13:4514:30, H. Peter Gumm (PhilippsUniversität Marburg)
A weak pullback preservation property of weakly monadic set functors
Inspired by recent investigations on congruence modularity, we show that a weakly monadic functor F (the associativity law is not required) weakly preserves pullbacks of constant maps if and only if F(1)=1.
Coffee break
15:0015:45, Jirí Tuma (Karlova Universita Praha)
Antichains and business intelligence
We will present an example how useful elementary theory of ordered sets can be in business intelligence. It is a case study of collaboration between algebra group at Charles University in Prague and a startup company, also based in Czech Republic.
15:5516:40 Ferdinand Ihringer (JustusLiebigUniversität Gießen):
ErdösKoRado Sets in Finite Geometries
Let $N = \{ 1, \ldots, n \}$. An \emph{Erd\H{o}sKoRado sets} (EKR set) of $N$ is a family $Y$ of $d$sets of $N$ which pairwise intersect nontrivially. A famous result by Erd\H{o}s, Ko and Rado shows \begin{align*} Y
\leq \binom{n1}{d1} \end{align*} for $n \geq 2k$. There are many natural generalizations of EKR sets to other structures such as permutation groups or combinatorial designs. This talk will put its focus on EKR sets in vector spaces and polar spaces. Let $V$ be an $n$dimensional vector space over a finite field of order $q$. A \emph{polar space} of rank $d$ over $V$ is defined by a nondegenerate, reflexive sesquilinear form (or some quadratic form) and consists of all subspaces which vanish on this form. The largest of these subspaces are called \emph{generators} and have dimension $d$. An EKR of a polar space is a set of pairwise nontrivially intersecting generators. Here the largest size of EKR sets in some Hermitian polar spaces is still open. The talk will start with discussing original EKR sets and its various proofs, which are already accessible to high school students, and will end with current research in vector spaces and finite classical polar spaces.
There will be no registration. Though, if you plan to attend, notice to logik(at)mathematik.tudarmstadt.de would be wellcome.
Friday, July 17, 2015, at 13:30 in S215201
Angeliki KoutsoukouArgyraki
Approximate common fixed points and rates of asymptotic regularity for oneparameter nonexpansive semigroups
In this talk we present a recent work [1] that involves an application of proof mining to xed
point theory and to nonlinear semigroups. This is joint work with Ulrich Kohlenbach.
We extract quantitative information on the approximate common xed points of a one
parameter strongly continuous semigroup of nonexpansive mappings (nonexpansive semi
group) T(t) : t ≥ 0 on a subset C of a Banach space E, under the assumption that for each
x ∈ C the mapping t → T(t)x from [0;∞] into C is uniformly continuous on each compact
interval [0;K] for all K ∈ N and moreover given a b ∈ N it has a common modulus of uni
form continuity for all x ∈ C such that x ≤ b. This is achieved by logical analysis of the
proof of a theorem by Suzuki in [2]. We then apply our result to extract rates of asymptotic
regularity for the nonexpansive semigroup T(t) : t ≥ 0 on a convex C ⊆ E with respect to
the Krasnoselskii iteration.
[1]Kohlenbach,U. and KoutsoukouArgyraki,A: Approximate common fixed points and rates of asymptotic regularity for oneparameter nonexpansive semigroups, Submitted preprint.
[2]Suzuki,T. : Common fixed points of oneparameter nonexpansive semigroups, Bull. London Math. Soc. 38 10091018(2006).
An isomorphism from a graph G to a graph H can be seen as a permutation matrix X satisfying the equation AX = XB, where A and B are adjacency matrices of G and H, respectively. This means that Graph Isomorphism Problem can be stated in terms of 01 integer linear programming. The solutions to the relaxation of this integer program are called fractional isomorphisms. In 1994, Ramana, Scheinerman and Ullman proved that there is a fractional isomorphism between two graphs, if and only if the graphs cannot be distinguished by naive vertex classification, which is a widely used, intuitive method for isomorphism testing. I am going to present a new elementary proof of this theorem. There are other characterisations of the same concept, including a logic and a game characterization. I will briefly talk about them and give a new combinatorial characterization: two graphs are fractionally isomorphic, if and only if there is a so called migration from one to the other.
