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The PCP theorem has recently been shown to hold as well in the real number model of Blum, Shub, and Smale (Baartse and Meer, 2015). The proof given there structurally closely follows the proof of the original PCP theorem by Dinur (2007). In this paper we show that the theorem also can be derived using algebraic techniques similar to those employed by Arora et al. (Arora et al., 1998; Arora and Safra, 1998) in the first proof of the PCP theorem. This needs considerable additional efforts. Due to severe problems when using low degree algebraic polynomials over the reals as codewords for one of the verifiers to be constructed, we work with certain trigonometric polynomials. This entails the necessity to design new segmentation procedures in order to obtain well structured real verifiers appropriate for applying the classical technique of verifier composition.
We believe that designing as well an algebraic proof for the real PCP theorem on one side leads to interesting questions in real number complexity theory and on the other sheds light on which ingredients are necessary in order to prove an important result as the PCP theorem in different computational models.
Mathematical Logic Quarterly
(2017)
Journal of Complexity
(2017)
Journal of Complexity
(2016)
Mathematical Logic Quarterly
(2016)
In a recent work, Gandhi, Khoussainov, and Liu [7] introduced and studied a generalized model of finite automata able to work over arbitrary structures. As one relevant area of research for this model the authors identify studying such automata over particular structures such as real and algebraically closed fields.
In this paper we start investigations into this direction. We prove several structural results about sets accepted by such automata, and analyze decidability as well as complexity of several classical questions about automata in the new framework. Our results show quite a diverse picture when compared to the well known results for finite automata over finite alphabets.
This book constitutes the refereed proceedings of the 10th Conference on Computability in Europe, CiE 2014, held in Budapest, Hungary, in June 2014. The 42 revised papers presented were carefully reviewed and selected from 78 submissions and included together with 15 invited papers in this proceedings. The conference had six special sessions: computational linguistics, bio-inspired computation, history and philosophy of computing, computability theory, online algorithms and complexity in automata theory.
An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if the prefix-free Kolmogorov complexity of each initial segment of X is the same as the complexity of the sequence of 0s of the same length, up to a constant. We study the gap between the minimum complexity K(0 n ) and the initial segment complexity of a nontrivial sequence, and in particular the nondecreasing unbounded functions f such that
(⋆)
for a nontrivial sequence X, where K denotes the prefix-free complexity. Our first result is that there exists a $\varDelta^{0}_{3}$ unbounded nondecreasing function f which does not have this property. It is known that such functions cannot be $\varDelta^{0}_{2}$ hence this is an optimal bound on their arithmetical complexity. Moreover it improves the bound $\varDelta^{0}_{4}$ that was known from Csima and Montalbán (Proc. Amer. Math. Soc. 134(5):1499–1502, 2006).
Our second result is that if f is $\varDelta^{0}_{2}$ then there exists a non-empty $\varPi^{0}_{1}$ class of reals X with nontrivial prefix-free complexity which satisfy (⋆). This implies that in this case there uncountably many nontrivial reals X satisfying (⋆) in various well known classes from computability theory and algorithmic randomness; for example low for Ω, non-low for Ω and computably dominated reals. A special case of this result was independently obtained by Bienvenu, Merkle and Nies (STACS, pp. 452–463, 2011).
Today's multi-cores and future many-cores are NUMA architectures with complex cache hierarchies and multiple memory channels. Depending on the topologies of these memory networks we find everything from true data sharing with shared caches to distributed memory architectures which just pretend to be physical shared memory systems. In fact, most many-cores are hybrid systems that exhibit the characteristics of both distributed systems and SMPs. In this paper we argue in favor of middleware platforms for many-cores. We will discuss the needed functionality in contrast to common distributed system middleware and present micro benchmarks on several architectures to substantiate our claims.
Many-core processors combine fast on-chip communication
with access to large amounts of shared memory. This
makes it possible to exploit the benefits of distributed as well
as shared memory programming models within single parallel
algorithms. While large amounts of data can be shared in the
memory and caches, coordinating the activities of hundreds
of cores relies on cross core communication mechanisms with
ultra low latency for very small messages. In this paper we
discuss two communication protocols for the Intel SCC and
compare them to the MPI implementation of the SCC. Our
micro-benchmark results underline that special purpose protocols
for small messages make much finer levels of parallelism possible
than general purpose message passing systems.
Index Terms—many-core, message passing, shared memory
Real Computational Universality: The word problem for a class of groups with infinite presentation
(2009)
Modularity is a widely used quality measure for graph clusterings. Its exact maximization is prohibitively expensive for large graphs. Popular heuristics progressively merge clusters starting from singletons (coarsening), and optionally improve the resulting clustering by moving vertices between clusters (refinement). This paper experimentally compares existing and new heuristics of this type with respect to their effectiveness (achieved modularity) and runtime. For coarsening, it turns out that the most widely used criterion for merging clusters (modularity increase) is outperformed by other simple criteria, and that a recent multi-step algorithm is no improvement over simple single-step coarsening for these criteria. For refinement, a new multi-level algorithm produces significantly better clusterings than conventional single-level algorithms. A comparison with published benchmark results and algorithm implementations shows that combinations of coarsening and multi-level refinement are competitive with the best algorithms in the literature.
Fragestellungen aus Mathematik und theoretischer Informatik bei der Konstruktion von Getrieben
(2008)