languages in TC0 =

Using Duality in Circuit Complexity

\mathcal{L}(Maj[<,Bit])

Universal Covers, Color Refinement, and Two-Variable Counting Logic: Lower Bounds for the Depth

and

A Team Based Variant of CTL

\mathcal{L}(Maj[<])