Proof interpretations are tools in mathematical logic with a wide range of applications: consistency results, independence results, and extraction of computational content from proofs, just to name a few. They are usually applied only in the context of logic or arithmetic. There is the dream of applying them even to set theory. Now the dream is fulfilled with a novel proof interpretation: Krivine's classical realisability. In this talk I present my personal digest of Krivine's classical realisability and one of its main applications to set theory: set theory ZF plus the statement "there is a set S between N (the set of natural numbers) and R (the set of real numbers) not equinumerous to its cartesian square S x S" is a consistent theory where both the axiom of choice and the continuum hypothesis fail. As a corollary we get two classical and celebrated results: ZF does not prove the axiom of choice nor the continuum hypothesis. This is a short, simple, and sweet talk, focusing on the ideas and setting aside technicalities.
* INRIA ParisRocquencourt, pi.r2, Univ Paris Diderot, Sorbonne Paris Cite, F78153 Le Chesnay, France. Financially supported by the French Fondation Sciences Mathematiques de Paris.
A guarded recursive definition is one where the recursion variable only occurs in positions guarded by a delay operator. In this talk I will present a new model of guarded recursion and motivate it using two different examples: first as a model for computations with coinductive types such as streams, and second as a model of a metalanguage useful for defining models of programming languages with higherorder store. The model is a simple presheaf model (over the ordered natural numbers) and I will describe the computational intuition. If I have time I will describe work in progress on constructing models of intentional type theory with guarded recursion. Most of the talk describes joint work with Lars Birkedal, Kristian Stoevring and Jan Schwinghammer published at LICS 2011.
Modern software tends to undergo frequent requirement changes and typically is deployed in many different scenarios. This poses significant challenges to formal software verification, because it is not feasible to verify a software product from scratch after each change. It is essential to perform verification in a modular fashion instead. The goal must be to reuse not merely software artifacts, but also specification and verification effort. In our setting code reuse is realized by deltaoriented programming, an approach where a core program is gradually transformed by code "deltas" each of which corresponds to a product feature. The deltaoriented paradigm is then extended to contractbased formal specifications and to verification proofs. As a next step towards modular verification we transpose Liskov's behavioural subtyping principle to the delta world. Finally, based on the resulting theory, we perform a syntactic analysis of contract deltas that permits to automatically factor out those parts of a verification proof that stays valid after applying a code delta. This is achieved by a novel verification paradigma called "abstract symbolic execution."
We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations. Certain strategy profiles are proved to be subgame perfect equilibria using a novel proof principle of predicate coinduction which is shown to be sound by reducing it to Kozen's metric coinduction. We characterize all subgame perfect equilibria for the dollar auction game. The economically interesting feature is that in order to prove these results we do not need to rely on continuity assumptions on the payoffs which amount to discounting the future. In particular, we prove a form of onedeviation principle without any such assumptions. This suggests that coalgebra supports a more adequate treatment of infinitehorizon models in game theory and economics. I will also sketch the possibilities that this (and more generalized) mathematics opens up for reflexive economics, which takes it serious that modeling in social science takes place within the modeled system and that essentially depends on selfreferential paradoxes and corecursion and similar mathematical constructions. This opens the door to a proper treatment of the Lucas critique, evolutionary, institutional, agend based, computational, numerical economics, econometrics of non experimental data, money, value theory, political economics and management theory and also to a unification and cross fertilization between systems sociology, computer science and biology.
