252 research outputs found
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Infinite loop spaces and nilpotent K-theory
Using a construction derived from the descending central series of the free
groups, we produce filtrations by infinite loop spaces of the classical
infinite loop spaces , , , , , and
. We show that these infinite loop spaces are the zero
spaces of non-unital -ring spectra. We introduce the notion of
-nilpotent K-theory of a CW-complex for any , which extends the
notion of commutative K-theory defined by Adem-G\'omez, and show that it is
represented by , were is the -th term of
the aforementioned filtration of .
For the proof we introduce an alternative way of associating an infinite loop
space to a commutative -monoid and give criteria when it can be
identified with the plus construction on the associated limit space.
Furthermore, we introduce the notion of a commutative -rig and show
that they give rise to non-unital -ring spectra.Comment: To appear in Algebraic and geometric topolog
The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)
We show that there is a homotopy cofiber sequence of spectra relating
Carlsson's deformation K-theory of a group G to its "deformation representation
ring," analogous to the Bott periodicity sequence relating connective K-theory
to ordinary homology. We then apply this to study simultaneous similarity of
unitary matrices
Braided injections and double loop spaces
We consider a framework for representing double loop spaces (and more
generally E-2 spaces) as commutative monoids. There are analogous commutative
rectifications of braided monoidal structures and we use this framework to
define iterated double deloopings. We also consider commutative rectifications
of E-infinity spaces and symmetric monoidal categories and we relate this to
the category of symmetric spectra.Comment: 34 pages, 4 figures, minor correction
An effective characterization of the alternation hierarchy in two-variable logic
We characterize the languages in the individual levels of the quantifier
alternation hierarchy of first-order logic with two variables by identities.
This implies decidability of the individual levels. More generally we show that
the two-sided semidirect product of a decidable variety with the variety J is
decidable
Topological Hochschild homology of Thom spectra and the free loop space
We describe the topological Hochschild homology of ring spectra that arise as
Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for
stable spherical fibrations. To do this, we consider symmetric monoidal models
of the category of spaces over BF and corresponding strong symmetric monoidal
Thom spectrum functors. Our main result identifies the topological Hochschild
homology as the Thom spectrum of a certain stable bundle over the free loop
space L(BX). This leads to explicit calculations of the topological Hochschild
homology for a large class of ring spectra, including all of the classical
cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p
and HZ.Comment: 58 page
Block products and nesting negations in FO2
The alternation hierarchy in two-variable first-order logic FO 2 [∈ < ∈] over words was recently shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. In this paper we consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment of FO 2 is defined by disallowing universal quantifiers and having at most m∈-∈1 nested negations. One can view as the formulas in FO 2 which have at most m blocks of quantifiers on every path of their parse tree, and the first block is existential. Thus, the m th level of the FO 2 -alternation hierarchy is the Boolean closure of. We give an effective characterization of, i.e., for every integer m one can decide whether a given regular language is definable by a two-variable first-order formula with negation nesting depth at most m. More precisely, for every m we give ω-terms U m and V m such that an FO 2 -definable language is in if and only if its ordered syntactic monoid satisfies the identity U m ∈V m. Among other techniques, the proof relies on an extension of block products to ordered monoids. © 2014 Springer International Publishing Switzerland
Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words
We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels
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