Timon Barlag’s research while affiliated with Leibniz Universität Hannover and other places

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Publications (7)


A logical characterization of constant-depth circuits over the reals
  • Article

September 2024

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4 Reads

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2 Citations

Journal of Logic and Computation

Timon Barlag

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In the eighties, Immerman showed that the class of languages definable by first-order formulae coincides with the class of languages decidable by unbounded fan-in Boolean circuits of constant depth and polynomial size. We show an analogous result for real-valued computation, i.e. we define circuits of unbounded fan-in operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on real valued structures. Our characterization holds both non-uniformly as well as for many natural uniformity conditions.


Figure 1. Example graphs that satisfy the imposed conditions of Example 28.
The sequence d(8, 2, 2)
Logical characterizations of algebraic circuit classes over integral domains
  • Article
  • Full-text available

May 2024

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15 Reads

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1 Citation

Mathematical Structures in Computer Science

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the ACR\mathrm{AC}_{\mathbb{R}} and NCR\mathrm{NC}_{\mathbb{R}}^{} classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the ACR\mathrm{AC}_{R} and NCR\mathrm{NC}_R^{} hierarchy. Those generalizations apply to the Boolean AC\mathrm{AC} and NC\mathrm{NC} hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.

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Computing Repairs Under Functional and Inclusion Dependencies via Argumentation

March 2024

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3 Reads

Lecture Notes in Computer Science

We discover a connection between finding subset-maximal repairs for sets of functional and inclusion dependencies, and computing extensions within argumentation frameworks (AFs). We study the complexity of existence of a repair, and deciding whether a given tuple belongs to some (or every) repair, by simulating the instances of these problems via AFs. We prove that subset-maximal repairs under functional dependencies correspond to the naive extensions, which also coincide with the preferred and stable extensions in the resulting AFs. For inclusion dependencies one needs a pre-processing step on the resulting AFs in order for the extensions to coincide. Allowing both types of dependencies breaks this relationship between extensions and only preferred semantics captures the repairs. Finally, we establish that the complexities of the above decision problems are NP {\textbf {NP}}-complete and Π2P\boldsymbol{\mathrm {\Pi }}^{ {\textbf {P}}}_2-complete, when both functional and inclusion dependencies are allowed.


Unified Foundations of Team Semantics via Semirings

September 2023

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11 Reads

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1 Citation

Timon Barlag

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[...]

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Semiring semantics for first-order logic provides a way to trace how facts represented by a model are used to deduce satisfaction of a formula. Team semantics is a framework for studying logics of dependence and independence in diverse contexts such as databases, quantum mechanics, and statistics by extending first-order logic with atoms that describe dependencies between variables. Combining these two, we propose a unifying approach for analysing the concepts of dependence and independence via a novel semiring team semantics, which subsumes all the previously considered variants for first-order team semantics. In particular, we study the preservation of satisfaction of dependencies and formulae between different semirings. In addition we create links to reasoning tasks such as provenance, counting, and repairs.


Logical Characterization of Algebraic Circuit Classes over Integral Domains

February 2023

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11 Reads

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the ACR\mathrm{AC}_{\mathbb{R}} and NCR\mathrm{NC}_{\mathbb{R}}-classes for this setting. We give a theorem in the style of Immerman's theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer and we show characterizations for the ACR\mathrm{AC}_{R} and NCR\mathrm{NC}_{R} hierarchy. Those generalizations apply to the Boolean AC\mathrm{AC} and NC\mathrm{NC} hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.


A Logical Characterization of Constant-Depth Circuits over the Reals

October 2021

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8 Reads

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2 Citations

Lecture Notes in Computer Science

In this paper we give an Immerman Theorem for real-valued computation, i.e., we define circuits of unbounded fan-in operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformly as well as for many natural uniformity conditions.


A Logical Characterization of Constant-Depth Circuits over the Reals

May 2020

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16 Reads

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.

Citations (3)


... Gates corresponding to atomic formulae are input gates and are labelled with an appropriate part of the input enc(D, α) determined by the ordering used in it. The argument that there is a DLOGTIME algorithm that describes Cn, given n, is the same as in the classical case for FO (see [7] for a similar proof for FO R and DLOGTIME-uniform AC 0 R (+, ×)). ...

Reference:

Rewriting Consistent Answers on Annotated Data
A logical characterization of constant-depth circuits over the reals
  • Citing Article
  • September 2024

Journal of Logic and Computation

... As part of the study of provenance in databases, Grädel and Tannen [15] gave semiring semantics to FO-formulae in negation normal form NNF (i.e., all negation symbols are "pushed" to the atoms) by using the notion of an interpretation, which is a function that assigns semiring values to atomic or negated atomic facts. Here, we give semiring semantics to FO-formulae on a K-database D by, in effect, considering a particular canonical interpretation on D; a similar approach was adopted by Barlag et al. in [6] for K-teams, which can be viewed as K-databases over a schema with a single relation symbol. Appendix A contains a more detailed discussion of semiring semantics via interpretations. ...

Unified Foundations of Team Semantics via Semirings
  • Citing Conference Paper
  • September 2023