The computability of functions by PourEl and Richards requires that Sequential Computability and Effective Continuity. So, a computable function must be continuous. The theory of Walsh series is based on the Fine metric, which is stronger than Euclidean metric. A function is Fine continuous if and only if it is continuous at dyadic irrationals and right continuous at dyadic rationals. So, Rademach functions and Walsh functions are fine continuous. The Fine computability is defined by means of the Fine topology instead of the Euclidean topology.
The fixed point property on treelike Banach spaces
A Banach space $X$ is said to have the fixed point property if for every weakly compact convex set $K\subseteq X$ and every nonexpansive $T:K\to K$ (i.e. $\TxTy\\leq \xy\$ for all $x,y\in K$), the map $T$ has a fixed point. This property is closely related to the geometry of the space under consideration. If the geometry of the space is "nice", i.e. if $X$ possesses normal structure, then $X$ has the fixed point property. On the other hand, a space without normal structure may fail this property (e.g. the space $L_1$) or it may enjoy this property (e.g. the space $c_0$).
In this lecture we will consider the important class of separable Banach spaces not containing $\ell_1$ with nonseparable dual. Our purpose is to study these spaces with respect to the fixed point property. In particular, after discussing some known members of this class, we will present a Banach space belonging to the class under consideration which has the fixed point property even though it does not possess normal structure.
Nonprincipal ultrafilters, program extraction and higher order reverse mathematics
Abstract: We investigate the strength of the existence of a nonprincipal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a nonprincipal ultrafilter on N exists and let ACA_0^w be the higher order extension of ACA_0. We show that ACA_0^w + (U) is Pi^{1}_2conservative over ACA_0^w and thus that ACA_0^w + (U) is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly \Pi^{1}_2 statement \forall f \exists g A(f,g) in ACA_0^w + (U) a realizing term in G"odel's system T can be extracted. This means that one can extract a term t\in T, such that \forall f A(f,t(f)).
Descriptive complexity is a bridge between complexity theory and logic. It asks for the logic resources that are necessary to define the computational resources such as time, space or the amount of hardware that are necessary to decide a property. Horn and Krom logic have their origins in propositional logic. Some researchers extend them to second logic, named SOHORN and SOKROM logic. We define revised and extended versions of these logics and get some results. We show that SOHORN$^r$ and ($\exists$)SOKROM$^r$ capture P and NL on ordered structures, respectively, and both SOEHORN and $\Pi^{1}_{1}$KROM capture coNP.
November 4, 2011 at 15:00 in S215201
Jaime Gaspar
Goedel functional interpretation
Abstract: We give a quantitative version of a strong nonlinear ergodic theorem for (a class of possibly even discontinuous) selfmappings of an arbitrary subset of a Hilbert space due to R.~Wittmann and outline how the existence of uniform bounds in such quantitative formulations of ergodic theorems can be proved by means of a general logical metatheorem. In particular these bounds depend neither on the operator nor on the initial point. Furthermore, we extract such uniform bounds in our quantitative formulation of Wittmann's theorem, implicitly using the prooftheoretic techniques on which the metatheorem is based. However, we present our result and its proof in analytic terms without any reference to logic as such. Our bounds turn out to involve nested iterations of relatively low computational complexity. While in theory these kind of iterations ought to be expected, so far this seems to be the first occurrence of such a nested use observed in practice. October 28, 2011 at 13:30 in S215201
Abstract: Every nonempty open set in a computable metric space contains a computable point. In contrast, the NonBasis Theorem asserts that a nonempty coc.e. closed set in Cantor space (hence, even in the real line) can avoid any computable points. Recent exciting progress in Computable Analysis naturally raises the question whether NonBasis Theorems exist for connected coc.e. closed sets. In this talk, we show global and local noncomputability results concerning connectivity, local connectivity, simply connectivity, and contractibility for planar continua.
Since then many other aperiodic tile sets have been found, not only on the discrete plane, but also on the continuous plane (the most famous being probably Penrose's "kite and darts" set of two tiles).
In this talk, I will describe yet another construction of an aperiodic tile set. Although the resulting tile set will produce tilings quite similar to those of Robinson's (an infinite hierarchical structure of embedded squares) the local constraints will be presented in a (hopefully) more natural way: we will start from simple geometrical figures and organize them step by step by adding new rules progressively to enforce the global structure.
At the seminar we will discuss the origins, basic ideas and notions of
synthetic topology, and some proofs. Depending on time, we'll speak about the logical and computational aspects of the theory.
Abstract: Hillam's theorem gives a characterisation of the convergence of fixed point iterations. We present a quantitative version of Hillam's theorem where we give an explicit rate of metastability (a kind of "finitary rate of convergence") for the fixed point iteration. We explain how this result is obtained by methods of proof theory used in the proof mining program, called functional interpretations, which extract computational content from proofs.
February 11, 2011 at 14:00 in S215201
Alexander Kartzow
Firstorder logic on generalisations of pushdown graphs
Abstract: In their search for a topological characterization of relatively (i.e. oracle) computable relations (aka multivalued functions), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Based on the Henkin quantifier, we propose a notion of uniform continuity and explore its relations to weak/strong continuity as well as to relative computability.
Abstract: It is known that determinacy of infinite games characterize many subsystems of second order arithmetic.
In this talk, we treat relatively stronger systems, beyond socalled "big five systems", and consider the possibility of determinacy as a tool to investigate strong systems of second order arithmetic.
We observe that the notion of nowhere density is essentially the earlier stability theoretic notion of superflatness, introduced by Podewski and Ziegler in 1978. With this, we show that for a subgraphclosed class C of (not necessarily finite) graphs, the following are equivalent:
1) C is nowhere dense
2) C is stable
3) C is dependent
The result can be generalised to classes of coloured digraphs. It fails if C is not subgraphclosed. For arbitrary graph classes, dependence is the most general notion, generalising both nowhere density and stability, as well as (locally) bounded cliquewidth.
This is joint work with Hans Adler.
MAX(f) = sup_{0<x<1} f(x) , INT(f) = int_0^{1} f(t) dt .
It has been shown by Friedman and Ko (1982) that for the restricted an be approximated up to an error 2^{n} on polynomialspace in n. In above's framework of computation, each computable real function f : [0,1] > R is essentially represented as a type2 function (in the sense of Weihrauch), mapping a rational sequence to a rational sequence, i.e., f \in (N>Q) > (N>Q). This enables us to discuss the computational complexity of a realvalued function on classical Turing machines with oracle access. Here, a TM M computing f is presented with an oracle phi \in (N>Q) for x in [0,1] and a precision n in N, such that M produces the output M^\phi(n) \in Q with
M^\phi(n)  f(x) < 2^{n}.
When it comes to the computation of the functionals MAX and INT, the encoding of an input for a Turing machine becomes even more critical: Instead of representing an x \in [0,1] by a fast converging sequence \phi \in (N>Q), we now have to encode a continuous realvalued function f : [0,1] > R. The first goal of this thesis is to devise, compare and explore different ways of accessing an unknown continuous real function via blackbox queries that model the actual computation of f as supported by certified numerics like interval arithmetic or the iRRAM. Building on this, the second goal of this Master's Thesis is to determine the query complexity of the MAXfunctional with respect to different protocols. In combination with an complexity analysis of the protocols themselves, this should yield more explicit quantitative and parameterized both upper and lower bounds (compared to those presented in the beginning) on the complexity of MAX (and INT).
Computation with infinitely presented groups
Abstract: We give an overview of the current methods for computing with a class of infinitely presented groups, the socalled finitely $L$presented groups. Theses infinite presentations generalize finite presentations and they allow interesting applications. In particular, our methods allow to compute the lower central series quotients and the Dwyer quotients of the Schur muliplier.
Computable fields and their algebraic closures
Abstract: We present recent results about computable fields F, especially those algebraic over their prime subfields, and embeddings of them into their algebraic closures. The discussion begins with the basic definitions and a review of Rabin's Theorem, and continues with more exact computabilitytheoretic comparisons, by Steiner and by the speaker, among the set of irreducible polynomials in F[X], the set of polynomials there with roots in F, and the image of F within computable presentations of its algebraic closure. Finally, we will discuss the BlumShubSmale model of computation on the real numbers, and explain how it changes the array of tools available for examination of these questions. back to the top of the page
Proof interpretations with truth
Abstract: A proof interpretation I of a theory S in a theory T is a function mappingThis is based on a joint work with Paulo Oliva. back to the top of the page
The monotone completeness theorem in constructive reverse
Abstract: We investigate the monotone completeness theorem: MCT. Every bounded monotone sequence of real numbers converges within a subsystem of elementary analysis EL as a formal framework of constructive reverse mathematics aiming at classifying various theorems in intuitionistic, constructive recursive and classical mathematics. We distinguish a Cauchy sequence with modulus from a Cauchy sequence without modulus.We give a logical principle which is stronger than LPO, and show that it is equivalent to the statement:
(1) Every bounded monotone sequence of real numbers is a Cauchy sequence without modulus.
We also give a function existence axiom which is weaker than the numbernumber choice for Pi^{0}_1 formulas, and show that it is equivalent to the statement:
(2) Every Cauchy sequence without modulus is a Cauchy sequence with modulus.
Therefore MCT is divided into the logical statement (1), the function existence statement (2), and the Cauchy completeness of the reals which is constructively provable without countable choice.
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Finite axiomatization of quasivarieties of relational structures
Abstract: Much attention was paid to the finite axiomatization of varieties and quasivarieties of algebras in the past. But the problem for relational structures was appearing sporadically. I will review what is known about this and add one new theorem. Namely, we prove that the graphs of monoids or groups (as relational structures) almost always generate non finitely axiomatizable quasivarieties.On the expressive CNF formulas of bounded Tree and CliqueWidth
Abstract: We study the expressive power of certain families of polynomials over a field K. The latter are defined via Boolean formulas in conjunctive normal form. We attach in a natural way a graph to such a formula and study the resulting polynomial families when certain parameters of that graph like tree and cliquewidth are assumed to bounded. back to the top of the page
Exakte reelle Arithmetik  reelle Zahlen aus der Sicht der Informatik
Abstract: Die Menge der reellen Zahlen ist, anders als etwa die viel leichter algorithmisch greifbaren natürlichen Zahlen, überabzählbar. Als Konsequenz ist bereits die Definition des Begriffs "Berechenbarkeit" für reelle Zahlen deutlich komplexer als die klassische Berechenbarkeitsdefinition für natürliche Zahlen mit Turingmaschinen.Implementierungen dieses Berechenbarkeitsbegriffes werden oft als "exakte reelle Arithmetik" bezeichnet; Basis ist dabei in der Regel eine Intervallarithmetik mit mehrfach genauen Zahlen, ergänzt um eine automatisierte Genauigkeitssteuerung. Damit sind numerische Algorithmen möglich, die in der endlichen Welt der doubleprecisionZahlen nicht sinnvoll wären.
Im Vortrag stellen wir einige der Prinzipien einer solchen Implementierung vor und werden dabei insbesondere als Beispiele ihre Anwendung (und ihr Verhalten) bei der "logistic map" und einfachen Differentialgleichungen betrachten. back to the top of the pageSlash and completeness
Abstract: Gödel's incompleteness theorems gave a fatal blow in Hilbert's program. Despite this, there were some attempts to salvage the program. One of them, by Hilbert himself, was to add omegarule (if A(n) is provable for each fixed n, then "for all n, A(n)" is provable) to Peano arithmetic. This rule evades Gödel's incompleteness and gives a complete arithmetical theory. Proof interpretations are prooftheoretic tools to prove consistency results, conservation results, closure under rules and to extract computational content from proofs. The slash is a very simple toy case of a proof interpretation. However, the omegarule boosts the slash, resulting in some interesting applications. We will talk about one of them: a very simple proof of the completeness of Peano arithmetic plus omegarule. back to the top of the page
Martin Otto, TU Darmstadt
Acyclicity in hypergraph covers and applications
Abstract: Which degrees of acyclicity can be achieved in FINITE hypergraph covers? While actual acyclicity can always be had in generally infinite unfoldings, finite covers can only guarantee certain bounded forms of acyclicity at best. I want to discuss new constructions of weakly Nacyclic covers (joint work with Vince Barany and Georg Gottlob, Oxford) and Nacyclic covers with a view to their use in the finite model theory of guarded logics. back to the top of the page
Expressiveness and Computational Complexity of Euclidean Quantum Logic
Abstract: Quantum logic generalizes, and in dimension one coincides with, Boolean logic. The satisfiability problem of quantum logic formulas is shown NPcomplete in dimension two. We then extend these considerations to three and higherdimensional Euclidean spaces R^d and C^d. For fixed d>2, quantum satisfiability turns out polytimeequivalent to the real feasibility of a multivariate quartic polynomial equation: a wellknown problem lying (probably strictly) between NP and PSPACE. We finally investigate the problem over INdefinite finite dimensions and relate it to the real feasibility of quartic NONcommutative *polynomial equations. back to the top of the page
Comparison of Complexity over the Real vs. Complex Numbers
Abstract: We compare the complexity of certain problems over the reals with their corresponding versions over the complex numbers. We argue in particular that the problem of deciding whether a system of (real) polynomial equations has a real solution is strictly harder than asking for a complex solution, since it cannot be solved by a complex BSSmachine. On the other side we describe a family of problems, whose real versions are in the model of algebraic decision trees strictly easier than its complex versions. For this end we prove a new lower bound for the decision complexity of a closed algebraic set X in C^n in terms of the sum of its (compactly supported) Betti numbers b_c(W), which is for the first time better than logarithmic in b_c(W).
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Topology in computable analysis
Abstract: Computable Analysis studies computability over the real numbers and related spaces. Since computable functions are continuous in some sense, topology plays an important role in this theory. The cartesian closed category QCB of quotients of countably based topological spaces turns out to be an appropriate category for modelling the topological aspects of higher type computation.
In this talk, we discuss topological properties of the KleeneKreisel continuous functionals and their surprising relationship to an open problem in exact realnumber computation. This problem is whether two approaches in functional programming on higher type computability over the reals coincide.
We show that the sequential topology on the space N^(N^N), which is the space of KleeneKreisel continuous functionals of type 2, is neither zerodimensional nor regular. Moreover, we establish nonregularity of further function spaces formed in the cartesian closed category of sequential spaces. This is one of the categories from which QCB inherits its cartesian closed structure. back to the top of the page
Sequential game theory: a formal and abstract approach
Abstract: In a game from game theory, a Nash equilibrium (NE) is a combination of one strategy per agent such that no agent can increase his payoff by unilaterally changing his strategy. Kuhn proved that all (treelike) sequential games have NE. Osborne and Rubinstein abstracted over these games and Kuhn's result: they proved a *sufficient* condition on agent *preferences* for all games to have NE. I will present a *necessary and sufficient* condition, thus accounting for the gametheoretic frameworks that were left aside. The proof is *formalised* using Coq, and contrary to usual game theory it adopts an inductive approach to trees for definitions and proofs. By rephrasing a few gametheoretic concepts, by ignoring useless ones, and by characterising the prooftheoretic strength of Kuhn's/Osborne and Rubinstein's development, the talk also intends to clarifies sequential game theory.
Then I will sketch an alternative proof. Both proofs so far preprocess the preferences by topological sorting, which is surprising since it decreases the number of NE. Finally I will sketch two other proofs that preprocess the preferences only by transitive closure. The four proofs are all constructive and by induction, and I will compare them when seen as algorithms. The four proofs also constitute diverse viewpoints on the same concepts, the trial of diverse techniques against the same problem, and reusable techniques for similar structures.
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Proof interpretations with truth
Abstract: Proof interpretations are prooftheoretic tools to extract computational content (such as computable choice functions) from proofs. In some cases this only works (at least without strengthening the theories involved) for proofs of theorems of a particular logical form, but we wish to extract computational content from proofs of arbitrary theorems. Interpretations with truth have been developed to cope with this. We present two new methods on how to do hardwire truth and apply them to new proof interpretations.
In a first part we make a brief introduction to proof interpretations and the extraction of computational content from proofs. Then in a second part we present our two methods. Finally in a third part we apply them in some case studies.
This is joint work with Paulo Oliva. back to the top of the page
Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability
Abstract: It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f:A>B, additional structural information about the input x in A (that is any kind of promise that x belongs to a certain subset A' of A) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A*x=0 for a given singular real NxNmatrix A is possible when knowing rank(A)=0,1,...,N1; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a BASIS of eigenvectors of) a given real symmetric NxNmatrix is possible when knowing the number of distinct eigenvalues: an integer between 1 and N (the latter corresponding to the nondegenerate case). And again we show that Nfold (i.e. roughly log N bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas for finding SOME SINGLE eigenvector of A, providing the truncated binary logarithm of the leastdimensional eigenspace of Ai.e. Theta(log N)fold adviceis sufficient and optimal. back to the top of the page
Effective Descriptive Set Theory and Applications in Analysis
Abstract: Descriptive Set theory is the area of mathematics which studies the structure of definable sets in separable and complete metric spaces. A central part of this area is Effective Descriptive Set Theory (for shortness Effective Theory) which uses ideas from Logic and in particular from Recursion Theory. This area provides a deeper understanding of the subject and has applications in Analysis. We begin this lecture with a brief summary of the basic concepts and we give a new approach to Effective Theory which yields some surprising results. Then we present some connections with General Topology; all theorems we talk about involve notions from both areas. Finally we give some results in Banach space Theory. Here all theorems involve only classic (i.e., noneffective) notions but still Effective Theory is essential for their proof. back to the top of the page
Ionads: a generalised notion of topological space
Acyclicity in hypergraph covers
On the Computational Content of BolzanoWeierstraß
Abstract: We will apply the methods developed in the field of 'proof mining' to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in a proof of a combinatorial statement. Using a simple proof based on weak König's lemma and arithmetical comprehension we provide an explicit solution of the Gödel functional interpretation as well as the monotone functional interpretation of BW for the product space \prod_i \in N [k_i, k_i] for a given sequence of rational numbers (k_i) \subseteq Q (with the standard product metric). In fact, we will use it to get optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for BW applied to fixed sequences in \prod_i \in N [k_i, k_i]. back to the top of the page
Logics with Rank Operators
Abstract: It is a longstanding open problem in descriptive complexity theory whether there is a logic that expresses exactly the polynomialtime decidable properties of unordered structures. It has been known since the early 90s that fixedpoint logic with counting (FPC) is not sufficient and there have been several proposals for more expressive logics. Taking our inspiration from recent results that show that natural linearalgebraic operations are not definable in FPC, we propose FPR  an extension of fixedpoint logic with an operator for matrix rank. We show that this is strictly more expressive than FPC and can express the polynomialtime problems that have been shown to be inexpressible in FPC. We also show that FO+R, the extension of firstorder logic with rank operators is surprisingly expressive. It can express some forms of transitive closure and, on ordered structures, captures the complexity classes ModpL.
In this talk, I will give the history and general background to the question and explain the context of the new results. I will also point to several interesting open problems. (Joint work with Martin Grohe, Bjarki Holm and Bastian Laubner).
back to the top of the pageTechnische Universität Darmstadt
Department of Mathematics
Logic Group
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